Encyclopedia of Physical Science and Technology, 3e, Physical Chemistry

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Encyclopedia of Physical Science and Technology

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Chemical Kinetics, Experimentation Terence J. Kemp University of Warwick

I. II. III. IV. V. VI. VII. VIII. IX.

Introduction Mathematical Manipulation of Rate Data Computer Modeling of Reaction Systems General Methodology of Reaction Kinetics Flow Systems Pulse and Shock Methods Relaxation Methods Spectral Line-Broadening Methods Electrochemical Methods

GLOSSARY Chemiluminescence Emission of electromagnetic radiation in the ultraviolet, visible, or infrared regions from the products of a chemical reaction proceeding in the dark. Fluorescence Spontaneous emission of light by a molecule from an electronically excited state that has the same total electronic spin as the ground state. Laser Optical device achieving the stimulated emission of electromagnetic radiation from a molecule, ion, or chemical system following the generation of a nonBoltzmann “population inversion” between the ground and an excited state by means of some source of excitation (e.g., an electric discharge or a powerful flash of white light (optical pumping)).

Monochromator Optical device based on dispersion of white light by one or more prisms or diffraction gratings into its constituent wavelengths, which are utilized in turn. Phosphorescence Spontaneous emission of light by a molecule from an electronically excited state that has a different total electronic spin from the ground state. Photomultiplier tube Electronic device contained within an evacuated glass envelope, enabling the photoelectric conversion of incident light at a metallic surface into electric current via a number of stages of amplification. Reaction layer That layer of solution adjacent to an electrode within which a stationary distribution of electroactive species is established as the result of homogeneous reaction.

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Relaxation Process where by a molecule returns from a high-energy state to its normal state of lower energy. Relaxation time Time for the fraction 1/e of excited molecules to return by relaxation to their normal state.

etc., indicate the concentrations of the various reactants; α, β, etc. are the orders of reaction with respect to [A], [B], etc., respectively; α + β + · · · represents the total reaction order; and k is the rate constant. The units of k are dependent on the total reaction order; from Eq. (3) it can be seen that they are

EXPERIMENTATION IN CHEMICAL KINETICS is the methodology of measuring the rates of chemical reactions to yield rate constants, reaction orders, and activation parameters with the ultimate goal of establishing the mechanism of a particular reaction. It covers classical approaches based on conventional sampling and analytical methods, and the use of high pressures as well as the impressive array of fast reaction methods now available through developments in magnet technology, electronics, lasers, and small dedicated computers. These are grouped under flow systems, pulse and shock methods, relaxation methods, analysis of spectroscopic line-broadening, and electrochemical methods.

(mol dm−3 sec−1 )/(mol dm−3 )α+β+···

I. INTRODUCTION The task of the experimentalist in dealing with chemical systems is to determine first the rate of the reaction concerned and then its rate constant before proceeding to the ultimate goal of elucidating the reaction mechanism. Such a reaction rate will be determined under a set of closely specified conditions, particularly reactant concentrations, catalyst, temperature, solvent, gas pressure, ionic strength, nature of the reactor surface, radiant light intensity, etc., depending on the type of system under investigation. The accuracy and precision of the figure arrived at will also need to be specified. Many investigators will attempt to factorize the rate constant for the reaction into its components via the Arrhenius equation, k = Ae−E/RT

(1)

by measuring rates (and hence k) over as wide a temperature range as will give reasonable accuracy to the individual values for A and E, or, in the language of transitionstate theory, the enthalpy and entropy of activation, H = and S = , respectively: kB T S = /R −H = /RT e (2) e h It should be noted that in translating reaction rates into rate constants, the experimentalist needs the rate equation, which typically takes the form k=

v = k[A]α [B]β [C]γ · · ·

(3)

where v signifies the rate as measured by loss of a reactant, e.g., −d[A]/dt or formation of a product; [A], [B],

and for total reaction orders of zero, one, two, and three the units of k are mol dm−3 sec−1 , sec−1 , dm3 mol−1 sec−1 , and dm6 mol−2 sec−1 , respectively. It is important to note that while the rate of reaction depends on the concentrations of the reactants, the rate constant is independent of these and is the parameter commonly referred to in discussion of reaction kinetics. While for normal reaction systems the rate constant is naturally independent of time, for systems featuring an initially inhomogeneous distribution of reactants—as, for example, along the track of an α-particle or laser pulse immediately after discharge—the rate constant varies with time until homogeneity is achieved.

II. MATHEMATICAL MANIPULATION OF RATE DATA When a reaction is mechanistically simple, it is easy to determine its rate, reaction order, and rate constant. For example, if a molecule A is decomposing in an inert solvent in a unimolecular process, then the rate law is −d[A]/dt = k[A]

(4)

which can be rearranged to −d[A]/[A] = k dt

(5)

Equation (5) can be integrated: −ln[A] = kt + constant

(6)

The integration constant is found by noting that at time zero the value of [A] can be regarded as [A]0 ; thus the constant equals ( −ln[A]0 ) and Eq. (6) may be rewritten as Eq. (7): ln[A] − ln[A]0 = −kt

(7)

Thus a graphical plot of [A] versus time will give a straight line of slope, −k and an intercept at time zero of [A]0 . For a second-order reaction, the rate law could take the form −d[A]/dt = k[A]2

(8)

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or, where we are dealing with two different reactants, A and B, −d[A]/dt = −d[B]/dt = k[A][B]

(9)

The experimentalist will normally select concentrations such that [A] = [B], and then Eq. (9) becomes identical with Eq. (8). Rearranging Eq. (8), −d[A]/[A]2 = k dt

(10)

and then integrating, 1/[A] = kt + const

(11)

As before, the integration constant is determined by noting that at time zero the concentration of A can be written as [A]0 , and the constant therefore equals 1/[A]0 . Equation (11) can now be rewritten as 1/[A] − 1/[A]0 = kt

(12)

Hence a plot of 1/[A] versus time will give a straight line with a slope of k. Equations (7) and (12) not only provide the vehicle for extracting rate constants from experimental data, they also provide the test for whether the disappearance of a particular reactant follows first- or second-order kinetics. To apply this test with rigor, it is desirable to test the adherence of the data to a particular mathematical law for several half-lives, i.e., for 75–95% of total reaction. (One half-life is the time for the concentration of A to fall from [A]0 to [A]0 /2.) To cope with kinetically more complex reactions, the experimentalist needs to rely on one or more of several well-tested procedures for simplifying the analysis of data by manipulating the starting concentrations of the reactants. The best known of these devices is the “isolation method,” due originally to Ostwald. If a reaction involves several reactants or retardation by a product, then the resulting rate law may be too complex, mathematically, to test in the normal situation when all the reactant concentrations are changing simultaneously. However, if every reactant concentration is kept large except for one (say A), then during the reaction only [A] will change appreciably, the equivalent amounts of B, C, etc., being removed representing only a tiny fraction of [B]0 , [C]0 , etc., so that [B], [C], etc., can be regarded as remaining essentially unchanged throughout reaction. For example, if we have a rate law given by Eq. (13), v = k[A]2 [B][C]2

(13)

then if [B] ≈ [C] [A], v = k[A]2 [B]0 [C]20

(14)

or v = k [A]2

(15)

where k = k[B]0 [C]20 . Thus Eq. (15) takes the form of Eq. (8) and can be dealt with similarly. It is also possible to determine reaction orders by this approach; thus if [B]0 in Eq. (14) is doubled, then k is doubled, indicating a firstorder dependence on [B], while if [C]0 is doubled then k

is quadrupled, indicating a second-order dependence on [C]. In other examples, of course, the “simplified” rate equation might take the form v = k [A]

(16)

when the test given by Eq. (7) would apply. The other main device for dealing with highly complex reactions is the method of initial rates. When it is impossible to investigate a reaction by some simple mathematical test such as Eq. (7) or (12), which normally depends on following at least one of the reactants for up to 90% of its course of disappearance, then the method of initial rates is utilized. This depends on the idea that if a reaction is followed for only 5 or 10% of its total course, then a plot of the concentration of that reactant disappearing most quickly versus time will approximate to a straight line. (All reactant–time plots, except for zero-order reactions, are curved with −d[A]/dt decreasing as [A] decreases; however, the graph of first few percent of any such reaction approximates to the tangent to this curve at time zero.) Such a straight line yields an immediate value for k if [A], [B], and [C] are taken as [A]0 , [B]0 , [C]0 , . . . . β

γ

−d[A]/dt = k[A]α0 [B]0 [C]0 · · ·

(17)

The effect of variation of [B], [C], etc., separately on −d[A]/dt will then yield rate data enabling evaluation of β, γ , etc.

III. COMPUTER MODELING OF REACTION SYSTEMS This takes various forms, depending on the complexity of the reaction system. At its simplest, programs are available in the principal programming languages for any computer operating system to enable the testing of raw data in terms of the simplest types of kinetic rate law, Eqs. (7) and (12), and it is now universal for microcomputers interfaced with apparatus to be capable of recording, storing, and processing kinetic data in digital form or after analog-to-digital conversion. Programs giving complete solutions have also been written for more complex kinetic situations, of which the following are well-known examples.

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1. Series first-order reactions: k1

k2

A→B→C

(18)

2. Reversible first-order reactions: k1

A

B

(19)

k−1

(A notable example is conformational interchange by NMR line-broadening techniques; see Section VIII.) 3. Reversible reactions involving second-order steps in both directions. The following examples are taken from the many studies based on line-broadening analyses from nuclear magnetic resonance (NMR) and electron spin resonance (ESR) spectroscopy, respectively. (a) Proton transfer: k1

A + H3 O+

 AH+ + H2 O

(20)

k−1

(b) Electron transfer: k1

A + B±

 A± + B

(21)

k−1

4. Parallel second-order reactions occurring in two different zones of polymeric solids without mutual interference: k1

zone 1: R· + R· → RR k2

zone 2: R· + R· → RR

(22)

(23)

5. Relaxation from two excited states in equilibrium (both stages first-order): ∗A

k2

k1 k −1

A

∗B

k2 A

+ B

B

k1 k −1

1. Two parallel terms: k = A1 e−E1 /RT + A2 e−E2 /RT

(24)

∗AB

k3 A + B

(25)

7. Kinetics of photobleaching of a system of absorbance A under steady illumination (making allowance for increased transmission, and therefore reduced rate of photolysis, as photoreaction proceeds): −d A/dt = k(1 − 10−A )

(27)

This is often recognized from a plot of ln k against 1/T , which, instead of exhibiting a single line as for Eq. (1), will show two linear regions intersecting in a particular, narrow temperature range. 2. For solution reactions investigated over very wide temperature ranges, especially cryogenic temperatures when molecular diffusion can be “frozen” at a limiting temperature T0 : k = Ae−[B/(T −T0 )]

k3

6. Relaxation from two excited states in equilibrium (one stage second-order): ∗A

where dozens of processes are involved, some chemical and others, such as diffusion or turbulence, being physical. Here the task is not to express the situation exactly, as in Eqs. (18)–(26), but rather to approximate to the observed state by examining the effects of gradually introducing particular component terms. The computer is used to generate a numerical solution from a set of given rate equations, rate constants, and reaction concentrations for different reaction times, and a screen graphical presentation is often employed. This procedure is illustrated in Fig. 1 for the concentrations-time profiles for reactants and products on a photoirradiated mixture of trans-but-2ene, NO, NO2 , and air. The relevant equations are given in Fig. 2. Computer modeling is also helpful in unravelling the temperature dependence of complex systems. Fitting to the Arrhenius equation [Eq. (1)] is simple, but the number of parameters involved increases, and therefore the programs become lengthier, with the complexity of the system. Well-known variants of the simple Arrhenius equation to which data can be fitted by computer are as follows:

(28)

IV. GENERAL METHODOLOGY OF REACTION KINETICS In relatively simple systems, where reactants proceed directly to a small number of products without the intervention of intermediate species (and in most cases even when these do play some role), the main problem is that of determining the concentration of one or more reactants or products as a function of time. This becomes essentially an exercise in analytical chemistry, in that virtually all methods of chemical and physicochemical analysis have been applied at some time in determining reaction rates.

(26)

Computers prove invaluable in dealing with exceedingly complex kinetic situations such as are found in combustion or the chemistry of planetary atmospheres,

A. Conventional Methods The design of specialized kinetic apparatus is largely determined by the magnitude of the reaction rates concerned.

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FIGURE 1 Experimental (upper) and computer predictions (lower) for the photoreaction of a mixture of trans-but-2ene, NO, NO2 , and air. [From Kerr, J. A., Calvert, J. G., and Demerjian, K. L. (1972). Chem. Br. 8, 252.]

When reactions are “slow,” i.e., with half-lives of the order of minutes, hours, or longer, then conventional sampling methods are very appropriate; thus a thoroughly mixed reaction system is made up after thermostating the separate reactants to the desired temperature, and samples are taken at suitable, measured time intervals and submitted to analysis for a particular reactant or product. Because reaction continues within the sample while sampling is occurring, it is usually the case that one needs to “quench” the reaction either by cooling the sample rapidly to “freeze” reaction or by mixing the sample with another reagents, which halts reaction by consuming one reactant virtually instantaneously. This type of procedure is still widely used because of its simplicity, particularly when more sophisticated apparatus is unavailable. The analysis itself is typically by titration, ultraviolet (UV)–visible or infrared (IR) spectrophotometry, or gas-liquid chromatography. An important variant of this method is to use a nondestructive analytical method in which the composition

of the entire reaction mixture with respect to one or more components is monitored throughout the reaction, from time to time or even continuously, by some physical method such as spectrophotometry, conductivity, potentiometry, or NMR. Commercial spectrophotometers and magnetic resonance spectrometers are now widely used in which a kinetic mode is available for automatically determining optical or radiofrequency absorbances at preset band maxima as a function of time: the resulting data, sometimes running to thousands of points, can be stored and analyzed at leisure. Polymerization kinetics can be followed very simply using a dilatometer (a reaction vessel consisting of a bulb with a capillary side-arm) to measure total reaction volume, which decreases gradually but markedly as reaction proceeds. Many “slow” gas phase reactions have been studied simply by measuring the total pressure of a “static” reaction system at different times. Such an approach can be dangerously simplistic, however, and it is best

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FIGURE 2 Reaction schemes for the photoxidation of trans-but-2-ene in the presence of NO and NO2 . (Left) Initiation of O atoms or ozone. (Right) Initiation by hydroxyl radicals. (These are incorporated into the computer model employed in Fig. 1.) [From Kerr, J. A., Calvert, J. G., and Demerjian, K. L. (1972). Chem. Br. 8, 252.]

supplemented by an in situ analytical method such as spectroscopy or by sampling tiny quantities into a mass spectrometer, if necessary via a gas chromatograph. While the static system has the advantage that it is well-defined in terms of temperature, concentration, surface area, and material of the containing vessel, it suffers from the comparatively small amounts of material available for analysis. Flow systems for gases provide plenty of material but are less well-defined otherwise. An often useful compromise is found in the capacity-flow technique, in which reactants are conducted into the vessel, efficient stirring is provided, and the reacting mixture is withdrawn at such a rate as to set up a steady state.

Kinetic processes occuring within solids are inherently more difficult to monitor, but ESR spectroscopy provides a nondestructive, highly sensitive method for investigating free radical processes. The development in NMR spectroscopy of magic-angle spinning (MAS) enables solid-state transformations to be determined, conventional NMR having been limited previously to bulk physical processes such as the transition through the glass temperature of polymers. One nice application of MAS-NMR is the kinetics of the hardening of wet cement by following the 29 Si resonances. Optical methods are most easily applied to thin films of materials— for example, the kinetics of polymer degradation can be

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determined from the development of the IR peak of the carbonyl group frequency in degraded polyalkenes. B. Steady-State Methods Many complex reactions proceed at a relatively slow rate as measured by the gross disappearance of reactant or appearance of product, and yet the measured rate, characterized by an “experimental” rate constant k, can include components relating to very fast processes. This is the case especially for chain reactions, where the number of reaction chains existing at a given moment is very small, but the individual processes associated with each center occur extremely rapidly. To resolve such a composite rate constant into its individual components normally requires some fast-reactions technique that concentrates on the initiation or termination stages. Another such example is given by fluorescence quenching. If a molecule A yields a fluorescent excited state A∗ that is partially quenched by an added material Q, then a simple kinetic analysis of the quenching data will reveal the second-order quenching constant as follows. Consider the reaction scheme:

kF [A∗ ] (kF + kNR )[A∗ ]

(38)

The ratio of the fluorescence yields, and therefore measured fluorescence intensities, IF and IF0 , in the presence and absence, respectively, of the quencher molecule is given by ϕF0 I0 (kF + kNR + kQ [Q])[A∗ ] = F = ϕF IF (kF + kNR )[A∗ ]

(39)

which can be simplified to IF0 /IF = 1 +

(kQ [Q])[A∗ ] (kF + kNR )

(40)

Thus a plot of IF0 /IF versus [Q] will yield a straight line of slope kQ /(kF + kNR ). This is called the Stern–Volmer constant, and it can be measured using a conventional spectrofluorimeter. Stern–Volmer constants can be measured for the quenching of a single emitting species by a range of quenchers and then factorized into their components simply by measuring the lifetime τ0 of the emitting state A∗ in the absence of quencher by some fast-reactions technique (Section VI). The term τ0 is related to kF and kNR by τ0 = 1/(kF + kNR )

Rate Ia

(29)

kF

kF [A∗ ]

(30)

kNR

kNR [A∗ ]

(31)

kQ

kQ [A∗ ][Q]

(32)

C. High-Pressure Methods

(33)

The rate of a chemical reaction is sensitive not only to temperature [Eq. (1)] but also to pressure. Simple thermodynamic arguments lead to Eq. (42):

hν

A −→ A

∗

A∗ −→ A + hvF A∗ −→ A Q + A∗ −→ A + Q

The rate of formation of A∗ is given by d[A∗ ]/dt = Ia −d[A∗ ]/dt = (kF + kNR + kQ [Q])[A∗ ]

(34)

In the steady-state situation, these two rates are equal, i.e., Ia = (kF + kNR + kQ [Q])[A∗ ]

(35)

Now the fluorescence quantum yield is given by the ratio of the number of photons emitted per second to the number of photons absorbed per second: ϕF =

kF [A∗ ] Ia

(36)

Substituting Eq. (36) into Eq. (35) yields Eq. (37): kF [A∗ ] (kF + kNR + kQ [Q])[A∗ ]

(37)

In the absence of any quencher, the fluorescence yield, denoted ϕF0 , is given by Eq. (38):

(41)

Another type of steady-state method is that of linebroadening of magnetic resonance spectra, covered in detail in Section VIII.

ln(k/k0 ) = − pV = /RT

The rate of disappearance of A∗ is given by

ϕF =

ϕF0 =

(42)

Here k and k0 refer to the rate constants of the reaction at pressures of p and l atm, respectively, and V = denotes the volume of activation. Accuracy in values of V = requires the use of pressures of up to several thousand atmospheres, as the effect is generally rather small (a few cubic centimeters per mole), and the reaction chamber is usually of thick stainless steel. The term V = is considered in the case of solution reactions to be composite, i.e., the sum of (1) the change in volume of the reactants in forming the transition state and (2) the change in volume of the solvating molecules during this process; the latter is particularly important when ions are involved as the reactants. When a process of molecular dissociation is involved, V = will be positive, while when molecules associate to form the transition state, V = is negative; these effects have been used diagnostically in solution kinetic studies of inorganic complexes, e.g., by Henry Taube.

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Recent technical developments of the method involve the use of the NMR method (Section VIII) at high pressures, both in steady-state and stopped-flow applications. The sample is contained in an open glass sample tube situated within a metallic pressure chamber; the metal has to be nonmagnetic in this case, and beryllium–copper alloy or titanium are the favored materials.

V. FLOW SYSTEMS These have been used to great effect in both solution and gas-phase studies. A. Continuous Flow (Solutions) The first successful use of continuous-flow systems in solution was due to Hartridge and Roughton; the principle here is that if the reaction time is of the order of a few milliseconds to a few seconds, then it is impossible to mix the reactants manually and then record optical changes. Instead, the two reactants are fed by gravity or under pressure through narrow-bore tubes to a mixing chamber, after which observations are made at various points remote from the mixing point in order to ascertain the extent of reaction. Alternatively, the flow rate can be varied to display varying time domains of the reaction at a fixed observation point. The disadvantage of the continuous-flow configuration is that considerable quantities of reactants

are used and observations are needed at different points in the mixed stream. A useful variant of the continuous-flow method is that in which the stream of mixed reactants is conducted into a quenching medium, which terminates reaction by eliminating one reactant virtually instantaneously. An adaptation of this idea is to pass the reacting solution into a Dewar vessel containing liquid nitrogen at 77 K when the reaction is literally frozen, and the extent of reaction, or the presence of some intermediate, is ascertained by a solid-state technique such as ESR spectroscopy. B. Stopped-Flow Methods The disadvantages of the continuous-flow method are largely overcome in the stopped-flow technique. After the reactants have been forced from their “drive syringes” into the mixing chamber, adjacent to the optical cell where an analyzing light beam is situated, the emerging mixed solution runs into another vertically mounted “stop syringe,” which rises as it fills until it is arrested by a mechanical stop (Fig. 3). Flow then ceases, but the fast reaction occurring in the optical cell continues to completion, normally in a time of between 10 msec and 10 sec. In recent adaptations the analyzing light, instead of passing into a monochromator–photomultiplier tube–oscilloscope–microcomputer assembly, is passed to a diffraction granting, which disperses the light over the faces of a series of photomultiplier tubes or photodiodes

FIGURE 3 Schematic diagram of stopped-flow apparatus. A, reservoir syringes; D, drive syringes; R, hydraulic ram; 1–5, valves; S, stopping syringe; L, light source; MO, monochromator; C, quartz windows; M, mixing point; PM, photomultiplier tube; HT, power supply to PM; MS, microswitch; W, waste; MA, microammeter; CRO, cathode-ray oscilloscope; TR, transient recorder; CR, computer.

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FIGURE 4 Time-resolved spectra (0–2 sec after mixing, taken at 40 msec intervals) for the reaction of Zn2+ with 2,2 :6 ,2

–terpyridine (L) in methanol:water (80%:20% v/v) at 25◦ C (Zn2+ ] = 10−4 mol dm−3 , [L] = 10 − 5 mol dm−3 ). [From Priimov, G. U. Ph.D. thesis, University of Warwick, 1999.]

(in the so-called photodiode array), each of which is separately powered and from which the individual outputs feed into a multiplexer. The output from the letter is presented on an oscilloscope screen as a complete optical spectrum, each data point corresponding to the signal from one of the photodetectors. The use of electronic delays enables a complete time-resolution of the spectrum to be built up (Fig. 4): An important development in the stoppedflow technique has been the incorporation of Fouriertransform NMR spectroscopy as the detection system. Another development of stopped-flow spectrophotometry is the quenched-flow system. Here, two solutions A and B are mixed to produce a highly reactive intermediate, which persists until it encounters a third reactant C at a second mixer. The resulting reacting mixture is analyzed optically by the usual optical train (Fig. 5).

C. Gas-Flow Methods The possibility of flowing reactant gases at low pressures down long tubes has long been recognized as a means of controlling a fast reaction. An additional feature is the use of some excitation source, often a microwave discharge, at a suitable location in order to generate atoms or small molecules such as ·CN, ·OH, ·Cl, or ·H. Detection systems include optical spectroscopy (especially for chemiluminescent species), electron spin resonance spectroscopy, and atomic resonance fluorescence. Figure 6 illustrates the use of mass spectrometric detection in a gas-flow system. High-capacity pumps remove most of the gas flow,

with only a small fraction entering the very low pressure (∼10−6 torr) mass spectrometer. One example of the type of reaction that can be examined is that between ethyl radical ( C2 H5 ) and oxygen, for which the stages are x

(i) Generate chlorine atoms in microwave discharge: Cl2 −→ 2Cl

(43)

x

(ii) React Cl with ethane: x

Cl + C2 H6 −→ HCl + C2 H5 x

x

(44)

(iii) React the generated ·C2 H5 with oxygen: C2 H5 + O2 −→ C2 H5 O2

x

x

(45)

The C2 H5 radicals are ionized in the mass spectrometer to yield C2 H5 + ions which are detected at a mass/charge ratio of 29. The moveable injector controls the relative concentration of ethyl radicals. Resonance fluorescence detection is a highly sensitive method for observing atomic species. A trace of gas such as hydrogen is mixed with helium and passed into a microwave discharge in which some hydrogen is dissociated: x

H2 −→ 2H

x

(46)

Some of the H atoms are excited and emit their typical fluorescence ∗

H −→ H + hvF x

x

(47)

The so-called resonance emission enters the reaction zone (Fig. 7) where ground-state H atoms become excited

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distribution as possible by means of a velocity selector (a set of slotted rotating discs). Better velocity selection can be achieved by using a supersonic nozzle in which the gas expands from a high pressure through a small nozzle into a vacuum. A beam of molecular ions can replace one of the neutral molecular beams. The detection system relies on the ionization of the scattered fragments as they strike the surface. Alternatively, the scattered species can be submitted to electron bombardment followed by mass spectrometric examination of the resulting ions. The internal energy state of the product molecule is often determined by laserinduced fluorescence, although in some cases infrared chemiluminescence can be observed directly from vibrationally excited states of the products. The geometry of the experimental arrangement is illustrated in Fig. 8.

VI. PULSE AND SHOCK METHODS These depend on the virtually instantaneous generation of new chemical species, normally of high reactivity, in a system by delivery of a burst of energy sufficient to generate electronically excited states or to break chemical bonds to generate radical species. FIGURE 5 (Top) Side and (bottom) plan views of a quenchedflow apparatus. Key: RES, reservoirs (three); DS, drive syringes (three); MS, microswitch; R, rack; P, pinion; M, motor; TI, thermostating fluid inlet; TO, outlet; M1, first mixer; M2, second mixer; Q, quartz windows (two); W, waste outlet; BN, brass nut (three); PS, motor power supply; L, lamp; MO, monochromator; FL, focusing lens; PM, photomultiplier tube; MA, milliammeter; CB, control box; CRO, cathode-ray oscilloscope; DR, data recorder; CR, chart recorder; 1–6, taps; shaded area, PTFE blocks. [From Goodman, P. D., Kemp, T. J., and Pinot de Moira, P. (1981). J. Chem. Soc., Perkin Trans. 2, 1221.]

H + hvF −→ ∗ H

x

A. Flash Photolysis In the technique of flash photolysis, later developed to laser flash photolysis, an intense flash of light is absorbed by molecules in the system to give excited states: hν

M → M∗

These then lose energy either by light emission [Eq. (50)], by nonradiative decay [Eq. (51)], by dissociation to give radical species [Eq. (52)], or by attacking the solvent, denoted SH [Eq. (53)]:

(48)

The concentration of H atoms in the reaction cell can thus be monitored by their fluorescence in a repeat of Eq. (47), and their reactions with the various molecules in the reaction cell determined. D. Crossed Molecular Beams Extremely detailed kinetic information for gas-phase processes is gained by allowing two streams of molecules emanating from small ovens to intersect at right angles and by examining the scattering intensity and velocity distribution of the resulting fragments as a function of the scattering angle. The experiment is carried out at high vacuum (10 − 6 to 10 − 7 torr), and the molecules in one or both of the streams are placed in as narrow a velocity

(49)

M∗ → M + hν1

(or hνp )

∗

M → M.

(51)

M∗ → A· + B· ∗

(50)

·

M + SH → MH + S

(52) ·

(53)

By monitoring the optical or ESR absorption of the species A· , B· , MH· , or S· , it is possible to determine their kinetics for such processes as dimerization, as in Eqs. (54) and (55). S· + S· → S—S

(54)

MH· + MH· → HM—MH

(55)

or reaction with oxygen, S· + O2 → SO·2

(56)

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FIGURE 6 Schematic diagram showing combination of gas-flow tube with mass spectrometric detection. [Reproduced with permission from Pilling, M. J., and Seakins, P. W. (1995). “Reaction Kinetics,” Oxford University Press, Oxford.]

While many types of experimental layout have been used successfully, that shown in Fig. 9 is quite typical. A pulsed excimer laser generates light at a wavelength that can be controlled by using different halogens or rare gases. The uv pulse, of duration that can also be controlled, can be used either directly or to pump a dye laser, giving extensive control of the wavelength range. The emerging pulse is then fed by mirrors, prisms, or an optical fiber to the sample cell where it excites the molecule of interest (in fluid solution or in gaseous form). Species generated by Eq. (49)–(56) are monitored by means of an analyzing beam from a high-pressure pointsource lamp, the output of which passes through the sample cell and thence through a preset monochromator to a photomultiplier tube or spectography or diode array. The signal from the former, which is related to the intensity of the incident light, is taken to a cathode-ray oscilloscope where it is displayed and stored as a function of time immediately following the firing of the laser. The stored signal is then either photographed for manual analysis or taken via an analog-to-digital converter to a microcomputer, where it is stored and subsequently has largely been processed to yield rate constants. The other main instrumental development involving lasers has been the evolution of lasers pulsed to picosecond or even femtosecond time intervals by mode-locking. These yield rather weak pulses and are normally used in association with devices enabling data accumulation. The Nobel prize for chemistry for 1999 was awarded to Ahmed Zewail for his work in extending flash photolysis

into the femtosecond (fs) regime, enabling, for example, observation of a transition state in the breaking of the I-C bond in ICN. The concept underlying this work is schematized in Fig. 10. The laser system generates a pulse which is split into two components, a pump pulse which instigates chemical processes and a probe pulse which monitors them. The probe pulse is delayed behind the pump pulse by a few femtoseconds by lengthening its light path (Fig. 10). The probe pulse causes fragments generated by the pump pulse to emit light, the characteristics of which provide dynamical information. An example of the fundamental character of this type of investigation is illustrated in Fig. 11. Ion pairs Na+ I− are excited by the pump pulse to the excited form [Na I]∗ . The bond distance is very short at the moment of formation, and the excited molecule has a covalent character. As the molecule vibrates, charge moves toward the I atom, and at the furthest point of the ˚ apart, with the stretching vibration the nuclei are 10–15 A bonding now becoming fully ionic [Na+ · · ·I− ]. The atoms continue to vibrate between these two forms: [Na − l]∗

 [Na+ . . . l− ]∗

(57)

˚ the energies of the exAt the critical distance of 6.9 A, cited and ground states are very close and there is a 20% chance that the excited state will dissociate to give separate Na and I atoms. The femtosecond technique detects a decaying signal from the excited state and an increasing signal as bursts of Na atoms appear from each vibration (Fig. 11).

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FIGURE 7 Schematic diagram illustrating the various processes occurring during reaction of H atoms with resonance fluorescence detection. [Reproduced with permission from Pilling, M. J., and Seakins, P. W. “Reaction Kinetics,” Oxford University Press, Oxford.]

Lasers are particularly well suited for the timeresolution of light emission [Eq. (50)], especially in the form of time-correlated single-photon counting (Fig. 9). In this technique, the weak output from a repetitively pulsed lamp (pulse width ∼1 nsec) or from a sub-nanosecond pulsed laser is divided such that part of the pulse is taken directly to one photodetector (PM 1 in Fig. 9) while the other part is taken to the sample cell. The signal detected by

PM 1 is conducted to a time-to-amplitude converter (TAC), which is triggered or “started” on its arrival. The photon arriving at the sample cell excites a potentially luminescent molecule [Eq. (49)], which, after some time, fluoresces [Eq. (50)] or phosphoresces [Eq. (50)]. The photons due to luminescence activate the second photodetector (PM 2 in Fig. 9), from which a pulse is fed to the TAC, which is then “stopped.” The time interval between “start” and

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by discharging a bank of capacitors through a quartz tube filled with Kr or Xe: this produces a broad-continuum flash of between a few microseconds and ∼100 µsec duration. The time-resolution of this apparatus is orders of magnitude inferior to that of laser-flash photolysis, but larger energies are available in view of the greater spectral width and timespan of the exciting source, i.e., hundreds or thousands of joules rather than fractions of a joule, and so the microsecond technique still enjoys considerable use for particular problems. B. Pulse Radiolysis

FIGURE 8 Schematic diagram illustrating laser flash photolysis. Laser pulses pass through the light guide to the sample cell. Optical monitoring of post-pulse events can be kinetic or spectrographic.

“stop” is stored in the multichannel pulse-height analyzer and the process is repeated for many thousands, if not hundreds of thousands, of times per second. The accumulated data yield the averaged luminescent response of the sample molecule and, after computer processing, can lead to lifetimes with very small standard deviations. In the more traditional form of flash photolysis, originated by R. W. Norrish and G. Porter, the flash is generated

The technique of pulse radiolysis is closely related to that of flash photolysis: the optical monitoring system is the same, but the source of light activation is replaced by one of high-energy radiation in the form of a short pulse of a few nanoseconds or microseconds of fast electrons (typically 3 MeV). The effect of such high-energy radiation is to excite and ionize the material in the sample cell, which may be in liquid, gaseous, or even (transparent) solid form. In the case of water, the initial act of radiolysis is given by Eqs. (58)–(60): H 2 O → H 2 O+ + e − x

+

·

H2 O + H2 O → OH + H3 O

+

− e− + H2 O → eaq

(58) (59) (60)

Optical studies reveal the presence of the solvated electron − eaq as a broad, intense absorption maximizing at ∼700 nm.

FIGURE 9 Block diagram of time-correlated single photon counting apparatus. Key: LS, lamp; LH, lamp housing; S, slits; HRM, high-radiance monochromator; C, sample cell; SH, sample housing; L, lenses; PM 1, PM 2, photomultiplier tubes; DISC 1, DISC 2, discriminators; TAC, time-to-amplitude converter; MCAPH, multichannel pulse-height analyzer.

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FIGURE 10 Schematic diagram illustrating layout of femtosecond flash photolysis technique. The pump and probe pulses are separated in time by adjusting the light path of the latter. Beams of molecules in the sample tube are excited or dissociated by the pump pulse, and the fragments monitored by the probe pulse on its path to the detector. Reproduced with permission from Scientific American (see Bibliography).

Addition of materials capable of reacting with electrons − , enabling kinetics of its fast rereduces the lifetime of eaq actions to be determined. The powerful oxidant ·OH (the hydroxyl radical) has only a weak absorption in the ultraviolet, and its reactivity is best measured by a competition method based on its very fast oxidation of thiocyanate ion − CNS− to yield the intensely absorbing (CNS)2 ion (λmax 472 nm). Addition of a second substrate X will provide competition for · OH, and the intensity of the absorption of − (CNS)2 will be systematically reduced as [X] is increased, enabling a rate constant to be derived. While optical methods remain the favored means of analysis in both flash photolysis and pulse radiolysis, other methods of detection have been used with great effectiveness from time to time, including conductivity and ESR spectroscopy. The latter technique, in association with flash photolysis in particular, has led to the observation of ESR signals with anomalous intensities, for example, appearing totally in emission, a phenomenon described as chemically induced dynamic electron polarization or CIDEP. x

x

Other developments include extension of the optical range into the near IR and the use of cryogenic equipment to examine radiolysis at temperatures as low as 4.2 K. C. Shock Tubes Shock tubes are applied to the study of gas phase reactions on the microsecond to millisecond timescale at temperatures of several thousand kelvins. The shock wave is generated by breaking a diaphragm that separates the “driver” gas, usually H2 or He at several atmospheres pressure, and the reactant gas diluted in argon at a few torr (1 torr = 10−6 mm Hg). The normal configuration for the apparatus is a hollow tube (Fig. 10), typically measuring 15 cm diameter and 6 m in length, which contains, in the reactant chamber, sensors for measuring the velocity of the shock wave, and an observation point. The shock front generated on rupture of the diaphragm travels at supersonic speed (several Mach) toward the reactant gas zone, compressing this and heating it to very high temperatures (which can be calculated), in much less than

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FIGURE 11 Femtosecond flash photolysis of gaseous sodium iodide. As the excited ion-pair vibrates, it gradually decays (lower curve) and the resulting free atoms are detected (upper curve).

1 msec. Detection of the species in the reaction zone is normally optical, especially as many of these emit light from their excited states, although absorption spectra can also be measured. Strong shocks in CH4 and CH4 − NH3 yield, respectively, the emission bands of C2 and CN, while Al2 O3 dispersed as a dust yields AlO. Alternatively, the reaction zone can be sampled by allowing materials to leak through a pinhole into a mass spectrometer.

VII. RELAXATION METHODS These were developed initially by Eigen and depend on the application of a small disturbance to a chemical system at equilibrium, normally by the dissipation of a pulse of energy in the temperature-jump method but also on occasion by a sudden change in pressure (pressure-jump) or electric field. The system then adapts (Fig. 11) to its new situation, normally a slightly higher temperature and hence a changed equilibrium constant, at a rate that can be measured either optically or conductimetrically. The

duration of the pulse should be much shorter than the relaxation time τ of the system and should approximate to a step-function as in Fig. 11. Interestingly, if the relative change in concentrations of the reactants is very small, then, whatever the rate law, the reaction curve will follow first-order kinetics described by the relaxation time τ , the time for A (the change in absorbance at time t) to fall to a value (l/e)(A0 ) (where A0 is the final total change in absorbance). The mathematics of rather complex systems undergoing relaxation have been established and are given by Hague. While the single-pulse methods described above are widely used, another approach is to apply the perturbation as a sine wave, in which the response will be in the form of another sine wave lagging consistently behind the perturbation to a degree related to the relaxation time of the system. The frequency of the applied perturbation is critical: if it is too slow, then the chemical response will “catch up” and no lag will be detected, but if it is too fast the chemical response will never attain a measurable amplitude before it must change direction. Only

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in the all-important intermediate situation will the amplitude of the chemical response be sufficient to enable determination of the phase lag. Perturbations can take the form of high-frequency oscillating electric fields, but the use of sound waves (to give the ultrasonic method) is that most widely adopted. The attenuation of the sound wave at a particular frequency is related to the chemical relaxation processes occurring in the solution. The excitation is via a quartz crystal generator, and detection is by measuring optical dispersion of light transmitted through the sample.

VIII. SPECTRAL LINE-BROADENING METHODS

N (υB) CH3

H3C

O





N

C H

O C

H3C

H (62)

and will resonate at different frequencies νA and νB giving two lines. Rotation about the N C bond is slow at room temperature, because of the partial double-bonding [Eq. (62)], but as the temperature is increased rotation becomes faster and finally so fast that the signals coalesce to a single line. Between these two extreme situations (known as slow and fast exchange, respectively) lies a continuum of intermediate situations with line shapes given by Eq. (63), g(ν) = [(1 + τ π )P + QR]/(4π 2P 2 + R 2 )

If the lifetime of a molecule is very short, say δt, then the uncertainty principle predicts a broadening of its absorption line δν (in hertz) given by Eq. (61). δν ≈ 1/(2π δt)

(υA ) CH3

(61)

If δt is ∼0.1 sec, then δν is ∼1 Hz, i.e., in the region of NMR spectroscopy. Accordingly, if a molecule is undergoing rapid (∼0.1 sec) interchange between two conformations, or is participating in a fast exchange reaction such as proton-transfer, then its NMR spectral linewidth provides a unique source of kinetic information. We cannot develop the theory of NMR spectroscopy here, but state simply that the nuclei of a sample in a strong magnetic field are excited by radiofrequency radiation of a particular frequency for each type of nucleus (called the Larmor frequency) into excited spin states, from which they return to lower spin states on losing spin energy by nuclear relaxation. Various factors contribute to such relaxation, especially the motion of surrounding solvent molecules designated (even in liquids) as the “lattice”; this process is called spin–lattice relaxation, and the time for l/e part of the excited nuclei to relax this way is denoted T1 . The additional process titled “spin–spin relaxation” is denoted T2 . Both of these are normally of several seconds duration in liquids, and, in accordance with Eq. (61). NMR lines are accordingly very narrow in liquid samples. Chemical pathways for reducing the lifetime reduce T2 and hence broaden the lines. Analysis of such broadening leads to values for the kinetics of the chemical pathway. Another factor influencing the appearance of NMR spectra is the exchange of nuclei (usually protons) between positions with different Larmor frequencies. In the case of dimethylformamide, the protons in the two methyl groups in the planar conformation are inequivalent.

(63)

where P = [0.252 − ν 2 + 0.25δν 2 )τ + /4π

(64)

Q = [−ν − 0.5( pA − PB )δν]τ

(65)

R = 0.5( pA − pB )δν − ν(1 + 2π τ )

(66)

The symbols are defined as follows: τ = τA τB /(τA + τB ), and τA and τB are the average lifetimes of the nuclei in positions A and B, respectively; pA and pB are the mole fractions of A and B, respectively; δν = νA − νB ;  is the width (at half-height) of the signal in the absence of exchange; and ν is the variable frequency. The lineshape calculations are exemplified in Fig. 12. The ESR spectrum of a dilute solution of a radical anion such as naphthalene consists of many lines. Addition of further naphthalene results in electron exchange [Eq. (67)], which broadens the lines: k2

÷

C10 H÷ 8 + C10 H8  C10 H8 + C10 H8

(67)

k−2

This is a consequence of reducing the lifetime of a particular spin state: an analysis of the lineshape yields the rate constant k2 , which equals k−2 . Radicals able to undergo conformational change show a spectral phenomenon called the alternating linewidth effect. A temperaturedependence study of this effect will yield activation parameters [energy and entropy, Eq. (2)] for this first-order process.

IX. ELECTROCHEMICAL METHODS Electrochemical methods depend on a competition between an electrode reaction, e.g., Eq. (68),

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removed from the system. Hence the (measured) cell current is proportional to the rate at which species X arrives at the electrode surface. Motion of X is due to (1) diffusion, (2) convection, and (3) migration, and conditions are arranged such that diffusion dominates. Diffusion occurs because of the depletion in the concentration of X near the electrode, resulting in a concentration gradient. If X is participating in a dynamic equilibrium in the bulk electrolyte solution. Eq. (69) where Y is electrochemically inactive and with the equilibrium far to the right, i.e., k1 k−1 , k1

X

 Y, K = k1 /k−1

(69)

k−1

FIGURE 12 Computed nuclear magnetic resonance spectra for the exchange process A

 B as a function of the parameter τ . [From Gunther, ¨ H. (1980). “NMR Spectroscopy,” Wiley, Chichester.]

X + e− → R

(68)

and the diffusion of the electroactive species X to the electrode surface. The potential of the electrode is selected so that Eq. (68) is essentially rapid and irreversible, resulting in all species X encountering the electrode being

then the rate of arrival of X at the electrode depends on the rate of generation of X from Y, i.e., the situation is under kinetic control. The mean lifetime τ of X is given by 1/k1 . Species X will diffuse a distance µ = Dτ during its mean lifetime, where D is the diffusion coefficient. Thus, a fraction 1/e of species X within distance µ of the electrode will be removed, while those farther away will be transformed to Y before they can be removed. The layer of solution of thickness µ is termed the reaction layer (Fig. 13). The current at the electrode is then determined by the rate at which species X is generated in the reaction layer. A forced convection electrode is exemplified in the technique of polarography, the cathode consisting of a growing mercury drop at the tip of a fine capillary, which provides a constantly renewed, reproducible electrode surface. A steadily increasing potential is applied to the electrode and the resulting current measured, to give a polarographic

FIGURE 13 Steady-state concentration distribution (reaction layer) in the case of a chemical reaction preceding an electrode reaction. [Adapted from Koryta, J., and Dvorak, J. (1987), “Principles of Electrochemistry,” Wiley, Chichester, England.]

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wave. The relation between the diffusion current, i d and the reaction current i r provides the kinetic information via Eq. (70) 1/2

ir Ck1 t 1/2 = id − ir K 1/2

(70)

where C is a constant of approximately unity and t is the drop time.

SEE ALSO THE FOLLWING ARTICLES ELECTROCHEMISTRY • KINETICS (CHEMISTRY) • NUCLEAR MAGNETIC RESONANCE

BIBLIOGRAPHY Andrews, D. L. (1992). “Lasers in Chemistry,” 2nd ed., Springer–Verlag, Berlin. Ashfold, M. N. R., and Baggott, J. E. (1989). “Bimolecular Collisions,” Royal Society of Chemistry, London. Baxendale, J. H., and Busi, F. (1981). “The Study of Fast Processes and Transient Species by Electron Pulse Radiolysis,” Reidel, Dordrecht. Bensasson, R. V., Land, E. J., and Truscott, T. G. (1983). “Flash Photolysis and Pulse Radiolysis. Contributions to the Chemistry of Biology

and Medicine,” Pergamon, Oxford. Billing, G. D., and Mikkelsen, K. V. (1996). “Molecular Dynamics and Chemical Kinetics,” Wiley, New York. Demas, J. N. (1983). “Excited State Lifetime Measurements,” Academic Press, New York. Espenson, J. H. (1995). “Chemical Kinetics and Reaction Mechanisms,” 2nd ed; McGraw-Hill, New York. Greef, R., Peat, R., Peter, L. M., Pletcher, D., and Robinson, J. (1985). “Instrumental Methods in Electrochemistry,” Ellis Horwood, Chichester. Hammes, G. G., ed. (1974). “Techniques of Chemistry,” 3rd ed., Vol. VI. Part II, Wiley (Interscience), New York. House, J. E. (1997). “Principles of Chemical Kinetics,” W. C. Brown, Dubuque, Iowa. Lewis, E. S., ed. (1974). “Techniques of Chemistry,” 3rd ed., Vol. VI. Part I, Wiley (Interscience), New York. Ng, C.-Y., Baer, T, and Powis, I. (1994). “Unimolecular and Bimolecular Ion-Molecule Reaction Dynamics,” Wiley, Chichester. Pilling, M. J., and Seakins, P. W. (1995). “Reaction Kinetics,” Oxford University Press, Oxford. Sandstr¨om, J. (1982). “Dynamic NMR Spectroscopy,” Academic Press, New York. Steinfield, J. I., Francisco, J. S., and Hase, W. L. (1989). “Chemical Kinetics and Dynamics,” Prentice-Hall, Englewood Cliffs, New Jersey. Wilkins, R. G. (1991). “Kinetics and Mechanism of Reactions of Transition Metal Complexes,” VCH, Weinheim, Germany. Wright, M. R. (1999). “Fundamental Chemical Kinetics,” Horwood Publishing, Chichester. Zewail, A. (1990). “The Birth of Molecules,” Scientific American, December issue, 40–46.

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Chemical Thermodynamics J. Barthel and R. Neueder University of Regensburg

I. Fundamentals II. Fundamental and Caloric Equations of Thermodynamics III. Partial Molar Quantities IV. Fugacities and Activities V. Thermodynamic Equilibrium

GLOSSARY Activity coefficient Ratio of the activity to the mole fraction of a component Yi in a mixture or solution; a measure of the departure from ideal behavior. Affinity Driving force of chemical reactions; equals zero at the equilibrium state. Chemical potential Content in Gibbs energy of 1 mol of a component Yi of a mixture or solution, that is, the change in the total Gibbs energy of the system at constant temperature and pressure when 1 mol of component Yi is added to an infinite amount of the system. Electrochemical system System consisting of electrically conducting phases. The electric potentials of the phases are called Galvani potentials. Excess property Difference between the actual property of a system and its hypothetical value calculated for an ideal mixture or solution at the same temperature, pressure, and mole fraction composition. Extensive property (variable) Property (variable) of a system that is proportional to mass. Heterogeneous system System consisting of phases that

are homogeneous systems and phase boundaries at which the intensive properties show discontinuities. Homogeneous system System of which all intensive properties are continuous functions of position throughout the system. Ideal mixture or solution Mixture or solution that shows no change in volume, enthalpy, or heat capacity when it is made up from its initially separated components. Intensive property (variable) Property (variable) of a system that is independent of mass. Partial molar quantity Increase in any extensive thermodynamic property of a system at constant pressure and temperature when 1 mol of component Yi is added to an infinite amount of the system. The chemical potential of component Yi is also the partial molar Gibbs energy of compound Yi . Phase See Heterogeneous system. Reversible process Process yielding the maximum of usable work. Thermodynamic equilibrium State of a system at which no measurable changes of its intensive properties occur and no measurable flow of matter or energy takes place during the period of observation.

767

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768 CHEMICAL THERMODYNAMICS is a phenomenological science. The object of investigation is the macroscopic properties of chemical systems in thermodynamic equilibrium. Today Gibbs formalism is generally used to yield the framework of thermodynamic compatibility throughout the phenomenological sciences and technology. A multitude of methods, partially based on extrathermodynamic assumptions, are applied to rationalize the observations of chemical systems and for the procurement of basic data for chemical thermodynamic calculations.

I. FUNDAMENTALS A. Thermodynamic Systems and Properties A chemical system is an accumulation of chemical materials limited by a boundary. The physical world outside this boundary constitutes the surroundings of the system. The system may (open system) or may not (isolated system) exchange mass and energy with its surroundings. A system that exchanges energy but not mass is called a closed system. A macroscopic property of a system may (for extensive properties) or may not (for intensive properties) depend on its mass. Examples of extensive properties are the volume of the system or its heat capacity; examples of intensive properties are permittivity, density, and specific heat. A system is considered homogeneous if all intensive properties are continuous functions of position throughout the system; otherwise it is considered heterogeneous. In a heterogeneous system the discontinuities of properties are situated on surfaces enclosing homogeneous regions, which are called phases of the heterogeneous system, with the surfaces being the phase boundaries. Phases are characterized throughout this article by Greek superscripts (e.g., V (α) , V (β) ), phase boundaries by symbols such as V (α/β) . A homogeneous system or a phase of a heterogeneous system can be a gaseous, liquid, or solid system. Systems showing electrical potentials (Galvani potentials) of their phases are called electrochemical systems. Galvani potentials are the result of movable charges (ions, electrons, etc.) within the phases. For electrochemical systems electrical potential differences are observed between electrically conducting phases (e.g., batteries). A system is in a state of thermodynamic equilibrium if during the period of observation (1) no measurable changes of its intensive properties occur and (2) no measurable flow of matter or energy takes place. The system is in a steady state if only the first condition is fulfilled. Electrochemical systems in equilibrium show constant Galvani potentials throughout each phase. Chemical thermodynamics uses observable quantities for the definition of the state of a system. The state of

Chemical Thermodynamics

a homogeneous system (phase) in equilibrium is defined unequivocally by a set of variables of state such as p (pressure), T (temperature), and n 1 , n 2 , . . . , n k (amount of substance). The mole number n i is a measure of the amount of substance of component Yi . The number of components of a system is the minimum number of independent chemical compounds Yi of which the system investigated can be made up. A system built by k components Y1 , Y2 , . . . , Yk is called a k-component mixture. Pure systems contain only a single component. The mole number n i relates the mass wi of component Yi to its molar mass Mi . n i = wi /Mi

(dimension of n i , moles).

(1)

According to the International Union of Pure and Applied Chemistry (IUPAC) Commission on Symbols, Terminology and Units, a mole is the amount of substance of a system that contains as many elementary entities as there are carbon atoms in 0.012 kg of 12 C. The elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles, as shown in the following tabulation: Elementary entity

Mass of 1 mol (g)

Hg HgCl Hg2+ 2 e− (electrons) Fe0.91 S Mixture of 78.09 mol % N2 , 20.95 mol % O2 , 0.93 mol % argon, and 0.03 mol % CO2

200.59 236.04 401.18 5.460 × 10−4 82.88 28.964

If the complete set of variables ( p, T, n 1 , . . . , n k ) of a k-component mixture is known, every property P of the system in its equilibrium state is unequivocally defined: P = P( p, T, n 1 , . . . , n k )

(2)

that is, an identical system can be prepared with the same properties by the specified set of variables. This statement is the basis of analytical chemistry. Infinitely small changes in the variables of state entail infinitely small changes in properties P:     ∂P ∂P dP = dp + dT ∂ p T,ni ∂ T p,ni +

 k   ∂P i=1

∂n i

dn i

(3)

p,T,n j =n i

In Eq. (3) the partial derivatives indicate the changes in P with one of the variables of state, the others being unchanged; dP is an exact differential as a consequence of

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the unequivocally defined, continuous, and continuously differentiable function P( p, T, n 1 , . . . , n k ). Applying to Eq. (3) the cross-differentiation identities known as Schwarz relations yields the relations ∂ 2 P/∂ T ∂ p = ∂ 2 P/∂ p ∂ T

(4a)

∂ 2 P/∂ n i ∂ p = ∂ 2 P/∂ p ∂n i

(4b)

∂ P/∂ n i ∂ T = ∂ P/∂ T ∂n i

(4c)

2

2

tional composition variables, molality (m i , moles of solute Yi per kilogram of solvent), demality (also called molonity, m˜ i , moles of solute Yi per kilogram of solution), and molarity (ci , moles of solute Yi per cubic decimeter of solvent). If in a mixture component Y1 is considered to be the solvent, Y1 = S, and Y2 , . . . , Yk are the solutes, the composition variables of the solutes are defined by the relations m i = n i /n s Ms  k  m˜ i = n i n i Mi

where (i = 1, 2, . . . , k). As a consequence of Eqs. (3) and (4) every finite change in property P is independent of the process applied (path of integration) when passing from an arbitrary intial state I to an arbitrary final state II,  II dP = PII − PI (5)

I

ci = n i /V

(6)

II

n s = n s1 + n s2 + · · · , Ms = xs1 Ms1 + xs2 Ms2 + · · · ,

In technological calculations the amounts of substance n i in the basic set defined in the preceding section are often replaced by other composition variables such as the weights of substance wi which are also extensive variables, or by intensive variables such as weight percent (wt. %), volume percent (vol %), or mole percent (mol %). A commonly used intensive composition variable in fundamental and applied research on mixed systems is the mole fraction xi :  k k   xi = n i ni ; xi = 1 (7) i=1

TABLE I Conversion of Concentration Scales in Binary Systemsa From To

x2

c2

m2

m˜ 2

x2 =

x2

M 1 c2 d + (M1 − M2 )c2

M1 m˜ 2 1 + (M1 − M2 )m˜ 2

c2 =

d x2 M1 + (M2 − M1 )x2 x2 M1 (1 − x2 ) x2 M1 + (M2 − M1 )x2

M1 m 2 1 + M1 m 2 dm 2 1 + M2 m 2

m˜ 2 =

(9)

where s1, s2 denote the components making up the mixed solvent. Electrochemistry uses mean composition variables based on the mole numbers of the ions in the solution produced by the electrolyte compounds that are the solutes. If the set of extensive composition variables (n 1 , . . . , n k ) is replaced by a set of intensive composition variables, for example, (x1 , . . . , xk ), where xi = 1, only the intensive properties of the system investigated are defined; extensive properties then are converted to appropriate intensive properties, which are the corresponding molar quantities (i.e., the amount of the extensive property per mole of substance).

Mixed systems distinguishing solvent and solute components are called solutions. Solution chemistry uses addi-

m2 =

(8c)

In Eq. (8c) the volume V of the solution is given in cubic decimeters. Conversion formulas for the frequently used scales of composition variables are given in Table I. If in a mixture the solvent itself is a mixture of components (mixed solvent), the quantities n s and Ms can be defined as “solvent quantities” when using the relations

B. Composition Variables

i=1

(8b)

i=1

I

and equals zero for a cyclic process,   II  I dP = dP + dP = 0

(8a)

c2 c2 d − M 2 c2 c2 d

d m˜ 2

m2

m˜ 2 1 − M2 m˜ 2

m2 1 + M2 m 2

m˜ 2

a Subscript 1 denotes solvent; subscript 2 denotes solute; d denotes density of the solution; M1 and M2 denote molar masses.

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C. Forms of Energy Energy can be transferred to and from a system and can be stored in a system in different forms: 1. 2. 3. 4. 5. 6. 7.

Mechanical energy, W mech Electrical energy, W el Electromagnetic energy, W elm Radiant energy, W rad Heat, W heat = Q Latent heat, W lat. heat = Q lat Chemical energy, W chem

tion of silver nitrate where silver deposition and zinc dissolution are observed. The chemical energy of the same process is used to a maximum as electrical energy by the help of an appropriate galvanic cell if the electrical current in the cell is kept infinitely small. This example illustrates the following features of reversible processes: 1. They are ideal limiting cases. 2. They are infinitely slow, passing from the initial to the final state through a series of equilibrium states. D. The Fundamental Laws of Thermodynamics

These can be converted totally or partially from one to another. By definition, any contribution that increases the energy of a system is counted positive. Any differential element of work, DW, is not an exact differential; that is, Schwarz relations are not applicable, and integration from the initial to the final state of the system depends on the path of integration (equation defining the particular process). If a system is subjected to a uniform, normal pressure p, the work executed on this system by an infinitesimal increase dV of its volume is DW mech = − p d V

(10)

If in a galvanic cell the infinitesimal increase of charge dq is subject to a Galvani potential , the electrical work is DW = dq el

(11)

The amount of heat passing to a system during a process is calculated from the change in temperature dT with the help of the heat capacity C of the system D Q = C dT.

(12)

Heat applied to a system for a change of phases (e.g., melting) without an increase in temperature is called latent heat. Energy produced in chemical reactions is chemical energy. Electromagnetic energy, radiant energy, as well as nuclear energy will not be discussed here. Since all forms of energy do not yield exact differentials, balances of energy forms require the specification of the path of the process studied—for example, isothermal process (T = const) or isobaric process ( p = const). A process in which the maximum of usable work is produced is called a reversible process, in contrast to irreversible processes, which waste usable energy. The usable chemical energy of the reaction Zn + 2AgNO3 (aq) ← → Zn(NO3 )2 (aq) + 2Ag

(13)

is wasted by transforming it to heat (unusuable energy) when a rod of solid zinc is inserted into an aqueous solu-

Thermodynamics is an empirical science. The quintessence of all practical knowledge is condensed in three fundamental laws (axioms), which are valid because so far they have not been found to be false. (Some textbooks quote an additional “zero law” concerning the definition of thermodynamic temperature.) 1. First Law In 1847 Helmholtz formulated his statement concerning the conservation of energy and the equivalence of work and heat: “Although energy may be converted from one form to another, it cannot be created or destroyed.” As a consequence, (dU )isolated system = 0 where dU =



DW i ;

i = mech, el, . . .

(14)

(15)

The differential dU of the total or internal energy U is an exact differential, in contrast to the differential elements DW i of the different forms of energy. The change in internal energy depends only on the initial and final states and not on the process path. 2. Second Law The second law of thermodynamics as stated by Clausius in 1854 concerns the spontaneous evolution of a system subject to processes. It defines the entropy of a system, which is related to the change in heat during the process and the temperature by the relation d S = D Q rev /T

(16)

Clausius’s formulation (d S)isolated system ≥ 0

(17)

is the most appropriate formulation of the second law for chemical thermodynamics.

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All the symbols in Table II have been explained in Section I, except µi (i = 1, . . . , k), which is the chemical potential of component Yi . In Table II electrical energy is separated from mechanical, heat, and chemical forms of energy, since its extensive variable q (charge) is not independent of mole numbers n i and its introduction as a variable of state would require the use of compatibility equations. To summarize, the set of variables of state ( p , T , n 1 , . . . , n k ) or any other set obtained by replacing one of the variables by its conjugate variable can be used to express the internal energy of the system as a function of its state. Only two functions are relevant in thermodynamics:

3. Third Law The third law is a summary of conclusions concerning the zero-temperature behavior of systems. Its formulation, that the zero-point entropy of perfect crystalline substances is zero, S0 = 0

(18)

as announced in 1923 by Lewis and Randall, indicates by simple application of thermodynamic equations that the heat capacity of such systems vanishes at the zero point of the thermodynamic temperature scale. Consequently, the zero point of temperature cannot be attained experimentally since the thermodynamic functions expressing maximum reversible work of a system also vanish with horizontal tangent. The internal energy and its Legendre transforms (see Section II) are not determined at absolute zero temperature.

U = U (S , V , n 1 , . . . , n k )

(20) U = U (T , V , n 1 , . . . , n k )

(caloric equation)

(21)

The other functions can be obtained by appropriate transformations of Eqs. (20) and (21).

II. FUNDAMENTAL AND CALORIC EQUATIONS OF THERMODYNAMICS

B. Fundamental Equation

Thermodynamics generally is based on two equations, the fundamental and the caloric equations.

1. Homogeneous Systems Equation (20), in which all variables of state are extensive variables, can be written using the notation of Table II,

A. Variables of State

U = U (ξi );

The basic set of variables of state can be extended by reflecting on the forms of energy, each of which is given by the product of an intensive (η) and an extensive (ξ ) variable [see Eqs. (10) and (11)]. Forms of energy yield differential elements DW i that are not exact differentials, but mathematics postulate that for such elements there exists at least one integrating factor that converts the nonexact differential form to an exact differential; for example, the integrating factor of heat is T −1 , yielding entropy as the exact differential d S, d S = T −1 D Q rev . Using this concept, every differential element DW i can be expressed with the help of the conjugate intensive and extensive variables of the corresponding form of energy (see Table II), where DW i = ηi d ξi .

(fundamental equation)

ξi = V, S, n i , . . . , n k

(22)

yielding the exact differential of the internal energy  k+2   ∂U dU = dξi (23) ∂ξi ξ j =ξi i=1 which entails the relations (∂U/∂ V ) S,ni = − p

(24a)

(∂U/∂ S)V,ni = T

(24b)

(∂U/∂n i ) S,V,n j =ni = µi

(24c)

because the combination of Eqs. (15) and (19) yields the relation

(19)

TABLE II Conjugated Extensive and Intensive Variables of Forms of Energy Energy form Variable

Arbitrary form of energy W i

Mechanical energy

Heat energy

Chemical energy

Electrical energy

Extensive Intensive

ξi ηi

V

S

n1 , . . . , nk

q

–p

T

µ1 , . . . , µ k



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dU =

k +2 

ηi d ξi

(25)

i =1

to which Eq. (23) can be compared. Because the internal energy U [Eq. (22)] is a homogeneous function of degree 1 of the variables ξi , Euler’s theorem of homogeneous functions yields  k +2  k +2   ∂U U= ξi = ξi ηi , (26) ∂ξi ξ j =ξi i =1 i =1 which, as compared with Eq. (25), yields the Gibbs– Duhem–Margules equation: k   ξi d ηi = −V dp + S dT + n i d µi = 0 (27) i =1

The replacement of an arbitrary number of extensive variables ξi (i = 1, . . . , m) by their conjugate intensive variables ηi yields new energy functions U (m) of the system U (m) = U (m) (η1 , . . . , ηm , ξm +1 , . . . , ξk +2 ).

(28)

These functions can be understood as the Legendre transforms of Eq. (25): U (m) = U (ξi ) −

m 

ηi ξi

(29)

i =1

The exact differential of Eq. (28) is  m   ∂U (m) (m) dU = d ηi ∂ηi ξi ,η j =ηi i =1

 k +2   ∂U (m) + d ξi ∂ξi η j ,ξ j =ξi i =m +1

and entails the Maxwell relations  (m)  ∂U = −ξi ∂ηi ξ j ,η j =ηi  (m)  ∂U = ηi ∂ξi η j ,ξ j =ξi

(∂U /∂ S)V ,ni = (∂ H /∂ S) p,ni = T (∂U /∂ V ) S ,ni = (∂ A /∂ V )T ,ni = − p (∂ T /∂ V ) S ,ni = −(∂ p /∂ S)V ,ni

(∂G /∂ T ) p,ni = (∂ A /∂ T )V ,ni = −S (∂G /∂ p)T ,ni = (∂ H /∂ p) S ,ni = V

(∂ S /∂ V )T ,ni = (∂ p /∂ T )V ,ni

(∂ S /∂ p)T ,ni = −(∂ V /∂ T ) p,ni

(∂ T /∂ p) S ,ni = (∂ V /∂ S) p,ni

µi = (∂U /∂n i ) S ,V ,n j =ni = (∂ H /∂n i ) S , p,n j =ni = (∂ A/∂n i )T,V,n j =ni = (∂G/∂n i )T, p,n j =ni

(35)

The exact differentials dU [Eq. (23)] and dU (m) [Eq. (30)] entail Schwarz relations. Maxwell relations and Schwarz relations are summarized in Table III. 2. Heterogeneous Systems The description of the state of a heterogeneous system requires a set of variables of the type given in Table II for every phase α. As a consequence, the internal energy of a heterogeneous system made up by ν phases is  U = U (α) ξi(α) = U (ξ j ) (36) when numbering the variables ξ either per phase, ξi(α) (i = 1, . . . , k + 2; α = 1, . . . , ν) or throughout the system, ξ j [ j = 1, . . . , ν(k + 2)]. The exact differential

(30)

dU =

ν  k+2 

ηi(α) dξi(α) =

α=1 i=1

(31a)

(31b)

The Legendre transforms currently used in thermodynamics are enthalpy, Helmholtz energy, and Gibbs energy: H = U + pV ;

d H = V dp + T d S + µi dn i

A = U − T S;

d A = − p d V − S dT + µi dn i (33)

G = U + pV − T S;

TABLE III Partial Derivatives of Energy Functions and Maxwell Relations

(32)

dG = V dp − S dT + µi dn i (34)

The replacement of all extensive by intensive variables again yields the Gibbs–Duhem–Margules equation. The chemical potentials µi introduced as the conjugate variables of mole numbers n i are given by Eqs. (24c) and (31b):

ν(k+2) 

η j dξ j

(37)

j=1

is of the type given by Eq. (25). Hence, all equations for homogeneous systems can be used for the phases of heterogeneous systems. Equations (36) and (37) neglect the contributions of the phase boundaries U (α/β) to the total energy U of the heterogeneous system. 3. Electrochemical Systems The internal energy of phase α in an electrochemical system depending on the charge q (α) of phase α and the Galvani potential (α) ,  (α) (α) U (α) = U (α) S (α) , V (α) , n (α) (38) 1 , . . . , nk , q yields the exact differential dU (α) = − p (α) d V (α) + T (α) d S (α) + µi(α) dn i(α) + (α) dq (α)

(39)

in which dq (α) is dependent on mole numbers n i(α) and charges F z i of the single species Yi . In Eq. (40) F is the

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Faraday number (charge of 1 mole of electrons) and z i is the electrical valence of particle Yi : dq (α) = F z i dn i(α)

(40)

intensive variable. The partial derivatives in Eq. (43) are material properties such as (∂U/∂ T )V = Cv

(heat capacity at constant volume) (44)

Combining Eqs. (39) and (40) gives dU (α) = − p (α) d V (α) + T (α) d S (α) +

k 

µ ˜ i(α) dn i(α) (41)

or functions of material properties [see Eq. (46)], (∂U/∂ V )T = (α/β)T − p

i=1

where µ ˜ i(α)

=

µi(α)

+ z i F

(α)

(42)

is the electrochemical potential of species Yi in the phase α. C. Caloric Equation The function U (T, V, n 1 , . . . , n k ) [Eq. (21)] yields the exact differential     ∂U ∂U dU = dT + dV ∂ T V,ni ∂ V V,ni +

 k   ∂U i=1

∂n i

dn i

in contrast to the partial derivatives of the fundamental equation, which are the positive or negative conjugate variables, respectively. Partial derivatives (∂ f /∂ xi )x j =xi can be transformed to (∂ f /∂ yi ) y j = yi with the help of functional determinants (Jacobi transformation) if the functions xi = xi (y j ) are known. For practical use all partial derivatives of energy functions U , H , A, and G and of entropy S are reduced to functions of the tabulated material properties α (thermal expansivity coefficient), β (isothermal compresibility coefficient), and C p (heat capacity at constant pressure):

(43)

α = (1/V )(∂ V /∂ T ) p

(46a)

β = −(1/V )(∂ V /∂ p)T

(46b)

C p = (∂ H/∂ T ) p

T,V,n j =n i

and the related Schwarz relations. The function U (T, V, n 1 , . . . , n k ) is not a homogeneous function since T is an

(46c)

The results of these calculations are compiled in Table IV for practical use.

TABLE IV Reduction of Frequently Used Partial Derivatives to Material Propertiesa (∂U/∂ T )V = C p − T V α 2 /β

(∂U/∂ T ) p = C p − V pα

(∂ H/∂ T )V = C p + V α/β − T V α 2 /β

(∂ H/∂ T ) p = C p

(∂ A/∂ T )V = −S

(∂ A/∂ T ) p = −S − V pα

(∂G/∂ T )V = −S + V α/β

(∂G/∂ T ) p = −S

(∂ S/∂ T )V = C p /T − V α 2 /β

(∂ S/∂ T ) p = C p /T

(∂ p/∂ T )V = α/β

(∂ p/∂ T )s = C p (T V α) 

(∂ V /∂ T ) H = C p β + V α − T V α 2 (1 − T α)

(∂ V /∂ T )s = −C p β/(T α) + V α

(45)

(∂U/∂ p)V = C p β/α − T V α

(∂U/∂ p)T = pVβ − T V α

(∂ H/∂ p)V = C p β/α + V − T V α

(∂ H/∂ p)T = V − T V α

(∂ A/∂ p)V = −Sβ/α

(∂ A/∂ p)T = pVβ

(∂G/∂ p)V = V − Sβ/α

(∂G/∂ p)T = V

(∂ S/∂ p)V = C p β/(T α) − V α

(∂ S/∂ p)T = −V α

(∂ V /∂ p) S = T V 2 α 2 /C p − Vβ

(∂ V /∂ p) H = V 2 α(T α − 1)/C p − Vβ

(∂ T /∂ p) H = V (T α − 1)/C p (∂U/∂ V ) p = − p + C p /(V α)

(∂U/∂ V )T = − p + T α/β

(∂ H/∂ V ) p = C p /(V α)

(∂ H/∂ V )T = (T α − 1)/β

(∂ A/∂ V ) p = − p − S/(V α)

(∂ A/∂ V )T = − p

(∂G/∂ V ) p = −S/(V α)

(∂G/∂ V )T = −1/β

(∂ S/∂ V ) p = C p /(T V α)

(∂ S/∂ V )T = α/β

a The material properties here are heat capacity C , thermal expansion coefficient α, and p isothermal compressibility β.

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The caloric equation does not contain entropy as a variable of state and hence cannot be used for providing information either on thermodynamic equilibria or on chemical potentials, in contrast to the fundamental equation.

III. PARTIAL MOLAR QUANTITIES A. Definitions Extensive thermodynamic quantities Z ( p, T, n 1 , . . . , n k ) of the type given by Eq. (47), such as volume V , entropy S, enthalpy H , heat capacity C p , and Gibbs energy G, yield partial molar quantities: Z i = (∂ Z /∂n i ) p,T,n j =ni .

(47)

Partial molar quantities Z i are the contributions per mole of the components Y1 , . . . , Yk to the total quantity Z p,T (n 1 , . . . , n k ) of the phase at constant pressure and temperature: k  Z p,T (n 1 , . . . , n k ) = ni Z i . (48) i=1

Equation (48) results from Euler’s theorem, since Z p,T is a homogeneous function of degree 1; it entails the relations d Z p,T =

k 

Z i dn i

(49a)

FIGURE 1 Determination of partial molar quantities Z2 and apparent molar quantities z from measurements of extensive thermodynamic quantities Z p,T,n (n2 ) (also Z2 from Z; see Section III.B).

Figure 1 illustrates the notation; beginning with n 1 moles of pure component Y1 , the system is made up by the addition of arbitrary amounts n 2 of component Y2 . The measurable quantity Z of the phase is given by the relation Z = n1 Z 1 + n2 Z 2

From Fig. 1 it follows that Z 2 is given by the slope of the tangent at point n 2 , which in turn permits the determination of Z 1 via Eq. (52) or (49b). Solution chemistry also makes use of another type of molar quantity, the so-called apparent molar quantity Z of the solute Y, according to Z = n 1 Z 1∗ + n 2 Z

i=1 k 

n i d Z i = 0.

(49b)

i=1

Equation (49a) is the exact differential of quantity Z [Eq. (3)] at constant pressure and temperature. Equation (49b) is a Gibbs–Duhem–Margules type of equation, indicating the mutual dependence of partial molar quantities. Using the set of variables (x1 , . . . , xk ) entails the relations Z¯ p,T (x1 , . . . , xk ) = xi Z i d Z¯ p,T = Z i d xi

Z l∗ = lim Z i ; xi →1

 or Z Y∞ = lim Z Y . xs →1

Z 1 = Z 1∗ + n 2 (∂ Z /∂n 1 ) p,T,n 2 ; Z 2 = Z + n 2 (∂ Z /∂n 2 ) p,T,n 1

Z 2∞ = lim Z 2 xl →1

(51)

(54)

and the limiting value at infinite dilution Z 2∞ = lim Z 2 = lim Z = ∞ z xl →1

where Z¯ p,T is the mean value of quantity Z p,T per mole of phase, Z¯ p,T = Z p,T / n i . The following discussion is limited to two-component systems (binary systems) of components Y1 and Y2 , mixtures of Y1 and Y2 , or solutions of Y2 (solute Y) in Y1 (solvent S). The quantity Z per mole of pure component Yi will be called Z i∗ and that per mole of Y2 (or Y) in the infinitely diluted solution in solvent Y1 , (or S) will be called Z 2∞ (or Z Y∞ ):

(53)

Y1 being the solvent. The apparent molar quantity Z is also shown in Fig. 1. The advantage of apparent molar quantities is their direct access from experimental data,

Z = (Z − n 1 Z 1∗ )/n 2 Comparison of Eqs. (52) and (53) yields the relations

(50)

xi d Z i = 0

(52)

xs →1

(55)

Figure 2 illustrates the features of mean molar properties Z¯ p,T [Eq. (50)] for a system of completely miscible components Y1 and Y2 . At mole fraction x2 the quantity Z¯ p,T is given by the relation Z¯ p,T = x1 Z 1 + x2 Z 2

(56)

yielding Z 1 = Z¯ p,T − x2 (∂ Z¯ /∂ x2 ) p,T ; Z 2 = Z¯ p,T + x1 (∂ Z¯ /∂ x2 ) p,T

(57)

Equation (57) shows that the tangent at Z¯ p,T at point P(x2 ) yields intercepts with the axis at x2 = 0 and x2 = 1, which produce the partial molar quantities Z 1 and Z 2 .

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is the change in Z 2 accompanying the transfer of Y2 from the pure state to the infinitely dilute solution in Y1 ; Z 2∞ is determined by appropriate extrapolation methods. Combining Eqs. (60b), (61b), and (62) yields the relation  ∞ ∞ mix Z = n 2 rel

rel (63) Z + Z 2 ; Z = Z − Z where − rel Z is the balance for the dilution process from concentration c2 of Y2 to infinite dilution FIGURE 2 Determination of the partial molar quantities Z1 and Z2 as a function of mole fraction x2 from mean molar quantities ¯ see Section III.B). Z¯ p,T (x2 ) (also Z1 and Z2 from  Z:

Besides partial molar and apparent molar quantities, thermodynamics makes use of relative quantities such as (Z i − Z i∗ ), (Z i − Z i∞ ), or ( Z − ∞ Z ), resulting from mixing or solution processes. B. Mixing Processes Formally, property Z before and after mixing of the components Y1 and Y2 can be written as Z init = n 1 Z 1∗ + n 2 Z 2∗ Z fin = n 1 Z 1 + n 2 Z 2

(58)

or Z fin = n 1 Z 1∗ + n 2 Z

(59)

yielding the balances for the mixing process mix Z = n 1 Z 1 + n 2 Z 2 ; or

Z i = Z i − Z i∗ (60a)

 mix Z = n 2 Z − Z 2∗



  ∞ ∗

Z − Z 2∗ = Z − ∞ Z + Z2 − Z2

(61b)

In Eqs. (61a) and (61b) the quantity Z 2∞ = Z 2∞ − Z 2∗

(62)

(64)

(e.g., heat of dilution). Dilution from concentration c2 , to c2 , c2 < c2 is then given by the relation mix Z (c2 )/n 2 − mix Z (c2 )/n 2 = (c2 ) − (c2 )

(65)

C. Thermodynamic Relations Using partial molar quantities Z i , the differential d Z of the thermodynamic quantity Z at arbitrary pressure and temperature is     k  ∂Z ∂Z dZ = dp + dT + Z i dn i (66) ∂ p T,ni ∂ T p,ni i=1 entailing Schwarz relations of the type given by Eqs. (4a)– (4c), where P ≡ Z . Hence, the relations existing for the extensive quantities Z can also be used for their partial molar quantities: for example, (∂G i /∂ T ) p = −Si (∂G i /∂ p)T = Vi  (∂(G i /T )/∂ T ) p = − Hi /T 2 (∂ Hi /∂ T ) p = C pi

(60b)

Here, mix Z is the integral effect of mixing: Z i = Z i − Z i∗ are the differential effects. For enthalpies and Gibbs energies, Z = H or G, only the differential effects Z i can be determined by experiments and not the partial molar quantities Z i themselves, in contrast to volumes and heat capacities, where both quantities are available. Since the mixing process may be accompanied by a change of state of component Y2 (e.g., solid or gas state of pure Y2 may change to liquid state in the mixture), it is advantageous to separate the differential effect Z 2 into two steps:   Z 2 = Z 2 − Z 2∞ + Z 2∞ − Z 2∗ (61a) and

∞ dil Z /n 2 = − rel Z (c2 ) = Z − Z (c2 )

(67a) (67b) (67c) (67d)

Equations (67a)–(67d) show the particular role of partial molar Gibbs energies G i . By their definition, quantities G i are both partial molar quantities and chemical potentials as defined by the fundamental equation of thermodynamics: G i ( p, T ) = (∂G/∂n i ) p,T,n j =ni = µi ( p, T )

(68)

D. Molar Quantities of Pure Compounds and Molar Quantities at Infinite Dilution Molar volumes of pure compounds Vi∗ and their temperature and pressure dependence are available from density measurements:

Vi∗ = Mi di∗ (69a)   ∗  ∂ ln di∗ ∂ Vi = αi∗ Vi∗ = −Vi∗ (69b) ∂T p ∂T p  ∗   ∂ Vi ∂ ln di∗ = −βi∗ Vi∗ = −Vi∗ (69c) ∂p T ∂p T

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Molar heat capacities C ∗pi are known from calorimetric measurements; their temperature and pressure dependence is tabulated for practical use. Molar entropies Si∗ are tabulated at the standard state ◦ ◦ as standard entropies S298 (Yi ), S298 (Yi ) = Si∗ (1 atm, 298.15 K). They are defined by the relation (◦ , standard)  298.15 ◦ S298 (Yi ) = c∗◦ (70) pi d(ln T ) 0

Molar enthalpies and Gibbs energies cannot be determined by experiments (third law of thermodynamics); only differences with respect to some reference state can be defined. The concept of formation reactions is frequently used to replace molar enthalpies Hi∗ and Gibbs energies G i∗ in thermodynamic balances by the standard enthalpies and standard Gibbs energies of formation, f H ◦ [Yi ] and f G ◦ [Yi ], respectively. The formation reaction at standard conditions (1 atm, 298.15 K) of a pure chemical compound Yi (e.g., CH3 OH) requires the formation of Yi from chemical elements in their most stable state at standard conditions; here, C(graphite) + 2H2 (g) + 0.5 O2 (g) = CH3 OH(1)

(71)

According to this definition, the formation of an element in its most stable state at standard conditions equals ◦ ◦ zero. Hence, f H298 [H2 (g)] = 0: f H298 [O2 (g)] = 0; ◦ ◦ f H298 [C(graphite)] = 0; but f H298 [C(diamond)] = 1.90 kJ mol−1 , this quantity being the heat of phase transition, C(graphite) → C(diamond), at standard conditions. Standard heats of formation and phase transition are tabu◦ lated as values f H298 (Yi ). They can be determined with high precision by combustion of compounds Yi in special calorimeters. Tabulated standard Gibbs energies of formation f G ◦298 (Yi ) are based on the same concept of a formation, reaction at standard conditions. The temperature and pressure dependence of Hi∗ and ∗ G i are given by the relations  ∗ ∂ Hi ∂ T p = C ∗pi (72)   ∗ ∂ Hi ∂ p T = Vi∗ 1 − αi∗ T 

∂G i∗ ∂ T p = −Si∗  ∗ ∂G i ∂ p T = Vi∗

Z i∞

(73a) (73b)

at standard conditons as used in solution Quantities chemistry are also tubulated. Standard heats of formation ◦ of ions in aqueous solutions, f H298 [Yi (aq)], include the heat of formation of the pure compound Yi under standard conditions and the heat of transfer of pure compound Yi from its pure state to infinite dilution in solvent S, that is, the quantity Hi∞ − Hi∗ . The tables of single-ion quantities in aqueous solutions are based on the additional assumption that f H ◦ [H+ (aq)] = f G ◦ [H+ (aq)] = S◦

[H+ (aq)] = 0, thus avoiding the ambiguity resulting from electrolyte dissociation.

IV. FUGACITIES AND ACTIVITIES A. Chemical Potential of Pure Gases On the basis of Eq. (68), the chemical potential of an ideal pure gas is obtained by integration of Eq. (73b) after replacing Vi∗ with the help of the equation of state of an ideal gas, Vi∗id = RT / p: µi∗id ( p, T ) = µi∗id ( p + , T ) + RT ln( p/ p + ).

(74)

In Eq. (74) p + is an arbitrary reference pressure. The chemical potential of a real gas is obtained when using an appropriate equation of state of a real gas. Vi∗real , such as the van der Waals, the Redlich–Kwong, or the virial equation:   p RT µi∗real ( p, T ) = µi∗id ( p, T ) + Vi∗real − d p (75) p p+ In Eq. (75) the reference pressure p + is chosen so low ( p + → 0) that the reference potentials µi∗ ( p + , T ) of ideal and real gases are equal. The integral expression in Eq. (75) is related to the so-called fugacity coefficient φi∗ by the help of the relation   p RT RT ln φi∗ = Vi∗real − d p, (76) p 0 permitting Eq. (75) to be written in the form

 µi∗real ( p, T ) = µi◦ (T ) + RT ln pφi∗ p ◦ .

(77)

Fugacity coefficients of numerous pure gases are tabulated from low (φi∗ = 1) to high pressures for practical use. In Eq. (77) p ◦ is the standard pressure, p ◦ = 1 atm; quantity pφi∗ is called the fugacity of the pure gas Yi . Comparison of Eqs. (74) and (77) shows that ideal and real gases can be treated as formally equal when pressures are replaced by fugacities. Using an appropriate equation of state of a real gas, for example, the virial equation, pVi∗ = RT + Bi∗ (T ) p + · · ·

(78)

the fugacity coefficient φi∗ can be approximated with the help of the relation 

φi∗ = exp Bi∗ p RT (79) B. Chemical Potential in Gas Mixtures In a gas mixture of ideal gases Y1 , . . . , Yk , at pressure p and temperature T , component Yi is at partial pressure pi (Dalton’s law) and Eq. (74) can be appropriately used to

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yield the chemical potential µi ( p, T ) of Yi in this mixture, since µi ( p, T ) = µi∗ ( pi , T ): µiid ( p, T ) = µi∗id ( p, T ) + RT ln( pi / p)  = µi◦ (T ) + RT ln pi p ◦

(80)

Equation (80) exhibits the dependence of µi on the phase composition ( pi = xi p, where xi is the mole fraction): µiid ( p, T )

=

µi∗id ( p, T )

+ RT ln xi

(81)

Equations of state of real gas mixtures, for example, the virial equation of state of a binary gas mixture, p V¯ = RT + B¯ p + · · · B¯ = x12 B1∗ + x22 B2∗ + 2x1 x2 B12

(82a) (82b)

permit the calculation of fugacities φi of components Yi in the real mixture   p RT RT ln φi = Vireal − dp (83) p 0 with the help of Eq. (56), which yields the partial molar quantities needed for estimating the integral expression in Eq. (83). In Eq. (82b) B1∗ and B2∗ are the second virial coefficients of the pure gases Y1 and Y2 ; the cross term B12 , resulting from the interaction of unequal particles in the gas phase, can be obtained from statistical models. A rough approximation is due to Lewis: B12 = (B1∗ + B2∗ )/2. The chemical potential of component Yi in a real gas mixture is related to that in the ideal mixture by the relation µireal ( p, T ) = µiid ( p, T ) + RT ln φi

(84)

Combination of Eqs. (75), (76), (81), and (84) yields the appropriate expression of the chemical potential µi in real gas mixtures:  µireal ( p, T ) = µi∗real ( p, T ) + RT ln xi + RT ln φi φi∗ (85) for the dependence of the chemical potential on the composition and molecular interactions. Equation (85) relates the chemical potential µireal ( p, T ) of component Yi in the gas mixture to that of its pure state, µi∗real ( p, T ); the quotient of the fugacities φi and φi∗ is called the activity coefficient f i ; the product of activity coefficient f i and mole fraction xi is the activity ai of component Yi in the mixture:

f i = φi φi∗ ; ai = xi f i (86) Combining Eqs. (85) and (86) yields the generally used expression µi ( p, T ) = µi∗ ( p, T ) + RT ln ai = µi∗ ( p, T ) + RT ln xi + RT ln f i (87)

The superscript “real” is omitted in Eq. (87) because this equation is also valid for ideal mixtures where f i = 1 (φi = φi∗ ). The activity coefficient of a pure phase (xi = 1, φi = φi∗ ) equals unity, in contrast to its fugacity coefficient [φi∗ ; see Eq. (76)]. C. Chemical Potentials in Condensed Phases The knowledge of equations of state for gas phases permits the calculation of activity coefficients via fugacity coefficients. Equations of state for general practical use such as the virial equation (and others) are not known for condensed phases (liquids and solids). However, as shown by Planck and Schottky, the passage from the gaseous to the liquid or solid state does not change the structure of Eq. (87) and leads to the general formulation for the chemical potentials, µi ( p, T ) = µi∗ ( p, T ) + RT ln xi + RT ln f i

(88a)

µi∗ ( p, T )

(88b)

= lim [µi ( p, T ) − RT ln xi ] xi →1

lim f i = 1

(88c)

xi →1

which can be used in gaseous, liquid, and solid phases. For solutions, the chemical potentials of the solutes (Y2 , . . . , Yk ), when referred to infinite dilution instead of pure phase, can be written µi ( p, T ) = µi∞ ( p, T ) + RT ln xi   ∗  µ − µi∞ + RT ln f i exp i RT entailing the definition of an activity coefficient:   f 0i = f i exp µi∗ − µi∞ RT .

(89)

(90)

The superscript (∗ ) in Eqs. (89) and (90) indicates that the solute Yi and the solution are considered in equal states of aggregation, which is not necessarily the stable state of the pure compound Yi with chemical potential µi∗ . The general definition of the chemical potential of a solute Yi in solution (Y1 is the solvent S) is based on the activity coefficient of the type given by Eq. (90). µi ( p, T ) = µi∞ ( p, T ) + RT ln xi + RT ln f 0i

(i = 2, . . . , k)

µi∞ ( p, T ) = lim [µi ( p, T ) − RT ln xi ] xs →1

lim f 0i = 1

xs →1

(91a) (91b) (91c)

In solutions, Eqs. (88a)–(88c) are used for the solvent and Eqs. (91a)–(91c) for the solutes. Identity within restricted

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concentration ranges for Eqs. (88a)–(88c) if f i = 1 and for Eqs. (91a)–(91c) if f 0i = 1: µiid ( p , T ) = µi ( p , T ) + RT ln xi .

Equation (96b) must be integrated from infinite dilution of component Y2 (x2 = 0, f 02 = 1) to the arbitrary composition x2 to yield  x2 (ln f 02 )x =x2 = d ln f 02

(92)

According to IUPAC rules the superscript  indicates an arbitrary reference state, here ∗ or ∞. Quantities µi ( p , T ) −

µiid ( p , T )

=

µiE ( p , T )

= RT ln f i

x2 =0



=−

(93a)

µi ( p , T ) − µiid ( p , T ) = µiE ( p , T ) = RT ln f 0i (93b)

(94)

The preceding considerations are based on the use of the mole fraction scale. Chemists and chemical engineers who use other scales for the composition of mixtures and solutions, for example, weight percent and mole percent for mixtures or molality, molonity, and molarity for solutions (see Section I.A) must convert chemical potentials and activity coefficients to these scales. Conversion is based on the fact that changes in composition scales do not change the chemical potential, for example, conversion from the mole fraction scale (µi∞ , f 0i ) to the molality scale (µi∞(m) , γi ): µi∞ + RT ln xi + RT ln f 0i

d µi∞ = 0,

and x1 d ln x1 + x2 d ln x2 = 0

(95)

the following equations are obtained for mixtures and solutions, respectively: x1 d ln f 2 = − d ln f 1 (96a) 1 − x1

= µi∞(m) +RT ln m i + RT ln γi

and

x1 d ln f 02 = − d ln f 1 (96b) 1 − x1 Integration of Eq. (96a) from the pure component Y2 (x2 = 1, f 2 = 1) to an arbitrary composition x2 yields the activity coefficient at this composition:  x2 (ln f 2 )x =x2 = d ln f 2 =−

x1 =1−x2 x1 =0

x1 d ln f 1 1 − x1

µi∞(m) = µi∞ + RT ln Ms ;

γi = xs f 0i

(97)

From µ∞ 2

µ∞ 2 =

µ∞ 2

∞(c)

µ∞(c) 2 ∞(c)

µ2 M1 d1

µ∞ 2 + RT ln

∞(m)

=

µ∞ 2 + RT ln M1

µ2

˜ ∞(m)

=

µ∞ 2 + RT ln M1

µ2

µ2 µ2

µ∞(m) 2 d1 M1

∞(c)

=

µ2

+ RT ln

(100)

Yi (i = 2, . . .) being the solutes and S the solvent. For frequently needed conversion formulas of mole fractions x, molarities c, molalities m, and molonities m˜ and corresponding activity coefficients v, y, γ , and β, see Tables V and VI.

TABLE V Conversion of Reference Chemical Potentials of Binary Systemsa

To

(99)

Using Eq. (91c) and the conversion formula from xi to m i (Table I) entails

x2 =1



(98)

D. Conversion of Reference Potentials and Activity Coefficients

In Eq. (94) the chemical potentials can be expressed with the help of Eqs. (88a)–(88c) for mixtures and with Eqs. (91a)–(91c) solutes in solutions. Since d µi∗ = 0,

x1 =1

x1 d ln f 1 1 − x1

If the activity coefficient of component Y1 is known as a function of composition, the integrals in Eqs. (97) and (98) can be evaluated graphically or analytically to yield the activity coefficients f 2 and f 02 , respectively.

are referred to as excess chemical potentials. According to the Gibbs–Duhem equation the chemical potentials and hence the activity coefficients of a mixture or solution are not independent; for a binary system at constant pressure and temperature we have x 1 d µ1 + x 2 d µ2 = 0

x1 =1−x2

µ2

˜ µ2∞(m)

∞(m)

− RT ln M1

µ2

∞(m)

− RT ln d1

µ2

µ2

µ2

∞(c)

+ RT ln d1

µ2

∞(c)

+ RT ln d1

µ2

˜ ∞(m)

− RT ln M1

˜ ∞(m)

− RT ln d1

∞(m)

µ2

˜ ∞(m)

∞(m)

µ2

˜ ∞(m)

a Subscript 1 denotes solvent; subscript 2 denotes solute; d denotes density of pure solvent; M denotes molar 1 1 mass of solvent.

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Chemical Thermodynamics TABLE VI Conversion of Activity Coefficients of Different Concentration Scales (Binary Systems)a From To

f 02

y2

γ2

β2

f 02 =

f 02

d + (M1 − M2 )c2 y2 d1

(1 + M1 m 2 )γ2

[1 + (M1 − M2 )m˜ 2 ]β2

y2 =

d1 (M1 x1 + M2 x2 ) f 02 d M1

y2

(1 + M2 m 2 )d1 γ2 d

d1 β2 d

γ2 =

x1 f 02

γ2

(1 − M2 m˜ 2 )β2

β2 =

M1 x 1 + M2 x 2 f 02 m1

(1 + M2 m 2 )γ2

β2

d − M 2 c2 y2 d1 d y2 d1

a Subscript 1 denotes solvent; subscript 2 denotes solute; d denotes density of pure solvent; d denotes density of 1 solution; M1 and M2 denote molar masses.

E. Chemical Potentials of Electrolyte Compounds Electrochemistry uses chemical potentials of the type given by Eqs. (91a)–(91c) for single ions. The link to thermodynamics is established by the help of mean mole fractions x± and mean activity coefficients f ± . For a binary z+ z− electrolyte as the solute, Y2 = Cv+ Av− (z + , valent cation: z − , valent anion), for example, Na2 SO4 (where z + = +2, z − = −1, v+ = 2, v− = 1), the chemical potential µ2 ( p, T ) consists of the chemical potentials of cations and anions, µ+ ( p, T ) and µ− ( p, T )

In Eq. (103) the symbol δ indicates virtual processes. A virtual process is a process that is realizable, not depending on time, for which entropy is defined in every state of the system; δS is an infinitesimally small variation and in this respect equivalent to a differential of first order. B. Generalized Forces and Internal Variables

A. Equilibrium Conditions

Figure 3 illustrates the situation of a system in equilibrium; entropy S is represented as a function of an internal variable ζ . Internal variables ζ and their conjugate quantities, generalized forces , are produced by the system in nonequilibrium states. Generalized forces such as gradients of concentration, temperature, or pressure or affinities of chemical reactions are the driving forces to equilibrium. Generalized forces are functions of the intensive variables of state of the system and equal zero at equilibrium,  = 0. Their conjugate internal variables ζ are functions of the extensive variables of state and of supplementary conditions characterizing the process studied. An example is given in Section V.F.1. Neither the generalized forces nor internal variables are variables of state; the system in its equilibrium state does not recognize them. The use of the notation of internal variables is exemplified in Fig. 3 by an equilibrium of the distribution of

According to Gibbs, thermodynamic equilibrium is defined by one of the following conditions: (δS)U,V,n ≤ 0 (103a) or (δU ) S,V,n ≥ 0 (103b) Equation (103a) refers to closed, Eq. (103b) to open, systems. Conditions equivalent to Eq. (103b) are obtained with the help of Legendre transformations of the fundamental equation (δ H ) S, p,n ≥ 0; (δ A)T,V,n ≥ 0; (103) (δG)T, p,n ≥ 0

FIGURE 3 Explanation of Gibbs’ equilibrium condition δS ≤ 0. For details see text.

µ2 ( p, T ) = v+ µ+ ( p, T ) + v− µ− ( p, T )   ∞ = v+ µ∞ + ( p, T ) + v− µ− ( p, T ) + v RT ln x± + v RT ln f =

(101)

In Eq. (101) the mean quantities are given by the relations   1/v 1 x± = x+v+ x−v− ; f ± = f +v+ f −v− ; (102) v = v+ + v−

V. THERMODYNAMIC EQUILIBRIUM

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a chemical compound Y in two immiscible solvents in (2) contact, S1 and S2 , n (1) Y and n Y moles of Y being the amounts of Y dissolved in S1 and S2 , respectively. Then the internal variable ζ is represented as the ratio of the (2) mole numbers, ζ = n (1) Y /n Y . If the state of equilibrium is attainable from both sides, ζ > ζ eq and ζ < ζ eq , the tangent at ζ eq is horizontal, (∂ S/∂ζ )eq = 0, and δS equals zero, δS = (∂ S/∂ζ )eq δζ = 0. If, however, compound Y is insoluble in one of the two solvents, the tangent at ζ eq is defined only from one side by the limit of (∂ S/∂ζ ), which is a negative quantity, and δS ≤ 0 (Gibbs’ equilibrium condition). Gibbs criteria of equilibrium, of course, are in agreement with the second law of thermodynamics, which gives evidence of the variation of entropy in spontaneous processes (entropy increase) but gives no explicit evidence on the state of equilibrium itself. C. Stability of Equilibrium Eqs. (103a) and (103b) do not provide information on the stability of equilibrium. Stability of equilibrium is recognized only when taking into account the variations of degrees greater than unity, δ 2 S, δ 3 S, . . . or δ 2 U , δ 3 U , . . . in the total variations, S = δS + δ S + δ S + · · ·

(104a)

U = δU + δ U + δ U + · · ·

(104b)

2

3

2

3

where δU =

 ∂U i

∂ξi

Mathematics provide a general criterion for stability. The quadratic form δ 2 U is positive for all considerable variations of the parameters ξi if, and only if, all the roots λ of the equation     U12 ... U1,k+2  U11 − λ    U21 U22 − λ . . . U2,k+2   . ..  .  = 0 (107) .  .    ∼  Uk+2,1 ... Uk+2,k+2 − λ are greater than zero (i.e., if the quadratic form is positive definite). Variations of order higher than 2 generally must not be considered. D. Equilibria at Phase Boundaries A heterogeneous system made up of two phases, α and β, and k components Yi is in equilibrium [Eq. (103b)] if   k  (α) (α) (α) (α) (α) (α) − p δV + T δS + µi δn i i=1

 + − p δV (β)

(β)

+T

(β)

δS

(β)

+

k 

 (β) (β) µi δn i

≥0

i=1

(108) All variations in Eq. (108) are under the constraints δS = 0;

δV = 0;

δn i = 0;

i = 1, . . . , k

(109)

yielding the relations dξi

(105a)

δS (α) = −δS (β) ;

δV (α) = −δV (β) ;

(β)

1  Ui j dξi dξ j 2 i j     ∂ ∂U Ui j = ∂ξ j ∂ξi ξ j =ξi

δ2U =

δn i(α) = −δn i ; (105b)

(105c)

ξi =ξ j

(110)

i = 1, . . . , k

which permit us to write Eq. (108) in the form   − p (α) − p (β) δV (α) + T (α) − T (β) δS (α) +

k  

(β)

µi(α) − µi

δn i(α) ≥ 0

(111)

i=1

and so on. The equilibrium position is a position of stable equilibrium if δ2 S < 0

or

δ2U > 0

(106a)

or

δ2U = 0

(106b)

or of unstable equilibrium if δ2 S > 0

or

δS (α) = 0;

δn i(α) = 0

(112a)

B:

δV (α) = 0;

δn i(α)

=0

(112b)

C:

δS

δV

=0

(112c)

A:

of undetermined equilibrium if δ2 S = 0

Equation (111) is the basic equation for the discussion of equilibrium conditions at phase boundaries. For this purpose, three sets of variations are studied:

δ2U < 0

(106c)

Also metastable equilibria can be observed (e.g., supercooled liquids). Metastable equilibria are stable only with regard to infinitely neighboring states.

(α)

= 0;

(α)

Set A entails the reduction of Eq. (111) to  − p (α) − p (β) δV (α) ≥ 0

(113)

Equation (113) is fulfilled if the following conditions are met:

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1. Both variations, δV (α) > 0 and δV (α) < 0, are possible and p (α) = p (β) (deformable phase boundary). 2. Variation δV (α) is impossible, δV (α) = 0, and p (α) > p (β) , p (α) = p (β) , or p (α) < p (β) (undeformable or rigid phase boundary). 3. Variation δV (α) is possible only in one direction, δV (α) ≥ 0 and p (α) ≤ p (β) (semideformable phase boundary). Similar considerations concerning sets B and C yield the conditions for heat-conducting (T (α) = T (β) ), nonconducting (T (α) > T (β) , T (α) = T (β) , or T (α) < T (β) ), and semi(β) conducting (T (α) ≥ T (β) ) and for permeable (µi(α) = µi ), (β) (β) (β) (α) (α) (α) impermeable (µi > µi , µi = µi , or µi < µi ), and (β) semipermeable (µi(α) ≥ µi ) phase boundaries. E. Phase Equilibria 1. Gibbs Phase Rule Each of the v phases of a heterogeneous system of k components is subjected to a Gibbs–Duhem–Margules equation [see Eq. (27)]: k+2 

ξi(α) dηi(α) = 0;

α = 1, . . . , v.

(114)

i=1

Hence the phase diagram of a heterogeneous system is defined by v equations of the type given by Eq. (114), showing in total v(k + 2) variables. If all phase boundaries are deformable, heat conducting, and permeable for all components Yi , the number of independent variables is reduced to f =k+2−v

(115)

FIGURE 4 Schematic phase diagram of a pure compound. The diagram exhibits a section around a triple point T where solid (s), liquid (l), and gaseous phases (g) are in equilibrium; C is the critical →g is the vapor pressure curve, s ← →g is the sublimation point; 1 ← →1 is the fusion pressure curve. pressure curve, and s ←

A one-component system k = 1, to begin with, exhibits three types of regions in its phase diagram depending on the variables of state p and T (Fig. 4): divariant (v = 1, f = 2), univariant (v = 2, f = 1), and nonvariant (v = 3, f = 0) regions. Divariant regions are the fields of solid, liquid, and gaseous states. One equation (v = 1) of type Eq. (116) shows that in these fields the chemical potential µ( p, T ) is a function of p and T , a well-known feature. Univariant regions, as given by the curves (evaporation, 1 ← → g; sublimation, s ← → g; fusion, s ← → 1) in Fig. 4 (for phase transitions s ← s, see Figs. 5 and 6), indicate → equilibrium of two phases. These curves are obtained by resolution of the appropriate system of two (v = 2) Gibbs– Duhem–Margules equations: −Vi(α) d p + Si(α) dT + dµi = 0 (β)

−Vi

(β)

d p + Si dT + dµi = 0

(117a) (117b)

Equations (117a) and (117b), can be transformed to yield the Clausius–Clapeyron differential equation,

Number f is the number of degrees of freedom: Eq. (115) is the Gibbs phase rule. If f = 0, 1, 2, the system is called nonvariant, univariant, or divariant, respectively. 2. Theory of Phase Diagrams The Gibbs–Duhem equations [Eq. (114)] for a heterogeneous system in which all phase boundaries are deformable, heat conducting, and permeable for all compounds can be written k 

 xi(α) −Vi(α) d p + Si(α) dT + dµi = 0;

i=1

α = 1, 2, . . . , v. (116) In Eq. (116) Vi(α) and Si(α) are the partial molar volume and entropy of component Yi in phase α. The set of v equations (116) is the basis for the theory of phase diagrams.

FIGURE 5 Stable (solid lines) and metastable (dashed lines) phase diagrams of sulfur. For explanation see text.

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diagrams, an arbitrary phase (β) is chosen to be the reference phase: its k + 1 independent variables, p, T , (β) (β) x1 , . . . , xk−1 are spanning the phase-diagram space. In order to transform the set of k + 2 variables of Eq. (116) to these reference variables, the chemical potential of component Yi in phase β is written  (β)   (β)  ∂µi ∂µi (β) dµi = dp + dT ∂p ∂T T,x j



+

=

d xs(β)

(β)

s=1

FIGURE 6 Phase diagram of water. Ice modifications: I, II, III, V, VI, VII. At the scale of this figure, representation is not possible for both the vapor pressure curve, beginning at triple point T (273.16 K. 6.03 × 10−6 kbar) and ending at critical point C (647.2 K. 0.218 kbar), and the sublimation pressure curve.

p,x j

 (β) k−1  ∂µi

(β) Vi

∂ xs

dp −

p,T,x j =xs

(β) Si

dT +

 (β)  k−1  ∂µi

d xs(β)

(β)

∂ xs

s=1

p,T,x j =xs

(119) (β)

and since dµi(α)

(α→β)

S − Si(α) Hi dp = i(β) = (α→β) (α) dT Vi − Vi T Vi

(118)

Nonvariant regions are points in the phase diagram (triple points T , v = 3) indicating equilibrium of three phases (g, 1, s or s, s, 1 or s, s, g, etc.). The vapor pressure curve (equilibrium: 1 ← → g) terminates in a critical point C. Above the critical temperature, liquid and gaseous states are indistinguishable (supercritical fluid state). Phase diagrams of pure compounds contain these elements in various combinations. Figure 5 (phase diagram of sulfur) and Fig. 6 (phase diagram of water) are examples. The phase diagram of sulfur shows the phenomenon called allotropy, occurring when the compound studied may exist in two different solid states (here monoclinic and rhombic). Starting at state A, very slowly increasing temperature yields the transition from rhombic to monoclinic sulfur at point B and the transition from monoclinic to liquid sulfur at point C, with every transition being (α→β) accompanied by a change in molar enthalpy (Hi ), (α→β) (α→β) ), entropy (Si ), and heat capacity volume (Vi (α→β) (c pi ). Transitions of this type are called transitions of first order, in contrast to transitions of second order, which show changes only in heat capacity (e.g., Curie point transition of iron, etc.). The phase diagram of stable equilibrium states (solid lines) of sulfur is superimposed by a phase diagram of metastable states (dashed lines) in which direct transition from rhombic to liquid sulfur can be formed by rapidly increasing the temperature from point A to C. This transition takes place at point D situated on the fusion pressure curve of the metastable rhombic sulfur, the corresponding metastable triple point being Tm (Fig. 5). Multicomponent heterogeneous systems require rather complex calculations. In a general treatment of their phase

(β) = dµi , Eq. (119) can be inserted into Eq.

(116) to yield k 

k   (β)  (β) xi(α) Vi − Vi(α) d p − xi(α) Si − Si(α) dT

i=1

+

i=1 k−1 

(β) G αβ s d x s = 0;

α = 1, . . . , v

(120a)

s=1

where G αβ s

=

k 

 xi(α)

i=1

(β)

∂µi

 (120b)

(β)

∂ xs

p,T,x j =xs

Since Eq. (120a) reduces to identity if α = β, Eq. (120a) is a system of v − 1 differential equations in k + 1 vari(β) (β) ables ( p, T, x1 , . . . , xk−1 ). All p, T , x diagrams can be reproduced with the help of Eqs. (120a) and (120b). F. Chemical Reactions 1. Generalized Forces and Internal Variables in Chemical Reactions The internal variable of chemical reactions is the “extent of reaction” ζ , defined by the variables of state n i (amount of substance Yi ) and supplementary conditions, which are given by the stoichiometry of the chemical reaction investigated, ω1 Y1 + ω2 Y2 + · · · ← → ω n Yn + · · · + ω k Y k

(121)

which in a more condensed form can be written ωi Yi = 0

(122)

In Eq. (122) the stoichiometric coefficients ωi are positive for the final and negative for the initial products. The

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internal variable ζ of chemical reactions is given by the relation dζ = dn i /ωi

(123) 3. Heat Balance of Chemical Reactions

yielding the expression dU = − p d V + T d S − A dζ

(124)

for the internal energy, where A=−

k 

ωi µi

(125a)

i=1 k v  

ωi µi(α)

(125b)

α=1 i=1

is the generalized force, called affinity, of a homogeneous [Eq. (125a)] or a heterogeneous [Eq. (125b)] reaction. 2. Chemical Equilibrium The Gibbs equilibrium condition entails the relations A=−

k 

ωi µi = 0

i=1

or A=−

v  k 

ωi µi(α) = 0

(126)

α=1 i=1

if equilibrium of the chemical reaction is attainable from both sides. Using the activity concept (Section IV), Eq. (126) is transformed to k 

ωi µi ( p, T ) = −RT ln K a ;

i=1 k   eq ωi Ka = ai

(127)

i=1 v  k 

At constant pressure and temperature the Gibbs energy of reaction  R G equals the negative affinity [Eqs. (125a) and (125b)], yielding the relation R G =

k  i=1

or A=−

be taken into account with the help of the corresponding transfer quantities, H (α→β) and V (α→β) , respectively.

ωi µi(α) ( p, T ) = −RT ln K a ;

α=1 i=1 v  k   (α)eq ωi Ka = ai

ωi µi =

k 

 ωi µi + RT ln xi f i

(129)

i=1

The enthalpy of reaction R H is obtained from the derivative of R G/T with respect to temperature:   k k   ∂ ln f i  2 R H = ωi Hi − RT ωi (130) ∂T p i=1 i=1 Replacing Hi at 1 atm by the enthalpy of formation (see Section III. D) yields the basic equation of thermochemistry,   k k   ∂ ln f i R H ◦ = ωi f H ◦ − RT 2 ωi (131) ∂T p i=1 i=1 where in most cases the effect resulting from the temperature dependence of the activity coefficients can be neglected when compared with the large enthalpies of formation. G. Equilibrium of Electrochemical Systems For electrochemical systems, electrochemical potentials µ ˜ i ( p, T ) (Section II) are used instead of chemical potentials. Under the action of driving forces, both chemical reactions (e.g., reaction in a galvanic cell) and charge transport (e.g., electron transport outside the cell from the anode to the cathode) may take place. The scheme   Cu(1) | Zn(2) | ZnSO4 (aq)(3)  CuSO4 (aq)(4)  Cu(5) (132) shows an example of a Galvanic cell. The cell reaction

(128)

α=1 i=1

In Eqs. (127) and (128) K a is the equilibrium constants of eq (α)eq the chemical reaction and ai and ai are the activities at equilibrium concentrations of the reactants. Using the concept of standard reaction enthalpies, standard Gibbs reaction energies, and standard entropies (Section III), the quantities µi ( p, T ) can be calculated with the help of tabulated standard values (at 25◦ C and 1 atm) and c p or Vi functions. Phase transitions on the path of integration must

→ (5) + ZnSO(3) Zn(2) + CuSO(4) 4 ← Cu 4

(133)

is accompanied by the transport of two electrons from phase 1 to phase 5, which materially are identical. At cell equilibrium the driving force equals zero: v  k 

 (1) ωi µi(α) + n µ ˜e −µ ˜ (v) =0 e

(134)

α=1 i=1

In Eq. (134) µ ˜ (α) e are the electrochemical potentials of the electrons, and n is the number of electrons transferred from the anode (phase 1) to the cathode (phase 5). Since

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the terminal phases are materially identical (here copper) they differ only in their Galvani potentials (α) .  (1) µ ˜ (1) (135) ˜ (v) − (v) = −F E. e −µ e = F

used for the determination of activity coefficients from vapor-pressure measurements:

(140) f i(1) = pi xi(1) pi∗

In Eq. (135) E is the measured potential difference of the cell at zero current, the electromotive force of the cell. From Eqs. (134) and (135) follows the wellknown Nernst equation:   v−1  k   (α) ωi ◦ RT E=E − (136a) ln ai nF α=1 i=1

Starting with the equilibrium condition for a solution and its gaseous phase,   ◦(g) d µi∞(1) + RT ln ai(1) = d µi + RT ln pi φi (141)

−n F E ◦ =

v−1  k 

ωi vi(α)

(136b)

α=1 i=1

H. Use of Equilibria for the Determination of Activity Coefficients Fugacity coefficients and hence activity coefficients can be calculated with the help of appropriate equations of state (see Section IV). This is possible, however, only for the gas phase (van der Waals equation, Redlich–Kwong equation, virial equation); for condensed phases no useful general equations of state are available. Experimental determination of activity coefficients in condensed phases is based on the study of equilibria. There are numerous methods, but only typical examples will be given.

and equivalent approximations yield the relation

f 0i(1) = pi xi(1) K

(142)

where K is Henry’s constant. An ideal mixture ( f i = 1) or dilution ( f 0i = 1) reduces Eqs. (141) and (142) to pi = xi(1) pi∗

and

pi = xi(1) K

(143)

known as Raoult’s and Henry’s laws, respectively. Figure 7 illustrates the calculation of f i and f 0i according to Eqs. (140) and (142) as the deviations from ideality. 2. Liquid–Liquid Equilibria In osmotic pressure measurements a solution is separated from the pure solvent S by a nondeformable membrane permeable only to the solvent. The pressure of the pure

1. Vapor–Liquid Equilibria Equilibrium of a gaseous and a liquid phase in contact (permeable, deformable, and heat-conducting boundary) entails the conditions   ◦(g) d µi∗(1) + RT ln ai(1) = d µi + RT ln pi φi (137) which at constant temperature can be transformed with the help of Eq. (67b), showing the pressure dependence of the chemical potentials, to yield the relation   xi(1) f i(1) RT d ln = −Vi∗(1) d p (138) pi φi∗ Integration from xi(1) = 1 (where Pi = pi∗ ) to an arbitrary mole fraction xi(1) (partial pressure in the gas phase pi ) yields the relation   ∗ pi Vi∗(1) pi φi (1) (1) xi f i = ∗ ∗ exp dp (139) pi φi RT pi The exponential term in Eq. (139), commonly called the Poynting correction, differs little from unity for temperatures not near the critical temperature. Assuming ideal behavior of the gas phase, fugacities can be replaced by pressures. Then Eq. (139) yields the relation commonly

FIGURE 7 Activity coefficients f 1 , f 2 (mixture), and f 02 (solution) from vapor-pressure measurements on binary systems of components Y1 (chloroform) and Y2 (acetone). p id , p1id , p2id : hypothetical total and partial vapor pressures of the ideal system ( piid , Raoult’s law); O1 E1 = p1∗ , O 2 E2 = p2∗ , vapor pressures of pure compounds Y1 (chloroform) and Y2 (acetone). p real , p1real , p2real : measured total and partial vapor pressures of the binary system chloroform–acetone. p2id.sol : hypothetical partial vapor pressure of the ideally dilute solution of Y2 (acetone, solute) in Y1 (chloroform, solvent) (Henry’s law); O 2 F = K (Henry’s constant). Activity coefficients [see Eqs. (140) and (142)]: f 1 = AC1 /AB 1 ; f 2 = AC 2 /AB 2 ; f 02 = AC 2 /AD.

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solvent is p, while in osmotic equilibrium the solution is subject to an additional pressure , the osmotic pressure, yielding the equilibrium condition µs ( p + , T ) = µ∗s ( p, T )

(144)

Using the pressure dependence of the chemical potential and assuming that the partial molar volume is independent of pressure yields the relation  p+ µs ( p + , T ) = µs ( p, T ) + Vs d p p

= µs ( p, T ) + Vs

(145)

By combining Eqs. (144) and (145), one can determine the activity coefficient of the solvent from the osmotic pressure xs f s = exp(−Vs /RT ) (146) The solvent activity xs f s can be expressed with the help of the molal osmotic coefficient which is defined as ln xs f s

=− (147) m Ms where m is the molality of the solute and Ms is the molar mass of the solvent. Using Eq. (146) yields the following relationship between the osmotic coefficient and the osmotic pressure: Vs

= (148) m Ms RT The measurement of osmotic coefficients combined with the Gibbs–Duhem–Margules equation is a wellestablished method for the determination of the activity coefficients of solutes.

account, and the integral of Eq. (151) can be solved analytically to yield the activity coefficients f s from freezing point depressions. Another method using liquid–solid equilibria determines solute activity coefficients from temperaturedependent solubility data. The pure solute Yi is in equilibrium with the saturated solution. With reference to the state of the infinitely dilute solution [Eqs. (91a)–(91c)], the equilibrium condition is given by the relation µi∗(s) ( p, T ) = µi∞ ( p, T ) + RT ln(xi f 0i )sat

(152)

The differential of Eq. (152) can be expressed with the help of the temperature dependence of the chemical potentials, yielding at constant pressure Hi∞ − Hi∗(s) Hi∞ dT = dT (153) 2 RT RT 2 In Eq. (153) the quantity Hi∞ is the change in enthalpy for the transfer of component Yi from the pure solid state to the infinitely dilute solution. This quantity can be obtained from heat of solution experiments. If the temperature-dependent solubility measurements are started at a temperature T  where the component Yi is very slightly soluble (xi ) and the activity coefficient f 0i approaches unity, Eq. (153) can be integrated to yield     T Hi∞ xi ( f 0i )sat,T = exp dT (154) 2 xi sat T  RT d ln(xi f 0i )sat =

4. Electrochemical System Equilibria 3. Liquid–Solid Equilibria In freezing point experiments an equilibrium between liquid solution and its pure solid solvent S is achieved. The equilibrium condition at constant pressure is     ∗(1) µ∗(s) µ ( p, T ) ( p, T ) s s d =d + d(R ln xs f s ) T T (149) The temperature dependence of the chemical potential [Eq. (67c)] can be used in Eq. (149) to yield Hs∗(1) − Hs∗(s) fus Hs dT = dT (150) RT 2 RT 2 where fus Hs is the molar enthalpy of fusion of the pure solvent at temperature T . Equation (150) is integrated from xs = 1, where T = T ∗ (freezing temperature of pure solvent) to an arbitrary concentration xs with freezing temperature T to yield  T fus Hs ln xs f s = dT (151) RT 2 T∗ d(ln xs f s )

Using the appropriate molar heat capacities, the temperature dependence of the enthalpy of fusion can be taken into

Electromotive force (emf) measurements are frequently used to determine activity coefficients of electrolyte solutions. Equation (136a) relates the emf to the activities of the reacting cell components. From concentrationdependent measurements the standard potential E ◦ of the cell reaction and the activity coefficients can be obtained. As an example, according to Eq. (136a), the emf of the Galvanic cell Pt(s) | K(Hg) | KCl (conc. in S) | AgCl(s) | Ag(s) | Pt(s) (155) can be written as 2RT (156a) ln(cy± ) F RT (156b) E ◦ = E ◦ + ln aK(Hg) F Here, y± is the mean activity coefficient of the electrolyte on the molar scale. In Eq. (155) K(Hg) is a potassium amalgam electrode connected to a solution of KCl at concentration c in solvent S; AgCl(s)/Ag(s) is the silver/silver chloride reference electrode. E = E ◦ −

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786 Data analysis of emf measurements according to Eq. (156a) yields the standard potential of the cell reaction and the activity coefficients at each concentration. Suffice it to mention that such measurements could also be used to determine the activity of potassium in the amalgam phase. A final remark: The IUPAC has changed the standard pressure from 1 atm to 105 Pa (105 Pa = 1 bar = 0.98692 atm). The numerical values of the tabulated standard data at 1 atm of condensed components generally are not affected, owing to their small pressure dependence in contrast to gases which undergo a slight change.

SEE ALSO THE FOLLOWING ARTICLES BATTERIES • CRITICAL DATA IN PHYSICS AND CHEMISTRY • ELECTROCHEMISTRY • ELECTROLYTE SOLUTIONS, THERMODYNAMICS • POTENTIAL ENERGY SURFACES

Chemical Thermodynamics

BIBLIOGRAPHY Denbigh, K. (1981). “The Principles of Chemical Equilibrium,” 4th ed., Cambridge Univ. Press, London. Gibbs, J. W. (1957). “Collected Works,” Vol. I, Yale Univ. Press, New Haven, CT. Guggenheim, E. A. (1986). “Thermodynamics,” 8th ed., North-Holland, Amsterdam. Haase, R. (1956). “Thermodynamik der Mischphasen,” Springer-Verlag, Berlin and New York. Honig, J. M. (1999). “Thermodynamics,” Academic Press, San Diego. Kirkwood, J. G., and Oppenheim, I. (1961). “Chemical Thermodynamics,” McGraw-Hill, New York. Klotz, I. M., and Rosenberg, R. M. (1994). “Chemical Thermodynamics—Basic Theory and Methods,” 5th ed., Benjamin, New York. Kondepudi, D., and Prigogine, I. (1998). “Modern Thermodynamics,” Wiley, New York. Lewis, G. N., and Randall, M. (1961). “Thermodynamics,” 2nd ed., McGraw-Hill, New York. Planck, M. (1964). “Vorlesungen u¨ ber Thermodynamik,” 11th ed., de Gruyter, Berlin. Pitzer, K. S. (1995). “Thermodynamics,” 3rd ed., McGraw-Hill, New York.

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Computational Chemistry Matthias Hofmann

Henry F. Schaefer III

Ruprecht-Karls-Universit¨at

Center for Computational Quantum Chemistry

I. II. III. IV.

History of Computational Chemistry Methods Used in Computational Chemistry Applications of Computational Chemistry Outlook for Computational Chemistry

GLOSSARY ab initio Latin term meaning from the beginning. In the context of computational chemistry, computations from first principle without any empirical input except fundamental physical constants. Density functional theory (DFT) Theory based on the electron density as the crucial property rather than the wave function in traditional ab initio methods. Force field A set of equations describing the potential energy surface of a chemical system. Hamiltonian operator An operator that describes the kinetic and potential energy of a system treated by wave mechanics. Molecular mechanics Theoretical treatment of molecules by a force field based on classical mechanics and electrostatics. Molecular modeling Branch of computational chemistry concerned with computer-aided molecular design. Orbital Function to describe a single electron. Molecular orbitals (MOs) build the total wave function of a system and are expanded in terms of atomic orbitals, AOs (basis functions). Orbitals can be occupied or virtual. Quantum mechanics Mathematical treatment based on the wavelike nature of small particles.

Schr¨odinger equation A differential equation for the quantum-mechanical treatment of a system. Self-consistent field (SCF) method Method used to solve mathematical equations which depend on their own solution. Semiempirical Making use of experimental results to derive parameters for approximations made in quantummechanical methods.

COMPUTATIONAL CHEMISTRY is the scientific discipline of applying computers to gain chemical information. It is the link between theoretical and experimental chemistry. Theoretical chemistry is mainly concerned with the development of mathematical models which allow one to derive chemical properties from calculations and to interpret experimental observations. The mathematical models developed in theoretical chemistry are usually validated by comparison with experiment. Theoretical chemistry existed before the arrival of electronic computers. Computational chemistry, however, relies heavily on powerful microelectronics to cope with huge computational tasks. It focuses on the application of theoretical methods which require calculational treatments which are by far too large to be done without fast computers.

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488 Of course, the strict separation of theoretical, computational, and experimental chemistry is of an academic nature. In practice, theoreticians often not only develop a new method but also need to design more efficient algorithms to make the method applicable. Before computational results can be interpreted, computational chemists need to undertake benchmark studies to determine the limitations of a method. Without the knowledge about the accuracy of the applied mathematical model, any computational study is without scientific significance. Likewise, many experimentalists use computer programs to support or complement their experimental studies.

I. HISTORY OF COMPUTATIONAL CHEMISTRY As early as 1929, only 3 years after Schr¨odinger’s formulation of the fundamental equation that bears his name, Dirac stated correctly, The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.

Hence, the further development of quantum chemistry was aiming at approximate solutions of the Schr¨odinger equation by simplifying the required mathematical treatment. In the 1930s the basics for a wide range of computational methods based on quantum mechanics were laid by the development of the Hartree-Fock method. In 1951 Roothaan, for the first time, considered molecular orbitals as a linear combination of analytic atomic oneelectron functions, shifting the mathematical task from the numerical solution of coupled differential equations to the evaluation of integrals over basis functions. Introduction of approximations for the most difficult integrals through the use of suitable parameters led to the development of semiempirical methods beginning in the 1950s. The more rigorous ab initio methods benefited from the use of Gaussian- instead of Slater-type basis functions, as pointed out by Boys in 1950 but generally accepted only two decades later. Configuration interaction (CI) was the first theoretical level used to include electron correlation and was widely applied during the 1970s. In the late 1970s many-body perturbation theory (Møller-Plesset methods), and during the 1980s coupled cluster methods, became more popular because they are more economical and more rapidly convergent, respectively, than CI. The 1990s can be considered the decade of density functional theory, which by that time had become so-

Computational Chemistry

phisticated enough to be useful for applications in chemistry. Molecular mechanics emerged in the mid-1960s and has become more sophisticated and more useful with time. Due to the number of various approximations, early computations performed to try to reproduce experimental findings yielded varying degrees of success. Computational chemistry could become a recognized scientific discipline only after a real predictive power was established. Perhaps the first case in which theory proved to be accurate enough to challenge experiment was the structure determination of methylene (CH2 ). From a spectroscopic investigation the ground state of this molecule was first concluded to be linear. However, this interpretation had to be revised after reliable computations predicted a significantly bent structure in 1970. The development of different methods and their efficient implementation is only one reason for the success of computational chemistry. Another factor is the dramatic development of computer technology (i.e., computational speed as well as the amount of core memory and of disk storage). Today’s personal desktop computers provide many times the computer power of early “supercomputers” at a fraction of the price. The combined development of both software and hardware allowed computational chemistry to become for chemical research an indispensable tool which allows one to plan experiments more carefully and hence to optimize the use of laboratory resources. The importance of computational chemistry was honored when the 1998 Nobel Prize in Chemistry was awarded to two pioneers of the field, J. A. Pople and W. Kohn.

II. METHODS USED IN COMPUTATIONAL CHEMISTRY The methods used in computational chemistry can be classified according to the sophistication of the underlying model (Fig. 1). Molecular mechanics methods are based on classical mechanics and are computationally the fastest. Semiempirical methods are based on a wave function description, in which some integrals are approximated by means of parameters and many others are neglected to reduce the computational cost. Ab initio methods use only fundamental physical constants but no further experimental results; Hartree-Fock (H F) theory is the starting level, which can be improved upon by accounting for electron correlation in various ways. Density functional methods are also quantum mechanical but are based on the electron density to describe chemical systems. They are often considered “ab initio” although some empirical parameters enter the energy functionals.

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FIGURE 1 Classification of computational chemistry methods. AMBER, assisted model building with energy refinement; CHARMM, chemistry at Harvard molecular mechanics; GROMOS, Groningen molecular simulation; CNDO, complete neglect of differential overlap; INDO, intermediate neglect of differential overlap; NDDO, neglect of diatomic differential overlap; MNDO, modified neglect of differential overlap; AM1, Austin model 1; PM3, parametric method number 3; HF, Hartree-Fock; MP2, Møller-Plesset, second order; MP3, Møller-Plesset, third order; MP4, MøllerPlesset, fourth order; CISD, configuration interaction singly and doubly excited; CISD(T), configuration interaction singly, doubly, and triply (estimated) excited; CCSD, coupled cluster singly and doubly excited; CCSD(T), coupled cluster singly, doubly, and triply (estimated) excited; LDA, local density approximation; GGA, generalized gradient approximation; BLYP, Becke/Lee, Yang, and Parr; B3LYP, Becke three-parameter/Lee, Yang, and Parr.

A. Force Field Methods A force field (FF) is a set of equations describing the potential energy surface of a chemical system. A molecular mechanics (MM) method uses a force field based on a classical mechanical representation of molecular forces to calculate static properties of a molecule (e.g., structure and energy of an energy minimum structure). Molecular dynamics (MD) also implements a force field but generates dynamic properties (e.g., evolution of an structure in time) by calculating forces and velocities of atoms. In MM methods atoms are treated as “balls” of different masses and sizes, and bonds are “springs” connecting the balls without an explicit treatment of electrons. The main advantage of this simple classical approach is the small computational cost, which allows one to treat very large molecules. FFs are typically constructed to yield experimentally accurate structures and relative energies. Some FFs are generated to accurately compute other properties such as vibrational spectra. The observation that properties of chemical functional groups are normally transferable from one compound to another validates the MM approach. The most basic component in a FF is the atom type and one element usually contributes several atom types. Each bond is characterized by the atom types involved and has a “natural” bond length since the variation with the chemical environment is relatively small. Similarly, bond angles between atom types have typical values. The energy absorptions in in-

frared (IR) spectroscopy associated with a certain bond stretch or angle deformation also fall in narrow ranges, which demonstrates that the variation of force constants is also relatively small. The existence of an increment system for heats of formation, for example, shows that the energy behaves additively as well. Hence, in MM the energy is expressed classically as a function of geometric parameters. 1. Energy Terms Advanced force fields distinguish several atom types for each element (depending on hybridization and neighboring atoms) and introduce various energy contributions to the total force field energy, E FF : E FF = E str + E bend + E tors + E vdW + E elst + · · · , where E str and E bend are energy terms due to bond stretching and angle bending, respectively; E tors depends on torsional angles describing rotation about bonds; and E vdW and E elst describe (nonbonded) van der Waals and electrostatic interactions, respectively (Fig. 2). In addition to these basic terms common to all empirical force fields there may be extra terms to improve the performance for specific tasks. Each term is a function of the nuclear coordinates and a number of parameters. Once the parameters have been defined, the total energy, E FF , can be computed and subsequently minimized with respect to the coordinates.

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FIGURE 3 (Curve a) Harmonic potential, E(R) = k(R0 − R)2 ; (curve b) third-order polynomial anharmonic potential, E(R) = 2  3 k(R0 − R) √ + k (R0 − R) ; (curve c) Morse potential, E(R) = k/2D(R −R ) 2 0 ) . D(1 − e

with θ = θ0ABC − θ ABC . If higher accuracy is desired (e.g., for computing IR frequencies), a third-order term can be included with an anharmonicity constant set to be a fraction of k ABC . FIGURE 2 Most basic energy terms included in empirical force field (FF) methods.

a. Stretch energy. The harmonic approximation gives the stretch energy of a bond between atom types AB A and B, E str , as AB E str (R) = k AB R 2 ,

where k AB is the force constant and R = R0AB − R AB is the bond length deviation from the natural value, R0AB , for AB which E str is defined to be zero. Further improvement can be achieved by including higher anharmonic terms to the equation. While these expressions describe the potential well for R close to R0 , the energy goes to infinity for large distances (Fig. 3). In contrast, a morse potential allows the energy to approach the dissociation energy, D, as R increases: E Morse (R) = D(1 − e

c. Torsion energy. The torsional potential, due to the rotation of bonds A–B and C–D about bond B–C, is periodic in the torsional angle ω, which is defined as the angle between the projections of A–B and C–D onto a plane perpendicular to B–C. The torsional energy therefore is expressed as a Fourier series:  ABCD E tors (ω) = Vn cos(nω), n

which allows the representation of potentials with various minima and maxima (Fig. 4). Three terms are enough to model the most common torsional potentials.

√ k/2DR 2

) ,

but it is much more expensive in terms of computational cost. b. Bending energy. The harmonic approximation ABC for the bending energy, E bend , due to the deformation of the angle between the A–B and B–C bonds, is sufficient for most purposes: ABC E bend (θ ) = k ABC θ 2

FIGURE 4 A three-minimum potential  (bold line) represented as a three-term Fourier series: E(ω) = 3n=1Vn cos(n ω).

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d. van der Waals energy. The van der Waals term, E vdW , covers nonelectrostatic interactions between nonbonded atoms. The van der Waals energy is positive (repulsive) and very large at short distances, zero at large distances, but slightly negative (attractive) at moderate distances due to temporarily induced multipole attraction (dispersion force), the most important attractive contribution of which (dipole–dipole interaction) has an R −6 dependence. The Lennard-Jones potential for E vdW includes a repulsive term, which is set proportional to (R −6 )2 to grow faster than R −6 :  12 6   AB R0AB  AB R0 E LJ (R) = ε . −2 R R εAB determines the energy depth of the minimum. Alternatively, a Buckingham or Hill potential can be used that employs an exponential function for the repulsive term. For each atom type a van der Waals radius, R0 , and the atom softness, ε, have to be determined, from which the diatomic parameters are calculated according to √ R0AB = R0A + R0B and ε AB = ε A ε B . e. Electrostatic energy. The electrostatic energy, E elst , is due to the electrostatic interactions arising from polarized electron distributions based on electronegativity differences. It can be modeled by Coulomb interactions of point charges associated with individual atoms: QA QB , εR AB ε being a dielectric constant, which can be used to model the effect of the same or other molecules present (e.g., solvent). The atomic charges, Q, are commonly obtained by fitting to the electrostatic potential as calculated by an electronic structure method. An E elst description based on dipole–dipole interactions between polarized bonds can alternatively be employed. Hydrogen bonds are nonbonded interactions between a positively charged hydrogen atom and an electronegative atom with lone electron pairs (mostly oxygen or nitrogen) and can be adequately modeled by appropriately chosen atomic charges. Although a single hydrogen bond is a very weak interaction, the large number occurring in biomolecules (e.g., proteins) makes hydrogen bonding a very important factor. In the large size limit, the bonded interactions increase linearly with the system size, but the nonbonded interactions show a quadratic dependence and determine the computational cost. The van der Waals interactions quickly fall off with the distance (R −6 dependence) and may be neglected for large separations. The electrostatic interaction (proportional to R −1 ) is much more far reaching and E elst (R AB ) =

needs to be considered out to very long distances. Fast multipole methods (FMMs) can be applied to reduce the computational cost of evaluating E elst . f. Other energy contributions. So that the performance can be improved, force fields include further parameters to take care of special cases. For example, cross terms account for the interplay between different contributions (e.g., longer bonds for small angles). Correction terms may be introduced to describe substituent effects (e.g., anomeric effect). Additional terms may be introduced to adequately treat special cases like pyramidalization of sp2 hybridized atoms. Hydrogen bonding may be treated explicitly (in addition to the electrostatic interaction) with a special set of van der Waals interaction parameters. Pseudo atoms maybe introduced to model lone pairs. In addition, atoms in unusual bonding situations (three-membered rings, molecules with linearly conjugated π -systems, aromatic compounds, etc.), which are not described adequately by the normal parameters, can be defined as new atom types. The force field energy, E FF , corresponds to the energy relative to a molecule with noninteracting fragments. Therefore, only energies for molecular structures built from the same fragments (conformers) can be compared directly. So that energy between different molecules (isomers) can be compared, the energy scale is converted to heats of formation by adding bond increments (estimated from bond dissociation energies minus the heat of formations of the atoms involved) and possibly group increments (e.g., methyl group): Hf = E FF +

bonds 

H AB +

groups 

H G .

2. Parametrization Determining the parameters for a force field is a substantial task. In general, not all necessary data are available from (accurate) experiments. Modern electronic structure computations can provide unknown data relatively easily and with sufficient accuracy. Another problem is the large number of parameters: for a force field with N atom types, the number of different types of bonds, bond angles, and dihedral angles scales as N 2 , N 3 , and N 4 , respectively, each requiring several parameters. So that the number of parameters can be reduced, the atom dependency can be reduced (e.g., the torsional parameters may be treated as dependent on the B–C central bond only and not on the atom types A and D). The parametrization effort can be reduced further by defining “generic” parameters to be used for less common bond types or when no reference data are available. This, of course, reduces the quality of a calculation. By deriving the di-, tri-, and tetra-atomic parameters

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from atomic data (atom radii, electronegativities, etc.), universal force fields (UFFs) allow one to include basically all elements. The performance, however, is relatively poor. The kind of energy terms, their functional form, and how carefully (number, quality, and kind of reference data) the parameters were derived determine the quality of a force field. Accurate force fields exist for organic molecules (e.g., MM2, MM3), but more approximate force fields (e.g., with fixed bond distances) optimized for computational speed rather than accuracy [e.g., AMBER (assisted model building with energy refinement), CHARMM (chemistry at Harvard molecular mechanics), GROMOS (Groningen molecular simulation)] are the only practical choice for the treatment of large biomolecules. The type of molecular system to be studied determines the choice of the force field. One limitation of force field methods is that they can describe only well-known effects that have been observed for a large number of molecules (this is necessary for the parametrization). The predictive power of these methods is limited to extrapolation or interpolation of known effects. 3. Quantum-Mechanical and Molecular-Mechanical (QM/MM) Method Another limitation of MM is the inability to investigate reactions. While force field methods are capable of describing conformational changes, for which all bonds remain intact along the reaction coordinate, they are by construction not capable of treating reactions in which bonds are broken and/or formed. The classical model is not designed to describe the electronic rearrangement associated with bond breaking and bond formation. Such problems are better treated by electronic structure methods discussed below. For large systems, a combined quantummechanical and molecular-mechanical (QM/MM) method can be applied. In this approach the reactive part of the molecule to be studied is described by a quantummechanical (semiempirical, ab initio, or DFT) method while the rest of the system is treated by a force field. The problem with this approach is the “communication” between classical and quantum-mechanical potential (i.e., how the atoms close to the QM/MM border are treated). The total energy, Etot , may be computed as follows: Etot = EQM + EMM + EQM/MM , where the quantum-mechanical contribution, EQM , and the molecular-mechanical contribution, EMM , are defined by a QM method and a MM method, respectively. The coupling term, EQM/MM , includes parameters that can be fitted

to reproduce experimental results and are specific to the chosen combination of QM and MM methods. Alternatively, the total energy, Etot , may be extrapolated from QM and MM calculations on a small part and on the whole of a suitably partitioned system (IMOMMintegrated molecular orbital, molecular mechanics method) Etot = EQM (small) + EMM (whole) − EMM (small) B. Wave Function Quantum-Mechanical Methods The explicit treatment of electrons in atoms and molecules requires quantum mechanics, which invokes a wave function, , to describe the system of electrons and nuclei. The square of the wave function represents the probability of a particle’s being at a given position. The central goal becomes the solution of the (time-independent) Schr¨odinger equation, H = E , which relates the wave function, , to the energy, E, of the system. The Hamiltonian operator, H, consists of the kinetic (T) and the potential energy (V) operators: H = T + V. The fact that electrons instantly adjust to changes in nuclear positions due to the much greater masses of the nuclei allows the motions of electrons and nuclei to be separated (Born-Oppenheimer approximation). The electronic wave function depends on only the nuclear position, not on the nuclear momenta. The electronic Hamiltonian, He , in atomic units is given by He = Te + Vne + Vee + Vnn = − +

Nucl.  Elec.  a

+

i

Nucl.  Nucl.  a

b>a

 1 Elec. ∇i2 2 i

Elec.  Elec.  Za 1 + |Ra − ri | |ri − r j | i j>i

Za Zb , |Ra − Rb |

where r and R represent the electronic and nuclear coordinates, respectively, and the Laplacian is defined as   ∂2 ∂2 ∂2 2 ∇i = + 2+ 2 . ∂ xi2 ∂ yi ∂z i The nucleus–nucleus repulsion, Vnn , is constant for a given geometry, and the kinetic energy, Te , and the electron– nucleus attraction, Vne , are easy to evaluate. The electron– electron repulsion, Vee , however, depends on the distances between electrons and is the reason why the Schr¨odinger

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equation cannot be solved exactly for systems with more than one electron. The energy can be computed as the expectation value of the Hamiltonian operator:

∗

H dτ |H|

E= ∗ = , |

dτ where the common bra-ket notation is used. The variational principle states that any trial wave function will give an energy equal to or higher than the exact value because the real system will adopt the best possible wave function (which corresponds to the exact energy). Thus, a trial wave function constructed in terms of a number of parameters can be improved by minimizing the energy with respect to the parameters (MO coefficients). A meaningful trial wave function should approach zero as r goes to infinity; it should be normalized, that is,

For example, a Slater determinant for the ground state of the hydrogen molecule can be written as follow:   1 φg (1) φg (1) (1, 2) = √   2 φg (2) φg (2) 1 = √ [φg (1)φg (2) − φg (2)φg (1)] = −(2, 1), 2 where φg represents the bonding molecular orbital (MO), the 1σg orbital. The electronic Hamiltonian can be written as sums of one-electron (hi ) and two-electron (gi j ) operator plus the constant nuclear–nuclear repulsion:   H= hi + gi j + Vnn i

with

and gi j =

E=

N 

α | β = β | α = 0

to give spin orbitals φα and φβ (or φ and φ¯ for short). This is appropriate for closed-shell species with paired electrons, but open-shell species with unpaired electrons cannot be expected to have identical α and β orbitals. The unrestricted Hartree-Fock (UHF) method allows a different spatial function for each electron. However, UHF wave functions can suffer from spin contamination (i.e., the spurious mixing of higher spin states into the desired one [more formally, the expectation value of the S2 operator is larger than the correct value of S(S + 1), S being the total spin]). Restricted open-shell Hartree-Fock (ROHF)– based methods avoid the problem of spin contamination but do not allow spin polarization.

φi |hi |φi +

i

N  N 1 ( φi φ j |gi j |φi φ j 2 i=1 j=1

− φi φ j |gi j |φ j φi ) + Vnn =

N 

hi

i

1. Hartree-Fock Method

α | α = β | β = 1;

1 . |ri − r j |

The Hartree-Fock energy expression becomes

+

Hartree-Fock theory employs a single Slater determinant. In the restricted Hartree-Fock (RHF) method, one spatial function φi is multiplied by an α (representing spin up, spin quantum number m s = + 12 ) or β (representing spin down, m s = − 12 ) spin function with the properties

j>1

Nucl.  Za 1 hi = − ∇i2 − 2 |R − ri | a a

|

= 1 (meaning the probability that the system is located somewhere in space is one); and it should comply with the Pauli principle. The latter states that two electrons must differ in at least one quantum number. Furthermore, the wave function should be antisymmetric (i.e., it should change sign when two electrons are interchanged). This is a characteristic property of electrons. Antisymmetry can be ensured by using Slater determinants with one-electron functions (orbitals) φi in columns and electrons (1, 2, . . .) in rows.

i

N  N 1 (Ji j − K i j ) + Vnn , 2 i=1 j=1

or E=

N  i

φi |hi |φi +

N  N 1 ( φ j |Ji |φ j 2 i=1 j=1

− φ j |Ki |φ j ) + Vnn . Ji j is called a Coulomb integral because it corresponds to the electronstatic repulsion of the charge distributions due to φi2 and φ 2j ; the exchange integral, K i j , has no classical equivalent. Ji and Ki are Coulomb and exchange operators, respectively. To find a minimum energy, one can vary the orbitals under the condition that they remain orthogonal by using the method of Lagrange multipliers. This leads to the definition of the Fock operator, Fi , which describes the kinetic energy, the nuclear attraction energy, and the electron repulsion energy of one electron in the field of the other electrons: N  Fi = hi + (J j − K j ). j

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A set of Hartree-Fock equations is obtained, Fi φi =

N 

λi j φ j ,

j

which can be simplified by a unitary transformation, which does not change the total wave function, to make the Lagrange multipliers diagonal: Fi φi = εi φi . These special MOs φi are called canonical MOs and the εi are the corresponding MO energies, the expectation values of the Fock operator in the MO basis: εi = φi |Fi |φi . According to Koopman’s theorem the orbital energy corresponds to the ionization energy for a particular electron (neglecting orbital relaxation). The Hartree-Fock equations can be solved only iteratively because the Fock operator depends on all the occupied MOs by means of the Coulomb and exchange operators. The molecular orbitals, φi , are constructed as a linear combination of atomic orbitals (LCAO), χα , which form the basis set (see below), φi =

M  α

cαi χα .

This leads to the Roothaan-Hall equations, which correspond to the Fock equations in the AO basis: FC = SCε.

χα χγ |g|χβ χδ ≡ χα χγ | χβ χδ ≡ αγ | βδ 1 ≡ χα (1)χγ (2) χβ (1)χδ (2) dr1 dr2 . |r1 − r2 | The Roothaan-Hall equations give the orbital coefficients as eigenvectors of the Fock matrix. So that the Fock matrix can be constructed, however, the density matrix, D (i.e., the orbital coefficients), has to be known. To start the iterative procedure, one must make an initial guess (e.g., from another calculation, or D is just set to zero), from which a Fock matrix can be derived. Diagonalization of the Fock matrix gives new (improved) orbital coefficients which allow one to build a new density matrix and a new Fock matrix. The procedure must be continued until the change is less than a given threshold and a self-consistent field (SCF) is generated (Fig. 5). In the large basis set limit the Hartree-Fock method formally scales with the fourth power of the number of basis functions due to the two-electron integrals. In practice, computations have a more favorable scaling. Modern algorithms are close to linear scaling because the Coulomb part of the electron–electron interaction which contributes the most to the computational effort (the distances where exchange becomes negligible is relatively short) can be replaced by a multipole interaction for large distances (fast multipole method, FMM). In a conventional HF implementation the two-electron integrals are computed and stored on disk. In contrast, in the direct SCF method, the integrals are recomputed whenever they are needed

The elements of the overlap matrix, S, are defined as Sαβ = χα |χβ , and the Fock matrix elements are given by Fαβ = χα |h|χβ +

occ.MO 

χα |J j − K j |χβ .

j

The energy in terms of integrals over basis functions is given by E=

M  M  α

+

Dαβ χα |h|χβ

β M  M  M  M 1 Dαβ Dγ δ ( χα χγ |g|χβ χδ 2 α β γ δ

− χα χγ |g|χδ χβ ) + Vnn , which introduces the density matrix elements, Dγ δ , as Dγ δ =

occ.MO 

c γ j cδ j .

j

The two-electron integrals are often written without the g operator, and for further simplification only the indices are given:

FIGURE 5 Schematic representation of the self-consistent field (SCF) procedure. MO, molecular orbital.

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to avoid the storage bottleneck and the slow input–output operations. It is also possible to effectively screen for integrals which contribute only negligibly and thus can be discarded. The use of symmetry, if present, also reduces the computational cost considerably. 2. Semiempirical Methods The HF method represents a point of departure in electronic structure theory. One direction involves improvement of the accuracy by including electron correlation (see Section II.B.3.). Semiempirical methods, however, try to provide moderate accuracy, but at much lower cost than that of ab initio methods. Therefore, only valence electrons are treated explicitly and core electrons are replaced by an effective core (covering nucleus plus core electrons) and a minimal basis of orthogonal Slater-type orbitals (usually only s and p types) is chosen to describe the valence electrons. The two-electron integrals require the main computational effort in a HF calculation and their number is significantly reduced in semiempirical methods by the zero differential overlap (ZDO) approximation. This basic semiempirical assumption sets products of functions for one electron but located at different atoms equal to zero (i.e. µA (1)νB (1) = 0, where µA and νB are two different orbitals located on centers A and B, respectively). The overlap matrix, S, is set equal to the unit matrix, Sµν = δµν , and the two-electron integrals µν | λσ are zero, unless µ = ν and λ = σ , that is, µν | λσ = δµν δλσ µµ | λλ , where δi j = 0 for i = j and δi j = 1 for i = j. All three-and four-center two-electron integrals vanish automatically. One-electron integrals involving three centers are also set to zero. The remaining integrals are handled as parameters which partly compensate the errors introduced by the ZDO approximation. The parameters are derived from experimental data on atoms or are fitted to reproduce experimental results for molecules. The various semiempirical methods introduce different approximations for the one- and two-electron parts of the Fock matrix elements, Fµν = µ|h|ν +

AO  AO  λ

σ

Dλσ ( µν | λσ − µλ | νσ ),

with the one-electron operator  Z  1 1 a h = − ∇2 − = − ∇2 − Va , 2 2 a |Ra − r| a where Z a denotes the charge resulting from the nucleus plus the core electrons.

a. Complete neglect of differential overlap. The complete neglect of differential overlap (CNDO) approximation is the most rigorous: only the one- and two-center Coulomb terms among the two-electron integrals survive: µA νB | λC σD = δAC δBD δµλ δνσ µA νB | µA νB . µA νB | µA νB are independent of the orbital type (to guarantee rotational invariance) and there are only two parameters, µA νA | µA νA = γAA and µA νB | µA νB = γAB , for the two-electron integrals. The γAB depends only on the nature of the atoms A and B and the distance between them and can be interpreted as the average electrostatic repulsion of one electron at center A and one electron at center B. The integral γAA is the average repulsion of two electrons at one atom. The one-electron integrals are µA |h|νA = −δµν

Nucl. 

µA |Va |µA .

a

The Pariser-Pople-Parr (PPP) method is a special case of CNDO, restricted to the treatment of π electrons. b. Intermediate neglect of differential overlap. In the intermediate neglect of differential overlap (INDO) approximation the two-electron integrals are limited to the Coulomb integrals. One-electron integrals involving different orbitals of one center and Va operator from another have to disappear to guarantee rotational invariance. The one-electron integrals are the same as in the CNDO approximation and the two-electron integrals are given by µA νB | λC σD = δµA λC δνB σD µA νB | µA νB and parametrized as γAB and γAA . INDO is comparable to CNDO in computational cost but has the advantage that electronic states of different multiplicities can be distinguished. MINDO/3 (modified intermediate neglect of differential overlap) was the first successful semiempirical method to give reasonable predictions of molecular properties. The main improvement over earlier methods was the use of molecular data rather than atomic data for the parametrization. However, the number of parameters to be determined in MINDO/3 increases with the square of the number of atoms included because one parameter depends on the type of bonded atoms. c. Neglect of diatomic differential overlap. Many of the shortcomings of MINDO/3 are corrected in the neglect of diatomic differential overlap (NDDO) approximation, which includes no further approximations beyond ZDO. Thus, all integrals involving any two orbitals on one

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center with any two orbitals on another center are kept, which increases the number of integrals dramatically. The one-electron integrals are  

 µA |h|νB = µA − 12 ∇ 2 − VA − VB νB and

  

 µA |h|νA = δµν µA − 12 ∇ 2 − VA µA − µA |Va |νA . a=A

The two-electron integrals are given by µA νB | λC σD = δAC δBD µA νB | λA σB . d. Modified NDDO. The more successful semiempirical methods, MNDO (modified neglect of differential overlap), AM1 (Austin model 1), and PM3 (parametric method number 3), are all based on NDDO but differ in the treatment of core–core repulsion and how the parameters are assigned. There are only atomic parameters, no diatomic parameters as in MINDO/3. The “modified” NDDO methods calculate the overlap matrix, S, explicitly rather than using the unit matrix. The MNDO method tends to overestimate the repulsion between atoms separated by approximately the sum of their van der Waals radii. To correct for this deficiency, AM1 modifies the core–core term by Gaussian functions. PM3 is essentially equivalent to AM1 but uses (automated) full optimization of the parameter set against a much larger collection of experimental data while the AM1 parameters are tuned by hand. PM3 therefore on average gives results in somewhat better agreement with experiment. Extending the basis set to include d functions as in MNDO/d and PM3(tm) raises the number of integrals (i.e., the number of parameters) tremendously but allows a larger variety of applications, for example, those including transition metal compounds (albeit with variable accuracy) or hypervalent molecules. Semiempirical programs usually report heats of formation calculated from the electronic energies less the calculated energies for the atoms plus the experimental heat of formations for the atoms: atoms  Hf = E calc. (molecule) − E calc. (atom) +

atoms 

Hf (atom).

The semi ab initio model 1 (SAM1) is another modified NDDO method, but it does not replace integrals by parameters. The one- and two-center electron repulsion integrals are explicitly calculated from the basis functions [employing a standard STO-3G (Slater-type orbital from three Gaussian functions) Gaussian basis set] and scaled by a function which has to be parametrized. SAM1 calculations take about twice as long as AM1 or PM3 calculations do.

3. Electron-Correlated Methods In the Hartree-Fock approach the real electron–electron interaction is replaced by an interaction with an averaged field. This means HF suffers from an exaggeration of electron–electron repulsion. The difference between the energy obtained at the HF level and the exact (nonrelativistic) energy (for a given basis set) is defined as the correlation energy. The name reflects that this energy difference is connected to the correlated movement of electrons which is not considered in the HF method and which reduces the electron–electron repulsion. The HF description typically allows electrons to be unrealistically resident in the internuclear region. This leads to an underestimated nuclear–nuclear repulsion and to bond lengths that are too short. As a consequence, stretching force constants and harmonic stretching frequencies computed at the HF level are too large. Likewise, the polarity of bonds is overestimated (the more electronegative atom tolerates a higher electron density in the HF picture) and computed dipole moments are often too large. Dynamic electron correlation, which is connected to the correlated movement of electrons, can be distinguished from static (near-degeneracy) electron correlation, which deals with the insufficiency of the one-determinant approach. HF usually provides a suitable description of closed-shell molecules in their electronic ground state. However, the homolytic dissociation of such a molecule generates two electronic states which are very close in energy. This situation requires a description by more than one Slater determinant (i.e., at least a two-configuration method). The energy difference between the HF method and a multiconfigurational method is the static correlation energy. Accounting for electron correlation is essential for quantitative answers from electronic structure calculations. Different post-HF methods which attempt to recover all or part of the correlation energy are discussed in the following text. Within the closed-shell HF picture, molecular orbitals are occupied by either exactly two or exactly zero electrons represented by the variationally best one-determinant wave function. Correlated levels give a different electron density which cannot be represented by a single Slater determinant. A logical starting point to account for electron correlation is to expand a multideterminantal wave function with the HF wave function as a starting point: 

= a0 HF + ai i , i=1

where a0 usually is close to 1. Because this is analogous to expanding one MO in terms of AOs, one speaks of the basis set as the one-electron basis (responsible for the one-electron functions, the MOs) while the number

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sity matrix) may be used, which promise faster convergence of the CI expansion. In general, several Slater determinants are contracted linearly to form eigenfunctions of the spin operators SZ and S2 and which are called spin adapted configurations, or configuration state functions (CSFs): 

CI = ai i . i=0

FIGURE 6 Examples of singly, doubly, triply, and quadruply excited determinants derived from a Hartree-Fock (HF) reference.

of determinants included in a correlated wave function builds the many-electron basis. In closed-shell HF theory there is only one determinant which has the lowest MOs occupied (occupation number = 2). The remaining orbitals are empty or “virtual” (occupation number = 0). Additional determinants are generated by exciting one or more electrons from an occupied MO into an unoccupied (virtual) MO. According to the number of excited electrons, one speaks of singles, doubles, triples, quadruples, and so forth (S, D, T, Q, respectively; Fig. 6). The larger the basis set the more virtual MOs and the more excited Slater determinants can be generated. The quality of a calculation is determined by both the size of the basis set and the number of excited determinants that are considered. If all possible determinants together with an infinite basis set could be used, one would get the exact solution of the nonrelativistic Schr¨odinger equation within the Born-Oppenheimer approximation. Because a different chemical environment mostly affects the valence electrons, but does not influence the core electrons, the frozen core approximation includes only determinants with excited valence electrons. Also the highest virtual orbitals may be left unoccupied in all determinants (frozen virtuals). a. Configuration interaction. In the configuration interaction (CI) procedure the trial function is constructed as a linear combination of the ground (reference configuration) and excited Slater determinants. The MO coefficients remain fixed throughout the calculation and are usually taken from the HF orbitals. Alternatively, natural orbitals (which are defined as diagonalizing the one-electron den-

The expansion coefficients, ai , are then determined variationally to give the minimum energy. For a full CI (FCI) the number of determinants grows factorially with the size of the system. A full CI recovers all of the electron correlation energy (for a given basis set) but can be applied only to obtain benchmark results for very small molecules to assess the performance of more economical methods. For applications to larger molecules, the CI expansion has to be truncated to make the computation feasible. CIS, CISD, CISDT, and CISDTQ correspond to expansions through singly, doubly, triply, and quadruply excited CSFs, respectively. According to Brilluoin’s theorem, the CI matrix elements of a closed-shell restricted HF wave function with singly excited CSFs vanish. Hence, CIS does not improve the description of the ground state. Doubles are found to contribute most to the correlation energy and consequently CISD (including only singly and doubly excited determinants) is the most widely applied CI method because inclusion of triples and quadruples is typically computationally too demanding. FCI is size consistent, but truncated CI methods are not. This means the energy computed for two noninteracting molecules is not identical to the sum of the energies computed for the individual molecules. This unphysical behavior is a major drawback of any truncated CI. So that CISD can be made approximately size consistent, the Davidson correction can be applied in which the contribution of quadruples, E Q , is estimated from the correlation energy given at the CISD level, E CISD , and the coefficient of the reference configuration, a0 : E Q = (1 − a0 )E CISD . CISD was also extended to quadratic CISD (QCISD) by the inclusion of some higher-order terms to yield a sizeextensive method. b. Multiconfiguration self-consistent field. The HF method does not give a good first-order description when more than one nonequivalent resonance structure is important for the electronic structure of a molecule. A multiconfiguration self-consistent field (MCSCF) calculation may be used instead. Not only the coefficients for the determinants are optimized in MCSCF, but also the MO coefficients simultaneously. The selection of configurations

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is not trivial. One easy way to construct an MCSCF is the complete active space self-consistent field, or CASSCF (also called full optimized reaction space, or FORS). Instead of choosing configurations, one must select a set of “active” (occupied and unoccupied) orbitals and all possible (symmetry-adapted) configurations within this “active space” are automatically included in the MCSCF. The method is called restricted active space self-consistent field, or RASSCF, when subsets of the active orbitals are restricted to have a certain (minimum or maximum) number of electrons to reduce the computational cost. The MCSCF provides a good first-order description covering the static electron correlation due to degeneracy problems. Dynamic electron correlation should be addressed with the MCSCF wave function as a reference. The multireference configuration interaction, or MRCI, generates excited determinants from all (or selected) determinants included in the MCSCF. The complete active space perturbation theory, second order (CASPT2) is a more economical approach. Both methods can be applied to compute excited states. c. Many-body perturbation theory. Perturbation theory assumes that somehow an approximate solution to a problem can be found. The missing correction, which should be small, is then considered as a perturbation of the system. When the perturbation is to correct for the approximation of independent particles the method is called many-body perturbation theory, or MBPT. In electronic structure theory the Hamiltonian operator, H, is written as a combination of a reference Hamiltonian, H0 , which can be solved for, and a perturbation H : H = H0 + λH , λ being the perturbation parameter (0 ≤ λ ≤ 1) which determines the strength of the perturbation. The energy, E, and wave function, , are expanded as Taylor series in λ: E = E 0 + λE 1 + λ2 E 2 + λ3 E 3 + · · · and

= 0 + λ 1 + λ 2 + λ 3 + · · · . 2

3

The Schr¨odinger equation, H = E , gives 0 and E 0 as the solution in the absence of any perturbation (λ = 0). E 1 , E 2 , etc., and 1 , 2 , etc., are the first-, second-, etc., order corrections to the energy and wave function, respectively. For λ > 0 the Schr¨odinger equation becomes   (H + λH ) 0 + λ 1 + λ2 2 + · · ·    = E 0 + λE 1 + λ2 E 2 + · · · 0 + λ 1 + λ2 2 + · · · .

Because this equation has to be true for all values of λ, the terms connected to the same power of λ can be separated: λ0: H0 0 = E 0 0 , λ1: H0 1 + H1 0 = E 0 1 + E 1 0 , λ2: H0 2 + H1 1 + H2 0 = E 0 2 + E 1 1 + E 2 0 , and so forth, which gives the zeroth-, first-, second-, etc., order perturbation equations. If one chooses the intermediate normalization condition, 0 | 0 = 1;

0 | i = 0,

i > 0,

simple energy expressions are obtained: E 0 = 0 |H0 | 0 , E 1 = 0 |H | 0 , E 2 = 0 |H | 1 , and so forth. Knowledge of wave function corrections up to order i allows calculation of the energy up to order (2i + 1). This relationship is known as the Wigner theorem. In Møller-Plesset (MP) perturbation theory the unperturbed Hamiltonian, H0 , is taken as the sum over n Fock operators (n = number of electrons) giving a total of twice the average electron–electron repulsion energy and the perturbation operator becomes the difference between the exact electron–electron repulsion and twice the average electron–electron repulsion. With this choice of H0 the zeroth-order energy is just the sum of MO energies and the first-order energy equals the Hartree-Fock energy. The second-order correction, E(MP2), is the first to contribute to the electron correlation energy and can be calculated from the two-electron integrals over MOs: E(MP2) =

virt occ   [ φi φ j | φa φb − φi φ j | φb φa ]2 i< j a
εi + ε j − εa − εb

.

MP2 is a very economical method but often overcorrects for electron correlation effects. The MP2 energy calculation scales only with N 4 , but the transformation of AO to MO integrals is an N 5 step (Table I). The next step in the series, MP3, also includes only contributions from doubly excited determinants but scales with N 6 . Full MP4 involves singly, doubly, triply, and quadruply excited determinants and is an N 7 method. In applications, triples are sometimes left out (MP4SDQ is N 6 ) to make the calculation affordable. MPn methods with n > 4 are not used routinely because they are both very complex and expensive in terms of resources. MP theory is size extensive but not variational (i.e., there is no guarantee that the correct energy is lower than

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Scaling

Size extensive?

Variational?

HF MP2 MP3

N4 N5 N6

Yes Yes

Yes No

Yes

No

MP4SDQ

N6

MP4SDTQ MP5

N7 N8

Yes Yes

No No

MP6 CCSD

N9 N6

Yes Yes

No No

Yes

No

CCSD(T)

N7

CCSDT CISD

N8 N6

Yes Yes

No No

No

Yes

CISDT

N8

No

Yes

CISDTQ

N 10

No

Yes

a HF, Hartree-Fock; MP, Møller-Plesset (numbers 2– 6 refer to second through sixth order; S, singly excited; D, doubly excited; T, triply excited; and Q, quadruply excited; CC, coupled cluster (S, D, and T as for MP except T in parentheses is estimated); CI, configuration interaction (S, D, T, and Q as for MP).

an MPn energy). This is no problem because usually only relative energies are of interest. However, the MPn series does not necessarily converge. When the HF reference provides a poor description of the electronic structure, the MPn series may become divergent and produce even worse results than those of HF. d. Coupled cluster methods. Coupled cluster (CC) theory was originally formulated for nuclear physics and only later was applied to the electron correlation problem in quantum chemistry. Today it is the method of choice for highly accurate computations. CC theory uses an exponential expansion of a reference function 0 , usually the Hartree-Fock determinant (in contrast with the linear expansion of CI):

T = T1 + T2 + T3 + · · · T N . The excitation operators, Ti , generate all ith excited Slater determinants from the reference:

i

T2 0 =

tia ia ;

a

occ  virt  i< j a
ab tiab j i j ;

1 4 T 24

+ ··· =

shows that due to the exponential ansatz in CC, i.) a given excitation level in general is not due to just one excitation operator (there are “disconnected” terms in addition to the “connected” term; for example, for doubles, T21 and T2 , respectively). ii.) restricting T generates not only excitations up to a given level, but also higher ones (quartets, etc.) up to infinity. For example, for T = T2 , eT2 = 1 + T2 + 12 T22 + 16 T32 + · · · . This is in contrast with both perturbation and CI methods and therefore CC theory should provide a better description of electron correlation effects at a given truncation level. The Schr¨odinger equation becomes H eT 0 = E CC eT 0 , and the CC energy is given by  

 E CC = 0 H eT 0 . Because the Hamiltonian operator contains only one- and two-electron operators, only the first few terms of the exponential series give nonzero values:

   E CC = 0 |H| 1 + T1 + T2 + 12 T21 0 . Further simplification leads to a CC energy expression from the two-electron integrals over MOs: E CC = E 0 +

occ  virt  

a b b a tiab j + ti t j − ti t j



× ( φi φ j | φa φb − φi φ j | φb φa ).

where the cluster operator T is defined as

occ  virt 

∞  1 k T k! k=0     = 1 + T1 + T2 + 12 T21 + T3 + T2 T1 + 16 T31   1 4 + T4 + T3 T1 + 12 T22 + 12 T21 T2 + 24 T1 + · · ·

eT = 1 + T + 12 T2 + 16 T3 +

i< j a
CC = eT 0 ,

T1 0 =

The expansion coefficients, t, are called amplitudes. Substituting the exponential function by a series,

···

CCSD is the only pure CC method that can be used in routine applications. Explicit treatment of triples (CCSDT) is usually too expensive. However, the contribution of triples can be estimated perturbatively in the CCSD(T) method. Brueckner (B) theory is a variation of CC theory which uses orbitals that make the singles contribution vanish. The accuracy and computational cost of BD is comparable to that of CCSD. Excited electronic states may be treated within the CC formalism by the equationof-motion (EOM-CC) approach.

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e. R12 methods. HF is not exact because of the approximate treatment of electron–electron repulsion. PostHF methods try to recover the electron correlation energy by expanding the N -electron wave function in terms of Slater determinants built from one-electron functions (orbitals). But the convergence toward the exact solution of the Schr¨odinger equation is typically slow due to the poor description of the cusp region (r1 − r2 = 0) of the wave function. Faster convergence may be achieved by adding terms that describe electron correlation effects more directly than the product of one-electron orbitals (e.g., by including a function that explicitly depends on the coordinates of two electrons rather than just one electron: the interelectronic coordinate r12 ). The r12 -dependent wave function may then be used with CI, MBPT, or CC, giving R12 methods. The formula for computing the energy includes integrals over three- and four-electron coordinates, which are difficult to evaluate and which increase with N 6 and N 8 . However, insertion of a resolution of the identity allows three- and four-electron integrals to be written as sums over products of integrals involving only two-electron coordinates. The substitution is exact only for a complete basis set, but it is a very good approximation for fairly big basis sets, which have to be used for very accurate results anyway. R12 methods converge to the same basis set limit as conventional electron correlation methods do but faster. 4. Basis Sets A basis set is used to express the unknown MOs in terms of a set of known functions. The more basis functions used, the more accurate the description of the MOs. Any type of function can be used, but most efficient are basis functions with a physical behavior (e.g., approach zero for large distances between electrons and nuclei) and which make the integral evaluation easy. Slater-type orbitals (STOs) are related to the exact solutions for the hydrogen atom, but Gaussian-type orbitals (GTOs) are preferable for computational ease: STO χς,n,l,m (r, θ, ϕ) = NYl,m (θ, ϕ)r n−1 e−ςr

and GTO χς,n,l,m (r, θ, ϕ) = NYl,m (θ, ϕ)r 2n−2−l e−ςr ; 2

GTO χς,l (x, y, z) = N x lx y l y z lz e−ςr . x ,l y ,l z 2

N is a normalization constant and Yl,m are the spherical harmonic functions. STOs lack radial nodes, which are introduced by making linear combinations of STOs. They are primarily used for high-accuracy atomic and diatomic calculations and with semiempirical methods, which neglect all three- and four-center integrals (which cannot

FIGURE 7 Adding flexibility to basis sets: (a) Split valence basis sets provide more and less diffuse orbitals to adjust to different bonding situations (e.g., σ - and π -bonding). (b) Higher angular momentum basis functions allow for polarization to gain better overlap.

be evaluated analytically for STOs). GTOs do not have the proper behavior near the nucleus (in contrast with the “cusp” of the STOs) and fall off too rapidly far away from the nucleus. A better description is provided by contractions of several Gaussian functions. GTOs are generally used because of the computational efficiency. The number of basis functions employed for each atom determines the quality of the basis set. A minimum basis provides only as many shells as necessary to accommodate all electrons (i.e., 1s for H and He; 1s, 2s, and 2p for first-row elements, etc.) A double zeta basis set adds more flexibility for the description of different bonding situations (Fig. 7) by using two functions varying in the exponent ζ (a measure of the diffuseness) for each orbital of an occupied shell (i.e., 1s, 1s for H, He; 1s, 1s , 2s, 2s , 2p, 2p for first-row elements). Because bonding involves valence electrons only, core electrons are described by a single basis function and only the valence region is split in split valence or valence double zeta basis sets (1s, 1s for H, He; 1s, 2s, 2s , 2p, 2p for first-row elements). Basis sets can be improved by adding more functions: triple zeta (TZ), quadruple zeta (QZ), quintuple zeta (5Z), etc., have three, four, five, etc., times the number of basis functions of a minimum basis set, respectively. The exponents are usually determined variationally for the atoms. Each basis function may consist of several “primitive” GTOs. Contraction of several GTOs to one basis function (contracted GTO) is especially useful for the inner orbital to mimic the cusp at the nucleus. Additional higher angular momentum functions are usually added as polarization functions to allow for a polarized charge distribution (e.g., one additional p set for H and He; one d set for first-row elements). For

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correlated wave functions even more than one set is essential if high accuracy is desired (e.g., one d and one p set for H, He; one f set and two d sets for first-row elements). Anionic species or molecules with many lone pairs have very diffuse electron distributions and require an additional set of diffuse functions (often denoted by a “+”). The basis sets by Pople and coworkers are very popular: STO-3G is a minimum basis with three GTOs contracted to represent one atomic orbital. 3-21G is a split valence basis with core orbitals contracted from three GTOs and the valence region described by one basis function contracted by two primitives and another uncontracted function. 6– 31G∗ is constructed analogously but includes one set of polarization d functions on heavy atoms (other than hydrogen). Dunning’s correlation consistent basis sets (cc-p VXZ; X = D, T, Q, 5, 6, . . .) represent a series of basis sets converging to the basis set limit. This allows extrapolation to the infinite basis set limit. The cc-pVDZ (correlation consistent polarized valence double zeta) basis set has [3s, 2p, 1d/2s, 1p] contracted from (9s, 4p, 1d/4s, 1p) for first-row and hydrogen atoms, respectively. The next-better basis set, cc-p VTZ, has one basis function more of each type and adds one next-higher angular momentum function: [4s, 3p, 2d, 1f/3s, 2p, 1d]. Augmentation of one extra diffuse function for each type of angular momentum (1s, 1p, 1d for cc-pVDZ) is denoted by the prefix aug-. 5. Optimizing the Performance/Cost Ratio In any application a compromise between computational cost and accuracy has to be made. Accurate geometries are relatively easy to compute; MP2 with a polarized double zeta basis set usually gives satisfactory results. Accurate energetics require a better theoretical treatment and therefore are generally determined from a “single energy point calculation” at a higher level [e.g., CCSD(T) with a TZP basis set]. Whereas a single slash (/) is used to separate method and basis set specification, a double slash (//) means “at the geometry optimized at” (e.g., CCSD(T)/TZP//MP2/DZP). Zero-point vibrational energy (ZPE) corrections to relative energy are usually applied but need not be derived from a frequency calculation at the highest level of optimization. ZPEs are usually scaled by an empirical correction factor (depending on the theory level) to account for the overestimation of vibrational frequencies. To achieve chemical accuracy (i.e., ±1 kcal mol−1 ) for relative energies, for molecules of chemical interest, investigators devised interpolation schemes based on an additivity assumption. Most popular is the G2 method, a general procedure based on ab initio theory for the accurate prediction of energies of molecular systems, like

enthalpies of formation, bond energies, ionization potentials, electron affinities, and proton affinities. Starting from the MP4/6-311G(d,p)//MP2(fu)/6-31G∗ + 0.8929 ZPE(HF/6-31G∗ ) level, corrections for diffuse functions, higher polarization functions, and a more complete electron correlation treatment, as well as a “higher level correction” depending on the number of α and β valence electrons and an empirical factor, are included to extrapolate to the QCISD(T)/6-311 + G(3df,2p) level. This procedure gives a mean absolute deviation from experimental data of 1.21 kcal mol−1 for the “G2 test set,” a large number of various types of relative energies accurately known from experiment. Variations to the G2 method including the use of DFT methods have been proposed either to further increase the accuracy (e.g., G3) or to reduce the computational expense. C. Density Functional Theory The Hohenberg-Kohn theorem provides the inspiration for density functional theory (DFT): all ground-state properties of a system are functionals of the charge density. In particular the correct energy can be derived from the correct charge density. Conversely, an incorrect density will give an energy above the correct energy. In a DFT calculation the energy is optimized with respect to the density. It should be much simpler to handle the total electron density (which depends on three coordinates) than to treat all electrons explicitly (involving one spin plus three spatial coordinates per electron). Although Hohenberg and Kohn proved that the electron density determines the electronic ground-state energy, the functional to convert the electron density function into an energy value is unknown. Each contribution to the total energy—the kinetic energy, E T [ρ]; the nucleus–electron attraction, E ne [ρ]; and the electron–electron repulsion, E ee [ρ]—can be expressed as a functional of the total electron density: E[ρ] = E T [ρ] + E ne [ρ] + E ee [ρ]. In analogy to HF theory, E ee [ρ] can be divided into a Coulomb (E J [ρ]) part and an exchange part. The formula for E ne is exact: Nucl.  ρ(r) E ne [ρ] = − Za dr, |r − Ra | a and 1 E J [ρ] = 2



ρ(r1 )ρ(r2 ) dr1 dr2 |r1 − r2 |

holds true for electrons moving independently in the field caused by all electrons—approximations which are hoped to be corrected by a combined exchange and correlation

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term, E XC . The major task of DFT is to develop approximate but accurate functionals for E T and E XC . For the noninteracting uniform electron gas, the kinetic and exchange energies can be derived:

3 E TTF [ρ] = 10 (3π 2 )2/3 ρ(r)5/3 dr and E KD [ρ] =

  3 3 1/3 4 π



ρ(r)4/3 dr.

In Thomas-Fermi theory the total energy is expressed as E TF [ρ] = E ne [ρ] + E J [ρ] + E TTF [ρ], while the Thomas-Fermi-Dirac expression adds the exchange expression E kD [ρ]. The uniform electron gas is a model too crude to describe molecules: neither TF nor TFD gives bonding between atoms. As a way to improve on the expression for the kinetic energy, Kohn-Sham theory calculates the kinetic energy for noninteracting electrons and corrects for the error relative to the real kinetic energy by means of the exchange correlation term E XC [ρ]: E DFT [ρ] = E TS [ρ] + E ne [ρ] + E J [ρ] + E XC [ρ], where E TS [ρ] can be computed from a Slater determinant: E TS [ρ]

N 

 1 2  φi − 2 ∇ φi , = i

and E XC [ρ] is usually separated into an exchange contribution (E X [ρ]) and a correlation contribution (E C [ρ]). In the Kohn-Sham implementation of DFT, the density, ρ, is derived from a single Slater determinant with orthonormal orbitals, φi : occ  ρ(r) = |φi (r)|2 . i

The energy is then optimized by solving a set of oneelectron equations, the Kohn-Sham equations, but with electron correlation included: hKS φi = εi φi , where KS

h

Nucl.  Za 1 = − ∇2 + + 2 a |Ra − r|



ρ(r ) dr + VXC (r). |r − r |

The (Kohn-Sham) orbitals, φi , which are used to represent the electron density, ρ, can be determined numerically or variationally as an expansion of basis functions. The Kohn-Sham equations have to be solved iteratively because the Coulomb term depends on the density (i.e., the orbitals to be determined). The main advantage of DFT methods is that they include some treatment of electron correlation at a computational cost equivalent to that of the HF method. The main disadvantage of DFT, however, is that there is no hierarchy of increasingly better functionals. The performance of a given functional must be

assessed by comparison with experimental data and there is no consistent way to improve the quality of a given functional. This is in contrast with wave function methods, in which a more complete treatment of electron correlation (and a more flexible basis set) means a closer approach to the exact solution. 1. Local Density Approximation The local density approximation (LDA) assumes variations of the density to be slow and treats the local density as a uniform electron gas:  1/3 4/3 ρ (r) dr. E XLDA [ρ] = − 34 π3 The X α method is an example of LDA in which the correlation energy is neglected and the exchange energy expression is multiplied by a parameter α. A fairly accurate expression for the correlation energy of the uniform electron gas, the VWN (Vosko, Wilk, Nusair) functional, was derived by fitting it to Monte Carlo results. 2. Generalized Gradient Approximation As a way to better treat the nonuniform electron distribution of molecules, the exchange and correlation functionals were modified to include derivatives of the density in the gradient-corrected approximation, or generalized gradient approximation (GGA). Gradient-corrected exchange functionals were developed [e.g., by Perdew and Wang (PW86) and Becke (B)]. Popular correlation functionals are those of Lee, Yang, and Parr (LYP), Perdew (P86), Perdew and Wang (PW91), and Becke (B91). Hybrid methods (such as the popular Becke three-parameter functional, B3) use part of the exchange as computed by the HF method. The three parameters in B3 determine the mixing of LDA and exact exchange, as well as the gradient-corrected contributions to the exchange and correlation terms. The parameters are fitted to experimental thermodynamical data. Exchange and correlation functionals can freely be combined to give an arsenal of DFT methods (e.g., BLYP, BP86, B3LYP, etc.), but B3LYP is the most popular because of its consistently good performance. Recently, the DFT formalism has been extended to treat excited electronic states through the implementation of time-dependent DFT (TD-DFT).

III. APPLICATIONS OF COMPUTATIONAL CHEMISTRY When a chemical problem is to be studied computationally, an appropriate level of theory must first be chosen. Simple qualitative concepts such as the frontier molecular

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orbital (FMO) theory or orbital symmetry (WoodwardHoffman) rules, which were developed on the basis of primitive computations, can often successfully predict relative reactivities and selectivities. Qualitative concepts are very useful as they can provide chemical insight and understanding. For quantitative answers more sophisticated computational methods have to be employed and a compromise between desired accuracy and computational cost has to be made. In any case, it is essential to know about the possibilities and limitations of the method to be applied. A. The Potential Energy Surface The methods described in Section II provide expressions for the energy as a function of the atomic coordinates [i.e., they describe the potential energy surface (PES) for a given molecular formula] (Fig. 8). This allows one to search for an atomic arrangement which makes the energy a minimum (geometry optimization). More efficient optimization algorithms can be employed when the forces and the Hessian [first and second derivatives, respectively, of the energy with respect to (w.r.t.) the nuclear coordinates] are also known. One global and usually many additional local minima exist for a given formula and relative energies for isomeric structures can be obtained. The transition structure, which is the highest point on a minimum energy path connecting two minima, can also be localized. This reveals information about chemical reactions, such as mechanistic details, activation barriers, and so forth, which is difficult or impossible to deduce from experimental investigations. Minima and transition states are stationary points (i.e., the forces are zero). All eigenvalues of the Hessian are positive for minima while transition states have one negative eigenvalue. The eigenvector corresponding to the nega-

tive eigenvalue describes the transition mode. Following the minimum energy path in mass-weighted coordinates (intrinsic reaction coordinate, IRC) allows one to confirm which minima are connected by a transition state. Harmonic vibrational frequencies can be derived from the force constants and allow one to compute entropy and enthalpy values. The molecular geometry input can be provided in the form of a Z-matrix [i.e., be defined through internal coordinates (bond length, bond angles, and dihedral angles)]. Due to improved optimization algorithms (handling redundant internal coordinates) and the size of molecules that can now be computed routinely, the input is mostly provided as Cartesian coordinates, often generated with the help of a graphical user interface to the computational chemistry program package. The computational speed of the more approximate methods allows one not only to treat larger molecular systems, but also to address new problems. The function of a protein is determined by its three-dimensional structure. Hence, there is considerable interest in solving the protein folding problem by predicting the threedimensional (secondary and tertiary) structure of a protein based on the amino acid sequence (primary structure). The binding of a substrate (or inhibitor) is crucial for the catalytic activity of an enzyme. Possible binding modes are investigated by molecular docking, in which various structures of intermolecular complexes are generated and evaluated. B. Analyzing the Wave Function The wave function, , which describes the electron distribution around a given nuclear arrangement, not only can be used to compute the energy of the system, but also offers useful interpretation opportunities. 1. Wave Function Analysis in Terms of Basis Functions Each MO φi (used to build ) is constructed from basis functions χi located on the nuclei and is occupied by n i electrons (0, 1, or 2 for HF, but any number between 0 and 2 is possible for correlated wave functions). It is therefore possible to distribute the electrons to individual atoms. However, there is no unique prescription. Mulliken population analysis, for example, employs ρA =

FIGURE 8 A model potential energy surface showing a transition structure, TS, connecting two minima, MIN1 and MIN2; the activation barrier, E = , for the transformation of MIN1 to MIN2; and the relative energy, Erel , of MIN1 versus MIN2.

AO  AO 

Dαβ Sαβ

α∈A β

=

 AO  MO AO   α∈A β

i

 n i cαi cβi

 χα χβ

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to determine the number of electrons ρA associated with atom A. The atomic charge, Q A , is given as the difference between the nuclear charge Z A and ρA : Q A = Z A − ρA . Bond orders bAB can be defined on the basis of the sum of electrons shared between atoms A and B:  bAB = (DS)αβ (DS)βα . α∈A β∈B

It is arbitrary how the electron density arising from basis functions located on different atoms is divided. The Mulliken procedure can give unphysical charges and shows a strong basis set dependence. Because of these shortcomings an atom definition which does not depend on the basis set is desirable. 2. Wave Function Analysis Based on the Electron Density Bader’s theory of atoms in molecules (AIM) is based on the electron density ρ(r) (which can be computed by integrating the square of the wave function over the coordinates of all but one electron) and the gradient of the density, ∇ρ(r). Regions in space, so-called atomic basins, are defined as all points from which following the gradient of the density leads to a common attractor = nucleus. Integration of the electron density for each basin gives the number of electrons associated with the nucleus in that basin. However, AIM charges are counterintuitive in some cases. Neither AIM charges nor Mulliken charges reproduce the dipole or higher multipoles of a molecule. For points on the surface separating atomic basins, the derivative of the electron density along the normal vector equals zero. Points where the derivative perpendicular to the normal vector is zero as well are called bond critical points. They are the points of minimum electron density along the bond path, the path of maximum electron density connecting two nuclei. The electron density at the bond critical point correlates with the bond strength. 3. Localized MOs The canonical MOs obtained as eigenfunctions of the Fock operator have contributions from all basis functions and thus are delocalized over all centers. They do not reflect the common picture of localized bonds between two atoms. However, the orbitals may be freely transformed by making linear combinations without changing the total wave function. Hence, an orbital rotation matrix can be applied to transform the canonical into localized orbitals which reflect bonds between two atoms. Several localization schemes were proposed, but the natural bond

orbital (NBO) analysis has some advantages (e.g., has no strong basis set dependence, is computationally inexpensive, and can also be applied to electron correlated methods). In a first step, natural atomic orbitals (NAOs) are generated by a diagonalization of the atomic blocks of the density matrix giving pre-NAOs which are orthogonalized in several steps applying occupancy-weighted and normal orthogonalization procedures. Diagonal values of the density matrix in the resulting NAO basis correspond to orbital populations which can be summed up to give atomic charges. Off-diagonal blocks define bonds between atoms. The resulting localized MOs give a description in agreement with chemical intuition (core orbitals, lone pairs, and bonds) and can be analyzed in terms of bond polarity, bond bending, hybridization, and so forth. Effects such as hyperconjugation can be investigated by analyzing the interactions between formally occupied and formally empty orbitals within a localized orbital picture.

C. Computing Properties One-electron properties such as the electric dipole moment, the quadrupole moment, and the magnetic susceptibility can be evaluated from the wavefunction as the expectation value of an operator O which is a sum of n one-electron operators o:   |O|

= φi |o|φi = Dµν χµ |o|χν . i

µ

ν

Most properties represent the response of the system to a perturbation. The effect can be calculated by derivative methods, perturbation theory, or propagator methods. For example the wave function, , is changed by an electric field which enters the Hamiltonian as part of the potential energy term. The energy in the presence of an electric field F, E(F), can be written as a Taylor expansion: E(F) = E 0 − µ0 F − 12 αF2 − 16 βF3 − · · · , where the permanent dipole moment, µ0 , the polarizability, α, and the (first) hyperpolarizability, β, are the first, second, and third derivatives, respectively, of the energy with respect to the field F. The analogous expression for the presence of a magnetic field, B, is E(B) = E 0 − m0 B −

1 ζ B2 − · · · , 2µ0

where the magnetic moment, m0 , and the magnetizability, ζ , are the first and second derivatives of the energy with respect to the magnetic field (µ0 is the vacuum permeability).

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D. Dynamics and Modeling Solvation

E. Spectroscopic Data

Standard computations treat isolated molecules (i.e., model the low-pressure gas phase situation). Chemistry in the condensed phase, however, can be significantly different because ionic and polar species are specifically stabilized. Computations can consider solvation explicitly by including the solvent molecules or as a continuous medium effect (reaction field methods). When the solute is embedded in a bulk of solvent molecules, an “ensemble” (i.e., a large number of configurations of the molecular aggregation) has to be generated and properly averaged to derive macroscopic properties. Molecular dynamics (MD) creates a “trajectory”: the configurations are obtained by following the evolution of the system in time by applying the classical equations of motion. The alternative, Monte Carlo (MC) method produces the configurations randomly: a starting configuration is perturbed and the new configuration is accepted if the new energy is lower (E < 0) than the old one. If E > 0 the new configuration is accepted with a probability proportional to the Boltzman factor, exp(−E/kT ) (Metropolis method). The forces are needed for MD, but energies are sufficient for MC simulations. Meaningful averaging requires a huge number of conformations to be calculated, which leaves parametrized force fields as basically the only practible methods to be used in MD and MC simulations. Reaction field methods model solutions by placing the solute in a cavity of a polarizable medium. The electrostatic potential due to the solute molecule polarizes the surrounding medium which in turn changes the charge distribution of the solute. Hence, the electrostatic interaction has to be evaluated self-consistently (self-consistent reaction field, SCRF). A term for creating the cavity (calculated from the surface of the cavity) has a be added to the solvation energy. Explicit treatment of solvent molecules can be combined with a reaction field method. Dynamics can also be used to model the mechanics and rates of reactions at a fundamental level. While a complete potential energy surface is the ideal starting point of any type of accurate dynamical computation, it can be obtained for only very simple systems. Ab initio MD can be performed by repeated calculation of forces, generation of a new geometry, and convergence of the MO coefficients for the new geometry. This iterative scheme, however, is very time consuming because accurate MO coefficients are required at each point of the simulation. The CarParrinello method does not optimize the electronic (i.e., MO coefficients) and nuclear coordinates separately, but simultaneously, because it could be shown that the errors in the nuclear forces and in the electronic forces cancel out.

Vertical and adiabatic ionization energies are given as the energy difference between the neutral molecule and the corresponding cation in the neutral geometry and in the relaxed geometry, respectively. Analogous comparison with the anion gives electron affinities. The various ionization energies measured in PE (photoelectron) spectroscopy can be computed as differences between the neutral ground state and different electronic states of the cation. The energy difference between the electronic ground state and electronically excited states without the loss of an electron corresponds to the transitions observed in UV-VIS (ultraviolet-visible) spectroscopy. Many other properties measured by spectroscopic methods can be computed as derivatives of the energy. The force constants (second-derivative w.r.t. the nuclear coordinates, r) allow one to calculate harmonic vibrational frequencies and the corresponding normal modes. The derivatives of the dipole moment and of the polarizability w.r.t. the normal modes are proportional to the intensity of infrared absorptions and of Raman bands, respectively. First and second derivatives of the energy w.r.t. nuclear magnetic spin, I, give the hyperfine coupling constant g (measured by electron spin resonance, ESR, spectroscopy) and the nuclear coupling constants J of nuclear magnetic resonance (NMR) spectroscopy, respectively. The nuclear magnetic shielding constants, σ , are given as the mixed derivatives of the energy w.r.t. an external magnetic field B and the magnetic moments I of the nuclei. The NMR chemical shifts correspond to differences in σ for a nucleus in a given molecule and in a reference compound. Magnetic properties suffer from the “gauge origin” problem because the magnetic field is a vector potential, which is not uniquely defined. For finite basis sets, results depend on the choice for the origin. This problem can be largely overcome by using either the gauge-invariant atomic orbital (GIAO) method or the individual gauge for localized orbital (IGLO) method. Comparison of calculated and measured spectroscopic data can help to identify new molecules and to determine their structure. The wealth of properties (Fig. 9) that can be computed as derivatives of the energy stresses the importance of the derivative methods. The ability to derive expressions for at least first and second derivatives is always desirable for a new theoretical level.

IV. OUTLOOK FOR COMPUTATIONAL CHEMISTRY During the last decades computational chemistry has evolved into an indispensable tool for understanding and

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FIGURE 9 Some chemical information that can be gained from a computational investigation. NMR, nuclear magnetic resonance; IR, infrared.

predicting molecular properties and chemical behavior. The era foreseen by R. S. Mulliken in his acceptance speech for the 1966 Nobel Prize has arrived:

Kryspin, K. N. Kirschner, P. v. R. Schleyer, and A. Y. Timoshkin. M.H. is grateful to Professor R. Kr¨amer for support.

I would like to emphasize strongly my belief that the era of computing chemists, when hundreds if not thousands of chemists will go to the computing machine instead of the laboratory for increasingly many facets of chemical information, is already at hand.

SEE ALSO THE FOLLOWING ARTICLES

Future goals of computational chemistry will be to determine structural, spectroscopic, and dynamic properties of even larger molecular system, also in the condensed phase, with even better accuracy and speed. This will allow computational chemistry to also play an important role in fields related to chemistry such as biology and material science.

ACKNOWLEDGMENTS The authors are grateful to the following persons for comments and valuable suggestions: H. F. Bettinger, M. B¨uhl, B. Goldfuss, I. Hyla-

MECHANICS, CLASSICAL • MOLECULAR ELECTRONICS • PHOTOELECTRON SPECTROSCOPY • PROTEIN STRUCTURE • QUANTUM CHEMISTRY • QUANTUM MECHANICS

BIBLIOGRAPHY Dirac, P. A. M., (1929). Proc. R. Soc. Lond. Ser. A, 123, 719. Jensen, F. (1999). “Introduction to Computational Chemistry,” Wiley, Chichester, UK. Leach, A. R. (1996). “Molecular Modelling Principles and Applications,” Longman, Essex, England. Lipkowitz, K. B., and Boyd, D. B., eds. (1990–). “Reviews in Computational Chemistry,” Vol. 1–, VCH, New York. Mulliken, R. S. (1967). Science, 137, 13–24. Schleyer, P. V. R. et al., eds. (1998). “Encyclopedia of Computational Chemistry,” Wiley, Chichester, UK. Young, D. (2001). “Computational Chemistry,” Wiley, Interscience, New York.

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Dynamics of Elementary Chemical Reactions H. Floyd Davis Hans U. Stauffer Cornell University

I. II. III. IV.

Kinetics and Collision Theory Activated Complex Theory Unimolecular versus Bimolecular Reactions Statistical Theories of Unimolecular Decomposition V. Reactions in Solution VI. Experimental Techniques

GLOSSARY Complex-mediated reaction An elementary bimolecular reaction proceeding via a long-lived collision complex having lifetimes ranging from several vibrational periods (100 fs) to many rotational periods (>10 ps). If complex lifetimes exceed several rotational periods, product angular distributions from crossed beam reactions exhibit forward-backward symmetry in the center-of-mass frame of reference. Direct reaction An elementary bimolecular reaction proceeding via direct passage through the transition state region. The absence of long-lived intermediates in such reactions leads to anisotropic center-of-mass product angular distributions that often provide insight into the most favorable geometries for reaction. Free radical An atom or molecule possessing one or more unpaired electrons. Free radicals may either be

stable molecules (e.g., NO, O2 , or NO2 ), or highly reactive transitory chemical intermediates (e.g., H, Cl, CH3 ) that react on essentially every collision with stable molecules. Ionization The process by which one or more valence electrons are removed from an atom or molecule. Most often achieved by electron impact or absorption of one or more ultraviolet photons. Laser-induced fluorescence (LIF) A spectroscopic technique usually employing visible or UV laser light, in which the fluorescence emission from a gaseous, liquid, or solid sample is monitored. A fluorescence excitation spectrum is a plot of the total emitted fluorescence vs. excitation wavelength and provides information similar to an absorption spectrum. Molecular beam Collimated stream of gaseous molecules produced by expansion of a gas through an orifice into an evacuated chamber. A supersonic molecular 697

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beam is characterized by a velocity distribution much narrower than a Boltzmann distribution. Potential energy surface Schematic two- or threedimensional representation of the total potential energy of a chemical system as a function of internuclear coordinates. Transition state Region of the PES corresponding to the critical geometry through which a reacting system must pass for reactants to become products.

AN ELEMENTARY CHEMICAL reaction is any process involving bond fission and/or bond formation following a single collision between two reactants. Chemical reactions occur in all three phases of matter (gas, liquid, and solid), and at their interfaces. Under the experimental conditions most commonly used to carry out reactions, the overall reaction usually consists of a sequence of two or more elementary reactions. For example, the reaction of gaseous hydrogen with chlorine forming hydrogen chloride is represented by the following balanced chemical equation: H2 + Cl2 → 2HCl.

(1)

This reaction proceeds by a chain mechanism involving a repetitive sequence of elementary reactions involving the three stable molecules listed in Eq. (1), as well as two short-lived free radical intermediates, i.e., chlorine atoms (Cl) and hydrogen atoms (H). The most important elementary steps in the overall reaction mechanism are: Initiation :

Cl2 → 2Cl.

Propagation : Cl + H2 → HCl + H

Termination :

presence of a third body (M), which may be a molecule or the wall of the container. Note that the two elementary propagating reactions may be added like mathematical equations, yielding the overall chemical reaction (1). For an overall reaction such as that in Eq. (1) involving a sequence of elementary steps, the overall rate of formation of products may be a complex function of reactant concentrations, because products are formed by several different elementary processes. In the previous example, the HCl products are formed by reactions (3) and (4), each of which has its own rate law and rate constant. Thus, for a complex multistep process such as reaction (1), the rate law can only be determined through experiment. For an elementary bimolecular reaction A + B → C + D, the reaction rate is proportional to the concentrations (denoted by [A], [B], etc.) of the reactants: d[D] d[A] d[B] d[C] = =− =− = k[A][B]. (5) dt dt dt dt The proportionality constant, k, is called the reaction rate constant. Since an elementary reaction involves a single bimolecular collision between A and B, the maximum possible rate constant is usually the frequency of collisions between reactants. A few simple atom-transfer reactions (e.g., F + H2 → HF + H) actually do occur on nearly every collision, and are said to proceed at or near the “gas kinetic limit.”

I. KINETICS AND COLLISION THEORY

(2)

(3a)

H + Cl2 → HCl + Cl.

(3b)

H + Cl + M → HCl + M

(4a)

Cl + Cl + M → Cl2 + M

(4b)

H + H + M → H2 + M.

(4c)

The chain reaction is initiated by dissociation of Cl2 , a stable molecule, to form two highly reactive chlorine atoms (Cl). Since chemical bond fission requires the input of energy, initiation may be achieved by heating the sample (see Lindemann mechanism) or by ultraviolet irradiation (photodissociation). Following initiation, the two elementary bimolecular propagating reactions will continue until either or both of the reactants (H2 and Cl2 ) are consumed, at which time the termination steps end the chain reaction. Termination typically involves termolecular recombination of two radicals to form a stable molecule in the

In order to estimate the frequency of collisions between gaseous A and B molecules, consider a beam of molecules of incident flux I A (molecules/cm2 · s) impinging on a static cell containing molecules at a concentration [B] (molecules/cm3 ). The particles interact in a volume element V . The collision rate per unit time, Z , is given by Z = σI A [B]V.

(6)

Here, σ is the collision cross section, which may be estimated using a simple hard sphere model for colliding particles (Fig. 1). Two particles collide with a relative velocity vector, g, the magnitude of which is denoted by g, and impact parameter b, also known as the “aiming error” of the collision. A hard sphere collision will occur provided 0 ≤ b ≤ (r A + r B ). The collision cross section is therefore the area of a circle of radius d AB = r A + r B , i.e., 2 . The incident flux, I A = [A]g A , may then be σh.s. = πd AB substituted into Eq. (6). If the rate of reaction between A and B is simply the collision rate, then −

d[A] = Z = σh.s. g A [A][B]. dt

(7)

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Whereas the hard sphere cross section depends only on the sum of the radii of colliding particles, a reaction cross section may depend strongly on the energy of the collision and therefore on the relative velocity of the colliding particles, the magnitude of which is given by g. As illustrated in Fig. 2, the relative velocity vector, g, may be decomposed into two perpendicular components. The first is a radial component, gr = r˙ , i.e., the time derivative of r , the distance between the particle centers. Perpendicular to the radial axis is a tangential component, g⊥ = r θ˙ , where θ˙ is the time derivative of θ , the angle between g and the radial axis. As a result, the total kinetic energy of collision, E kin = 1/2 µg 2 , can be thought of as a sum of a radial kinetic energy, Er = 1/2 µgr2

(11)

and an energy associated with the perpendicular velocity component, 2 E ⊥ = 1/2 µg⊥ = 1/2 µr 2 θ˙ 2 =

FIGURE 1 Depiction of hard sphere collision cross section. Note that collisions only occur for impact parameters b < dAB .

Comparison of Eqs. (5) and (7) indicates that the collision rate constant, k, is related to the collision cross section and relative velocity of colliding particles by k(g) = σ g .

(8)

In deriving Eq. (8), it is assumed that molecules A and B collide with a single relative velocity g. In a real gaseous sample containing both A and B molecules at thermal equilibrium, the distribution of relative velocities is described by the Maxwell–Boltzmann Distribution Law:  3/2 2 µ − µg f (g) = 4πg 2 e 2k B T , (9) 2πk B T

L2 , 2µr 2

(12)

where L = µr 2 θ˙ = µgb is the magnitude of the angular momentum associated with the colliding pair. Thus, for interaction potentials that depend solely on the distance between the colliding pair, V (r ), only gr is effective in surmounting potential energy barriers such as those associated with the energy required to break and form bonds during reaction; g⊥ is associated purely with rotational motion of the two particles. One way to model the energy dependence of σ is to assume that reaction can only occur if the component

where µ = m A m B /(m A + m B ) is the reduced mass of the colliding particles, k B is Boltzmann’s constant, and T is the temperature in Kelvin. The hard sphere collision rate constant, kh .s. , is thus temperature dependent, and may be evaluated explicitly by integrating over all possible relative velocities, g:    ∞ 8k B T 1/2 2 kh .s. (T ) = σh .s. g f (g) dg = πd AB . πµ g=0 (10) Using typical molecular values of d AB ≈ 0.35 nm and µ = 14 amu for room temperature collisions between N2 molecules, one observes the magnitude of a hard sphere collision rate constant to be on the order of 2.6 × 10−10 cm3 /molecule · s.

FIGURE 2 Decomposition of relative velocity vector, g, into radial (gr ) and perpendicular (g⊥ ) components, with θ defined as the angle between g and the internuclear axis. At the moment of a hard sphere collision, sin θ = b/dAB .

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of kinetic energy along the axis of the collision (i.e., Er ), exceeds a critical energy, Vc . Note that during the course of a collision, θ varies from 0 to 180◦ as r changes from −∞ to +∞. However, if the particles are treated as hard spheres, the internuclear potential, V(r ), is zero for r > d AB , and there is no interaction between them until collision, at which time (sin θ) = b/d AB . Using the fact that gr = g cos θ , the radial kinetic energy at the moment of contact is   b2 1 2 2 2 Er = /2 µg cos θ = E kin (1 − sin θ ) = E kin 1 − 2 . d AB (13) In order for the reaction to be successful, Er ≥ Vc . For a given collision energy, E kin , this implies that the impact parameter must be smaller than a critical impact parameter, bc , defined such that   bc2 E kin 1 − 2 = Vc . (14) d AB Thus, the cross section for a reaction involving a critical energy, Vc , given by   Vc 2 σ = π bc2 = πd AB 1− (15) E kin is expected to increase with energy, as illustrated in Fig. 3. Note that this model predicts that the threshold for reaction occurs at E kin = Vc , and that the cross section reaches half the hard sphere value at E kin = 2Vc , asymptotically approaching the hard sphere value as E kin → ∞. In many cases, reaction cross sections for real systems differ considerably from that shown in Fig. 3. For example, some processes, such as charge exchange (e.g., A+ + B− → A + B), proceed with reaction cross sections far exceeding the hard sphere limit. Here, the long-range Coulomb potential causes reactants to be attracted to one another at large distances, considerably increasing the reaction cross section. Even for neutral–neutral interactions, the interaction potential, V (r ), often differs substantially from that of hard spheres. Long-range induced dipole-

FIGURE 3 Cross section (σ ) dependence on kinetic energy, Ekin , for the hard sphere model requiring an energetic threshold, Vc .

FIGURE 4 Schematic internuclear potentials for different models of atomic and molecular interactions. The hard sphere model exhibits only a repulsive component at small r ; more realistic potentials exhibit attractive and repulsive components.

induced dipole interactions (van der Waals’ interactions) result in an attractive region of the potential surface at longer bond distances even in cases when formal bonds between the interacting pair cannot be formed. When strong bonds can be formed between colliding particles, as in the case of two halogen atoms like Cl, a strongly attractive component of the potential is present over a wide range of internuclear separations. In such cases, the hard sphere potential would only be useful in modeling the interaction potential at small distances where electron– electron repulsion becomes dominant, as demonstrated in Fig. 4. However, the magnitudes of most reaction cross sections are controlled predominantly by the form of the attractive component of the potential at longer internuclear separations. We now discuss a relatively simple model for reactions involving gaseous particles interacting through a potential V (r ) operating at long range. Recall from Eq. (11) and (12) that E kin can be written as a sum of radial energy, Er , and energy associated with rotational motion of the interacting particles. Thus, the total energy, E = E kin + V (r ), of the colliding partners is E = Er +

(µgb)2 + V (r ), 2µr 2

(16)

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making it convenient to conceptualize an effective potential, Veff (r ) =

(µgb)2 + V (r ), 2µr 2

(µgb)2 + V (rt. p. ). 2µrt.2 p.

(18)

This may be rearranged to solve for b = bmax , the maximum impact parameter for a given relative velocity, g, that allows the collision partners to reach a fixed critical distance for reaction, rc :   2V (rc ) 1/2 bmax (g) = rc 1 − , (19) µg 2 which holds for g ≥ gmin , where, in order to be physically meaningful, gmin = [2V (rc )/µ] /2 1

=0

if V (rc ) > 0 if V (rc ) ≤ 0.

(20)

The reaction rate constant may in both of these cases be determined analytically by integration over all relative velocities exceeding gmin :  k(T ) =

∞

π {bmax (g)}2 g f (g) dg

gmin

 =

πrc2 

= πrc2

8k B T πµ 8k B T πµ

1/2 e

− Vk (rcT)

1/2 

if V (rc ) > 0

B

V (rc ) 1− kB T



k = Ae−Ea /k B T ,

(17)

that governs the radial motion of the particles. At large internuclear distances, the total energy of the system, E = 1/2 µg 2 , is simply the radial kinetic energy, since lim V (r ) = 0. However, at smaller values of r , where r →∞ eff Veff (r ) > 0, some of the initial kinetic energy is converted into energy associated with the effective potential. The distance of closest approach, or turning point, rt. p. , is reached when the magnitude of the effective potential is equal to the initial radial kinetic energy, and the radial velocity becomes zero: E = 1/2 µg 2 =

to the empirical Arrhenius expression found to satisfactorily model a large number of chemical reactions:

(21) if V (rc ) ≤ 0. (22)

In cases where V (rc ) is positive, the critical distance rate constant expression [Eq. (21)], is similar to the hard sphere collision rate constant [Eq. (10)]; however, an additional exponential term is present. This term represents the fraction of molecules at temperature T having sufficient energy to react. This temperature dependence is thus similar

(23)

where A is the Arrhenius preexponential factor, and E a is the Arrhenius activation energy. These quantities are most readily determined by plotting ln k vs. 1/T , which should be linear with a slope −E a /k B and intercept ln A. Note that this purely empirical relationship often holds for elementary as well as multistep reactions. The obvious similarity between Eqs. (21) and (23) suggests that E a is at least loosely related to the height of the potential energy barrier for the rate-limiting step in the reaction. However, the Arrhenius parameters are only phenomenological quantities derived from the temperature dependence of reaction rate constants. In fact, Arrhenius plots are in many cases found to be markedly nonlinear, suggesting the occurrence of a multistep reaction mechanism or a mechanism that changes at different temperatures. The critical distance model can be used to derive an explicit formula for the temperature dependence of the reaction rate constant for charge transfer reactions of the form A+ + B− → A + B. Such interactions are subject to long-range Coulomb attractions of the form V (r ) = −q 2 /4π ε0r , where q is the charge of an electron. Taking the critical distance, rc , to be the ionic–covalent curve crossing radius (R), which corresponds to the distance at which the Coulomb attraction between ions balances the energy required for electron transfer, one obtains by substitution into Eq. (22) the following expression for the charge exchange rate constant:     8k B T 1/2 q2 k(T ) = πR 2 1+ . (24) πµ 4πεo Rk B T This reaction rate constant expression bears some similarity to a hard-sphere rate constant; however, an additional term (q 2 /4π εo Rk B T ) results from the long range attractive interaction, and is in general the dominant contribution to the reaction rate constant for reactions of this type. For attractive potentials of the form V (r ) = −a/r s , Veff is given by Veff (r ) =

L2 a (µgb)2 a − = − s. 2 s 2 2µr r 2µr r

(25)

Provided s ≥ 3, Veff has a local maximum for a given impact parameter, b, at a radial distance, rmax , determined by 2−s  sa rmax = . (26) µg 2 b2 For close approach required for reaction, the two particles must overcome this maximum, Vmax = Veff (rmax ), as

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FIGURE 5 Relationship between Veff (r ) and V(r ) for a given impact parameter, b. Close approach necessary for reaction requires E >Vmax , where Vmax is the maximum in Veff (r ).

illustrated in Fig. 5. Classically, if the energy of the colliding reactants is exactly equal to Vmax , all of the initial kinetic energy is converted into Veff when r = rmax , and “orbiting” will occur; i.e., the interacting pair will rotate together for an infinite amount of time since both the radial kinetic energy and the centrifugal force acting on the particles (−d Veff /dr ) are zero. The orbiting impact parameter, bor b , defined as the impact parameter for a given initial collision energy, E = 1/2 µg 2 , at which orbiting occurs, can be determined by setting Vmax = E and rearranging to arrive at     s a(s − 2) 2/s 2 bor b = . (27) µg 2 s −2 Since smaller impact parameter collisions will result in a smaller value of Vmax , only collisions occurring with impact parameters less than bor b will lead to a close collision 2 and reaction. Thus, the reaction cross section, σ = π bor b, which depends on the relative velocity g, may be integrated over all relative velocities to derive the rate constant temperature dependence:  ∞ 2

k(T ) = π bor b g f (g) dg 0

=2

3s−4 2s

  1/2  π s −2 s−4 2/s 2/s , (s − 2) a (k B T ) 2s µ s (28)

where the gamma function, , is available in mathematical tables. This equation predicts that reactions involving quenching of an electronically excited state, which can be modeled using s = 3, will show weak inverse temperature dependence, k(T ) ∝ T 1/6 . Ion–molecule reactions, having s = 4, are predicted to have rate constants independent of temperature. Reactions dominated by

van der Waals interactions (i.e., s = 6) are expected to show a small positive temperature dependence (k ∝ T 1/6 ). The profound effect of the exact form of the internuclear potential on the interaction between particles can be observed in elastic scattering experiments. These studies allow determination of the angle of deflection of a particle from its original direction upon interaction with the second particle. Conceptually, the scattering process can be understood by considering the effect of a particle of mass µ colliding with an infinitely massive particle fixed in space. The deflection angle, χ , defined as the angle between the initial and final relative velocity vectors of the colliding pair, will depend on the form of potential, V (r ), and, based on the impact parameter at which a given collision occurs, the region of V (r ) that is sampled by the colliding pair. Recall from Eq. (17) that the effective potential, Veff , governs the radial motion of the colliding particles, and therefore determines the radial turning point, rt. p. , for a given magnitude of initial collision energy, E. Figure 6 shows three effective potentials (for impact parameters b = 0, b1 , b2 , and b3 , where 0 < b1 < b2 < b3 ) for an internuclear potential, V (r ), with both long-range attractive and short-range repulsive components. If the two particles collide with a fixed collision energy (e.g., that denoted by E 1 ), the radial turning point, and therefore the regions of V (r ) accessed during the collision, will depend strongly on the magnitude of b. For very large values of b, (b ∼ b3 ), the turning point lies at very large r , and the particle experiences little deflection upon approach (χ ∼ 0). For smaller values of b (b ∼ b2 ), the turning point moves to smaller values of r , and the interacting particles therefore sample more of the long-range attractive part of V (r ), resulting in a deflection toward more negative angles. As b becomes even smaller, however, the effects of the repulsive part of the potential begin to play a role, and χ

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FIGURE 6 Dependence of Veff on impact parameter, b, where 0 < b 1 < b 2 < b 3 . Larger magnitudes of b result in larger values of the radial turning point for a given collision energy.

reaches a minimum value, called the “rainbow” angle, χr , by analogy to the optical rainbow resulting from scattering in small water droplets. The impact parameter that results in this most negative degree of deflection is referred to as the rainbow impact parameter, br . For impact parameters less than br (b ∼ b1 ), short-range repulsion begins to dominate, and the deflection angle becomes less negative, reaching zero at the impact parameter at which the attractive and repulsive forces during the collision exactly offset, the so-called “glory” impact parameter, bg . For impact parameters smaller than bg , repulsion dominates, and the particle is scattered to positive angles, reaching a max-

imum of χ = π for direct head-on collisions (b = 0). A pictorial depiction of the dependence of χ on the impact parameter, known as the deflection function, is shown in Fig. 7, where particle trajectories during the course of a collision are shown for a wide range of impact parameters. Note that this figure depicts the dependence of the deflection angle, χ , on b∗ = b/re , where re is the internuclear separation where V (r ) reaches a minimum. Although collision theory has provided considerable insight beyond simple hard spheres, it cannot properly address questions such as what fraction of collision geometries are likely to lead to reaction. Such issues cannot be

FIGURE 7 Deflection angle, χ , dependence on impact parameter. Trajectories depict the degree of deflection of impinging particle resulting from attractive and repulsive components of the interaction potential. (From Levine, R. D., and Bernstein, R. B. (1987). “Molecular Reaction Dynamics and Chemical Reactivity,” Oxford University Press, New York.)

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properly accounted for unless the details of the molecular structure of reactants are considered. Often, the rate constant calculated from collision theory must be reduced by inclusion of an ad hoc “steric factor” which represents the fraction of collisions that have proper geometry for reaction. This additional factor is an empirical factor used to bring experimental observation into line with collision theory, and accounts for those important factors not addressed by collision theory.

According to transition state theory, a fast equilibrium exists between reactants and molecules at the transition state, denoted by AHB‡ : k1

ω

A + HB
AHB‡ → AH + B. k−1

(31)

The equilibrium concentration of molecules at the transition state is given by [AHB‡ ] = K ‡ [A][HB],

(32)

where K = k1 /k−1 . The overall rate of product formation depends on the rate constant, ω, with which the activated complexes cross over to products: d[AH] (33) = ω[AHB‡ ] = ωK ‡ [A][B]. dt A comparison of Eqs. (30) and (33) indicates that the rate constant k = ωK ‡ . The equilibrium constant for production of activated complexes K ‡ is related to molecular partition functions (Q), calculated using statistical mechanics: ‡

II. ACTIVATED COMPLEX THEORY Many models of chemical reactions are based on the concept of an “activated complex,” or “transition state,” which corresponds to the nuclear configuration with the highest potential energy of the system traversed during the course of the reaction. The transition state corresponds to a critical geometry of the reacting system marking the boundary between reactants and products. Consider the elementary reaction: A + HB → AH + B.

(29)

The overall rate for this elementary reaction is given by d[AH] = k[A][HB]. (30) dt The reaction coordinate for this H-atom transfer reaction may be considered to be translational motion of the H atom from B to A. A potential energy diagram for this process may be represented in 2D or in 3D, as shown in Fig. 8.

K‡ =

[AHB‡ ] Q AHB‡ −E/k B T e . = [A][HB] Q A Q HB

(34)

In the above equation, E = E AHB‡ − E A − E HB is the energy difference between the reactants and activated complex. Assuming that the reaction involves translational motion of the hydrogen atom from B to A, the rate of passage through the transition state is given by   v k B T 1/2 1 ω= = , (35) δ 2πµ δ

FIGURE 8 Schematic potential energy surface of A + HB → AH + B reaction, shown as a two-dimensional contour plot (left) and a three-dimensional surface plot (right). Trajectory on contour plot corresponds to lowest energy pathway from reactants to products and traverses the region corresponding to the reaction transition state, AHB.‡

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where v is the velocity through the transition state region and δ is the “width” of transition state along the reaction coordinate. The contribution to the partition function of the activated complex corresponding to motion along the reaction coordinate, in this case translational motion, is factored out of the transition state partition function, Q AHB‡ , to yield: (2πµk B T ) /2 δ  Q AHB‡ . = h (36) 1

Q AHB‡ =

qtrans,AHB‡ Q AHB‡

Note that the width parameter δ appears again in Eq. (36). This parameter ultimately factors out in the transition state theory rate constant, k TST , which is given by: k TST = ωK ‡ =

 k B T Q AHB‡ −E/k B T e . h Q A Q HB

(37)

k2

A ∗ → P 1 + P2 .

(38)

Under steady-state conditions, the concentration of the collisionally activated molecule, A∗ is constant, i.e., the rate of its formation is exactly balanced by the rate of its destruction: d[A∗ ] = k1 [A][M] − k−1 [A∗ ][M] − k2 [A∗ ] = 0. dt

(39)

Rearranging: [A∗ ] =

k1 [A][M] . k−1 [M] + k2

k2

A + B
AB∗ → P1 + P2 . k−1

(40)

(42)

Following formation of the AB∗ intermediate by bimolecular collision, the reaction dynamics are somewhat analogous to those in the Lindemann mechanism: an internally excited molecule may decay to products or reform reactants. In the absence of collisions, the dynamics are solely determined by the unimolecular dynamics of the complex. However, at high pressures, particularly if the lifetime of the AB complex is long, it may undergo collision with another body, possibly carrying away excess energy resulting in formation of stable AB: k3

Many early experiments showed that thermal decomposition of a molecule A, forming products P1 and P2 , often exhibits first order kinetics at high pressure, and second–order kinetics at low pressure. As first proposed by Lindemann, the mechanism involves collisional excitation of the reactant A: k−1

At high pressure, k−1 [M]  k2 , and d[P1 ]/dt = (k1 k2 / k−1 )[A], yielding first-order kinetics. Under low-pressure conditions, k−1 [M]  k2 and d[P1 ]/dt = k1 [A][M] resulting in second-order behavior. The important result is that reactions that appear to be unimolecular, exhibiting firstorder kinetics at high pressures, actually involve bimolecular processes. An important class of elementary bimolecular reactions are those that involve formation of persistent collision complexes, denoted AB∗ , that may ultimately form products C + D, or decay back to reactants:

AB∗ + M → AB + M.

III. UNIMOLECULAR VERSUS BIMOLECULAR REACTIONS

k1

k1 k2 [A][M] d[P2 ] d[A] d[P1 ] . = =− = k2 [A∗ ] = dt dt dt k−1 [M] + k2 (41)

k1

Although we have derived this equation assuming that the reaction coordinate for atom transfer corresponds to translational motion, the same expression is obtained if the reaction coordinate is assumed to be vibrational motion. According to Eq. (37), the reaction rate constant may be calculated using the relevant molecular partition functions, known from statistical mechanics, remembering that Q AHB‡ does not include the translational motion contribution to the transition state partition function.

A + M
A∗ + M,

The overall rate of product formation is

(43)

IV. STATISTICAL THEORIES OF UNIMOLECULAR DECOMPOSITION The Lindemann mechanism as well as reactions occurring via formation of long-lived complexes involve participation of highly internally excited intermediate species that may ultimately dissociate by one or more chemical channels. For example, the intermediate complex AB∗ in reaction (42) may form new products P1 + P2 , or decay back to reactants, A + B. The total rate constant for decay of AB∗ is the sum of the two rate constants, k−1 + k2 , and the relative importance of these competing processes is defined as the product branching ratio k2 /k−1 . Of key importance in understanding these reactions are the reaction rate constants k−1 and k2 . A number of theories have been developed to quantitatively predict rate constants for unimolecular reactions. Rice, Ramsperger, and Kassel developed a simple theory, now known as RRK theory, which contains many fundamental elements underlying most modern theories of unimolecular reaction. According to RRK theory, the

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energized molecule produced by bimolecular collision may be considered to consist of a group of s identical harmonic oscillators, each of frequency ν. If any oscillator accumulates sufficient energy E o = mhν, where m is an integer and h is Planck’s constant, the energized molecule will dissociate. An underlying assumption of RRK (as well as other related theories) is that energy may flow freely between the oscillators in the molecule. The total energy of all of the oscillators is denoted by E = nhν. The number of ways that n quanta can be placed in a molecule consisting of s oscillators is: wn =

(n + s − 1)! . n!(s − 1)!

(44)

The number of ways that n quanta can be placed in a molecule such that at least m quanta are in one oscillator is given by wm =

(n − m + s − 1)! . (n − m)!(s − 1)!

(45)

The probability, P, of dissociation is the ratio of these quantities: P=

wm (n − m + s − 1)!n! = . wn (n − m)!(n + s − 1)!

(46)

If n and m are large, then Sterling’s approximation (x! ≈ x x /e x ) may be applied, and, if s is small relative to (n − m), then (n − m + s − 1) ≈ n − m. The probability of decomposition then reduces to   n − m s−1 P= . (47) n The rate constant for unimolecular decomposition is the probability P multiplied by a frequency factor ν:     n − m s−1 E − E o s−1 kRRK = ν =ν . (48) n E For a given value of s, the reaction rate constant increases with increasing energy E above threshold, E o . If E  E o , the reaction rate constant approaches the frequency factor ν. On the other hand, for reactions involving two similar but different-sized molecules having the same E and E o , since the larger molecule has a greater number of oscillators, s, the reaction rate constant k is smaller (since (E − E o )/E < 1). In practice, to obtain agreement with experiment, it is necessary to use values of s in Eq. (48) which are approximately one half of the actual number of vibrational modes in the molecule. Of course, because RRK theory treats all oscillators as having the same vibrational frequency, the theory employs very simple equations that represent qualitatively but not quantitatively the behavior of real molecules.

With the development of computers, accurate calculations using theoretical models better able to represent the behavior of real molecules has become widespread. A very important extension of the original theory, due to Marcus, is known as RRKM theory. Here, the real vibrational frequencies are used to calculate the density of vibrational states of the activated molecule, N (E). The number of ways that the total energy can be distributed in the activated complex at the transition state is denoted W (E  ). Note that the geometry of the transition state need not be known, but the vibrational frequencies must be estimated in order to calculate W (E  ). In calculating the total number of available levels of the transition state, explicit consideration of the role of angular momentum is included. The RRKM reaction rate constant is given by: kRRKM =

W (E  ) , h N (E)

(49)

where h is Planck’s constant.

V. REACTIONS IN SOLUTION The density of molecules is substantially higher in liquids than in the gas phase. However, for reactions carried out in solution under relatively dilute conditions, the concentrations of reactants are not appreciably different from in the gas phase. Since reactant molecules A and B must undergo collision in solution in order to react, many of the same principles developed for gas-phase reactions also apply in solution. However, the presence of solvent molecules leads to important differences between reactions in solution and in the gas phase. In solution, the rate of diffusion often limits the rate of approach of molecule A to within a sufficient distance to B for reaction to occur. Once an encounter pair AB is formed, however, the solvent may act as a “cage,” effectively holding them in close proximity, thereby increasing the probability of reaction. In solution, the overall reaction may again be broken down into a sequence of elementary steps: kd

A + M
AB, k−d

kr

AB → P,

(50)

where, AB is an encounter pair, kd and k−d are rate constants for approach and separation of the reactant molecules by diffusion, and kr is the rate of conversion of encounter pairs to products. Applying the steady-state approximation to the concentration of encounter pairs, we obtain d[P] k d kr = [A][B] = k[A][B]. (51) dt k−d + kr If the activation energy for the reaction is large, kr  kd and the reaction rate constant k ≈ kd kr /k−d . Alternatively,

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if kr  kd , then k ≈ kd , and the reaction rate constant is the rate of diffusion of reactants to sufficiently close proximity to facilitate reaction. The rate of a diffusion controlled reaction is determined by the magnitude of the diffusion coefficients, D, for the reactants A and B: k = 4π(DA + DB )R.

(52)

The diffusion coefficients are related to the viscosity of the solvent, η, and hydrodynamic radius, r , of the diffusing species by the Stokes–Einstein Law: D=

kB T , πβηr

(53)

where β is a constant typically ranging from 4 to 6. Note that this derivation assumes that diffusing particles are equally likely to move in any direction. However, if the reactants are oppositely charged ions, Coulomb attraction substantially increases reaction rates relative to the rate of diffusion.

VI. EXPERIMENTAL TECHNIQUES It is well known from the empirical Arrhenius expression that reaction rate constants often increase with reactant temperature. One goal of molecular reaction dynamics is to understand the roles that different forms of reactant energy (translational, electronic, vibrational, and rotational) play in chemical reaction. Also, if a reaction is successful, how is the total available energy channeled into the various available degrees of freedom of the product molecules? Many of the microscopic details of a chemical reaction are related to the nature of the transition state. Most insight into the transition state region has been obtained through experiments focusing on the asymptotic limits of the reaction; i.e., reactants and products. Four general categories of experiments will be discussed here, all focusing on elementary gas-phase reactions. In the first type of experiment, the total cross sections for various chemical reaction channels are measured, usually as a function of collision energy. In the second type, the angular and velocity distributions of products from single reactive encounters are measured. In the third type of experiment, the quantum state distributions (vibrational and rotational) of products are measured using spectroscopic techniques. In this latter approach, velocity distributions may also be obtained using Doppler or velocity imaging methods. Finally, a relatively recent development is “transition state spectroscopy,” which focuses directly on the transition state itself, usually through spectroscopic measurements.

A. Cross Section Measurements The reaction cross section may be determined experimentally as a function of collision energy using a variety of methods. For the reaction H + D2 → HD + D, the reaction has been carried out in a flow cell containing mixtures of D2 and a stable H atom precursor such as HBr. The H atoms are produced by photolysis of HBr at various UV wavelengths. Reaction cross sections are determined by measuring the concentration of products relative to reactants directly. In the present case, this involves monitoring the relative concentrations of H and D atoms (i.e., reactants and products) as a function of time following photodissociation of reactant precursor. Both H and D atoms may be detected by laser induced fluorescence (LIF) excitation spectroscopy near 121 nm (Lyman-α). By choosing different photolysis wavelengths and H atom precursor molecules, reaction rate constants may be determined for different collision energies. In this case, reaction cross sections are determined directly from Eq. (8), since the relative velocity is well defined using photolytic reactants. In Fig. 9, experimental data from several different laboratories are shown as solid points surrounded by rectangles, and theoretical values are connected by a solid line. The reaction cross section is zero below 0.4 eV due to the presence of a potential energy barrier for reaction, as discussed in detail in Section VI.B. The reaction cross section increases with increasing energy above threshold, with behavior qualitatively similar to that predicted by Eq. (15).

FIGURE 9 Cross section for H + D2 → HD + D reaction vs. collision energy. Solid points surrounded by rectangles are experimental data and open points connected by solid line is theoretical calculation. (From Gerlach-Meyer, U., Kleinermanns, K., Linnebach, E., and Wolfrum, J. (1987). J. Chem. Phys. 86, 3047–3048.)

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708 In some cases, reaction may lead to more than one type of chemical product. For example, the reaction of chromium cations, Cr+ , with methane, CH4 , may lead + to production of CrCH+ 2 + H2 or CrH + CH3 . In order to study ion-neutral reactions such as these, a beam of mass-selected reactant ions, e.g., Cr+ , is accelerated to a well-defined laboratory kinetic energy. The ion beam encounters target molecules held in a gas cell, where bimolecular reaction occurs. The product ions are then extracted from the reaction volume, mass selected, and counted. A wide variety of reactions have been studied using such techniques. In Fig. 10, the cross sections for reactions of ground state Cr+ are shown using solid symbols. Due to the endoergicity of reaction, the formation of + CrCH+ 2 only occurs at energies above ∼2.3 eV, and CrH formation has a threshold just under 3.0 eV. Reactions of an electronically excited state of Cr+ with CH4 , on the other hand, have a large cross section for production of CrCH+ 2 even down to zero collision energy, as indicated by open symbols. Experiments such as this provide insight into the role of electronic state on chemical reactivity. Furthermore, by measuring energetic thresholds for reaction, thermodynamic quantities such as bond dissociation energies may be determined directly. B. Angular and Velocity Resolved Studies Crossed molecular beam reactive scattering facilitates experimental determination of the angular and velocity distributions of chemical products from elementary bimolecular reactions. The technique involves production of two molecular beams containing the reactants, initially moving at right angles relative to one another, in an evacuated

FIGURE 10 Experimental cross sections for Cr+ + CH4 reaction. Solid points denote reaction of ground electronic state Cr+ and open points denote reactions of electronically excited Cr+ . (From Armentrout, P. B. (1991). Science 251, 175–179.)

Dynamics of Elementary Chemical Reactions

FIGURE 11 Schematic diagram of crossed molecular beam apparatus. Beams cross at right angles; products are detected by a detector that may be rotated with respect to the reactant beams.

(<10−6 Torr) chamber (Fig. 11). The beams, denoted A and BC here, are usually generated by a supersonic expansion, in which a gas at a pressure near an atmosphere or greater passes through a pinhole into an evacuated chamber. Due to adiabatic cooling of the molecules during expansion, rotational temperatures are reduced to less than 100 K. Vibrational excitation is also cooled, although less efficiently. Velocity distributions of molecules in a supersonic beam are typically less than 10% full width at half maximum (FWHM); much narrower than in a Boltzmann distribution. One or more collimating devices (skimmers) are used to define the beams to angular divergences of several degrees. Since molecules within the beam move together at nearly the same velocity in the evacuated chamber, the probability of collisions within a beam or with a background gas molecule is negligible. A small fraction of reactants in one beam undergo a single collision with molecules in the other, forming chemical products, e.g., A + BC → AB + C. In a typical experiment, the beams are several millimeters in diameter and the detector is a mass spectrometer with electron impact ionizer that may be rotated in the plane of the beams. By measuring the arrival times of products at the detector at different angles, the laboratory angular and velocity distributions of the products are determined. Reactive and nonreactive scattering data are analyzed with the aid of a Newton diagram, illustrated in Fig. 12 for the hypothetical reaction A + BC → AB + C. This vector diagram in velocity space facilitates the transformation between the laboratory and center-of-mass (CM) reference frames. Laboratory angles are denoted by , with  = 0 and 90◦ defined by the directions of the A and BC beams, respectively. All laboratory velocities are denoted by the variable v, and have their origins in the lower left corner of

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FIGURE 12 Newton diagram for A + BC reaction. Note that velocity scale is indicated on right side of diagram. See text for details.

the Newton diagram. The relative velocity vector is given by vrel = vA − vBC = uA − uBC , with velocities in the CM reference frame denoted by u and angles in the CM frame denoted by θ. The laboratory velocity of the center-of-mass of the colliding partners, vcm , which in this case is oriented along  = 50◦ , may be calculated from the masses and initial velocities of the A + BC reactants. Note that AB products scattered from the collision volume will be traveling with a velocity in the laboratory reference frame that is simply vcm + uAB . Thus, a circle is shown in Fig. 12, centered at the tip of vcm . The radius of this circle corresponds to the maximum possible velocity of AB products in the CM reference frame, u AB , for all possible orientations of the vector uAB . From energy conservation, this maximum velocity corresponds to production of internally cold AB + C products. Thus, if the thermodynamics of the reaction are known or can be estimated, u AB may be calculated using energy and momentum conservation laws. According to Fig. 12, product AB molecules may be observed at laboratory angles  ranging from 30 to 70◦ ; no products can be observed outside of this range. Products formed with progressively greater internal energies, and hence smaller translational energies, are constrained to smaller Newton circles, which are always centered on the tip of vcm . A single CM recoil velocity vector uAB is shown in this figure, at θ = 45◦ , with θ = 0 and 180◦ corresponding to the ve-

709 locities of A and BC reactants, respectively, in the CM reference frame. However, depending upon the dynamics of the reaction, θ can range from 0 to 360◦ . For randomly oriented reactant molecules, CM angular distributions must be symmetric about the relative velocity vector, i.e., about θ = 0 and 180◦ . Among the earliest reactions to be studied using crossed molecular beams were those producing alkali atoms and alkali–halide molecules, due to their ease of detection using surface ionization methods. The reaction of potassium atoms (K) with molecular bromine (Br2 ) leads to formation of KBr + Br via the so-called “harpoon mechanism.” Cuts of the two relevant potential energy surfaces are illustrated in Fig. 13. In this reaction, the neutral K + Br2 reactants approach on the covalent potential energy surface. At a distance rc , termed the ionic–covalent curve crossing distance, the energy required to transfer an electron from K to Br2 , given by IPK − EABr2 , is exactly compensated by the Coulomb energy gained by formation of the K+ Br− 2 ion pair. Reaction is initiated by electron transfer, ˚ which in the present case occurs at long range (∼7.5 A), and is analogous to the throwing of a harpoon during a whale hunt. Because the electron is transferred into an antibonding orbital, the equilibrium bond length of Br− 2 is much greater than for Br2 and rapid Br–Br− stretching in the presence of K+ leads to immediate formation of the K+ Br− + Br products. This reaction has been termed a “spectator stripping” reaction, by analogy with some nuclear reactions. Here, the spectator is the bromine atom not involved in formation of the KBr product. Because of the rapid dissociation of Br–Br− while the K+ is still at relatively long range, the motion of the spectator Br atom remains essentially unperturbed throughout the course of the collision.

FIGURE 13 Ionic and covalent potential energy curves for K + Br2 reaction.

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FIGURE 14 Energy level diagram for OH + CO reaction.

Consequently, this spectator atom is produced with a CM velocity vector that is essentially the same as that of the initial Br2 molecule reactant. This leads to a highly anisotropic spatial distribution of products, with the Br fragment scattered at CM angles near θ = 180◦ relative to the direction of the K atom reactant, whereas the KBr counterfragment is scattered preferentially near θ = 0◦ . The KBr product is thus “forward scattered” with respect to the incoming K atom in the CM reference frame. Note ˚ for the K + Br2 system (Fig. 13). Since a that rc ≈ 7.5 A large fraction of those collisions involving electron transfer lead to reaction, the reaction cross section is essential˚ 2, ly the electron transfer cross section, σ = πrc2 ≈ 177 A which is much larger than typical hard sphere cross sections. The elementary reaction OH + CO → CO2 + H is the primary mechanism for formation of CO2 in combustion. This reaction, as well as its reverse, have been studied extensively using a variety of methods, including the crossed molecular beams method. The reaction is thought to involve the participation of long-lived HOCO intermediates, as illustrated in Fig. 14. Note that although OH + CO may form the intermediate complex with at most only a small potential energy barrier, a substantial barrier exists for H-atom loss forming CO2 . In a crossed molecular beams experiment using supersonic beams in which the OH and CO are in their lowest few rotational levels, the total angular momentum of the HOCO intermediate is primarily orbital angular momentum, L = µvr el b, with the vector L lying perpendicular to the plane containing the colliding species (Fig. 15). In many reactions, including this one, a relatively small fraction of total angular momentum appears as product rotational angular momentum. Consequently, the final orbital angular momentum, L  , for the H + CO2 products is approximately the total angular momentum, L, and the product CM velocities uAB and uC , must lie in a plane nearly identical to that of the incident reactants. Because the intermediate HOCO lifetime is long compared to its rotational period, uAB may lie in any direction in the plane of the collision.

Dynamics of Elementary Chemical Reactions

FIGURE 15 Schematic of one OH + CO collision event showing reactant velocities in CM frame. CO2 product velocities are constrained by angular momentum conservation to lie in plane near that of reactants.

The experimentally measured CO2 angular distribution, shown in Fig. 16, is broad with products scattered to both sides of the center-of-mass of the system. Such angular distributions are said to exhibit “forward–backward symmetry” in the center-of-mass frame of reference. From analysis of product translational energy distributions (not shown), the degree of CO2 vibrational excitation has been inferred for this reaction. Perhaps the most extensively studied elementary bimolecular reaction is the simplest: the hydrogen exchange reaction and its isotopic variants. For example, studies of reactions such as H + D2 → HD + D provide direct insight into reactive collisions. This reaction is slightly endothermic and exhibits a large (∼0.4 eV) potential energy barrier to product formation (Fig. 17). The reaction is direct, and proceeds via a near-collinear H–D–D transition state on a timescale short relative to rotation, leading to anisotropic D atom angular distributions. The experimental apparatus used in a recent H + D2 study is shown in Fig. 18. Two molecular beams containing HI and D2 are generated in separately pumped

FIGURE 16 CO2 laboratory angular distribution from crossed beams OH + CO reaction. (From Alagia, M., Balucani, N., Casavecchia, P., Stranges, D., and Volpi, G. G. (1993). J. Chem. Phys. 98, 8341–8344.)

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FIGURE 17 Energy level diagram for H + D2 reaction.

regions and allowed to propagate parallel to one another, 30 mm apart. The HI molecules are photodissociated by a pulsed 266-nm laser, producing ground state H atoms with a very narrow translational energy distribution. Some of these H atoms collide with the D2 molecules in the second beam, leading to chemical reaction forming HD + D. The D atom products from the reaction are “tagged” before they leave the interaction region by excitation to high-n Rydberg states using two tunable “probe” lasers. The first laser, operating at 121.6 nm, is tuned to the n = 1 → 2 transition, and the second laser, at 365 nm, excites from n = 2 → ∼40. The lifetimes of these highly excited D atoms are in excess of 100 µs. Since the velocities of the atoms are essentially unchanged during the tagging step, they evolve spatially with their nascent velocities to a detector located approximately 30 cm away from the reaction zone, where they are field ionized and collected. The detector may be rotated with respect to the fixed beams in order to map out the angular distribution of the products. By measuring the time of arrival at the detector following time zero for reaction, the velocity and hence translational energy distribution is obtained.

FIGURE 19 Lower figure: Newton diagram for H + D2 reaction. Circles show maximum CM velocities for differerent rotational levels of HD (v  = 0) products. Upper figure: D atom laboratory angular distribution; solid line shows theoretical simulation. (From Casavecchia, P., Balucani, N., and Volpi, G. G. (1999). Annu. Rev. Phys. Chem. 50, 347–376.)

At a collision energy, E coll , of 0.53 eV, the HD product may be produced in either the ground or first excited vibrational states (Fig. 17). A Newton diagram for H-atom products recoiling from HD in various rotational levels of the ground vibrational state (v  = 0) is shown in Fig. 19. The largest circle corresponds to the velocity of D atoms recoiling from internally cold HD (i.e., v  = 0, j  = 0). lSmaller circles correspond to recoil from HD molecules born with progressively greater internal energies. A measurement of the D atom velocity distribution at different

FIGURE 18 Experimental apparatus for studies of H + D2 reaction in crossed molecular beams. (From Schnieder, L., Seekamp-Rahn, K., Wrede, E., and Welge, K. H. (1997). J. Chem. Phys. 107, 6175–6195.)

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tion is based on exact trajectory calculations on high level ab initio potential energy surfaces. This close agreement between state-of-the-art theory and experiment is a particularly impressive achievement, demonstrating that the hydrogen exchange reaction is presently very well understood from a theoretical standpoint. Other related three atom systems that have been extensively studied include halogen atom reactions X + H2 → HX + H; X = F, Cl, and their reverse. Like the hydrogen exchange reaction, these reactions proceed via direct mechanisms over substantial potential energy barriers.

FIGURE 20 D-atom laboratory kinetic energy distribution from H + D2 reaction at 0.53 eV. (From Casavecchia, P., Balucani, N., and Volpi, G. G. (1999). Annu. Rev. Phys. Chem. 50, 347–376.)

angles thus provides a direct measure of the internal state distribution of the HD product. The D atom angular distribution is highly anisotropic, peaking at laboratory angles near zero degrees (Fig. 19). In Fig. 20, the translational energy distribution for the D product, obtained by time of flight measurements for the D atoms, is shown at a laboratory angle, , of 0◦ . Structure corresponding to varying degrees of HD (v  = 0) rotational excitation (denoted by j  ) is observed in the D atom kinetic energy distribution. Note that in addition to the experimental distributions, a calculated kinetic energy distribution, also shown, is found to almost exactly reproduce the degree of HD rotational excitation. This calculated distribu-

C. Product Detection Using Spectroscopic Methods Many different spectroscopic techniques have been applied to studies of elementary bimolecular reactions. Measurements of the infrared chemiluminescence emitted from reaction provides a direct measure of the product state (rotational and vibrational) distributions. Studies of many different types of reactions have led to the development of fundamental principles that facilitate an understanding of how various forms of energy are partitioned between reactants and products. Recall that the transition state is a region of the potential energy surface that must be traversed in order for reaction to occur. The location of the transition state relative to reactants and products has a significant effect on the dynamics of the reaction. As shown in Fig. 21, an analogy may be drawn between the motion of atoms during a chemical reaction and the motion of a ball rolling on a three-dimensional potential energy surface. If

FIGURE 21 Two-dimensional contour map for AB + C reaction. Figure on left denotes reaction involving “early” transition state, in which translation is more effective in promoting reaction than is vibration; figure on right denotes reaction having “late” transition state, where vibrational energy promotes reaction.

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the transition state geometry more closely resembles the reactants (AB + C), than the products (A + BC), the transition state is termed “early.” A “late” transition state, on the other hand, corresponds to the case where the transition state more closely resembles the products. For reactions involving an early barrier, reactant translational energy is more effective than vibrational energy in surmounting the barrier, and much of the available energy is deposited into product vibration. Note that the opposite is true for reactions involving late barriers. In this case, vibrational energy of the reactants is more effective than translational energy, and the products from the reaction tend to be vibrationally cold, with a large fraction of available energy appearing in translational energy. To illustrate these ideas, consider the reaction H + Cl2 → HCl + Cl. Studies of the infrared chemiluminescence emitted from the diatomic product HCl indicate that although the reaction is highly exothermic, only a small fraction of the total available energy is channeled into HCl vibrational and rotational energy. Since electronically excited Cl and HCl products are not energetically accessible, a large fraction of the available energy must be partitioned into relative translational energy. This reaction is therefore an example of one involving a late transition state. Measurement of product state distributions using various spectroscopic techniques has been applied to a wide variety of reactions, including H + CO2 . The H + CO2 → OH + CO reaction, the reverse of the reaction shown in Fig. 14, is substantially endoergic. However, UV photodissociation of a diatomic molecule such as HI or HBr can produce H atoms with more than enough kinetic energy to react. The OH products may be readily detected using fluorescence excitation spectroscopy in which the total OH fluorescence intensity is monitored as a function of OH excitation wavelength. This provides a direct measure of the OH quantum state distribution from the bimolecular reaction. In the upper panel of Fig. 22, the rotational distribution of OH (v  = 0, 1) from photodissociation of HBr in the presence of CO2 molecules is shown. For both OH vibrational levels, the most probable rotational level is near j  = 8. In the lower panel, a similar experiment is carried out, but in this case the reaction is initiated within weakly bound HBr · · · CO2 van der Waals complexes, produced by coexpanding HBr and CO2 in a dilute He mixture in a supersonic molecular beam. In this case, the HBr moiety of the complex is photodissociated, producing H atoms that react with CO2 to form OH + CO. As indicated in the lower panel, the OH rotational distributions are substantially “colder” for products from reactions initiated within complexes, with the most probable rotational level near j  = 4. The different behavior is believed to result from the presence of the halogen counterfragment

713

FIGURE 22 OH (v = 0 and 1) rotational distributions from H + CO2 reaction. Upper panel indicates distribution obtained from gas phase reaction. Lower panel indicates distribution obtained from reaction photoinitiated in CO2 · · · HBr van der Waals complexes. (From Wittig, C., Engel, Y. M., and Levine, R. D. (1988). Chem. Phys. Lett. 153, 411–416.)

near the reacting H + CO2 pair in the complex-initiated reaction. Spectroscopic methods can also be used to determine rates of elementary reactions. As noted earlier, the existence of HOCO intermediates persisting longer than their picosecond rotational timescales during the OH + CO reaction has been inferred from CO2 product angular distributions in crossed molecular beam experiments. More direct determinations of bimolecular reaction timescales are often very difficult because molecules must first collide before they can react. Due to the “random” nature of bimolecular collisions, both in time and in space, it is not usually possible to define the “time zero” for a bimolecular reaction to sufficient precision to permit a direct study of reaction timescales. These difficulties, however, have been surmounted by initiating the bimolecular reaction via photodissociation of van der Waals complexes such as HI · · · CO2 . Since the H + CO2 reactants are already in close proximity, and because the time zero for reaction can be accurately defined using an ultrashort laser pulse (<1 ps), it is possible to measure the reaction rate in real time by measuring the buildup of OH products following photodissociation of HI within a binary complex. Rate constants for H + CO2 → OH + CO obtained at different HI photodissociation wavelengths, and hence

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ics. For example, the reactivity of H2 O |04, having four quanta in one OH bond, was compared to that for H2 O |13, which has three quanta in one OH bond and one in the other. It was found that Cl + H2 O |04 produced primarily HCl + OH (v = 0), whereas reaction of Cl + H2 O |13 yielded primarily HCl + OH (v = 1). This illustrates that the OH bond not involved in the reaction acts much like a spectator, retaining its initial level of vibrational excitation in the chemical products. D. Spectroscopy of the Transition State

FIGURE 23 Experimentally determined bimolecular reaction rate constants for H + CO2 reaction photoinitiated in HI · · · CO2 van der Waals complexes. Solid line denotes calculated RRKM decomposition rate for HOCO reaction intermediates. (From Ionov, S. I., Brucker, G. A., Jaques, C., Valachovic, L., and Wittig, C. (1992). J. Chem. Phys. 97, 9486–9489.)

collision energies, are shown in Fig. 23. Note that the rate constant, k, for this elementary reaction, which is the inverse of the HOCO lifetime (τ ), increases monotonically with increasing energy, E † , in excess of the potential energy barrier for H + CO2 production. This behavior is in close agreement with that predicted by statistical theories such as RRKM theory, illustrated by a solid curve. Through the use of direct optical excitation of reactants, it has been possible in several cases to direct the course of reaction to favor certain product channels or product state distributions. Using an intense pulsed near-infrared laser, the very weakly absorbing OH vibrational overtones in small molecules like water may be excited relatively efficiently, depositing a substantial amount of vibrational energy into well-defined vibrational modes of reactants. Using this approach, the role of reactant vibrational excitation in bimolecular reactivity has been studied. Because of the local mode character of OH bonds in water, the vibrational excitation initially deposited into the molecule remains localized in a well-defined oscillator. For example, the reaction of Cl + HOD (4νOH ) (having four quanta in the OH stretch with zero in the OD stretch), leads to relatively efficient production of HCl + OD, but essentially no DCl + OH, as determined by LIF detection of OH and OD products (Fig. 24). In this example of bond specific chemistry, the vibrationally excited OH bond is highly reactive, whereas the unexcited OD bond is relatively inert. In other related studies of reactions of H2 O, mode specific chemistry has been observed, in which different excited H2 O vibrational states at nearly the same energy displayed somewhat different reaction dynam-

In the experiments described previously, much of the information about the reaction pathway is obtained through measurements of the velocity, angular, or internal state distributions of products, or through measurements of their cross sections or timescales for formation. In some cases, the effect of different forms of reactant energy in promoting chemical reactivity also provided insight into the thermodynamics of the reaction or on the nature of the transition state. A conceptually different and more recent development involves direct measurement of the transition state itself. Since, in most reactions, the transition state corresponds to an energy maximum along the reaction coordinate, this region is only accessed for a very short time, typically on timescales comparable to a vibrational period (<10−13 s). Consequently, the fraction of molecules in a reacting sample near the transition state at any given time

FIGURE 24 Fluorescence excitation spectrum for OD and OH products from HOD(4νOH ) + Cl reaction. (From Sinha, A., Thoemke, J. D., and Crim, F. F. (1991). J. Chem. Phys. 96, 372– 376.)

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tra, recorded at different experimental resolutions, show structure corresponding to antisymmetric stretching vibrational levels of the neutral IHI molecule. This approach has also been applied to studies of a wide variety of other reactive systems involving transfer of a light atom, such as F + CH3 OH → HF + CH3 O, and F + H2 → HF + H.

SEE ALSO THE FOLLOWING ARTICLES

FIGURE 25 Photoelectron spectra at two different resolutions from IHI− photodetachment. Peaks correspond to asymmetric stretching levels of neutral IHI product. (From Metz, R. B., and Neumark, D. M. (1992). J. Chem. Phys. 97, 962–977.)

is vanishingly small, making it difficult to study such species directly. Much of the recent progress in the area of transition state spectroscopy has therefore utilized photoinitiation qualitatively similar to that described above for HI · · · CO2 to produce molecules near the transition state. For example, the transition state regions for a number of hydrogen exchange reactions have been studied by first producing molecular beams containing stable negative ions, such as IHI− . The geometries of these stable negative ions often resemble that of the transition state for the neutral reaction, e.g., I + HI → HI + I . Photodetachment of the electron produces the neutral IHI species. As in conventional photoelectron spectroscopy, by measuring the kinetic energy distribution of the ejected electron, the vibrational energy levels of the molecular counterfragment, in this case IHI at nuclear geometries near the transition state, are obtained. In Fig. 25, two spelc-

ATOMIC AND MOLECULAR COLLISIONS • COHERENT CONTROL OF CHEMICAL REACTIONS • ENERGY TRANSFER, INTRAMOLECULAR • ION KINETICS AND ENERGETICS • KINETICS, CHEMICAL • MOLECULAR BEAM EPITAXY, SEMICONDUCTORS

BIBLIOGRAPHY Baer, T., and Hase, W. L. (1996). “Unimolecular Reaction Dynamics, Theory and Experiments,” Oxford Univ. Press, New York. Crim, F. F. (1999). “Vibrational state control of bimolecular reactions: Discovering and directing the chemistry,” Acc. of Chem. Res. 32, 877– 884. Johnston, H. S. (1966). “Gas Phase Reaction Rate Theory,” Ronald Press. Lee, Y. T. (1987). “Molecular beam studies of elementary chemical processes,” Science 236, 793–798. Levine, R. D., and Bernstein, R. B. (1987). “Molecular Reaction Dynamics and Chemical Reactivity,” Oxford Univ. Press, New York. Neumark, D. M. (1992). “Transition State Spectroscopy of Bimolecular Chemical Reactions” In Annual Review of Physical Chemistry (H. L. Strauss, G. T. Babcock, and S. R. Leone, eds.), Vol. 43, pp. 153–176, Annual Reviews Inc., Palo Alto, CA. Pilling, M. J., and Seakins, P. W. (1995). “Reaction Kinetics,” Oxford Science Publications, New York. Polanyi, J. C. (1987). “Some concepts in reaction dynamics,” Science 236, 680–690. Polanyi, J. C., and Zewail, A. H. (1995). “Direct observation of the transition state,” Acc. Chem. Res. 28, 119–132. Scoles, G. (1988). “Atomic and Molecular Beam Methods, Vol. 1,” Oxford Univ. Press, New York.

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Electrolyte Solutions, Thermodynamics J. Barthel W. Kunz R. Neueder University of Regensburg

I. II. III. IV.

Electrolytes and Solvents Thermodynamics of Electrolyte Solutions Statistical Approaches Thermodynamic Properties of Electrolyte Solutions

GLOSSARY Activity coefficient Ratio of the activity to the concentration of a component Yi in a mixture or solution; the activity coefficient represents a measure of the departure from ideal behavior. Aprotic solvent Solvent incapable of acting as a proton donor; an aprotic solvent may act as an electron donor or acceptor (Lewis acid or base). Born process Hypothetical process for calculating the energy of transfer of an electrolyte compound from a vacuum into a solvent where the electrolyte exhibits complete dissociation into ions. Chemical potential Content in Gibbs energy of a component Yi of a mixture or solution, that is, the change in total Gibbs energy of the system at constant temperature and pressure when 1 mol of component Yi is added to an infinite amount of the system. Electromotive force Difference in potential at the elec-

trodes of a galvanic cell when no electric current passes through the cell. Excess property Difference between the actual property and its hypothetical value calculated for an ideal mixture or solution at the same temperature, pressure, and mole fraction composition. Hamiltonian Energy function of a system expressing the total energy of the system as a function of the coordinates and momenta of its particles, molecules, and/or ions. Hamiltonian model Statistical mechanical model based on a Hamiltonian. Ideal mixture or solution Mixture or solution that shows no change in volume, enthalpy, and heat capacity when the mixture or solution is being made up from its initially separated components. Ionogene Electrolyte compound that in its pure phase exists as a neutral molecule and forms ions in solution by a chemical reaction with solvent molecules. 219

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220 Ionophore Electrolyte compound that in its pure phase exists as an ionic crystal and forms ions in solution by decomposition to the preformed ions. Medium activity coefficient Measure for the change in Gibbs energy when an electrolyte is transferred from one solvent to another solvent at infinite dilution. Osmotic pressure Additional pressure applied to a solution to maintain equilibrium with a pure solvent that is in contact with the solution through a membrane permeable only to solvent molecules. Partial molar quantity Increase in any extensive thermodynamic property of a system at constant pressure and temperature when 1 mol of component Yi is added to an infinite amount of the system; the chemical potential of component Yi is also the partial molar Gibbs energy of compound Yi . Protic (amphiprotic) solvent Solvent apt to react as a proton donor or acceptor (Br¨onsted acid or base). Solvation Process leading from the ions of an electrolyte, after complete dissociation in a vacuum, to the ions in solution surrounded by oriented solvent molecules. Structure-making (-breaking) ion Ion that by a solvation process increases (decreases) the structure of a pure solvent.

ELECTROLYTE SOLUTIONS are solutions of electrolyte compounds in pure or mixed solvents such that the solutions become electric conductors in which current is carried by the movement of ions. They exhibit specific properties due to the more or less complete dissociation of their solutes into ions. Aqueous electrolyte solutions are involved in numerous biological, biochemical, geological, and technical processes. Nonaqueous electrolyte solutions are intensively studied owing to unique properties for their application in various new technologies such as high-energy batteries, electrodeposition, nonemissive displays, solar cells, phase-transfer catalysis, or electroorganic synthesis.

I. ELECTROLYTES AND SOLVENTS A. Classification of Electrolytes Electrolytes can be classified into two categories, ionophores and ionogenes, independent of their stoichiometry. Stoichiometry entails formal arrangement in classes according to the valency of their cations and anz+ z− ions. An electrolyte of type Cν+ Aν− composed of a z + z+ valent cation C and a z − -valent anion Az− is referred to as a binary z + , z − electrolyte, for example, HCl (1,1), CdCl2 (2,1), and CdSO4 (2,2). Binary electrolytes contain only one type of cation and anion, in contrast to ternary

Electrolyte Solutions, Thermodynamics

(e.g., NaHSO4 ) or still more complex  electrolyte compounds. Electroneutrality requires νi z i = 0. 1. Ionophores Ionophores are substances that already exist as ionic crystals in their pure state, such as alkali metal halides. When dissolved in a solvent, ionophores are initially completely dissociated in the solution and their ions are solvated. However, association to ion pairs and higher ion aggregates with and without inclusion of solvent molecules may occur in many solvents depending on electrolyte concentration and solvent permittivity. Equation (1) shows the ionic species in an aqueous CdSO4 solution—solvated ions and ion pairs—which can be detected by appropriate methods and which determine the physical properties of this solution: → CdSO4 (s) −−−−→ Cd2+ (aq) + SO2− 4 (aq)← H2 O

→ [Cd(OH2 )SO4 ]0 ← → [CdSO4 ]0 . [Cd(OH2 )2 SO4 ]0 ← (1) Aqueous solutions of alkali metal salts exhibit almost no association, in contrast to solutions of 2,2 electrolytes, such as CdSO4 . Solutions of low permittivity contain ion pairs (equilibrium constant K A ), triple ions (equilibrium constant K T ), and higher associates, such as 1,2-dimethoxyethane

LiBF4 (s) −−−−−−−−−−→ Li+ + BF− 4 KA   0 → Li+ BF− ← 4 , KT   + − 0  − + − − → Li BF4 + BF− , 4 ← BF4 Li BF4 and



Li+ BF− 4

0

KT   + + → Li+ BF− + Li+ ← . 4 Li

(2a) (2b) (2c)

2. lonogenes Ionogenes, such as mineral or carboxylic acids, form ions only by a chemical reaction with solvent molecules, for example, KI   − 0 → CH3 COOH+ HClO4 + CH3 COOH ← 2 ClO4 KD

− → CH3 COOH+ ← 2 + ClO4 .

(3)

The solution of perchloric acid in glacial acetic acid [Eq. (3)] shows the typical two-step mechanism of ion formation of ionogenes. An ionization (equilibrium constant K I ) process is followed by a dissociation (equilibrium constant K D ) process. The overall constant K is given by the relationship K = K I K D /(1 + K I ).

(4)

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Solvents of type LH (amphiprotic solvents) such as water, alcohols, and carboxylic acids themselves behave as ionogenes in producing their cations LH+ 2 (lyonium ions) and anions L− (lyate ions) by autoprotolysis reactions:

capacity to interact with other species. Four main groups of solvents with typical donor–acceptor properties generally are distinguished:

− → LH+ LH + LH ← 2 +L

1. Protic solvents (e.g., water, alcohols, amides, carboxylic acids) 2. Aprotic solvents (e.g., aprotic amides, nitriles, ketones) 3. Low-permittivity electron donor solvents (e.g., ethers) 4. Inert solvents (e.g., hydrocarbons, chlorinated hydrocarbons)

(5)

→ H3 O+ + OH− ). The solution of acids and (e.g., 2H2 O ← bases in amphiprotic solvents in generally followed by a protolysis reaction as a consequence of the following equilibria: → A− + LH+ AH + LH ← 2

(6)

→ BH+ + L− , B + LH ←

(7)

and

where LH represents the amphiprotic solvent, AH the acid, and B the base. When these equilibria are shifted markedly → H3 O+ + Cl− ), the toward ionization (e.g., HCl + H2 O ← acids or bases are almost completely replaced by the lyate or lyonium ions of the solvent (leveling effect).

II. THERMODYNAMICS OF ELECTROLYTE SOLUTIONS

B. Classification of Solvents Two aspects determine the role of the solvent: its bulk properties and its proton or electron donor or acceptor capabilities. A solvent S may act by a basic center β (acceptor: βS), an acidic center α (donor: Sα), or both (βSα). Solvation of an electrolyte, Y = C+ A− , then occurs according to the scheme → C+ (βS)m + A− , Y + m(βS) ←

(8a)

→ C + (Sα)n A , Y + n(Sα) ←

(8b)

+

Despite such limitations as the overlapping of solvent classes or possible interactions evading the unambiguous classification of a solvent, such classifications are useful for understanding the properties of electrolyte solutions and for rationalizing the choice of appropriate solvents and solvent mixtures for particular investigations.

−

A. Fundamentals Electrolyte solutions commonly are one-phase systems of k components. The internal energy U of such systems expressed as a function of its extensive variables S (entropy), V (volume), and n i (amount of substance of component Yi , i = 1, 2, . . . , k) yields the fundamental equation of thermodynamics, dU = T dS − p d V +

k 

or

µi dn i ,

(9a)

i=1

→ C+ (βSα)m + (βSα)n A− . Y + (m + n)(βSα) ←

(8c)

Stabilization of cations and/or anions by the solvent molecules (solvation) is essential for the comprehension of the properties of electrolyte solutions. The mean number of solvating particles, n or m in Eqs. (8), depends on the nature of the solvent and solute and is specific for every solution. Theoretical and semiphenomenological approaches use the bulk properties or molecular quantities, dipole and quadrupole moments, polarizability, and so on, or mean force potentials in order to take these effects into account. Empirical methods take account of them with the help of empirical scales such as donor numbers, acceptor numbers, or correlation parameters. Various attempts have been made in the literature to classify solvents according to their acid–base properties, which can be used as a general chemical measure of their

and its Legendre transforms, dH = T dS + V dp +

k 

µi dn i ,

(9b)

i=1

dA = −S dT − p dV +

k 

µi dn i ,

(9c)

µi dn i ,

(9d)

n i dµi ,

(9e)

i=1

dG = −S dT + V dp +

k  i=1

and 0 = −S dT + V dp −

k  i=1

where T (temperature), p (pressure), and µi (chemical potentials, i = 1, 2, . . . , k) are the intensive variables of the system, which are conjugated to the extensive variables. H

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is the enthalpy, A the Helmholtz energy, and G the Gibbs energy. Equation (9e) is the Gibbs-Duhem equation. For use in thermodynamic calculations the chemical potential at constant temperature and pressure µi = (∂G/∂n i ) p,T,n j =ni , is split into two parts,     − (10) µi ( p, T ) = µ− i ( p, T ) + RT ln ai /ai , which relates µi ( p, T ) to the reference potential µ− i for  which the activity is a− i . The commonly chosen reference states for solutions are  ∗ the pure solvent µ− S ( p, T ) = µS and for the solute Y the  − state of infinite dilution µY ( p, T ) = µ∞ Y ( p, T ). The mole fraction xS of the solvent equals unity for both states: 

 µi ( p, T ) = µ− i ( p, T ) + RT ln x i f i ,

i =S

or

 = ∗ or ∞, −

Y,

(11a)

and µ− i ( p, T ) = lim [µi ( p, T ) − RT ln x i ], 

to rationalize thermodynamic properties for chemical engineering. Graphs of the Gibbs excess energy and thermodynamic functions derived from it such as the excess ex ex enthalpy, H = −T 2 (∂( G /T )/∂ T ) p , or excess enex ex tropy, S = −(∂ G /∂ T ) p , as a function of mole fraction x can be used for an easy determination of the partial molar quantities Z i = (∂ Z /∂n i ) p,T,n j=i [Eq. (38)] by the method of intercepts. The partial molar excess Gibbs energies are the chemical potentials µiex ( p, T ) of the studied solution. B. Activity Coefficients If the solute Y in a binary system is an electrolyte composed of a cation Cz+ of valency z + and an anion Az− of z+ z− valency z − (i.e., Y = Cν+ Aν− ), the appropriate form of the chemical potential of the solute is based on the single-ion chemical potentials of its cation and anion and is given by the relationship µY ( p, T ) = ν+ µ+ ( p, T ) + ν− µ− ( p, T ),

xS →1

lim f i = 1.

(11b)

xS →1

In Eqs. (11) the activites of solvent and solute are defined with the help of mole fractions xi and activity coefficients fi :  ni xi = n i (12a)

where µ+ ( p, T ) = µ∞ + ( p, T ) + RT ln x + f + , x + = ν+ x Y , µ− ( p, T ) = µ∞ − ( p, T ) + RT ln x − f − , x − = ν− x Y ,

(12b)

A solution for which the activity coefficients equal unity independent of the mole fractions is called an ideal solution; the activity coefficients of a real solution can be interpreted with the help of the excess chemical potentials µiex with regard to this ideal solution: µi ( p, T ) − µiid ( p, T ) = µiex ( p, T ) = RT ln f i .

(13)

The Gibbs-Duhem equation [Eq. (9e)] gives the mutual dependencies of the activity coefficients of a solution at constant pressure and temperature. Equation (13) permits the representation of the mean molar  Gibbs excess energy ex of the solution G = G ex / n i , in the form

G

ex

=

k  i=1

xi µiex = RT

k 

xi ln f i ,

(14a)

i=1

which, in turn, may be combined with the help of commonly used fitting equations, for example, for a binary mixture

G

ex

= x(1 − x)[A + B(2x − 1) + C(2x − 1)2 + · · ·], (14b)

(16a)

and

and ai = xi f i .

(15)

(16b)

are determined in agreement with Eqs. (11). In Eqs. (16) the mole fraction xY is not the usual stoichiometric mole fraction [Eq. (12a)] but the mole fraction that takes into account the dissociation of the electrolyte component into ions: nY nS xY = , xS = , n S + νn Y n S + νn Y (17) xS + νxY = 1. Combination of Eqs. (15) and (16) yields the chemical potential of the electrolyte Y: µY ( p, T ) = µ∞ Y ( p, T ) + ν RT ln x ± f ± , ν = ν+ + ν− ,

(18a)

and µ∞ Y ( p, T ) = lim [µY ( p, T ) − ν RT ln x ± ], xS →1

lim f ± = 1.

(18b)

xS →1

The so-called mean mole fraction x± and the mean activity coefficient f ± of the electrolyte are defined as follows:   1/ν 1/ν x± = x+ν+ x−ν− , f ± = f +ν+ f −ν− . (18c)

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The choice of concentration scales other than the mole fraction scale entails changes in the reference potentials and activity coefficients: Molality scale: µY ( p, T ) = µ∞(m) ( p, T ) + ν RT ln m ± γ± . Y

(19)

Molarity scale: µY ( p, T ) = µ∞(c) ( p, T ) + ν RT ln c± γ± . Y

(20)

In Eqs. (19) and (20) µ∞(m) and µ∞(c) are the reference Y Y potentials at infinite dilution, m ± and c± are the mean molality (moles per kilogram of solvent) and the mean molarity (moles per cubic decimeter of solution), and γ± and y± are the corresponding activity coefficients in the molality and molarity scales. All quantities are defined according to Eqs. (18c). The transfer from one concentration scale to another is executed with the help of conversion formulas: ∞(m) µ∞ ( p, T ) − ν RT ln MS Y ( p, T ) = µY   ∞(c) = µY ( p, T ) − ν RT ln 103 MS dS , (21)

x± = =

MS m± 1 + MS νm ± /Q 103 MS c± , d + 103 (ν MS − MY )c± /Q

(22)

µ ˜ i ( p, T ) = µi ( p, T ) + F z i ,

and f ± = (1 + MS νm ± /Q)γ± =

d + 10 (ν MS − MY )c± /Q y± , dS

trolyte compounds are components. As a consequence, there is no rigorous method for measuring the chemical potential or other thermodynamic properties of a single ionic species. Despite this drawback there is an interest in science and technology in single-ion quantities for rationalizing the discussion of electrolyte solution properties. Various methods have been developed for their estimation based on extrathermodynamic assumptions, such as the following: (1) The contributions of cation and anion are set equal for a salt composed of ions of equal charge and approximately equal radii; (2) the results of measurements on a series of homologous electrolytes are extrapolated with regard to ionic radii or ionic volumes to zero ion size or zero reciprocal radii; (3) the differences in conventional ionic properties are used for theoretically rationalized extrapolations; and (4) the properties of ions are compared with those of isoelectronic neutral molecules of similar chemical constitution and size. Besides the chemical potentials of single ions, electrochemical potentials are defined in electrochemistry. They are useful in electrode kinetics and related topics but are of only limited interest in the discussion of the thermodynamic properties of electrolyte solutions. Electrochemical potentials µ ˜ i ( p, T ) are related to chemical potentials with the help of the relationship

3

(23)

where Q = (ν+ν+ ν−ν− )1/ν ; MS and MY are the molecular weights of solvent and solute, respectively; and dS and d are the densities of the solvent and the solution, respectively. These conversion formulas are valid if SI units are used for all quantities except molar concentration c± , which traditionally is given as moles per cubic decimeter. At a pressure of 1 bar and a temperature of 298.15 K  the chemical potentials µ− i are called chemical standard potentials. Chemical single-ion potentials are referred to the chemical standard potential of the proton in aqueous solutions, which is arbitrarily set to zero, µ∞ H+ (aq)( p = 1 bar, T = 298.15 K) = 0. C. Single-lon Quantities The definition of single-ion chemical potentials [Eqs. (16)] and the properties derived from them are nonoperational. There is no way of adding to the solution ions of only one species. That is, the ions in a solution are not components according to the Gibbs definition; only the elec-

(24)

where F is the Faraday number, the Galvani potential of the liquid phase, and z i the valency of the charged species. They are equal to the corresponding chemical potentials for electrically neutral species, molecules, and neutral electrolyte compounds (z i = 0). Their definition results from formally  introducing the electrical charge of the phase, q = n i F z i , as a further extensive variable into Eq. (9a), which yields an additional term, dq.

D. Ionic Equilibria If the electrolyte component is not completely dissociated (α, degree of dissociation), as happens for ionogenic electrolytes, or if partial association of the ions to ion pairs takes place in the solution, Eq. (15) must be replaced by the relationship µY ( p, T ) = αν+ µ + ( p, T ) + αν− µ − ( p, T ) + (1 − α)µ x ( p, T ),

(25)

where µ + , µ − , and µ x are, respectively, the chemical potentials of “free” cations, “free” anions, and the species X, which is the nondissociated ionogene or the ion pair.

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The equilibrium between species X and “free” cations → X, yields the equilibrium and anions, ν+ Cz+ + ν− Az− ← condition µ x ( p, T ) − ν+ µ + ( p, T ) − ν− µ − ( p, T ) = 0.

(26)

Combining Eqs. (25) and (26) and taking into account the

definitions of c± and µ ∞(c) , c± = αc± and µ ∞(c) = µ∞(c) , Y Y leads to the chemical potential of the electrolyte Y in the form µY ( p, T ) = µ∞(c) ( p, T ) + ν RT ln αc± y± , Y

(27)

which is identical to Eq. (20) if we set y± = αy± .

(28)

The importance of Eq. (28) is that both degree of dissociation and activity coefficient of the “free” ions are parts of the excess Gibbs energy. The equilibrium condition [Eq. (26)] yields the relationship ∞(c) µ∞(c) ( p, T ) − ν+ µ∞(c) + ( p, T ) − ν− µ− ( p, T ) x

= −RT ln

cx yx

ν. (c± y± )

(29)

The left-hand side of Eq. (29) is the molar Gibbs energy

G ∞ A of formation of the species X from the initially infinitely separated ions; the right-hand side shows the equilibrium constant as the association (K A(c) ) or dissociation (K D(c) ) constant [the superscript (c) indicates the molar concentration scale], which for symmetric electrolytes (ν+ = ν− = 1) takes the form  −1 1 − α yx K A(c) = K D(c) = 2 . α c y± 2

(30)

The quantity yx is the acitivity coefficient of the species X; the other symbols are explained in the foregoing text. In dilute solutions yx equals unity. Incomplete dissociation and partial association of symmetric electrolytes are given here as simple but important examples of ionic equilibria in solution. Other equilibria, such as those resulting from the formation of solventseparated and contact ion pairs, triple-ion formation, multistep equilibria of unsymmetric electrolytes, or simultaneous complex ion formation, in the investigated solution can be treated in a similar way. They require in Eq. (25) the consideration of all identified species in the solution. E. Solvent Activity and Osmotic Coefficient In dilute solutions (xS ∼ = 1) both the activity and the activity coefficient of the solvent differ only very little from unity with varying electrolyte concentration, in contrast

to the related osmotic coeffcients, used as molal osmotic coefficient and the rational osmotic coefficient g:

= −(ln aS )/νm MS

(31a)

and

g = (ln aS )/(ln xS ). (31b) The Gibbs-Duhem equation [Eqs. (14)], when combined with Eqs. (11a) and (19), yields the relationship 1 d(ln aS ) + νm d(ln mγ± ) = 0, (32) MS which can be easily transformed with the help of Eq. (31a) to yield the basic equation for calculating solute activity coefficients from solvent activities and vice versa: d(m ) = dm + m d(ln γ± ).

(33)

Either or ln γ± is obtained by integration of Eq. (33) to yield

1 m

=1+ m d(ln γ± ) (34a) m 0 or

m 1− ln γ± = ( − 1) − dm. (34b) m 0 The peculiar behavior of electrolyte solutions that follow a limiting law of type 1 − ∝ Am 1/2 entailing divergency of the integrand in Eq. (34b) is taken into account by transforming this equation into the relationship

m 1− ln γ± = ( − 1) − 2 dm 1/2 . (35) m 1/2 0 Equations (34) are valid for both completely and incompletely dissociated electrolytes except that for incompletely dissociated electrolytes γ± is replaced by αγ± , the degree of dissociation being related to the association constant in the molal scale (dilute solution, γx = 1): K A(m) =

1−α 1 = K A(c) dS . α 2 m γ± 2

(36)

F. Partial and Apparent Molar Quantities All extensive thermodynamic properties Z (volume, enthalpy, entropy, Gibbs energy, heat capacity) when defined as functions of the set of the variables p, T , and n i , Z = Z ( p, T, n i ),

i = 1, . . . , k,

yield the partial molar (or molal) quantities  ∂Z Zi = ∂n i p,T,n j=i and Z=

k  i=1

ni Z i ,

(37)

(38)

(39)

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which are the contributions per mole of each component Yi to property Z of the multicomponent system at constant temperature and pressure. For a pure phase (onecomponent system) the molar quantity Z i is denoted by the symbol Z i∗ . The solution process of n Y mol of electrolyte Y in n S mol of solvent S is accompanied by a change in property Z :  

sol Z = (n S Z S + n Y Z Y ) − n S Z S∗ + n Y Z Y∗     = n S Z S − Z S∗ + n Y Z Y − Z Y∗ . (40) Another useful formulation of the extensive property Z is based on the definition of apparent molar quantities Z of the solute, which yields Z = n S Z S∗ + n Y Z .

(41)

According to Eq. (38) the partial derivative of Z [Eq. (41)] leads to the partial molar quantities Z S and Z Y of the solvent S and the electrolyte Y,  ∂ Z Z S = Z S∗ + n Y (42a) ∂n S p,T,n Y and

 Z Y = Z + nY

∂ Z ∂n Y

,

(42b)

p,T,n S

which relate the partial molar quantities Z S and Z Y to the apparent molar quantity Z . When the molal scale is used for determining Z as a function of molality, Eqs. (42) are converted to the relationships  MS m 3/2 ∂ Z ∗ ZS = ZS − (43a) 2 ∂m 1/2 p,T and ZY = Z +

 m 1/2 ∂ Z . 2 ∂m 1/2 p,T

(43b)

The advantage of using apparent molar quantities is the direct accessibility of Z from experimental results,

Z = (Z − n S Z S∗ )/n Y [Eq. (41)]. At the limit of infinite dilution, Eqs. (42) and (43) yield Z Y∞ = lim Z Y = lim Z = ∞ Z . n Y →0

FIGURE 1 Operations on electrolyte solutions and their appropriate translation into thermodynamic quantities: Z = Zfin − Zinit . Abbreviations: lat, lattice; solv, solvation; sol, solution; dil, dilution; tr, transfer from solvent S to solvent S. Z = V (volume), S (entropy), C p (heat capacity), H (enthalpy), G (Gibbs energy). [Reprinted with permission from Barthel, J., Gores, H.-J., Schmeer, G., and Wachter, R. (1983). Non-Aqueous Solutions in Chemistry. In “Topics in Current Chemistry” (F. L. Boschke, ed.), Vol. 111, pp. 33–144, Fig. 1. Springer-Verlag, Berlin. Copyright Springer-Verlag 1983.]

n Y →0

(44)

Hence, the balance of an extensive property, sol Z , in a solution process can be expressed by the relationship  

sol Z = sol Z Y∞ + Z − ∞ Z nY

(45a)

where

infinite dilution in the solvent S. The term Z − ∞ Z is often called the relative apparent molar quantity rel . Z Figure 1 summarizes the operations and notations used in solution and dilution processes. Applying the foregoing formalism to the Gibbs energy leads to the standard Gibbs energy of solution, sol G ∞ Y, which is the energy difference between the pure electrolyte component Y and the reference state at infinite dilution: ∞ ∗

sol G ∞ Y ( p, T ) = µY ( p, T ) − µY ( p, T ).

(46a)

According to the rules of thermodynamics, differentiation of sol G ∞ Y yields the molar solution properties

  ∂ sol G ∞ Y ∞ , (46b)

sol SY = − ∂T p,n i

  ∞ ∂

G sol Y

sol VY∞ = , (46c) ∂p T,n i

  ∞ ∞ 2 ∂ sol G Y T

sol HY = −T , (46d) ∂T p,n i

sol Z Y∞ = Z Y∞ − Z Y∗ .

sol Z Y∞

(45b)

The molar quantity corresponds to the transfer of 1 mol of electrolyte compound from its pure state to

and

sol C ∞ pY

  ∂ sol HY∞ = ∂T

. p,n i

(46e)

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Gibbs energies of solution, sol G ∞ Y , are obtained from solubility or electromotive force (emf) measurements,

sol HY∞ from calorimetric measurements, and sol VY∞ from density determinations. The other quantities are obtained through the temperature dependence according to Eqs. (46b) and (46e). G. Temperature and Pressure Dependence of Activity Coefficients Using Eq. (11a), we can write the activity coefficient of a component Yi as     RT − ln xi . ln f i = µi − µ− (47) i Since the mole fraction xi is independent of temperature and pressure, the derivatives of Eq. (47) yield   Hi − Hi− ∂(ln f i ) =− (48a) ∂T RT 2 p,n j and



∂(ln f i ) ∂p



Vi − Vi− . RT 

= T,n j

(48b)

Quantities of type Z i − Z− i are referred to as relative partial molar quantities. Conversion of the mole fraction scale to the molality scale does not change Eqs. (48), whereas the use of the molarity scale requires that the temperature and pressure dependence of molarity be taken into account:  ∂(ln c) = −α (49a) ∂T p,n j 

and



∂(ln c) ∂p

are often unreliable. Many problems, however, can be reduced to the comparison of solvation in different solvents based on the transfer quantities tr Z Y∞ , which are easily obtained from solution properties sol Z Y∞ (see Fig. 1). Gibbs energy of transfer tr G Y is the change in Gibbs energy when the electrolyte component Y is transported from solvent S solvent S. If we choose water as the reference solvent S , the Gibbs energy of transfer at infinite dilution defines the so-called medium activity coefficients m γY: ∞ ∞

tr G ∞ Y = sol G Y (S) − sol G Y (W) = RT ln m γY . (51) Then the Gibbs energy of transfer can be written at any concentration:  

tr G Y = RT ln m γY + RT ln aY(S) aY(W) . (52)

At distribution equilibrium of Y between the two solvents,

tr G Y = 0, the distribution coefficient aY(S) /aY(W) yields the transfer activity coefficient. Separation into ionic medium activity coefficients is executed with the help of the equation (m γY )ν = (m γ+ )ν+ (m γ− )ν− .

(53)

Transfer activity coefficients are used in the field of extraction processes, ionic equilibria and emf measurements, and analytical application. For example, the pH scale is transferred from water to other solvents with the help of transfer proton activity coefficients m γH+ : pH(S) = pH(W) + log m γH+ .

(54)

= β.

(49b)

T,n j

Here α and β are the thermal expansivity coefficient and the compressibility coefficient of the solution, respectively:  1 ∂V α= (50a) V ∂ T p,n j and 1 β=− V



∂V ∂p

.

(50b)

T,n j

H. Thermodynamic Quantities of Transfer Thermodynamic quantities for the solvation process

solv Z Y∞ are formally obtained from the solution and lattice quantities according to Fig. 1. However, lattice data are lacking for many compounds. Furthermore, the transfer of electrolyte from the gas phase to solution involves a rather complicated measurement, and the resulting data

I. Ion Solvation Ion solvation is the transfer of ions from a vacuum to an infinitely dilute solution in a solvent S. In order for us to represent solvation by models, the ion–solvent interactions are split into electrostatic, nonelectrostatic, and chemical contributions. Dielectric continuum models such as the Born model consider the solvent to be a structureless continuum of relative permittivity ε. The Gibbs energy of solvation of an ion, solv G i∞ , is calculated by the difference of the charging process in a vacuum (ε = 1) and in the solvent (ε):



0 NA (z i e0 )2 1 1 ∞

solv G i = λ dλ + λ dλ 4π ε0 ai ε 0 1  1 NA (z i e0 )2 1− =− . (55) 8π ε0 ai ε The integration is carried out with regard to a charging parameter λ, assuming that the ionic radii ai in the vacuum

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and in the solution are equal and the transfer of the uncharged ion does not affect the Gibbs energy of solvation. For an electrolyte component Y, the overall energy is found by summing over the ionic contributions: ∞ ∞

solv G ∞ Y = ν+ solv G + + ν− solv G − .

(56)

The Born model is only a rough approximation. Improvements of the method take into account a local permittivity ε∗ and effective ionic radii ai(eff) = ai + δi , where δi is the distance between an ion and an adjacent solvent dipole. More elaborate models include in the calculation the energy of formation of a spherical cavity in the pure solvent into which an ion and its solvation shells can be transferred from the vacuum. Further interactions that can be taken into account result from ion–quadrupole, ioninduced dipole, dipole–dipole, dispersion, and repulsion forces. For nonaqueous electrolyte solutions most of the molecular and structural data needed for this calculation of the solvation energy are unknown, and ab initio calculations have not so far been very successful. Actual information on ion solvation in nonaqueous solutions is based almost exclusively on semiempirical methods and/or the extrathermodynamic assumptions quoted in Section II.C. Extrapolation methods are based on the representation of the electrostatic ion–solvent interactions as a series expansion of type b j (ai )− j , where ai is one of the ionic radii. For example, in a series of electrolytes with a common cation and differing anions, the relationship ∞ ∞

solv Z Y∞ = solv Z + + solv Z − ∞ = solv Z + +

n 

b j (a− )− j

j=1 ∞ + solv Z − (neutral)

(57)

is used for determining single-cation solvation quantities from the measured solv Z Y∞ values. The extrapolation of a− toward infinite values, a− → ∞, yields the ∞ quantity solv Z + because the electrostatic contribution vanishes for the anion. The nonelectrostatic contribution ∞

solv Z − (neutral) is usually estimated from neutral reference molecules that are similar in size and chemical constitution. Other extrathermodynamic assumption methods are based on reference systems, such as large ion couples and their corresponding uncharged molecules (e.g., ferrocinium/ferrocene) or systems of large cations and anions of similar structure [e.g., (Ph4 As)+ /(BPh4 )− or (iAm3 BuN)+ /(BPh4 )− ]. The reference ion–molecule method assumes that the ion–molecule pair shows equal solvation in the investigated solvent; that is, the electrostatic contribution of the ionic species is neglected, and the nonelectrostatic contribution is considered to be the same

for the molecule and the ion. The reference electrolyte method proposes the partition of the measured electrolyte solvation into equal parts for the ions of the electrolyte; that is, the electrostatic and nonelectrostatic contributions of cations and anions are assumed to be equal. Also, unequal partition of solvation quantities between the Ph4 As+ cation and the BPh− 4 anion is used in the literature.

III. STATISTICAL APPROACHES A. Hamiltonian Models Establishing the relationships between the macroscopic properties of an electrolyte solution and the properties of its particles, ions, and solvent molecules is the task of statistical mechanics. Efficient statistical mechanical models are based on the Hamiltonian H , which expresses the total energy of the N -particle system as a function of the momenta pi and coordinates ri of the particles: H (r1 · · · r N , p1 · · · p N ) =

N N  N   pi2 + u i j (ri , r j ). 2m i=1 i=1 j≥i

(58)

In Eq. (58) limitation is made to models omitting molecular orientation. The first term on the right-hand side of the equation represents the kinetic energy, and the second term the potential energy U N (r1 · · · r N ), approximated by a sum of pairwise direct interaction potentials u i j (ri , r j ). Hamiltonian models are classified according to their level of approximation. The features of Schroedinger (S), Born-Oppenheimer (BO), and McMillan-Mayer (MM) level Hamiltonian models are exemplified in Table I by a solution of NaCl in H2 O. The majority of investigations on electrolyte solutions are carried out at the MM level. BO-Level calculations are a precious tool for Monte Carlo and molecular dynamics simulations as well as for integral equation approaches. However, their importance is widely limited to structural investigations. They, as well as the S-level models, have not yet obtained importance in electrochemical engineering. S-Level quantum-mechanical calculations mainly follow the Car-Parinello ab initio molecular dynamics method. B. Distribution Functions and Mean Force Potential Statistical mechanics when based on Liouville’s theorem yields a hierarchy of equations (BBGKY hierarchy) that makes use of the s-particle distribution function ρ (s) giving the probability of finding s particles, i = 1 . . . s, out of the N particles in the system in the positions r1 · · · rs and

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TABLE I Hamiltonian Models of Electrolyte Solutions Level Schroedinger (S) Born-Oppenheimer (BO) McMillan-Mayer (MM)

Methoda

Variables in the Hamiltonian

Mechanics

Coordinates and momenta of all the nuclei and electrons of all the H, O, Na, and Cl atoms Coordinates and momenta of the water molecules and the Na and Cl ions Coordinates of the Na and Cl ions only

Quantum

Approximating the wave function

Classical

Simulation: MD, MC; perturbation: PY, HNC MD, MC, MSA, PY, HNC, ChM

Classical

a Abbreviations: HNC, hypernetted chain equation; MC, Monte Carlo simulation; MD, molecular dynamics simulation; MSA, mean spherical approximation; PY, Percus-Yevick equation; ChM, chemical model.

with momenta p1 · · · ps , regardless of the position and momentum coordinates of the remaining particles: ρ (s) (r1 · · · rs , p1 · · · ps , t)

(s) =σ ρ(r, p, t) drs+1 · · · dr N dps+1 · · · dp N . (59) In Eq. (59) the coefficient σ takes into account whether identical or different particles are observed. ρ(r, p, t) is the density function of the N -particle system in a 6N dimensional space, the so-called  space. The symbol r stands for r1 · · · r N , and an analogous definition holds for p. The density function ρ(r, p, t) tends toward a timeindependent equilibrium distribution, (s)

ρ (eq) (r, p) =  

exp[−H (r, p)/kT ] , exp[−H (r, p)/kT ] dr dp

(60)

where H (r, p) is the Hamiltonian of Eq. (58), k the Boltzmann constant, and T the temperature. It can be shown with the help of the BBGKY hierarchy of equations that a knowledge of the ρ (1) and ρ (2) functions, together with the pertinent principle of superimposition, is sufficient to produce ρ (3) functions and so on. Molecular distribution functions f (s) (r1 · · · rs ) at equilibrium, indicating the probability of finding s particles in positions r1 , r2 , . . . , rs , are obtained by integrating the functions ρ (s) themselves with respect to the momentum coordinates p1 , p2 , . . . , ps : 

UN σ (s) f (s) (r1 · · · rs ) = (N ) exp − drs+1 · · · dr N Q kT (61) and 

UN Q (N ) = exp − dr. (62) kT The distribution function of interest is the pair correlation function gi j (ri , r j ), which is defined by the relationship gi j (ri , r j ) =

f (2) (ri , r j ) . f (1) (r j )

f (1) (ri )

(63)

The one-particle distribution function is independent of particle position and equals the number of particles per unit of volume (particle density): f (1) (ri ) = ρi = Ni /V.

(64)

The expression for gi j (r ) is easily obtained from Eqs. (61) and (63). The equilibrium distribution around a spherically symmetric ion depends only on the distance r between particles i and j, r = |ri j | = |r j − ri |:  2  exp(−U N /kT ) dr3 · · · dr N gi j (r ) = V . (65) exp(−U N /kT ) dr1 · · · dr N Values of gi j (r ) greater than 1 indicate attraction, and less than 1 repulsion, between a pair of particles. The particles, when represented as hard spheres of mutual minimum distance a = ai j , require the boundary condition lim gi j (r ) = 0,

r →a

lim gi j (r ) = 1.

r →∞

(66)

A related function of fundamental interest is the potential of mean (or average) force, which is defined such that its derivative is the force on the selected particle i averaged over the position coordinates of all the remaining particles in the system except those of particle j: ∂ Wi j = K i j (r ) ∂r  (∂U N /∂r ) exp(−U N /kT ) dr3 · · · dr N  = . (67) exp(−U N /kT ) dr3 · · · dr N Comparison of Eqs. (65) and (67) yields the link between functions gi j (r ) and Wi j (r ), Wi j (r ) = −kT ln gi j (r ),

(68)

defining the potential of mean force, which is different from the interaction potential u i j (r ) [Eq. (58)] as a consequence of the averaging procedure by Eq. (67). The pair correlation functions gi j (r ) can be experimentally determined. They are obtained by using scattering techniques such as X-ray or neutron scattering.

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Wide-angle neutron scattering yields them at the BO-level, whereas small-angle neutron scattering usually cannot detect the structure of the solvent and therefore may be regarded as the experimental analogue of the theoretical MM approach. The vanishing scattering angle in a so-called contrast experiment is even directly related to the osmotic coefficient. There are two usual routes for determining the thermodynamic properties of the N -particle system in equilibrium from its pair correlation function gi j (r ): 1. The pressure or virial equation:

 ∂u i j (r ) P ρi ρ j 4πr 3 =1− gi j (r ) dr. (69) ρkT ∂r i j 2. The energy equation:

E 3 1  = ρkT + ρi ρ j 4πr 2 V 2 2 i j   ∂u i j (r ) gi j (r ) dr, × u i j (r ) − (70) ∂(ln T )  where ρ = ρi , P is the pressure, E the internal energy, V the volume, and the sum is over all particles of interest in the system. Another, less frequently used relationship, the so-called compressibility equation, is an alternative to the pressure equation, but the virial equation is more convenient to use. C. McMillan-Mayer Solution Theory The evaluation of the pair correlation functions of both the solvent molecules and the solutes is feasible but troublesome. For electrolyte solutions, however, averaging over the solvent effects yields reliable approximations, as shown by McMillan and Mayer, who considered the solution in osmotic equilibrium with the pure solvent (Fig. 2). They stressed that solutes can be treated as an imperfect gas, provided that one uses the potential of mean force at infinite dilution. The calculation yields the osmotic pressure  = P − P0 (see Fig. 2) from the virial equation [Eq. (69)] in terms of the forces among the particles for the solution state (ρ, T ). The independent variables of

FIGURE 2 Osmotic equilibrium of a solution and its pure solvent; the separation M is permeable only to the solvent molecules.

the MM system are then (ρ, T, P0 ) and the solution is under pressure P =  + P0 . The osmotic coefficient in the MM system is found to be

= /ρkT.

(71)

Thermodynamic functions, however, are defined with the help of the independent variables (m, T, P1 ), where P1 is generally the standard pressure of 1 atm. Hence it is necessary to convert experimental data to the MM system when comparison is needed for model calculations at this level. For dilute solutions the resulting corrections are negligible. At the MM level any plausible model for the ion–ion interactions can be described by the direct potential u i j , which can be interpreted as the potential of mean force at infinite dilution: u i j (r ) = u i∗j (r ) + z i z j e02 4π ε0 εr. (72) In Eq. (72) the second term on the right-hand side is a coulombic potential, ε0 is the permittivity of vacuum, z i e0 and z j e0 are the charges on ions i and j, and ε is the relative permittivity of pure solvent. The first term, u i∗j (r ), is the short-range noncoulombic part of the direct potential, including repulsion, dispersion, or induction forces. In analogy to Eq. (72) the mean force potential, related to the pair correlation function at finite concentration by Eq. (68), can be split into two parts, Wi j (r ) = Wi∗j (r ) + Wielj (r ),

(73)

where Wielj (r ) and Wi∗j (r ) are again the contributions due to coulombic and noncoulombic interactions, respectively. D. The Chemical Model 1. Fundamentals The chemical model is an MM-level Hamiltonian model taking into account both long- and short-range forces. It has been used for investigating and calculating the properties of dilute electrolyte solutions of numerous salts in a great variety of solvents and has proved to be successful insomuch as all of the properties of an electrolyte solution investigated can be understood from the same set of interaction parameters. The chemical model subdivides the space around an ion into three main regions (Fig. 3): 1. r ≤ a, where a is the minimum distance of two oppositely charged ions, which is assumed to be the sum of effective cation and anion radii, a = a+ + a− . 2. a ≤ r ≤ R, the region of short-range interactions. For dilute solutions this region is occupied almost exclusively by paired states of oppositely charged ions.

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pair correlation functions by means of structure factors— is an integral of the type

∞ r 2 [gi j (r ) − 1] dr 0



R

= a

r 2 [gi j (r ) − 1] dr



∞

+

r 2 [gi j (r ) − 1] dr.

(75)

R

2. Ion-Pair Concept

FIGURE 3 Chemical model of electrolyte solutions. O, Observer: i , ion Xi ; j , ion X j in an arbitrary position r21 with regard to the ion Xi ; special positions (contact, separation by one or two oriented solvent molecules) are sketched with dashed lines; r, a, R, distance parameters; Wi j , mean force potentials; v j i and vi j relative velocities of ions Xi and X j .

3. r ≥ R, the region of long-range interactions. Region 1 is controlled by a hard-sphere potential. The po(II) tentials of mean force of regions 2 and 3, Wi(I) j and Wi j , are split into two parts according to Eq. (73). The electrostatic parts Wielj are obtained as solutions of PoissonBoltzmann differential equations and appropriate boundary conditions; the nonelectrostatic parts can be chosen ∗(II) as step potentials, that is, Wi∗(I) = 0. The j = const, Wi j total mean force potentials are compiled in Table II. The parameter κ is the reciprocal Debye length as defined by the relationship e02  2 κ2 = (74) ρi z i . ε0 εkT The link between the pair correlation function gi j and the experimentally determined property of the solution—with the exception of scattering experiments, which yield the TABLE II Mean Force Potential for the Chemical Model Region r ≤a a≤r ≤ R r≥R

Mean force potential ∞ e02 z i z j e02 z i z j κ − + Wi∗j 4π ε0 εr 4π ε0 ε 1 + κ R e02 z i z j 1 exp[κ(R − r )] 4π ε0 ε r 1 + κR

Solution chemists usually think of short-range interactions in terms of ion-pair formation. Models of the electrolyte solution allow the introduction of the ion-pair concept if a critical distance around the central ion can be defined within which the paired states of oppositely charged ions are considered ion pairs. The ion-pair concept for symmetric electrolytes can be easily introduced into the chemical model, assuming that the cutoff distance parameter R of the short-range forces equals the upper limit of ion association. The relationship (see Table II) 2κqκ T 2κqκ T (I) ∗ W+− + + W+− (r ) = − ; r 1 + κR q=

e02 |z + z − | 8π ε0 εκ T

(76)

yields the integral (a . . . R) of Eq. (75) in the form

  

R (I) W 2κq i j 2 r exp − dr = exp − kT 1 + κR a

× a

R

 ∗ W 2q +− − dr. r 2 exp r κT

(77)

At low electrolyte concentrations the activity coefficient of ion pairs may be set equal to unity and Eq. (77) is linked to the ion-pair association constant K A and the mean activity coefficient y± of the free ions by the relationships 1−α 1 = K A = 4000π NA α 2 c y± 2



R ∗ W+− 2q 2 × r exp − dr (78a) r kT a and

 y± = exp −

κq . 1 + κR

(78b)

The distance parameter R can either be determined by experiment or set by chemical evidence; α is the degree of dissociation.

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The chemical association concept is based on the equilibrium between “free” ions and ion pairs in the solution (see Section II.D). Equation (78a) can also be derived as an equilibrium constant in the framework of the statistical mechanical equilibrium theory based on the molecular partition functions Q + , Q − , and Q P of cations, anions, and ion pairs,  QP − E 0 K A = NA V (79a) exp Q+ Q− κT and (2π m i kT )3/2 V, (79b) h3 when the cations and anions are represented as charged hard spheres of masses m + and m − without internal int degrees-of-freedom Q int + = Q − = 1, and the internal molecular partition function of the ion pair is made up by integration over all configurations a ≤ r ≤ R of the paired states of a cation and an anion with variable distance pa∗ rameter r . E 0 = 2q/r − W+− /kT is the corresponding distance-depending activation energy per molecule of the reaction of ion-pair formation. The internal partition function thereby is based on an internal degree of translation of a particle of reduced mass m˜ = m + m − /(m + + m − ) representing the ion pair P(r ). Figure 4 shows the family tree of some theories which assume association. For example, the often-used Bjerrum Q i = Q itrans Q iint ;

Q itrans =

association constant and its appropriate activity coefficient ∗ are obtained by setting R = q and W+− = 0 in Eq. (78a). 3. Debye-Huckel ¨ Theory and Extensions A multitude of MM-level models can be found in the literature differing from one another by the underlying assumptions of short-range interactions. The oldest one is the Debye-H¨uckel theory, which does not recognize shortrange interactions and association. The Debye-H¨uckel theory yields the limiting laws of thermodynamic and transport properties based on the potential of mean force, Wi j (r ) =

e02 z i z j exp(−κr ) ; 4π ε0 ε r

0 < r < ∞,

(80)

which can be deduced from the data in Table II by assuming a = R = 0 and Wi∗j = 0. By adding pairwise short-range interactions β+− between cation and anion, Guggenheim obtained an activity coefficient which for 1,1 electrolytes is given by the relationship (dS : solvent density) √ Aγ m ln γ± = − √ + 2β+− m; 1+ m 3/2

e02 1/2 Aγ = (2π NA dS ) . (81a,b) 4π ε0 εkT

FIGURE 4 Family tree of association constants of the chemical model. For explanations see the text.

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It may be used in aqueous solutions up to molalities of 0.1 mol/kg. Among a variety of extensions of the Guggenheim equation, those of Pitzer are the most successful ones. For a single electrolyte in solution they read ln γ± = f (γ ) + m B (γ ) + m 2 C (γ ) ,

f

(γ )

√ Aγ I =− √ ; 3 1+b I

I =

ξ (i) =

(i)

2β αi2 I



√

1 m i z i2 , 2 i

1 − exp(−αi I ) 1 + αi

(82b,c)

s

√

I−

αi2 I 2

The parameters αi , b, β (0) , β (1) , β (2) , and C (γ ) must be fitted to experimental data. It turns out that αi and b are common parameters for large classes of electrolyte solutions. The short-range forces are introduced by Guggenheim’s concept. The term ξ (2) is relevant only at significant ion association. It is possible to correlate its numerical values with association constants stemming from calculations similar to those of the chemical model. Pitzer’s equations can be used for mixtures of electrolyes. Thermodynamic functions are obtained in the usual way as the derivatives of the chemical potential with respect to temperature or pressure. However, a considerable number of empirically adjusted parameters is needed to obtain satisfactory data description. The Pitzer approach is used as a self-standing data-reduction method, but it is also embedded by engineers in the so-called NRTL (nonrandom two liquid) electrolyte models. E. Integral Equation Methods

cis (r )h s j (r ) drs . (84)

0

u i j (r ) + gi j (r ) − 1 kT − ln gi j (r ) + Bi j (r ).

ci j (r ) = −

 .

∞

When used in connection with integral equations, it is considered the definition for the direct correlation function ci j . It can be shown by cluster expansion that the direct correlation function has the following closed analytical expression:

(82d)

(82e)

(85)

In Eq. (85), Bi j (r ) is the sum of the so-called bridge graphs. The diagrammatic representations and the interrelations of the correlation functions can be found in textbooks on liquid-state chemical physics. The integral equations themselves are obtained by inserting various approximations to the direct correlation function ci j (r ) into the Ornstein-Zernicke equation. It is common practice to omit the bridge graphs Bi j in these approximations. Once the integral equation is solved together with the Ornstein-Zernicke equation for a given potential u i j (r ), equations of, the type of Eqs. (69) and (70) may be used to obtain thermodynamic properties from the resulting gi j (r ) functions. At the MM-level, Eq. (69) yields the osmotic pressure  instead of the total pressure p of the solution. 2. Mean Spherical Approximation The lowest level of the integral equation approach treats the ions as charged hard spheres embedded in a dielectric continuum. It fulfills the conditions ci j (r ) = −z i z j e02 4π ε0 εkT for r > ai j (86a) and h i j (r ) = 1

1. Direct Correlation Function A quite different approach to thermodynamics of ionic solutions consists of solving the integral equations that relate the correlation functions and the pair potentials. For this purpose two new correlation functions are introduced: the total and the direct correlation functions. The total correlation function h i j (r ) is simply defined by the relationship h i j (r ) = gi j (r ) − 1.

h i j (r ) = gi j (r ) − 1 

= ci j (r ) + ρs

(82a)

B (γ ) = 2β (0) + ξ (1) + ξ (2) , and

ticle. This feature is reflected by the Ornstein-Zernicke equation:

(83)

It is made up of two contributions: the direct correlation ci j (r ) and the correlations transmitted through a third par-

for r ≤ ai j .

(86b)

Equation (86a) means that all short-range potential contributions u i∗j are neglected beyond the contact distance of ion i and ion j, ai j [u i∗j (r ) = 0 for r > ai j ; c.f. Eq. (72)], and that in Eq. (85) only the first term on the right-hand side is considered at larger distances. Equation (86b) means that the ions are hard spheres with infinitely large repulsion potential up to contact distance ai j [u i∗j (r ) = ∞ for r ≤ ai j ]. These conditions together with the Ornstein-Zernicke equation allow the calculation of ci j (r ) at short distances and gi j (r ) at long distances. This approximation is called

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mean spherical approximation (MSA). Because of its simplicity, obtaining analytical expressions for osmotic and activity coefficients is possible. For the restricted primitive model, which simply considers cations and anions as charged hard spheres of equal diameter a, these expressions are

=

1 + η + η2 − η3 3 − (1 − η)3 3πρ

(87)

(1 + 2η)2 2q − . (1 − η)4 1 + a

(88)

and ln γ± = ln

In each of these equations the first term on the right-hand side is the Carnahan-Starling expression for the interaction of the uncharged hard spheres, where π η = ρa 3 . (89) 6 It is superimposed to the electrostatic MSA contribution given by the second term, where q is given by Eq. (76). The key function  is the electrostatic screening parameter 1 √ 2 = [ 1 + 2κa − 1], (90) a where κ is the reciprocal Debye length [Eq. (74)]. MSA can be considered as an extension of the Debye-H¨uckel theory that correctly takes into account ion sizes. It yields satisfactory results for solutions of small to moderate ion densities and low coupling strength commonly expressed with the Bjerrum parameter, b = 2q/a, which is the ratio of the coulombic potential to the thermal energy of two charged hard spheres at contact distance a. The coupling strength is low at low ion charges or high ion concentrations, high temperature, or high solvent permittivity. In a more developed form for unequal ionic diameters and arbitrary charges, MSA yields analytical expressions even for mixtures. Figure 5 shows the efficiency of MSA for describing experimental data up to molar concentrations even in solvents of moderate permittivity by the use of only one adjusted parameter. A combination of the hard-sphere MSA concept and the chemical model (Section III.D) concept permits correct data reproduction over large concentration ranges, which makes this type of MSA a good candidate for the data reduction in practical problems treated by engineers. It may be expected that in the near future MSA can replace more and more the semiempirical methods commonly installed in process simulators. Despite its progress, MSA has some deficiencies. It sometimes yields unrealistic correlation functions for equally charged particles, and it seems doubtful whether, beyond its practical importance, structural information

FIGURE 5 Comparison of experimental (points) and calculated (lines) osmotic ( ) and activity (y± ) coefficients of NaI in (a) methanol and (b) acetonitrile at 25◦ C. The experimental data stem from vapor pressure measurements. MSA calculations are executed with one-parameter fits.

can be obtained. Because it considers only charged hard spheres at the MM level, the temperature dependence of u i j (r ) at short distances is not correctly taken into account, so that the internal excess energy is only poorly reproduced. 3. Hypernetted Chain and Percus-Yevick Approximations From a theoretical point of view the hypernetted chain (HNC) and Percus-Yevick (PY) equations are better approximations. Although both can be solved only by numerical methods, they offer the opportunity to study any model potential if appropriate computer facilities are available. The HNC approximation consists of setting Bi j (r ) equal to 0 in Eq. (85): ciHNC (r ) = − j

u i j (r ) + gi j (r ) − 1 − ln gi j (r ). kT

(91)

The PY equation can be obtained from Eq. (85) by dropping Bi j (r ) and linearizing the logarithmic function:    u i j (r ) ciPY (r ) = g (r ) 1− exp . (92) ij j kT

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When applied to the primitive model of electrolyte solutions (i.e., charged hard spheres of arbitrary diameters in a dielectric continuum), the HNC equation is superior to the PY equation because it preserves the correct long-range behavior. On the other hand, in fluids with only shortrange forces the PY equation can be successfully applied because some of the omitted terms cancel one another. A widely used, more refined potential was introduced by Friedman, who split the short-range part into three contributions: u i∗j (r ) = CORi j (r ) + CAVi j (r ) + GURi j (r ).

(93)

1. The repulsive term CORi j (r ) is deduced from crystal data of the electrolyte and is either of the type CORi j (r ) =

FM e02 (ai j /r )9 , 36πε0 ai j

(94a)

where FM is the ratio of Madelung’s constant to the coordination number, or of type   CORi j (r ) = kB ∗ exp (ai j − r ) R ∗ , (94b) where B ∗ and R ∗ are characteristic parameters for each salt. The COR term reflects the dominant effect at short distances. 2. The term CAVi j (r ) takes into account the effect of the local permittivity εc in the ionic cavities, usually set to the square of the refractive index of the pure solvent. This contribution, which is less important than the other terms, is defined as   [(ε − εc )/(2ε + εc )] ei2 a 3j + e2j ai3 CAVi j (r ) = . (95) 8πε0 εr 4 3. The contribution GURi j (r ) results from the overlap of the solvation spheres (Gurney spheres) if two ions have approached each other to distances less than the radii of their solvation spheres: GURi j (r ) = Ai j Vmu (ai + w, a j + w, r )/Vw

(96)

and

 2(a 3 + b3 ) a 2 − b2 Vmu (a, b, r ) = π − + 4r 3  r (a 2 + b2 ) r3 − + . 2 12

FIGURE 6 Ion-pair correlation functions gi j (r ) obtained from HNC calculations based on experimental small-angle neutron scattering spectra of a 0.3-M solution of n-tetrabutylammonium bromide in D2 O at 25◦ C.

(97)

In Eq. (96) Ai j is the molar free energy change if cosphere solvent is transferred from the solvate state to the bulk solvent, and Vw is the molar volume of the pure solvent in the same units as the overlap volume Vmu . The overlap volume is a function of the distance r , the ionic radii ai , and the cosphere radius w, which is usually calculated

on the base of one or two solvent molecules. It can be obtained by geometric considerations. The absolute value of the GURi j term may be of the same order of magnitude as that of the coulombic interaction term. Friedman’s model fits the thermodynamic excess functions (osmotic coefficient, excess volume, and excess energy) of aqueous solutions of alkali and earth alkaline halides up to 1-M ionic strength and of tetraalkylammonium halides up to 0.4 M. The variation of the Ai j parameters with the ionic parameters is chemically meaningful and permits the estimation of thermodynamic properties of unknown systems by the combination of the parameters known from appropriate other systems. Figure 6 shows pair correlation functions from HNC calculations of an aqueous tetraalkyl-ammonium salt solution. In contrast to the MSA calculations, the HNC calculations yield gi j (r ) functions that give a realistic picture of the solution structure. It can be inferred that no noticeable cation–anion association takes place, in agreement with chemical model calculations at lower concentrations. Also, no cation–cation association exists, which indicates that no hydrophobic interaction occurs between the large organic ions. Interpenetration of the cations is found, which can be reproduced because of the soft-sphere COR++ (r ) potential in Eq. (93). F. Semiempirical Equations In the field of chemical engineering there is a need for equations allowing the description of thermodynamic data of electrolyte solutions or their prediction over large temperature and concentration ranges. In contrast to the situation for nonelectrolyte solutions, the rigorous theoretical approaches cannot yet yield such equations with a reasonable amount of computing effort. A few pragmatic MSA

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modifications show promising aspects in the description of vapor pressures up to very high concentrations. Also, the combination of a pragmatic BO- and MM-level description was successfully applied (e.g., for the description of multicomponent mixtures based on the parameters of its constituting binary mixtures without any additional adjusted parameter). It is common to these pragmatic methods to use the permittivity of the solution as an empirical function of the salt concentration, or to vary the size of the solvated ions with salt concentration. Modified integral equation approaches have yet to enter into industrial process simulations and data descriptions, in contrast to the empirical extensions of the DebyeH¨uckel theory (see Section III.D.3). G. Other Approximation Methods Two computer simulation methods are widely used for ionic solutions: the Monte Carlo (MC) method and molecular dynamics (MD). The MC method consists of generating a set of molecular configurations by random displacements of the N particles in the model and is basically a multidimensional integration procedure that evaluates the integrals for the calculation of the pair correlation function [Eq. (65)] directly. It can be used to calculate the equilibrium properties of a system. In the MD method the classical Newtonian equations of motion of an assembly of particles are solved numerically and integrated to yield the evolution of the configurational and velocity distributions of the system. It is used to calculate both equilibrium and transport properties. Although the progress in computer science is continuously increasing the possibilities of computer simulations, it is currently not possible to use either method to deal with systems containing more than a few particles in a simulation box of reasonable dimensions. The simulation box is still much smaller than the range of the far-reaching coulombic interactions. Therefore care must be taken to avoid size and wall effects by the use of periodic boundary conditions and lattice methods such as the Ewald sum and so forth. Both MC and MD simulation, can be applied to MM and BO Hamiltonian models of electrolyte solutions. MD at the MM level is known as Brownian dynamics simulation. It has gained some importance for the study of large ions in solution. At the BO level only concentrated solutions can be considered due to the restricted number of solvent molecules per number of ions in the simulation box. Although thermodynamic properties can be inferred from computer simulations, these methods mainly aim at a realistic picture of the structure and, in the case of MD, of the dynamics of the solutions.

IV. THERMODYNAMIC PROPERTIES OF ELECTROLYTE SOLUTIONS A. Generalities Experimental investigations of thermodynamic properties are of importance for both examining theories and providing data for technology. The limiting values of the properties at zero concentration are of crucial interest; they yield the standard values of the solution. However, ideality of electrolyte solutions, y± = 1, appropriate for such extrapolations, would require experiments at such low concentrations that their execution would be either useless as a consequence of low precision or even impossible. Reliable investigations in the low-concentration region use computer-assisted data analysis of the experimental results based on appropriate statistical thermodynamic models. The chemical model (Section III.D) provides property equations of type E(c; p, T ) = E ∞ ( p, T ) + E (αc; R; p, T ), 1−α = KA, α 2 cy± 2 and − ln y± =

(98) (99)

κq , 1 + κR

(100)

where E(c; p, T ) is the property investigated and E ∞ its limiting value at zero concentration. The basic model parameters R (cutoff distance of the short-range interactions or upper limit of association) and K A (association constant) are found independent of the special thermodynamic or transport property being investigated. This is a feature of reliable Hamiltonian models at the MM level. Thermodynamic properties yield a set of values R, K A , and E ∞ , generally obtained by three-parameter least-squares fits based on statistical methods known to produce reliable data. Transport properties of dilute solutions yield consistent data for K A when R is set to the value determined by thermodynamic property equations. For example, conductance data yield sets of ∞ and K A , at every temperature and pressure. Limiting values of ionpair formation, HA∞ and VA∞ , can be determined with the help of the relationships  ∂(ln K A )

HA∞ = RT 2 (101) ∂T p and

VA∞ = −RT



∂(ln K A ) ∂p

(102) T

when temperature- and pressure-dependent conductance data are available.

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TABLE III Thermodynamic Data of Ion-Pair Formation in Aqueous Solutions KA (dm3 mol−1 )

∆H ∞ A (J mol−1 )

Rexp (nm)

Rcalc (nm)

MgSOa4 CaSOa4

161

5780

0.93

0.88

192

5570

0.95

0.92

CdSOa4

239

8390

0.96

0.91

CdSOb4

245

7900

—

0.91

Electrolyte

a

From heats of dilution. b From temperature dependence of conductance.

A comprehensive investigation of solution properties in various solvents and with a multitude of electrolytes has yielded an important result. Distance parameters can be obtained for all property equations from chemical evidence: R = a + ns, where a is the center-to-center distance of closest approach of cation and anion in the solution, s the dimension of an oriented solvent molecule, and n = 0, 1, or 2. An example is given in Table III, where Rexp is the distance parameter from heat of dilution measurements and Rcalc the quantity calculated from a configuration M 2+ (H2 O)2 SO2− 4 . B. Electrolyte Solution Data Collections and Databases Critically revised data of various electrolyte solution properties help scientists and engineers to overcome the time-consuming procedure of searching for reliable data for technical applications. Special knowledge-based databases undertake the interpolation, estimation, or simulation of data by theory-founded procedures differing fundamentally from those for nonelectrolytes. The reason for the difference is the essentially different reference states of electrolyte solutions which are the infinitely dilute solutions with at least three interacting components, namely solvent molecules, cations, and anions. In contrast, databases for nonelectrolytes always use the pure substances as the references. C. Vapor Pressure Measurements Vapor pressure measurements yield the activity and/or osmotic coefficients of electrolyte solutions. The equi(g) librium condition µ(l) S ( p, T ) = µS ( p, T ) for the solvent in the liquid (l) phase and gaseous (g) phase yields the relationship

 ∗(g)  ∗(l)  1 ∂ µS − µS d ln aS = dp RT ∂p T

=

∗(g) VS

− RT

VS∗(l)

dp

(103)

if the solute is a nonvolatile compound such as a salt. The symbols are explained in Section II.A. The integration of Eq. (103) gives the solvent activity aS as a function of the vapor pressure p of the solution at temperature T : 

p VS∗(l) − BS ln aS = ln ∗ + ( p ∗ − p) p RT p ∼ = ln ∗ . p

(104)

In Eq. (104) p ∗ is the vapor pressure of the pure solvent. The nonideality of the gas phase is taken into account with the help of the second virial coefficient of the solvent vapor; VS∗(l) is the molar volume of the pure solvent in the liquid phase. Neglect of the nonideality and of the pressure dependence of the activity yields the well-known approximation aS = p/ p ∗ . Vapor pressure measurements can be carried out by static, dynamic, or isopiestic methods. Static methods are based on direct measurements with the help of manometers. They require thorough outgassing of the solution, prevention of vapor condensation in any part of the measuring device, and highly precise temperature constancy. Differential measurements on solution and pure solvent are advantageous. The dynamic boiling point method (ebulliometry) studies the solution in a steady-state equilibrium of boiling and condensing under a constant pressure of an inert gas. The difference in boiling points of the solution and the pure solvent, T − T ∗ , at pressure p, ln aS =

vap H (T − T ∗ ), RT T ∗

(105)

where vap H is the enthalpy of vaporization, is converted to a change in pressure with the help of the Clausius-Clapeyron equation. The boiling point method at constant pressure and the static vapor pressure method at constant temperature are equivalent methods. The dynamic gas saturation method is based on the determination by weight of the amount of solvent transported by a current of an initially dry inert gas after passage through the solution where the gas is saturated. The content of solvent per unit volume of gas is proportional to the partial pressure of the solvent. The isopiestic method is a highly accurate but timeconsuming relative method. In a constant-temperature enclosure the solvent distills isothermally from a reference solution of known activity to a solution of unknown activity or vice versa, which entails a change in concentration. At equilibrium the solvent activities of both samples are equal and (R denotes reference solution)

= (νR m R /νm) R .

(106)

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The accuracy of the method depends on the long-time temperature stability and the accuracy of the reference data, usually those of aqueous KCl, NaCl, or H2 SO4 solutions. The method for obtaining osmotic coefficients of electrolytes from solvent activities is shown in Section II.E. D. Osmotic Pressure Measurements Solvent activities can also be obtained from measurements of osmotic pressures. The osmotic equilibrium µS ( p + , T ) = µ∗ ( p, T ) across a solvent-permeable membrane separating solvent and solution is maintained by the osmotic pressure . The osmotic equilibrium condition, where

p+ µS ( p + , T ) = µS ( p, T ) + VS d p p

= µS ( p, T ) + VS ,

(107)

yields the desired relationship ln aS = −VS /RT.

E. Molar Volumes Molar volumes of solute and solvent, VY and VS , are determined from apparent molar volumes V (Section II.F) by density measurements. In terms of the measured densities of the solution and the pure solvent, d and d ∗ , the apparent molar volume is given by (molarity and molality scale, respectively) d∗ − d MY d∗ − d MY + = + . 1000cd ∗ d∗ mdd ∗ d

(111)

exhibits a contribution that is inversely proportional to molality, thus tending toward infinity if molality decreases to zero. Assuming precise measurements of 0.001% relative error, both in density and in molality for aqueous NaCl solutions, the rapidly increasing error at low concentrations in

V is caused entirely by the second term on the right-hand side of Eq. (111);

V = 0.5 cm3 mol−1 at m = 0.01 mol kg−1 and

V = 5 cm3 mol−1 at m = 0.001 mol kg−1 . Extrapolation with the help of inappropriate methods would yield seriously erroneous values of ∞ V . This example illustrates the necessity of using theoretically well founded extrapolation methods. Thermodynamics yields the relationship

 ν RT m ∂(ln γ± ) ∞

V = V + dm, (112) m 0 ∂p T from which the limiting law of apparent molar volumes

(108)

Equation (108) is an approximation assuming a pressureindependent molar volume of the solvent. In practice, this method is limited to large solute particles such as polymers or biocolloids since it is difficult to find an impermeable membrane for small ions.

V =

     m   d  MY 1  − V  +  − −

V =  V ∗ d m md d

(109)

Then the molar volumes VY and VS are obtained with the help of the relationship (molality scale; see Section II.F)  MS m 3/2 ∂ V ∗ VS = VS − (110a) 2 ∂m 1/2 p,T

1/2

V = ∞ , V + SV c

1/2  NA2 e03 |z i z j | 1000 νi z i2 SV = ε 3 RT 4π ε03   ∂(ln ε) β × − , ∂p 3 T

(113a)

(113b)

is obtained by introducing the activity coefficient of the infinitely dilute solution (κ R  1) of statistical thermodynamics. Equations (113) are valid for a completely dissociated electrolyte. The treatment of partially associated electrolytes subdivides the space around every ion into a region populated by ion pairs (a ≤ r ≤ R) and a region of free ions (r ≤ R) (see Section III.D); the volume V of the electrolyte solution is then given by the relationship V = n S VS∗ + n Y [α V + (1 − α)VA ],

(114)

(110b)

where VA is the concentration-independent molar volume of the ion pair. Equation (114) can be transformed to yield the density equation for associated electrolytes:   ∗ ∗ 3/2 d = d ∗ + M Y − ∞ V d αc − SV d (αc)   + MY − VA d ∗ K A y± 2 (αc)2 . (115)

The main problem in calculating molar volumes from density measurements is the difficulty of measuring reliable densities at low electrolyte molalities. An estimation of the error of apparent molar volumes

Equation (115) is a property equation of the type given by Eq. (98) and is consequently combined with Eqs. (99) and (100) for data analysis. Even measurements of the highest obtainable precision (σd < 10−7 g cm−3 ) are not sufficiently precise for the simultaneous evaluation of

and VY = V +

 m 1/2 ∂ V . 2 ∂m 1/2 p,T

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Partial molar volumes of ions are discussed in terms of ion solvent interactions, that is, ∞ ∞ ∞ ∞ = Vcryst + Velect + Vdisorder + Vcaged . Vion

FIGURE 7 Apparent molar volume of aqueous CdSO4 solutions at 25◦ C in the conventional c1/2 plot. The full line (1) is calculated from the experimental data [10−2 < c (mol dm−3 ) < 5 × 10−2 ) by the help of the chemical model; (2) is the limiting law; the dashed lines (3) indicate the limits of error on

resulting from a precision of σd = 4 × 10−6 g cm−3 in density measurements. [Reprinted with permission from Schwitzgebel, G., Luhrs, ¨ C., and Barthel, J. (1980). Ber Bunsen-Ges. Phys. Chem. 84, 1220–1224.]

∞ V , A , K A , and R; K A and R must be known from independent measurements (e.g., conductance or heat of dilution measurements). Figure 7 shows the apparent molar volume of CdSO4 in water and the extrapolation to zero concentration based on Eq. (115). The quantity

VA = VA − ∞ V is the molar reaction volume of ionpair formation; it is related to the pressure derivative of the association constant of Eq. (102) and can also be obtained from the pressure dependence of the association constant of conductance measurements at various pressures, for example, VA (CdSO4 ) = 9.4 cm3 mol−1 from density measurements and 11.7 cm3 mol−1 from conductance measurements, which is fairly good agreement when the mutual limits of error are taken into account. z+ z− The molar volumes of electrolytes Y = [Cν+ Aν− ] at ∞ ∞ infinite dilution, VY or V , do not depend on ion– ion, but only on ion–solvent interactions and therefore should be additively composed of the ionic quantities (i.e., VY∞ = ν+ V+∞ + ν− V−∞ ). The partition of the VY∞ values into their ionic parts can be executed with the help of extrathermodynamic assumptions (see Section II.C). The assumption that the ratio of the molar ionic volumes in aqueous solutions, V+∞ /V−∞ , of Ph4 AsBPh4 equals the ra3 3 tio a+ /a− (a+ , a− : ionic radii) yields VH∞+ (aq) = −4.1 cm3 mol−1 ; equating the ratio to that of the ionic vander Waals volumes yields a value of −5.5 cm3 mol−1 ; and the use of a series of ions of various sizes yields VH∞+ (aq) = −4.5 cm3 mol−1 . Ionic vibration potential measurements, the only method for a direct determination of ionic molar volumes, yields VH∞+ (aq) = −5.4 cm3 mol−1 .

(116)

It is assumed that the “nonsolvated” part Vcryst , the crystal volume, is calculated from ionic crystallographic radii, ∞ Velect , the electrostrictive contribution, is due to solvent molecules adjacent to the ion, which are more closely packed than those in the bulk solvent. Highly structured solvents such as water exhibit a region of disordered solvent between the electrostricted region and the bulk sol∞ vent which gives rise to the Vdisorder term in the molar volume equation for an ion. Ions with large hydrophobic surfaces such as tertaalkylammonium ions in effect increase hydrogen bonding in the adjacent water structure, ∞ which entails a further contribution, Vcaged . If this effect prevails, the ion is a structure-making ion, in contrast to ∞ the structure-breaking ions with a large Vdisorder term. F. Heats of Solution and Dilution Heats of solution and dilution, sol HY and dil HY , respectively, are determined with the help of calorimetric measurements. According to Fig. 1 and Eqs. (45) the following relationships can be written:

sol H = sol HY∞ − dil HY∞ , nY ∞

dil HY∞ = − rel H = H − H .

(117)

Thermodynamics yields the relationship

m ∂(ln γ± ) rel 2 1

H = −ν RT dm, m 0 ∂T p

(118)

from which the limiting law of the heat of dilution of completely dissociated electrolytes 1/2

rel , H = SH c

(119a)

 NA2 e03 |z i z j |T 1000 νi z i2 SH = − ε 3 RT 4π ε03   α 1 ∂(ln ε) p+ + , × ∂T 3 T

1/2

(119b)

is obtained with the help of the activity coefficient of the infinitely dilute solution (κ R  1). Associated electrolytes are treated in the usual way by superimposition of the contributions of the free ions and rel rel ion pairs, rel H (FI) and H (IP); the quantity H (IP) is ∞ the molar enthalpy of ion-pair formation, HA , from the initially infinitely separated cation and anion:

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rel H = α H (FI) + (1 − α) H (IP), ∞

rel H (IP) = HA .

G. Electromotive Force (120)

Heats of dilution dil HY∞ are determined by the measurements of intermediate heats of dilution (see Fig. 1),

rel H (m 2 )

−

rel H (m 1 )

=

α2 rel H (FI, m 2 ) + (α1 −

−

α1 rel H (FI, m 1 )

α2 ) HA∞ ,

(121)

an appropriate extrapolation to infinite dilution. Heats of dilution yield precise information on association constants K A , heats of ion-pair formation HA∞ , and cutoff distances of the chemical model R. Entropies of ion-pair formation SA∞ can be calculated from K A and HA∞ by the usual thermodynamic relationships. Table III shows the agreement of HA∞ values from heat of dilution and temperature-dependent conductance measurements; Fig. 8 shows the dependence of heat of dilution on the concentration of a partially associated electrolyte. Heats of solution, sol HY = HY − HY∗ , are obtained by direct measurements and can be combined with the appropriate heats of dilution to yield the quantities sol HY∞ (Fig. 1), which in turn yield the enthalpies of solvation

solv HY∞ when combined with lattice energies lat HY . Single-ion enthalpies of solvation are obtained with the help of extrathermodynamic assumptions.

For a galvanic cell, M(Hg) | MX(c in S) | AgX(s) | Ag(s),

(122)

which involves the chemical reaction → MX(solv) + Ag(s), M(Hg) + AgX(s) ←

(123)

emf is given by the relationship −n FE =

ν  k 

(β)

(β)

ωi µi .

(124)

β=1 i=1

In Eqs. (122) and (123), M(Hg) is an alkali metal amalgam electrode, MX the solvated halide of the alkali metal M at concentration c in a solvent S, and AgX(s)/Ag(s) a silver halide–silver electrode. Equation (124) is the general expression for the electromotive force E of a galvanic cell without liquid junction in which an arbitrary cell reaction → ωi Yi + · · ·, takes place between ω1 Y1 + ω2 Y2 + · · · ← k components in ν phases. In Eq. (124) n is the number of moles of electrons transported during this process from the anode to the cathode through the outer circuit, F the (β) Faraday number, and µi the chemical potential of component Yi in phase β. Cells with liquid junctions require the electromotive force E in Eq. (124) to be replaced by the quantity E − E D , where E D is the diffusion potential due to the liquid junction. The standard potential E ◦ for the cell investigated by Eq. (122) is given by the relationship   ∗ sat FE ◦ = − µ∗Ag + µ∞ (125) MX − µAgX − µM(Hg) . That is, E ◦ would be the emf measured if all components were in their standard states. Hence, for the example given, Eq. (124) can be written as E = E◦ − = E ◦ −

FIGURE 8 Relative apparent molar enthalpy rel H of potassium iodide–propanol solutions at 25◦ C in the convetional c1/2 plot. The full line curve (1) is calculated from the experimental data [10−3 < c (mol dm−3 ) < 10−2 ] with the help of the chemical model; the dashed curve (2) depicts the limiting law. [Reprinted with permission from Barthel, J., Gores, H.-J., Schmeer, G., and Wachter, R. (1983). Non-Aqueous Solutions in Chemistry. In “Topics in Current Chemistry” (F. L. Boschke, ed.), Vol. 111, pp. 33–44, Fig. 4. Springer-Verlag, Berlin. Copyright 1983 Springer-Verlag.]

aMX RT ln F aM(Hg) 2RT ln (αcy± ) F

(126)

if the metal concentration of the amalgam is kept constant for a series of measurements. Equation (126) again is of the type given by Eq. (98). Data analysis on dilute solutions together with equations (99) and (100) yield E ◦ and K A as the desired information. The association constant K A obtained in this manner is in agreement with those from heats of dilution or conductance measurements. According to Eq. (126) the reference potential E ◦ at activity aM(Hg) is given by the relationship E ◦ = E ◦ +

RT ln aM(Hg) F

(127)

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and yields the standard potential E ◦ when the activity aM(Hg) is known. Hence the Gibbs energy of transfer of the electrolyte MX from the pure to the infinitely dilute state in the solvent S is available: ∗ ∞ ◦ ∗ ∗ µ∞ MX −µMX = sol G MX = −FE +µAgX −µMX . (128)

Measurements of emf as a function of temperature yield ∞

sol HMX in agreement with calorimetric data. Measurements of emf are also widely used for calculating activity coefficients at concentrations higher than the limit of validity of the model calculations. For this purpose Eq. (126) is used for the determination of y± , y± = αy± , after previous determination of E ◦ from dilute-solution data. H. Solubility Making solubility measurements is another method for the determination of Gibbs energies. A precipitated electrolyte when studied in equilibrium with its saturated solution sat µ∗Y (S) = µY (sat sol) = µ∞ Y + ν RT ln a±

(129)

sat yields the mean activity at saturation a± and hence of∞ fers the possibility of calculating sol G Y with the help of Eq. (129). Solubility measurements may advantageously replace emf methods for the determination of sol G ∞ Y values and related quantities of sparingly soluble electrolytes, such as the Gibbs energy of transfer from a solvent S to a solvent S:

∞ ∞

tr G ∞ Y (S → S) = µY (S) − µY (S )

= ν RT ln

sat a± (S ) . sat a± (S)

(130)

Frequent use of the solubility method is made for investigating activity coefficients of ternary systems where the variation of the activity coefficient γY of the electrolyte is studied as a function of the concentration of a second solute Z. Thermodynamics provides the relationship   ∂(ln γY ) ∂(ln γZ ) = . (131) ∂m Z ∂m Y m Z mY If the solute Z is a nonelectrolyte, the sign of the partial derivative (∂(ln γZ )/∂m Y )m determines whether Z is salted-in (−) or salted-out (+) by the electrolyte. Salting-in and salting-out of electrolytes is an important effect with regard to charged polymers and particularly

for biopolymers. The effectiveness of anions for saltingin and salting-out follows the so-called Hofmeister series. Despite numerous efforts a satisfactory explanation of this series by proper models is still unavailable. I. Phase Separation by Electrolytes Specific electrolyte solutions have been discovered and intensively studied which can produce a phase with a higher, and a phase with a lower salt concentration. The reasons for phase separation may be different. For aqueous solutions of large organic ions (e.g., tetra-n-pentylammonium bromide), the phenomenon is ascribed to hydrophobic interaction; for large ions in low permittivity solvents (e.g., tetra-n-pentylammonium picrate in 1-chloropentane), it is due to the long-range electrostatic interactions (coulombic phase separation). J. Nernst’s Distribution Law An important technical application of liquid–liquid equilibria uses Nernst’s law of phase distribution of a solute Y between two nonmiscible solvents to make up the two phases in contact, α and β. The equilibrium condition, (β) µ(α) Y ( p, T ) = µY ( p, T ) yields the relationship aY(α)

∞(β)

=

KY

= N, (132) (β) K Y∞(α) aY where aY means the activity of the solute and N is Nernst’s distribution coefficient which is the ratio of the Henry ∞(β) constants, K Y∞(α) and K Y , for the solution of Y in the two separate solvents. Distribution coefficients of water– octanol are widely used in toxicological and ecological studies.

SEE ALSO THE FOLLOWING ARTICLES CHEMICAL THERMODYNAMICS • ELECTROCHEMISTRY • ELECTROLYTE SOLUTIONS, TRANSPORT PROPERTIES • LIQUIDS, STRUCTURE AND DYNAMIC • PERMITTIVITY OF LIQUIDS • SOLVENT EXTRACTION

BIBLIOGRAPHY Barthel, J., Krienke, H., and Kunz, W. (1998). “Physical Chemistry of Electrolyte Solutions—Modern Aspects,” Steinkopf, Darmstadt/Springer-Verlag, New York. Blum, L. (1980). Primitive electrolytes in the mean spherical approximation. In “Theoretical Chemistry: Advances and Perspectives,” pp. 1–66, Academic Press, New York. Bockris, J. O., Reddy, A. K. N. (1998). “Modern Electrochemistry” (Vol. 1, Ionics), 2nd ed., Plenum, New York.

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Electrolyte Solutions, Thermodynamics DECHEMA Datenbank DETHERM (1989). Elektrolytdatenbank ELDAR, DECHEMA, Frankfurt a.M. Friedman, H. L., and Dale, W. D. T. (1977). Electrolyte solutions in equilibrium. In “Statistical Mechanics,” Part A, Equilibrium Techniques (B. J. Berne, ed.), pp. 85–135, Plenum, New York. Kreysa, G., ed. (1992 ff). DECHEMA Chemistry Data Series, Vol. XII,

20:28

241 Barthel, J. et al., Electrolyte Data Collection (12 partial volumes), DECHEMA, Frankfurt, to be continued. Pitzer, K. S. (1991). “Activity Coefficients in Electrolyte Solutions,” 2nd ed., CRC Press, Boca Raton, FL. Robinson, R. A., and Stokes, R. H. (1970). “Electrolyte Solutions,” 2nd rev. ed., Butterworth, London.

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Electrolyte Solutions, Transport Properties J. Barthel W. Kunz

P. Turq O. Bernard

University of Regensburg

University Pierre et Marie Curie

I. Thermodynamics of Irreversible Processes II. Statistical Mechanical Theory of Electrolyte Transport Properties III. Electrolyte Conductivity and Transference Numbers of Dilute Solutions IV. Viscosity V. Transport Equations for Concentrated Electrolyte Solutions

GLOSSARY Chemical diffusion Transport of matter caused by a gradient in concentration. Electrolyte conductivity Transport of electric charges by ions in the gradient of an electric potential. Also known as electrolyte conductance. Electroneutrality Compensation to zero of positive and negative charges in a finite volume of the solution. Electrophoretic effect Hindrance of the undisturbed movement of the ions in the solution, produced by hydrodynamic interactions of ions and solvent molecules. Equilibrium state State of the system in which, during the observation time, no measurable flows of matter and/or energy and no measurable changes of the intensive properties of the system take place.

Glass transition temperature Temperature at which transition of the undercooled solution to an amorphous solid state takes place. Hamiltonian Energy function of a system expressing the total energy of the system as a function of the position and momentum coordinates of its particles. Hamiltonian model Statistical mechanical model based on a Hamiltonian. Intensive property Property of the system that is independent of the mass of the system (e.g., density, viscosity, molar conductance). Migration Movement of particles under the action of driving forces. Mode Motion characterized by a time constant, due to specific effects. Phase space 6N -dimensional space defined by an

243

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244 orthogonal coordinate system of the 3N momentum and 3N position coordinates of the N particles of the system. Relaxation effect Hindrance of the undisturbed movement of the ions in the solution, produced in an external electric field by the nonspherical symmetry of the charge distributions around the ions. Self-diffusion Transport of matter caused by Brownian motion. s-Particle distribution function Probability of finding s particles out of the total number of particles of a system at time t with given momenta at given positions. Steady state State of a system in which, during the observation time, measurable flows of matter and/or energy may take place without changing the intensive properties of the system. AN ELECTROLYTE SOLUTION that is not in thermodynamic equilibrium is subject to generalized forces that control the irreversible processes taking place in such systems. Processes of this type are chemical reactions, including relaxation processes, and transport properties. Homogeneous chemical reactions do not presuppose spatial inhomogeneities, in contrast with transport phenomena requiring a gradient of concentration, electric potential, temperature, or drift velocity. Transport phenomena in electrolyte solutions can involve both types of irreversible processes, taking place simultaneously, since ion-pair formation in electrolyte solutions and formation of higher aggregates are related to chemical reactions. The models of electrolyte solutions and the fundamentals of thermodynamics and statistical thermodynamics are given in Electrolyte Solutions, Thermodynamics. Just as in the case of equilibrium thermodynamics, thermodynamics of irreversible processes yields phenomenological equations which in principle are independent of the molecular nature of the system. In contrast, statistical mechanics derives transport properties from the microscopic picture of the solution. Considerable progress has been made in this latter type of approach during recent years.

I. THERMODYNAMICS OF IRREVERSIBLE PROCESSES A. Definitions Gradients of properties produce flows that at low concentrations are proportional to these gradients. Concentration inhomogeneities entail chemical diffusion as the transport process. Concentration presupposes a system of at least two components for its definition. The law linking diffusion and flow is Fick’s law. Electric potential gradients produce ion migration and are manifest by electric

Electrolyte Solutions, Transport Properties

conductance according to Ohm’s law. Temperature gradients yield thermal conductance according to Fourier’s law. Inhomogeneities of drift velocity entail Newton’s law. Gradients of electric potential, temperature, and drift velocity are observed in both pure phases and mixtures. Coupled transport phenomena obey Onsager’s laws of irreversible processes, yielding the appropriate cross coefficients. However, only diffusion, viscosity, migration, electric conductance, and related topics will be considered in this article; little or no information on the other transport processes in electrolyte solutions can be found in the literature. The formulation of transport processes requires the use of time and space variables t and ri , in contrast with equilibrium thermodynamics which does not recognize time as a variable. The object of investigation is a phase with local inhomogeneities, composed of particles X 1 , X 2 , . . . , X i , . . . , molecules or ions, at particle number densities ρi (ri ; t) and particle velocities vi (ri ; t), the generalized forces being also functions of time and space. If particle X i is an ion of valency z i , the charge density at point ri would be e0 NA z i ci ; e0 is the proton charge, NA Avogadro’s number (e0 NA = F, Faraday constant), and ci the concentration of particle X i (ci = ρi /NA ). The amount of particles X i passing per interval of time dt in a perpendicular direction through an elementary surface dS of a volume V is given by the quantity ci vi dSdt if the solution does not move with regard to the chosen reference system, the “laboratory frame.” If the solution moves at velocity s v with regard to its reference system, this quantity is charged to ci (vi − s v) dSdt. In solutions the reference velocity s v is given by the relationship   s v= gi vi ; gi = 1, (1) i

i

where gi is the weight factor for particle X i . Various reference systems are used in the literature. The most important examples are the Hittorf reference system, in which the velocity of solvent particles v0 is chosen as the reference, and the Fick reference system, in which the mean volume velocity of the system is chosen. Investigations on transport properties commonly are based on diffusion current densities Ji (also, flux or flow):   Ji = ci vi − s v . (2) Then the total diffusion current φi of particles X i through an observed surface S enclosing volume V is given by the relationship   ∂ci − Ji dS. (3) dV = φi = ∂t (V )

(S)

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Equation (3) presupposes that there is no source of particles X i within volume V . Transformation of the surface integral to a volume integral by the help of Gauss’s theorem yields the relationship

K c , as usual in chemical kinetics. The determination of quantities σi is straightforward:

∂ci (4) = div Ji . ∂t In the theory of electrolyte conductance, reference is made to the density of charges e0 NA ci z i (C m−3 ) rather than to the density of particles. Then Eqs. (2) and (3) are transformed to   ji = e0 NA z i Ji = e0 NA z i ci vi − sv , (5a)  Ii = ji dS, (5b)

and

−

j=



ji = e0 NA

i

and



  z i ci vi − sv ,

(8a)

σp = +k1 c+ c− − k2 cp .

(8b)

The quantities c+ , c− , and cp are the molarities of the “free” cations, anions, and ion pairs in the solution which are related to electrolyte concentration cE by the degree of dissociation α of the ion pair:

j dS.

(5d)

In Eqs. (5), ji is the partial electric current density of ions X i , j the total current density, and Ii and I the partial and total electric currents (A). Electroneutrality within each elementary volume,  ci z i = 0, (6) i

entails that the electric current density j is independent of the reference system, in contrast with the partial current densities ji . B. Electrolyte Solutions in Irreversible Thermodynamic Processes The application of Eq. (3) or Eq. (4) to electrolyte solutions would require a solution made up of completely dissociated electrolytes. This restriction can be overcome by the introduction of sources that do not produce spatial inhomogeneities in the solution. A simple but important example of such sources is the chemical reactions of ionpair formation and decomposition, which produce particle densities per unit of time σi at rate constants k1 and k2 : z+

k1

−→ [Cz+ Az− ]0 + A ←− z−

k2

(7a)

and k1 = Kc. k2

(9a)

cp = (1 − α)cE ,

(9b)

Kc =

i

I =

c+ = c− = αcE ,

and (5c)



C

σ+ = σ− = −k1 c+ c− + k2 cp

(7b)

In Eqs. (7), Cz+ , Az− , and [Cz+ Az− ]0 are the z + -valent cation, the z − -valent anion, and the electrically neutral ion pair (z + = |z − |). The ratio of the rate constants k1 and k2 yields the concentration-dependent equilibrium constant

(1 − α) . α 2 cE

(9c)

This simple case of a nonconserved system is chosen for exemplifying the fundamental concept. Electrolyte solutions exhibiting more complex equilibria can be treated in the same way, the quantities σi always being functions of rate constants and concentrations of free particles in the solution. With the sources taken into account, Eqs. (3) and (4) must be transformed to     ∂ci − Ji dS (10) − σi d V = ∂t (V )

(S)

and ∂ci (11) + div Ji − σi = 0. ∂t Equations (10) and (11) apply to a system removed from equilibrium as well as to a system in equilibrium undergoing spontaneous fluctuations. For this purpose, superposition is assumed of the average values ci , Ji , and σi and the fluctuations δci , δJi , and δσi : ci (ri ; t) = ci + δci (ri ; t),

(12a)

Ji (ri ; t) = Ji + δJi (ri ; t),

(12b)

σi (ri ; t) = σi + δσi (ri ; t).

(12c)

and Introduction of expressions (12) into Eq. (11) permits the discussion of the relaxation to equilibrium if macroscopic heterogeneities are imposed on the initially equilibrated system; systems in equilibrium undergoing fluctuations: Ji = 0, σi = 0; and systems in steady states: ci (ri ; t) is constant in time but not in space, ci (ri ; t) = ci (ri ); Ji and σi are also constant in time but may have nonzero values.

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The electroneutrality condition,  ci z i = 0,

Di = ωi kT. (13)

reduces the number of independent species. In dilute solutions the solvent transport generally can be neglected. The problem of how to fulfill the average condition (13) both locally and instantaneously cannot be solved completely in a description of the electrolyte solution based on Eqs. (11). These equations, however, give the important information on the time that is needed by the system to return to the local electrostatic equilibrium (Debye relaxation time) after perturbation. The ionic representation of electrolyte solutions is not generally necessary since most of the transport processes in electrolyte solutions take place with a characteristic time larger than the Debye relaxation time. For moderately concentrated solutions of 1,1 electrolytes this time is in the nanosecond range. At longer times cations and anions diffuse with a common diffusion coefficient and the ionic equations can be replaced by component equations. The general method for solving Eqs. (11) consists of transforming the partial differential equations with the help of Fourier-Laplace transformations into a set of linear algebraic equations that can be solved by the standard techniques of matrix algebra. The roots of the secular equation are the normal modes. They yield the laws for the decays in time of all perturbations and fluctuations which conserve the stability of the system. The power-series expansion in the reciprocal space variables of the normal modes permits identification of relaxation, migration, and diffusion contributions. The basic information provided by the normal modes is that the system escapes the perturbation by any means at its disposal, regardless of the particular physical or chemical reason for the decay.

The total force Fi applied on particle X i is given as the gradient of the electrochemical (i = +, −) or chemical (i = p) potential: Fi = −

Ji = ci vi = ci ωi Fi

(14a)

vi = ωi Fi ,

(14b)

and where vi is the velocity of particle i (s v = 0, dilute solution), and ωi is the generalized mobility coefficient related to the diffusion coefficient Di by the Einstein relationship

 1 grad µi∞ + RT ln ci yi + e0 NA z i ψ , NA (16a)

i = +, −,

(16b)

and Fi = −

 1 grad µi∞ + RT ln ci yi . NA

(16c)

In Eqs. (16), µi∞ is the chemical reference potential at infinite dilution, yi the corresponding activity coefficient in the molarity scale, and ψ the electric potential acting on the ions. The “thermodynamic” forces grad(ln yi ) will be neglected in the following discussion for the sake of simplicity. The electric potential ψ is related to the electric field E by the relationship E = −grad ψ, N A e0  div E = ci z i , ε0 ε

(17a) (17b)

where ε0 is the permittivity of the vacuum and ε the relative permittivity of the solvent. The sources σi in Eqs. (11) are given by Eqs. (8). As a result, the diffusional transport of a symmetric electrolyte in a solvent in which ion-pair formation takes place is given by the following set of differential equations:

 ∂c+ z + e0 = D+ div grad c+ − c+ E ∂t kT − k 1 c+ c− + k 2 cp , 

∂c− z − e0 = D− div grad c− − c− E ∂t kT

C. Chemical Diffusion Diffusional transport of symmetric electrolytes superimposed by ion-pair formation kinetics, Eqs. (7), is chosen as an example for exemplifying the methods of irreversible thermodynamics. Time and space evolution of the observed system, symmetric electrolyte Cz+ Az− in a solvent, are given by Eqs. (11), one equation for each species i (i = +, − , p). The flows Ji are given by Eq. (2):

(15)

− k 1 c+ c− + k 2 cp ,

(18a)

(18b)

and ∂cp = Dp div grad cp + k1 c+ c− − k2 cp . (18c) ∂t This set of partial derivative equations can describe either a macroscopic diffusion phenomenon taking place in a measuring cell with a small concentration gradient or a fluctuation process [see Eqs. (12)], for example, as observed in dynamic light-scattering experiments. In the first case, the boundary conditions (fixed concentration gradients) are determined by the experiment; in the second case, small perturbations on the initial equilibrium are produced from thermal fluctuations. For a fluctuating system,

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the mass-action law applied only to the equilibrium concentrations [ σi = 0], k1 c+ c− − k2 cp = 0,

(19)

and the electroneutrality of the equilibrium distribution of the ions entails [ Ji = 0] e0 N A  div E = z i δci . (20) ε0 ε i The investigation of relaxation times and diffusion coefficients requires the determination of the eigenvalues of the matrix corresponding to the system of algebraic equations obtained from Eqs. (18) after Fourier-Laplace transformation (s, Laplace transform of time; q, Fourier transform of the space coordinate). The roots of the secular equation are s1 = 0,   s2 = − q+2 D+ + q−2 D− ,

(21a) (21b)

DNH =

  D+ D− q+2 + q−2 q+2 D+ + q−2 D−

s3 = −(k1 c+ + k1 c− + k3 ),

(21c)

where qi2 is the partial contribution of the ionic species Xi to the Debye parameter κ: e2 z 2 NA qi2 = 0 i ci

ε0 εkT

(22a)

κ 2 = q+2 + q−2 .

(22b)

and

The root s1 simply indicates that infinite distances are correlated with infinite time, s2 is the reciprocal of the Debye relaxation time, and s3 is the kinetic relaxation frequency of the system. Depending on the kinetic parameters of the chemical process, the kinetic relaxation frequency can be faster or slower than the Debye frequency of the system. If the kinetic relaxation frequency is much smaller than the Debye mode, it can be determined experimentally by conductance fluctuation analysis. The diffusional modes are the roots of the secular equation which are proportional to q 2 . The corresponding diffusion coefficient is given by the relationship ( c+ = c− ) D=

Dp DNH + , 1 + 2K A c+ 1 + 1/(2K A c+ )

(23)

where DNH is the chemical diffusion coefficient of the dissociated part of the electrolyte (free ions), which is related to the individual ionic diffusion coefficients by the Nernst-Hartley equation, and Dp that of the associated ions (ion pairs):

2D+ D− . D+ + D−

(24)

The chemical diffusion coefficient is a phase property where the reference velocity is chosen in the Fick reference system. Various methods are known for the experimental determination of chemical diffusion coefficients, such as open and closed capillary methods, dynamic light scattering, or the Taylor dispersion method. In contrast with chemical diffusion, the self- (or intraor tracer-) diffusion of ions simply is the manifestation of Brownian motion. It can be visualized by labeling a small amount of ions and observing their displacement in an environment of nonlabeled ions of the same type. Nuclear magnetic resonance (NMR) spin-echo experiments use spin labeling; radioactive tracers are used in closed capillary methods. At infinite dilution the self-diffusion Di∞ of an ion Xi is linked to its limiting ion conductivity λi∞ , Eq. (43), and its generalized mobility ωi , Eq. (15), (F = e0 NA , Faraday constant): Di∞ =

and

=

kT λ∞ = ωi kT. e0 |z i |F i

(25)

D. Reaction–Diffusion Coupling In this section interest is focused on the influence of the rate constants of chemical processes on observable transport coefficients, in contrast with the common point of view following the limitations of kinetics by the transport of matter, for example, the arrival of reactants or the departure of products. An example is given in Fig. 1 for the electrophoresis of ligand-exchanging transition-metal complexes. Again, time and space evolution of the observed system are given by a set of equations of the type of Eqs. (11), where the flows are Ji = ci vi = ci (ωi Fi + vi ).

(26)

In Eq. (26), vi is the velocity of the solvent at the position of particle i due to the average effect of the hydrodynamic interactions of solute and solvent (electrophoretic effect); the other symbols were explained in the preceding section. Exchange reactions of the type k12

−→ X z2 X 1z1 + L z1 ←− 2 k21

(27)

between the complexes X 1 of charge e0 z 1 and X 2 of charge e0 z 2 in a solution, containing an excess of the ligand L of charge e0 z 1 , are independent of the ligand concentration. This simple kinetic situation is exemplified in Fig. 1 by the reaction  − 2+   −→ RuNO NO− + RuNO NO3 ←− (28) 3 2 − NO3

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Again the term −q 2 D shows the diffusional broadening of the peaks; −k21 and −k12 indicate the diminution of the peaks by the relaxation factors exp[−k12 t] and exp[−k21 t], respectively; t is the observation time; and the terms iωe0 Eqz i characterize the migrational separation of the two peaks according to the effective values z 1 and z 2 and velocity vi (which is included in the effective values of z 1 and z 2 by the underlying theory). The high-field approximation does not exhibit the contribution due to the production of compounds by the exchange reactions to the electrophoretic pattern. This effect can be illustrated by the fast-exchange approximation which yields the roots s1 = −q 2 D + iωe0 Eq

z 2 k12 + z 1 k21 k12 + k21

(30a)

and s2 = −q 2 D +iωe0 Eq 2+ FIGURE 1 Paper electrophoresis of a mixture of RuNO(NO− 3) + and RuNO(NO− ) in 2-M HNO at various temperatures. Low 3 3 2 temperature: two peaks, separation of the two complexes. High temperature: one peak, no separation of the two complexes. x Axis: broadness of the Ru zone after migration (relative distance scale); y axis: Ru distribution after migration (relative concentration scale). [Orcil, L., Fatouros, N., Truq, P., Chemla, M., and Barthel, J. (1983). Zeitschrift Phys. Chem. N. F., Oldenberg Verlag, Munchen, ¨ 138, 129–136.]

in aqueous solutions of HNO3 . The figure shows the patterns at various temperatures for the paper electrophoresis + 2+ of a mixture of RuNO(NO− and RuNO(NO− 3) 3 )2 after migration under the following experimental conditions: initial concentration of ruthenium in the active drop about 10−3 mol dm−3 and initial concentration of NO− 3 in the supporting electrolyte (aqueous solution of HNO3 ) about 2 mol dm−3 . The separation into two peaks occurs only for slow-exchange reactions, that is, at low temperatures. Fast-exchange reactions do not permit the separation, except at much higher electrical fields which, however, cannot be applied to the highly conducting solutions used in this experiment. The experiment illustrates well the existence of the modes deduced from the secular equation of this problem. The separation of peaks takes place at high fields. Under this condition the roots of the secular equation are (D1 ≈ D2 = D) s1 = −q D + iωe0 Eqz 2 − k21 2

(29a)

Equation (30a) shows that the relevant peak undergoes a simultaneous migration and diffusion process but is free from kinetic attenuation. Equation (30b) shows a kinetically fast-attenuated mode which yields a negligible contribution to the electrophoretic pattern.

II. STATISTICAL MECHANICAL THEORY OF ELECTROLYTE TRANSPORT PROPERTIES A. Hamiltonian Models of Transport Processes Another type of approach to transport properties, also frequently used in the theory of electrolyte solutions, is based on statistical mechanics and starts with Liouville’s theorem. Electrolyte conductance is chosen to exemplify the features of this theory. In terms of statistical mechanics the electrolyte solution is made up of N solute particles (ions) at positions r1 , . . . , r N and N0 solvent particles r N +1 , . . . , r N +N0 with momenta p1 , . . . , p N and p N +1 , . . . , p N +N0 yielding the density function ρ of the configurational points in the 6(N + N0 )-dimensional  space: ρ = ρ N +N0 (r1 , . . . , r N , r N +1 , . . . , r N +N0 ; p1 , . . . , p N , p N +1 , . . . , p N +N0 ; t).

(29b)

(31)

Hamiltonian models are based on the Hamiltonian H , which expresses the total energy of the system as a function of the momenta pi and coordinates ri of all particles:

and s2 = −q 2 D + iωe0 Eqz 1 − k12 .

z 1 k12 + z 2 k21 −(k12 +k21 ). (30b) k12 + k21

H=

 p2 1  i + Ui j (ri , r j ; t). 2m i 2 i j i

(32)

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The first term on the right-hand side of Eq. (32) represents the kinetic energy; the second term is the potential energy U N +N0 (r1 , . . . , r N +N0 ; t) approximated by a sum of pairwise direct interaction potentials Ui j (ri , r j ; t); both momenta pi and coordinates ri are functions of time. The Liouville equation [cf. Eq. (4)],   ∂ H ∂ρ ∂ρ ∂ H ∂ρ − , (33) = −div(ρv) = ∂t ∂ri ∂ pi ∂ pi ∂ri i entirely determines the distribution function ρ N +N0 at time t = 0 once the initial state ρ N +N0 (r1 , . . . , r N +N0 ; p1 , . . . , p N +N0 ; 0) is given. In Eq. (33), the operator “div” is a 6(N + N0 )-dimensional operator, and vector v has the components (r1 , . . . , r N +N0 ; p1 , . . . , p N +N0 ). Averaging over all position and momentum variables of the solvent molecules and all momentum variables of the ions yields the N -particle distribution function D N of the ions: D N (r1 , . . . , r N ; t)  = ρ N +N0 dr N +1 · · · dr N +N0 dp1 · · · dp N +N0 . (34) The N -particle distribution function D N gives the probability of finding at time t the N ions in the positions r1 , . . . , r N regardless of the momenta and positions of the solvent molecules and the momenta of these ions. Hamiltonian models based on Hamiltonians averaged with regard to the coordinates of the solvent molecules are referred to as Hamiltonian models at McMillan-Mayer (MM) level. The velocity vi(N ) of ion i out of N ions is then given by the relationship vi(N ) (r1 , . . . , r N ; t)  −1 = DN vi (r1 , . . . , r N +N0 ; p1 , . . . , p N +N0 ; t) ρ N +N0 dr N +1 · · · dr N +N0 dp1 · · · dp N +N0 , (35) and the Liouville equation yields the continuity equation    ∂ DN (36) divi D N vi(N ) . =− ∂t i The ion velocity vi(N )

depends on the applied external electric field E (which is independent of the position variables), the ionic interactions, and the hydrodynamic coupling effects of ion i with the remaining ions: vi(N ) = v0 (ri ) + ωi Fi(N ) +

N 

) ωt χ ti F(N t .

(37)

t=1

In Eq. (37), v0 (ri ) is the velocity of solvent at position ri , and ωi and ωt are the generalized ion mobilities of ions i ) are the forces acting on and t [see Eq. (15)]. Fi(N ) and F(N t

ions i and t, and χ ti is the hydrodynamic coupling tensor ) of ions t and i. The forces F(N k (k = i or t) are given by the relationship ) F(N = e0 z k E − kT gradk (ln D N ) k

−

N 

gradk Ukm (rk , rm ),

(38)

m=1

where Ukm is the direct interaction potential of ions k and m [Eq. (32)], the term kT gradk (ln D N ) takes into account Brownian motion, and e0 z k E is the electric force acting on the charge e0 z k (ion k) in the external field E. Stepwise reduction of the N -particle distribution function by integration [cf. Eq. (34)] yields the s-particle correlation functions g (s) (s < N ),  (s) s g (r1 , . . . , rs ; t) = V D N drs+1 · · · dr N , (39) and the velocity vi(s) of ion i, (N ) v D N drs+1 · · · dr N (s) vi (r1 , . . . , rs ; t) = i , D N drs+1 · · · dr N

(40)

at arbitrary positions of the ions (s + 1) to N ; V is the volume of the solution. The use of s-particle correlation functions g (s) and velocities vi(s) permits the integration of Eq. (37) with regard to all coordinates except ri to yield the velocity of ion i regardless of the position of all other ions: vi(1) (ri ; t) = v0 (ri ) + ωi e0 z i E  N  − ωi n k gradi Uik gik drk k=1

+

N 

 ωk n k

χ ik gik Fk drk .

(41)

k=1

In Eq. (41), gik is the two-particle correlation function g (2) of ions i and k (see Fig. 2); the force Fk , depending on both the action of the external field E and the ion–ion interaction forces in the solution, is given by the relationship Fk = e0 z k E − kT gradk gik − gradk Uik  N  gikm − nm gradk Ukm drm , gik m=1

(42)

where gikm is the three-particle correlation function g (3) of ions k, i, and m; and n k and n m are the one-particle distribution functions of ions k and m, which are the particle densities of Section I in the molecular scale, ρi (Section I) = n i /NA . The velocities vi(1) and v0 of Eq. (41) are the particle velocity vi and the Hittorf reference velocity v0 as defined in Section I.

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of drift velocities. The approximation used for the pair potentials Ui j depends on the model underlying a conductivity equation. The most elaborated models use the superposition of long-range Coulomb Uicj and short-range potentials Ui∗j :

FIGURE 2 Explanation of the two-particle distribution and correlation functions, f i j and gi j . The probability of simultaneously finding an ion i in dV1 and an ion j in dV2 is given by the relationship. f i j (r1 , r21 ) = ni (r1 )n j (r2 )gi j (r1 , r21 ) = n j (r2 )ni (r1 )g j i (r2 , r12 ) = f j i (r2 , r12 ). The elementary volumes dV1 and dV2 , with ion- and solventpermeable walls, are immovably fixed in relation to the observer O at positions r1 and r2 in the solution. The quantities ni and n j are the concentrations of ions i and j in the solution (number of ions per unit of volume).

B. Single-Ion and Electrolyte Conductivity In a homogeneous and isotropic medium (solvent), the ion velocity vi is independent of position and yields the molar single-ion conductivity λi = e0 |z i |NA

|vi (t) − v0 | . |E|

(44)

Both single-ion and electrolyte conductivity depend on three effects: unperturbed ion migration [second term in Eq. (41)], relaxation effect [third term in Eq. (41)], and electrophoretic effect [fourth term in Eq. (41)]. Relaxation and electrophoretic effects are calculable with the help of Kirkwood’s superposition approximation for the three-particle correlation function gikm : gikm = gik · gmk .

(45)

The two-particle correlation functions gi j do not show the spherical symmetry of equilibrium functions as a consequence of the applied external field. The calculation of gi j functions requires a further approximation: the superposition of the unperturbed equilibrium function gi0j and a perturbation function gi j of axial symmetry, gi j (r1 , r12 ; t) = gi0j (r ) + gi j (r1 , r12 ; t).

(46)

The perturbation term equals zero for correlation functions of equal ions, gii = g j j = 0. The hydrodynamic coupling tensor χ ik is given by the Oseen or the Navier-Stokes equations for Newton’s law

e02 z i z j + Ui∗j . 4π ε0 εri j

(47)

Less elaborated theories neglect the short-range potentials. The theory of electrolyte conductivity is based on the two-particle continuity equation divi ( f i j vi j ) + div j ( f ji v ji ) = 0.

(48)

where vi j and v ji are the relative velocity of ion X j in the vicinity of ion X i and vice versa. Equation (48), often called the Onsager continuity equation, is obtained from the general continuity equation (36) by the usual averaging processes, that is, integration over all momenta and N − 2 space coordinates of the ions. The resulting two-particle continuity equation yields Eq. (48) for the stationary case, ∂ f i j /∂t = 0. The boundary conditions for the continuity equation are

(43)

Equation (43) is the basic relationship of electrolyte conductivity. An electrolyte C z+ A z− shows the molar electrolyte conductivity m = λ+ + λ− .

Ui j = Uicj + Ui∗j =

lim gi j = 0

|r12 |→∞

and

 (vi j − v ji )

ri j ri j

(49a)

 = 0,

(49b)

ri j=R

requiring attenuation to zero of the perturbation function at infinite ion distance, Eq. (49a), and zero normal component of the relative ion velocity at a distance R, Eq. (49b); only the tangential component is different from zero at R. For the chemical model, distance R is the cutoff distance of the short-range forces, which is also assumed to be the upper limit of ion-pair association. Paired states of unequally charged ions of symmetric electrolytes (z + = |z − |) at distances lower than R are considered to be nonconducting ion pairs. The integration of the continuity equation is a laborious and time-consuming task, even for simple models of vi j and gi j . Onsager’s limiting law is based on the correlation function gi j of the Debye-H¨uckel theory which is the first correct chemical model. It neglects the noncoulombic interaction potentials. Since it was developed many attempts have been made to put the concentration limit to higher values. During the last 20 years, a quantitative description of conductance and self-diffusion up to 1-M solutions has been achieved by the use of modern gi j functions coming from integral equation techniques such as the hypernetted chain (HNC) equation or mean spherical approximation

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(MSA). Some of these approaches will be discussed in detail in the following sections. The progress in equilibrium correlation functions has also been used to extend the validity range of relations of irreversible thermodynamics to higher concentrations. An example is the Nernst-Hartley equation, Eq. (24), for completely dissociated electrolytes to yield the relationship Q 1 D2∗ + Q 2 D1∗ , Q1 + Q2

(50)

  ci o z i Di + k B T z k ik k B T εεo k

(51a)

D= where Qi =

z i e02

FIGURE 3 Fundamentals of the moving boundary (mb) method.

and 2NA ik = ck 3η

∞ h ik (r )r dr.

(51b)

0

The term ik gives the hydrodynamic corrections on the moving ion X i due to all ions X k , h ik = gik − 1 is the total correlation function, and η is the solvent viscosity. There are two routes to derive the chemical (or mutual) diffusion coefficient. One has been taken for HNC calculations and the other for an MSA treatment. It should be noted that while the HNC calculation is generally regarded as more accurate, it does not lead to explicit analytical expressions, in contrast with the MSA which is simpler, but perhaps not as accurate, and leads to explicit formulas that are reasonably accurate when the energy route for the thermodynamic quantities is used.

III. ELECTROLYTE CONDUCTIVITY AND TRANSFERENCE NUMBERS OF DILUTE SOLUTIONS A. Transference Number and Conductivity Measurements The transference number ti of an ion X i is equal to the net number of faradays carried by this ion, free ion, or constituent of an ionic species, across a reference plane that is fixed with respect to the solvent, when one faraday collectively passes (total electric current I ; partial electric currents I+ and I− [see Eqs. (5)]: I+ I− ; t− = ; t+ + t− = 1. (52) I I Transference numbers ti are available by measurements, the most reliable method at low electrolyte concentrations being the moving boundary (mb) method (see Fig. 3). This method requires the formation of a sharp boundary + − − b between the solutions of two salts C+ 1 A and C2 A in the same solvent when the cationic transference number t+ =

+ − + − t+ (C+ 1 ) is to be determined, or of C A1 and C A2 for the − anionic transference number t− (A1 ). The passage of a quantity of electricity I × t through a fixed reference plane perpendicular to the direction of migration is accompanied by a boundary movement, observable by spectroscopic or electrical methods, which defines a volume V between the boundary positions b at time t = 0 and t. The transference number is given by the relationship

t+ = t+ (C+ 1)=

(1) VF c+ , It

(53)

(1) where c+ is the equivalent concentration of the cation + C1 and F the Faraday constant. The precision of transference numbers obtained from mb experiments approaches 0.02% for some systems. Conductivity measurements on electrolyte solutions are generally carried out with the help of bridge methods, a conductance cell filled with the electrolyte solution being one of the resistances in the arms of an ac Wheatstone bridge. Bridge and cell design is highly developed, and a precision of 0.01% is not unusual. Specific conductivities at temperature T, κ(T ), are obtained from the measured resistances of the electrolyte solutions R, after correction for the resistance of the pure solvent Rs by using the temperature-dependent cell constant A(T ):

κ(T ) = A(T )[1/R − 1/Rs ].

(54)

The cell constant is obtained from measurements on aqueous KCl solutions. z z Equivalent conductivities  of electrolytes Cν++ Aν−− ,  = m /n e , are related to specific conductivities κ by the relationship (n e , electrochemical valency) =

κ ; 1000n e c

n e = ν+ z + = ν− |z − |.

(55)

B. Dispersion of Conductance The time dependence of pair correlation functions gi j (r1 , r2 ; t) entails dispersion of conductivity when the

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applied external field is a high-frequency field. Finite relaxation times of the ions surrounding the moving central ion as a cloud of axial symmetry more or less prevent the ion cloud from taking on the configuration that it would have in a static external field, which thus diminishes the conductance-hindering relaxation effect. According to the discussion in the preceding section, molar conductivity can be expressed by an equation of type rel el  m = ∞ m +  m + m ,

(56)

showing the three contributions to conductivity: unperturbed ion movement in the external field at infinite dilution of the electrolyte compound, ∞ m; ; and relaxation effect, rel m electrophoretic effect, el m. At the lowest meaningful level of theory (limiting law at very low electrolyte concentrations), the relaxation term at frequency ν of the external field is given by the relationship rel m =

e02 z + z − κ∞ m f (ν) 12πε0 εkT

(57)

where f (ν) is a frequency-dependent function which gives rise to the relaxation time of the ion cloud, (z + z − )2 e02 NA  2 ∞ , 2 ∞ kT pκ 2 z + λ− + z − λ−

τ=

(58)

C. Conductivity Equations of Dilute Electrolyte Solutions Onsagar’s limiting law is obtained from Eq. (57) when in Eq. (56) the passage to zero frequency is made, ν → 0. Improved approximations in the calculation of pair correlation functions gi j and the introduction of short-range potentials have led to improved conductivity equations of the type given by Eq. (56) or its truncated series developments of type  = ∞ − Sc1/2 + Ec ln c + J1 c − J2 c3/2 .

(61)

In Eq. (61) every coefficient, S, E, J1 , and J2 , contains the contributions of the electrophoretic and relaxation effects, el and rel . The coefficients of the series-developed Fuoss-Hsia equation, including the so-called Chen effect (see Section III.D), are given in Table I. Onsager’s limiting law in terms of Eq. (61) would be given by the series development  = ∞ − Sc1/2 .

(62)

It should be stressed that the coefficients S and E are independent of short-range contributions, that is, independent of the model parameters a and R, in contrast with J1 and J2 . The literature shows contributions by Justice and Ebeling to the so-called echo or feedback effect dealing with the action of the electrophoretic and relaxation effects on the underlying pair correlation functions gi j . This

and at zero frequency has the value f (0) =

p √ ; 1+ p

p=

∞ λ∞ |z + z − | + + λ− ∞. |z + | + |z − | |z + |λ∞ − + |z − |λ+ (59a,b)

Quantity κ is the reciprocal radius of the unperturbed Debye ion cloud; it is given by the relationships (22). The electrophoretic term is given at this level by the relationship 2 2 el m = −(ν+ z + + ν− z − )

NA e02 κ. 6πη

(60a)

For symmetric 1,1 electrolytes (ν+ z + = ν− |z − | = 1) in the concentration range of the chemical model, Eq. (60a) is changed to el m =

NA e02 κ . 3πη 1 + κR

TABLE I Coefficients of the Conductivity Equation of Symmetric Electrolytes Λ = Λ∞ −Sc1/2 + Ec log c + J 1 (R)c − J 2 (R)c3/2 [z + = |z − | = z; SI units except c/(mol dm−3 )] S = S1 ∞ + S2 ;

S2 = 82.484 × 10−5 z 2 (εT )−1/2 η−1 E = E1

∞

− 2E 2 ;

J1 = σ1 ∞ + σ2 ;

J2 = σ3 ∞ + σ4 ;

(60b)

The frequency dependence of conductivity was postulated in 1928 by Debye and Falkenhagen from theoretical considerations; a satisfactory experimental proof of this effect is still lacking.

S1 = 0.82043 × 106 z 3 (εT )−3/2 E 1 = 6.7749 × 1012 z 6 (εT )−3 E 2 = 0.99750 × 103 z 5 (εT )−2 η−1

 κR 2b2 + 2b − 1  σ1 = 2E 1 + 0.9074 + ln 1/2 b3 c

 35 2 κR σ2 = E 2 + 2 − 2.0689 − 4 ln 1/2 3b b c

 4.4748 3.8284 κR  + σ3 = 1/2 E 1 0.6094 + c b b2

 34 κR 2 σ4 = 1/2 E 2 −1.3693 + − 2 c 3b b

κR = 0.50290 × 1012 z(εT )−1/2 ; c1/2 b = 16.709 × 10−6 z 2 (εT )−1 R −1 ; E 1 = 0.43429E 1 ; E 2 = 0.43429E 2

where

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effect is small and does not change the coefficients of the conductivity equation to a significant extent. Conductivity equations for unsymmetric electrolytes, such as MgCl2 or Na2 SO4 , are also given in the literature. The chemical model of electrolyte solutions introduces short-range interactions by means of potentials of mean force Wi∗j , which can be considered as contributions to ion-pair formation. For electrolyte conductivity, ion pairs of symmetric electrolytes are considered to be nonconducting, electrically neutral species. The appropriate expression of the thermodynamic equilibrium constant of ion-pair formation, C

z+

−→ [Cz+ Az− ]0 , + A ←− z−

(63)

introduced ad hoc into conductance theory, is given by the relationship    R Wi∗j 2q 2 K A = 4000π NA r exp − dr, (64a) r kT a q=

e02 |z + z − | . 8πε0 εkT

(64b)

The upper limit of ion-pair association and the cutoff distance of short-range interactions in electrolyte solutions are considered identical in the framework of the chemical model. Association constants are independent of the electrolyte property, thermodynamic property, or transport property for which they are determined. Table II proves this statement by comparison of association constants obtained from conductance measurements and measurements of the heat of dilution. The inclusion of the ion-pair concept in the statistical thermodynamic theory is made by setting the distance R of the boundary condition, Eqs. (49), equal to the upper limit R of the association integral. Taking into account that only the “free” cations and anions at concentrations

c+ = c− = αc are conducting species leads to an expression of the equivalent conductance: /α = ∞ − S(αc)1/2 + Eαc ln(αc)

KA =

+ J1 (R)αc − J2 (R)(αc)3/2 ,

(65a)

1−α 1 ; α 2 c y±2

(65b)

y±2 = y+ y− ,

and − ln y± =

κq . 1 + κR

(65c)

The coefficients J1 (R) and J2 (R) depend on the cutoff distance R and thus include the influence of the short-range forces on the transport phenomenon; for the activity coefficient y± of the chemical model, see Electrolyte Solutions, Thermodynamics. Figure 4 shows the typical concentration and temperature dependence of the conductivity of a moderately associated electrolyte in a nonaqueous solvent. The calculated conductivity curve [Eqs. (65)] reproduces the measured conductivities (open circles) with a precision of better than 0.02%. The broken line of 25◦ C is the limiting law, Eq. (62). There is no doubt that the Onsager limiting law can be confirmed by highly precise measurements, but its validity range is limited at very low concentrations. Data analysis of conductivity measurements yields the limiting conductivities ∞ and the association constants KA.

TABLE II Comparison of Association Constants from Calorimetric KA (∆H ) and Conductivity KA (Λ) Measurements at 25◦ C K A (∆ H) Solvent

Electrolyte

Water

MgSO4 CaSO4 NiSO4

2-Propanol

Pr4 NI Bu4 NI i-AM4 NI LiBr KI Bu4 NI

Acetone

mol−1

dm3

161 192 210 513 534 581 4593 164 242

K A (Λ) mol−1 dm3

mean value:

162 189 209 516 515 573 4530 177 232

FIGURE 4 Temperature dependence and concentration dependence of a moderately associated electrolyte, Bu4 NCIO4 [116 (−40◦ C) < K A /mol−1 dm3 < 904 (25◦ C)], in propanol [31.2 (−40◦ C) > ε > 20.5 (25◦ C)] solutions at various temperatures. Broken line: limiting law at 25◦ C.

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D. Theory of Transference Numbers of Dilute Solutions The definition of transference numbers, Eqs. (52), entails for 1, 1 electrolytes the relationships λi Ii = , I  λi∞ = ∞, 

ti = ti∞

(66a)

ti − 0.5 ∞ . = ti∞ − 0.5 ∞ + el

(66b)

and  = λ+ + λ− ,

The first is the contribution to the electrophoretic part of conductance, and the second results from the negligibly small space-dependent part of the interionic two-particle force, λel(2) ∼ = 0 (Chen effect). Combining Eqs. (66)–(68) yields the transferencenumber equation in the form

(66c)

where λi and  are the single-ion and the electrolyte conductivities and λi∞ and ∞ the corresponding limiting values. Single-ion conductivities contain relaxation (λirel = λi∞ × E/E) and electrophoretic (λiel ) contributions, just as electrolyte conductivity does; E is the change in the electric field caused by the ion charges in the solution:

 E λi = λel + λi∞ 1 + . (67) E The electrophoretic effects on anions and cations are equal. The electrophoretic contribution consists of two parts:

 E el el(1) λ =λ 1+ E

 E 2λel el(2) +λ 1+ (68) + ∞ . E 

(69)

Equation (69) indicates that the relaxation effect does not influence transference numbers of symmetric electrolytes. Using Eq. (60b) for el yields the relationships √ 

ti − 0.5 S2 αc −1 ∞ ∞  − + Bαc, (70a) =  ti∞ − 0.5 1 + κR KA =

1−α 1 , α 2 c y±2

and

(70b)

κq . (70c) 1 + κR In Eq. (70a) the coefficient B is not completely calculable; B results from the terms of improved theories that are neglected in Eq. (70a) and might include the consequence of unknown corrections needed in the experiments; S2 and κ R are given in Table I. Figure 5 shows the features of transference numbers. The symmetry of Eq. (70a) is obvious. If t+∞ > 0.5, the transference numbers increase with increasing concentration and decreasing temperature, and vice versa if t+∞ < 0.5. Transference-number measurement yield the − ln y± =

FIGURE 5 Temperature dependence and concentration dependence of cationic transference numbers of methanol solutions of Me4 NSCN ( ❤) and KSCN (♦) at various temperatures. The full lines are the computer plots according to Eqs. (70). [From Barthel, J., Stroder, ¨ U., Iberl, L., and Hammer, H. (1982). Ber. Bunsenges. Phys. Chem. 86, 636–645.]

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quantities ti∞ by extrapolation as the only characteristic quantities of the electrolyte solution; the small concentration dependence of the transference numbers for symmetric electrolytes does not permit the determination of association constants, in contrast with conductance measurements. The most important application of transference-number measurements for symmetric electrolytes is the determination of precise single-ion conductances. Transference numbers were also successfully applied in investigations of ion aggregates, which contribute to ion mobility by their charges, such as ion pairs of nonsymmetric electrolytes and triple ions of symmetric electrolytes. E. Triple-Ion and Higher-Ion Aggregate Formation For low-permittivity solutions, the highest concentration for which pairwise additivity of the potential functions is reasonable in MM-level Hamiltonian models is found at very low concentrations, for example, 10−4 M in Fig. 6. Figure 6 shows the dependence on concentration and temperature of the molar conductivity of 1,2dimethoxyethane solutions of LiBF4 from infinite dilution to saturation. The plots of  versus c1/2 show a minimum at moderate concentrations and a maximum at high concentrations. Although the minimum is only weakly dependent on temperature, the maximum exhibits a strong displacement. The minimum is a general feature of bilateral triple-ion formation: −→ [C+ A− C+ ]+ ; K + [C+ A− ]0 + C+ ←− (71a) T

and −→ [A− C+ A− ]+ ; A− + [C+ A− ]0 ←−

K T− ,

(71b)

where commonly the two formation constants K T+ and K T− are supposed to be equal. The maximum results from the competition of ion aggregates of various types. The conductivity equation of Fuoss and Kraus, y± c1/2 1 − S(∞ )−3/2 [c(1 − /∞ )]1/2 =

∞ 1/2 KA

+ λ∞ T

KT 1/2 KA

(1 − /∞ )c,

(72)

is the appropriate equation for reproducing the conductivity curve up to concentrations near the conductivity minimum (cmin = 2.14 × 10−2 mol dm−3 at 25◦ C in Fig. 6). In Eq. (72), y± is the mean activity coefficient of the free ions; S the limiting slope (see Table I); λ∞ T the limiting value of the triple ions C+ A− C+ and A− C+ A− ; and K A and K T = K T+ = K T− are the equilibrium constants of ionpair, Eq. (63), and triple-ion, Eqs. (71), formation. At concentrations far below the conductivity minimum (c  10−3 mol dm−3 in Fig. 6), triple-ion formation can be neglected. Data analysis is possible with the help of Eqs. (65), in agreement with pairwise additive potential functions, and yields the values of ∞ in Table III. The ion-pair association constants K A(1) of these plots agree well with the K A(2) determined independently at higher concentrations with the help of Eq. (72), which takes into account both ion-pair and triple-ion formation. No method is known for the determination of the values of λ∞ T ; data analysis yields only the product λ∞ K , in which the quanT T ∞ tity λ∞ is commonly estimated to be 2 /3. Both ion-pair T and triple-ion formation decrease with decreasing temperature in accordance with increasing solvent permittivity. The conductivity equation for electrolytes undergoing unilateral triple-ion formation, Eq. (71a) or Eq. (71b), is given by the relationship 2 (y±2 )2 c(1 − /∞ ) 1 − S(∞ )2 [c (1 − /∞ )]1/2 =

(∞ )2 KT c  ∞ ∞ + 2λT  − (∞ )2 (1 − /∞ ). KA KA (73)

TABLE III Limiting Conductivities and Ion-Pair and TripleIon Formation Constants of LiBF4 Solutions in 1,2Dimethoxyethane at Various Temperatures from Conductivity Measurements

FIGURE 6 Ion-pair, triple-ion, and higher-ion aggregate formation exemplified by measurements of the equivalent conductivity of LiBF4 in 1,2-dimethoxyethane solutions at 25◦ C and −45◦ C. [From Barthel, J., Gerber, R., and Gores, H.-J. (1984). Ber. Bunsenges. Phys. Chem. 88, 616–622.]

Temperatures [◦ C] ∞ /(S cm2 mol−1 )

−45 46.5

−25

−5

68.8

94.1

15

25

123

139

10−6 K A /(mol−1 dm3 )

0.57

1.7

5.1

14.4

10−6 K A /(mol−1 dm3 ) K T /(mol−1 dm3 )

0.53

1.7

5.0

13.9

23

28

30

(1) (2)

15

19

23

24

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Triple-ion formation is commonly restricted to lowpermittivity solvents (ε < 15), but it is also known in highpermittivity solvents as a consequence of noncoulombic interactions.

IV. VISCOSITY The concentration dependence of the viscosity η of electrolyte solutions up to moderate electrolyte concentrations is commonly represented by the equation √ η(c) − 1 : A c + Bc. ηs

(74)

In Eq. (74), ηs is the viscosity of the solvent, η(c) is the viscosity of electrolyte concentration c, and A and B are constants. Coefficient A is available from the theoretical z+ z− limiting law for Cν+ Aν− electrolytes: 

 √  1 p¯ 2 1 − p ∗ e02 NA η(c) − η = , κ p¯ + 4 √ 2 2 p¯ 1 + p ∗ 480π ν+ z + ν− z − (75) where 3 3 z+ z− p¯ = ν+ ∞ − ν− ∞ ; λ+ λ−

p¯ = ν+ 2

2 z+ ∞ λ+

2

+ ν−

2 z− ∞ λ−

2

(76a,b) and p∗ =

∞ ν+ z + λ∞ 1 + − ν − z − λ−    . 2 2 ∞ ν+ z + + ν− z − λ∞ + /z + − λ− /z −

(76c)

κ is given by Eqs. (22). A rough estimation based on the approximation ∞ z + λ∞ + = |z − |λ− shows that the relative viscosity increases proportionally to the ratio of the radii of ion and ion cloud. Coefficient B is an empirical quantity which reflects the effects of ionic size, solvent structure, and ion–solvent interactions. Commonly it is split into ionic contributions B = ν+ B+ + ν− B− ,

(77)

which are related to crystallographic radii and structure parameters of ion–solvent interactions known from molar volumes.

V. TRANSPORT EQUATIONS FOR CONCENTRATED ELECTROLYTE SOLUTIONS Today, empirical transport equations of concentrated electrolyte solutions are available, as well as equations which

are rigorous statistical mechanical approaches. Only a few of those that have attracted the interest of applied research and engineering science are treated here. Three classes of transport equations can be found in the literature: molten salt approaches, empirical extensions of the equations for dilute solutions, and empirical equations just for fitting measured data. Molten salt approaches such as the Vogel-FulcherTamman (VFT) equation have been used repeatedly for analyzing the temperature dependence of transport properties W (T ) such as diffusion, conductance, and fluidity, or of relaxation processes:   B . (78) W (T ) = A exp − R(T − T 0 ) Equation (78) can be deduced from the equilibrium distribution of an isothermal, isobaric ensemble of cooperatively rearranging domains in the liquid, which can undergo a transition to a new configuration without configurational change at and outside its boundary. At the glass transition temperature T 0 of the system the configurational part of entropy vanishes. It is assumed that the transition of the supercooled melt to the glass is a type of second-order transition to obtain Eq. (78), where B is a temperature-independent energy term of transport, R is the gas constant, and A is a temperature-independent quantity, depending on the composition of the solution. Equations based on empirical extensions of the equations for dilute solutions use the fact that the viscosity of the system is the most important effect on the transport properties and introduce appropriate viscosity corrections. A. Empirical Equations The representation of physical properties of electrolyte solutions by the use of fitting equations is commonly executed with polynomials of concentration, temperature, pressure, and so forth, or with mathematical functions known for the appropriate representation of the shape of the experimentally determined curves. One of the most useful expressions of this type is given by Amis and Casteel for the specific conductivity of concentrated solutions:   κ m m−µ 2 = b(m − µ) − a . (79) κmax µ µ It makes use of four parameters (κmax , µ, a, b) for the representation of the measured data over wide concentration ranges and reproduces well the maximum of specific conductivity κmax and its position µ (see Figs. 7 and 9); a and b have no physical meaning. The specific conductivity of concentrated electrolyte solutions and its temperature dependence are of crucial

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For data analysis of specific conductivities based on the VFT equation, Eq. (78), the glass transition temperature of the electrolyte solution at molality m, T 0 (m), is assumed to be T 0 (m) = T 0 (0) + am + bm 2 , T 0 (0) = lim T 0 (m). m→0

(81)

In Eq. (81), T 0 (0) is the glass transition temperature of the pure solvent. The temperature dependence of specific conductivity at molality m, κm (T ), is then given by the relationship   Bm(κ) (κ) κm (T ) = Am exp − . (82) R[T − T 0 (m)]

FIGURE 7 Specific conductivity of LiBF4 in propylene carbonate at various temperatures from dilute solutions to saturation. Points: experimental data; lines; Eq. (79). [From Barthel, J. (1985). Pure and Appl. Chem. 57, 355–367.]

A similar equation is obtained for viscosities and other transport properties. The glass transition temperature of the pure solvent T 0 (0), obtained by extrapolation (m → 0), is found to be independent of the solutes in a given solvent and equal to that from viscosity measurements, which shows that the glass transition temperature is the appropriate reference temperature for transport processes in the liquid state. Using this result in Eq. (82) yields the further important

interest in technology. Figure 7 shows the features of specific conductivity. The maximum of specific conductivity κmax is a feature of every electrolyte solution permitting sufficiently high solubility of the solute. It follows from the competition between the increase dc of the electrolyte concentration and the decrease d of the ion mobility when the electrolyte concentration increases. The variation dκ of the specific conductivity is given by the relationship dκ = dc + cd.

(80)

Equation (80), following from the definition of molar conductivity  ∝ κ/c [Eq. (55)], shows that maximum specific conductivity κmax is attained at a concentration µ, at which dκ equals zero. The maximum of specific conductivity κmax and the concentration µ at which it is attained are correlated. Figure 8 indicates linear correlations as observed for propylene carbonate solutions at various temperatures. This correlation is due to the existence of an energy barrier depending on the temperature and on the solvent parameters, particularly viscosity. At concentration µ, the electrolyte shows an activation energy for the transport process equivalent to that of the barrier. The concentration µ at maximum specific conductivity decreases with decreasing temperature, cf. Fig. 8, which proves that viscosity is the most important conductancedetermining factor.

FIGURE 8 Linear correlation of maximum specific conductivity κmax and corresponding values of µ exemplified by propylene carbonate solutions of various 1,1 electrolytes at 25◦ C, −5◦ C, and −35◦ C. (1) LiBF4 , (2) LiClO4 , (3) Bu4 NPF6 , (4) KPF6 , (5) Pr4 NPF6 , and (6) Et4 NPF6 . [From Barthel, J., and Gores, H.-J. (1985). Pure and Appl. Chem. 57, 1071–1082.]

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λi =

λi∞

δµel 1 + ∞i µi



δE 1+ E

 (83)

after the second-order effects are neglected. The first-order approximation to the electrophoretic effect can be inferred from the electrophoretic velocity correction δµiel . The MSA pair distribution functions permit an easy extension of Henry’s law of electrophoretic mobility, cf. Eq. (60b), δµiel  kT , ∞ =− ∞ µi 3π ηDi 1 + σ

FIGURE 9 Specific conductivity κ versus molarity m at 25◦ C for Bu4 NClO4 in mixed solvents propylene carbonate–acetonitrile (mole fraction of acetonitrile is indicated on the right side of each curve). Points: measured data; full lines: Eq (79); dashed lines: MSA equation.

feature that the temperature coefficients of conductivity for all electrolyte solutions at infinite dilution in a given solvent, and of viscosity, are equal at every temperature; that is, infinitely dilute solutions are corresponding states in terms of transport energies. Suffice it to note that the values extrapolated toward zero concentration of transport energies from highly concentrated solutions based on the VFT equation equal those obtained from the conductivity equations based on MM-level Hamiltonian models. B. Statistical Mechanical Approaches

(84)

where µi∞ is the electrophoretic velocity, Di∞ is the ionic diffusion coefficient at infinite dilution [deducible from λi∞ ; see Eq. (25)], and  is Blum’s screening parameter of the MSA: κ 2 = . (85) (1 + σ ) In Eq. (84) σ is an average ionic diameter calculated from the ionic diameters σi and the ion concentrations ci :  2 i z i σi ci σ =  . (86) z i2 ci κ is Debye’s parameter [Eqs. (22)]. The first-order relaxation effect is obtained by the solution of the continuity equation at this level, which yields β 2 e02 |z i z j | δE 1 =− E 4π ε0 ε 6kT σ (1 + σ )2 ×

β2

1 − exp(−2βσ ) , + 2β + 2 2 [1 − exp(−βσ )]

(87)

where 2 ∞ 2 ∞ e02 NA ci z i Di + c j z j D j . ε0 εkT Di∞ + D ∞ j

Extended laws are available for the variation with concentration of the transport coefficients of strong and associated electrolyte solutions at moderate to high concentrations. Like the CM calculations, this work is based on the Fuoss-Onsager transport theory. The use of MSA pair distribution functions leads to analytical expressions. Ion association can be introduced with the help of the chemical method. A simplified version of the equations, by taking average ionic diameters, reduces the complexity of the original formulas without really reducing the accuracy of the description and is therefore recommendable for practical use for up to 1-M solutions.

Association can be included in the extended theory with the help of the CM, which yields the association constant given by Eq. (65b). For the calculation of the portion (1 − α) of ion pairs, Eq. (65b) is subjected to an iteration process beginning with yIP = y± = 1:

C. Conductivity Equations

(89) The single-ion and the ion-pair activity coefficients of MSA are made up by an electrostatic part and a hardsphere contribution:

For a completely dissociated electrolyte, the  appropriate equivalent conductivity expression,  = λi , follows from the ionic conductivities

β2 =

(88)

1/2  2 yIP 1 4yIP yIP 1−α =1+ − + . 2 K A2 c2 y±4 2K A cy±2 4K A cy±2

ln yi = ln yiel + ln yihs ;

i = +, −

or

IP.

(90)

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The electrostatic part yiel is given by the expression ln yiel = −

1 e02 z i2  , 4πε0 ε kT 1 + σ

(91)

el that is, yIP = 0 for a symmetric electrolyte. For the hardsphere contribution, only the ratio  hs 2 y± 1 − 0.5ξ πσ 3 NA  cj = ; ξ = (92) hs (1 − ξ )3 6 yIP

is needed for the calculation of the association constant. The data analysis with the help of Eqs. (83), (84), (87), and (88) where association constants are used with MSA activity coefficients [Eqs. (90), (91), (92)] yields a good reproduction of experimental data up to molar concentrations. As an example, Fig. 9 shows the conductivity of Bu4 NClO4 in the mixed solvent system acetonitrile– propylene carbonate. Comparison is made of measured data with MSA and the Amis-Casteel equation, which both exactly reproduce the conductivity maximum at every solvent composition. D. Self-Diffusion The self-diffusion coefficient is computed from the interdiffusion coefficient of two isotropically different species or from tracer diffusion, where a labeled ion in tracer amount diffuses in a large excess of another electrolyte of the same ion. The diffusion coefficient Di of the species Xi is given by the expression

 δki , (93) Di = Di∞ 1 + ki where ki is the diffusive force acting on an ion Xi and δki is the relaxation force. In the framework of MSA, the first-order contribution to the relaxation effect for the ionic species Xi reads 2 z i2 e02 κ 2 − κdif δki 1  , =− 2 ki 4π ε0 ε 3k B T (1 + σ )2 κdif + 2(1 + σ )

(94) where 2 κdif =

e02 NA  cn z n2 Dn∞ . ε0 εkT n Di∞ + Dn∞

 is given by Eq. (85) and κ by Eqs. (22); σ is the average diameter, Eq. (86). E. Other Approaches Altenberger and Friedman have given a conductance equation based on an HNC approach which is also valid up to concentrations of about 1 mol dm−3 .

Ebeling and Kraeft developed a statistical theory for ion–dipole solutions (physical model) with the aim of taking into account ion–solvent interactions. Computer simulations such as molecular dynamics (MD) and Brownian dynamics (BD) permit the study of transport properties. Self-diffusion coefficients can easily be obtained by differentiation of mean-square displacements or by integration of the velocity self-correlation functions of the ion. In contrast, the evaluation of conductivity by means of cross-correlation functions is cumbersome and computer-time-consuming and can only scarcely be executed. The advantage of computer simulations is the possibility of obtaining transport data that cannot or can only barely be measured. It is possible in this way to simulate diffusion coefficients of solvent molecules in the ionic solvation shells and to compare them with those of the bulk solvent molecules and with those of the ions, or to study transport coefficients at different time scales.

SEE ALSO THE FOLLOWING ARTICLES CHEMICAL THERMODYNAMICS • ELECTROCHEMISTRY • ELECTROLYTE SOLUTIONS, THERMODYNAMICS • ELECTROPHORESIS • STATISTICAL MECHANICS

BIBLIOGRAPHY Barthel, J., Krienke, H., and Kunz, W. (1998). “Physical Chemistry of Electrolyte Solutions—Modern Aspects,” Steinkopf, Darmstadt/ Springer-Verlag, New York. Bernard, O., et al. (1992). J. Phys. Chem. 96, 398–403, 3833–3840. Bernard, O., Turq, P., and Blum, L. (1991). J. Phys. Chem. 95, 9508– 9513. Bockris, J. O., and Reddy, A. K. N. (1998). “Modern Electrochemistry,” Vol. 1, 2nd ed., Plenum, New York, London. Covington, A. K., and Dickinson, T., eds. (1973). “Physical Chemistry of Organic Solvent Systems,” Plenum, New York. Falkenhagen, H. (1971). “Theorie der Elektrolyte,” 2nd ed., Hirzel, Leipzig (Engl. ed., 1952). Friedman, H. L. (1985). “A Course in Statistical Mechanics,” Prentice Hall, Englewood Cliffs, N.J. Hansen, J. P., and McDonald, I. R. (1976). “Theory of Simple Liquids,” Academic Press, London. Justice, J. C. (1983). Conductance of electrolyte solutions. In “Comprehensive Treatise of Electrochemistry,” Vol. 5 (B. E. Conway, J. O. Bockris, and E. Yeager, eds.), pp. 233–337, Plenum, New York. Resibois, P. M. V. (1968). “Electrolyte Theory: An Elementary Introduction to a Microscopic Approach,” Harper & Row, New York. Robinson, R. A., and Stokes, R. H. (1970). “Electrolyte Solutions,” 2nd rev. ed., Butterworth, London. Spiro, M. (1984). Conductance and transference determinations. In “Physical Methods of Chemistry,” 5th ed. (B. W. Rossiter, J. F. Hamilton, eds.), Wiley (Interscience), New York. Turq, P., Barthel, J., and Chemla, M. (1987), “Transport, Relaxation and Kinetic Processes in Electrolyte Solutions,” Lecture Notes in Chemistry, Springer-Verlag, Heidelberg.

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Energy Transfer, Intramolecular Paul W. Brumer University of Toronto

I. II. III. IV. V. VI. VII.

Introduction Qualitative Dynamics Dynamics: Theory Statistical Approximations and Dynamics Experimental Studies Control of Dynamics Summary

GLOSSARY Adiabatic Process during which there is no change in the electronic configuration of the molecule. Constant of the motion Property of a system, expressible as a smooth function of the coordinates and momenta, whose numerical value remains unchanged during the course of the system dynamics. Ergodic Classical mechanical system in which a trajectory uniformly covers a specific surface in phase space. The physics literature utilizes this term to imply uniform coverage of the surface in phase space defined by fixed total energy. Integrable Also termed regular or quasiperiodic. Classical mechanical system characterized by the existence of a set of independent constants of the motion equal in number to the number of degrees of freedom of the system. Specific attributes of such systems are discussed in the text. Mixed state State of a system in which there is some information missing relative to a complete specification of the state.

Mixing Classical mechanical system that is ergodic and possesses additional properties associated with relaxation. Pure state State of a system in which all information about the state is known and specified. Relaxation Tendency of a system to evolve from a specific time-dependent initial state to a time-invariant final state. Statistical Qualitative term signifying models or dynamics with characteristics of ergodic and mixing behavior. Unimolecular decay Process in which an energized molecule breaks up into smaller molecular constituents. The molecular analog of nuclear fission.

INTRAMOLECULAR dynamics is the time evolution of rotational, vibrational, and electronic degrees of freedom of isolated individual molecules, that is, molecules in a collision-free environment. As in any mechanical system, one can consider the time evolution of any of a variety of system properties. Intramolecular energy transfer focuses

477

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I. INTRODUCTION Naturally occurring systems in the gaseous or liquid phase at typical temperatures are composed of vast numbers of atoms and molecules in constant motion. The well-defined system temperature is a reflection of the stored energy content, both in the form of intermolecular features such as molecule–molecule interactions and translational motion and in the form of motion internal to individual molecules. The latter, the motion of isolated molecules, is termed intramolecular dynamics. Included within this definition are both dynamics at energies below the dissociation energy of the molecule, in which case it remains perpetually bound, and dynamics at energies above dissociation, in which case the molecule can break up into different chemical products. Intramolecular energy transfer is the subset of intramolecular dynamics in which the focus of attention is on the flow of energy within the isolated molecule. As the simplest of models, one may imagine a linear molecule A B C as being three mass points coupled by springs. Initiating an oscillation of the model by stretching the A B spring and subsequently observing the time-dependent alteration of lengths of the A B and B C springs correspond to a simple model of an experiment on vibrational energy transfer. Indeed, modern experimental techniques allow for the creation of collision-free environments in which such isolated intramolecular dynamics may be studied and possibly externally influenced. The isolated molecule, comprised of a set of N atoms bound by interatomic forces, is a complex physical system. It possesses 3N degrees of freedom related to nuclear motion, three being associated with center-of-mass translation, three with rotation (or two if the molecule is linear), and the vast remainder with vibration. The molecule also has electronic degrees of freedom associated with the configuration of its electrons. Additional degrees of freedom, related to internal nuclear composition, are not readily altered at the energies of interest in molecular chemistry and physics and can therefore be neglected. Thus, even the simplest description of the dynamics of a typical small molecule such as benzene (C6 H6 ), which regards it as being composed of 12 atoms, involves the complex motion of 33 interacting degrees of freedom. It is convenient to visualize intramolecular dynamics in terms of three steps, not necessarily independent. First, the molecule is prepared in a time-dependent state by any of a variety of means (e.g., collisions, chemical reaction, laser excitation). Second, the molecule evolves in time in accor-

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dance with quantum mechanics. Third, the time-evolved state of the system is analyzed, that is, measured. Explicit emphasis in a measurement (and sometimes preparation) on the flow of energy among subcomponents of the molecule (e.g., chemical bonds or rotational degrees of freedom) constitutes the study of intramolecular energy transfer. It is advantageous to distinguish two qualitatively different types of intramolecular energy flow, which we shall term reversible and irreversible. In the former, energy flows from one part of the molecule to another, but then subsequently returns, reforming the initial state. That is, the system shows no long-term trend toward a final redistribution of energy among subcomponents of the molecule. In the latter, there is a transfer of energy within the molecule, with a general trend toward a stationary final state. The latter behavior is that associated with relaxation, or statistical, dynamics and has historically been assumed to occur in highly excited molecules. Conditions under which molecular systems appear to display reversible vs irreversible energy transfer are discussed in detail later. Intramolecular dynamics and intramolecular energy transfer have been, and continue to be, areas of intense scientific interest. Such interest falls into two categories, loosely termed “practical” and “fundamental.” From the practical viewpoint, note that the outcome of a molecular process is heavily linked to the flow of energy in the molecular participants. That is, an understanding of intramolecular energy transfer proves central to the interpretation of chemical processes and their dependence on system conditions. For example, unimolecular decay, in which an isolated energized molecule dissociates into chemical products (e.g., ABC → A + BC, where ABC, A and BC are arbitrary molecules), occurs via the concentration of sufficient energy in the A BC bond. Further practical interest arises from important developments in laser technology that permit the introduction of energy into molecules in a variety of controlled ways. This opens the possibility of externally influencing the outcome of a molecular process by varying the initial mechanism of preparation (e.g., producing AB + C, rather than A + BC, from ABC by judicious preparation of the initial state). Indeed, theoretical developments over the past three years have led to several proposals for controlling chemical reactions in this manner (see Section VI). From a fundamental viewpoint, intramolecular energy transfer links to three basic scientific issues: (1) reversible vs relaxation phenomena, (2) quantum vs classical chaos, and (3) quantum/classical correspondence. These are briefly introduced in Section II, where it becomes clear that intramolecular energy-transfer studies provide a useful laboratory for the study of fundamental questions in these areas.

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Experimental and theoretical studies of intramolecular energy transfer and intramolecular dynamics have a long history. This article, designed as an introduction rather than as a survey, focuses on recent directions in this field of research. Of specific interest are insights gained into general rules for understanding and measuring intramolecular dynamics. For this reason we cite only a few sample computations to illustrate relevant general features and provide only a brief qualitative discussion of experimental methods. References to more historical interests in the field are provided at the end of this article. The organization of this article is as follows. Section II is designed to provide qualitative insight into intramolecular energy transfer via two subsections. The first, Section II.A, discusses selected computational results on two molecular models, and the second, Section II.B, qualitatively introduces the fundamental problems alluded to above. The reader who is interested in a qualitative picture is urged to first focus on these sections. Section III contains a description of isolated molecule dynamics from both the quantum and the classical viewpoints. Emphasis here is on several general features of classical and quantum intramolecular dynamics. Finally, two brief sections, Sections IV and V, discuss statistical approximations to intramolecular energy transfer and the nature of modern experiments designed to probe molecular motion. Space limitations, coupled with the author’s intention to provide a useful introductory treatment, have led to restrictions on the material that can be covered. Thus, we focus throughout this article on adiabatic processes, that is, dynamics that take place without change in the electronic configuration of the molecule. When this is not the case, a remark to this effect is made. In addition, we assume throughout that the radiation field, be it associated with radiative absorption or emission, is sufficiently weak to be treatable as a perturbation. Further, the field of intramolecular dynamics is replete with model approaches that, albeit reasonable, have not been justified either theoretically or experimentally—the latter due principally to technological limitations. The modern focus on accurate dynamics is emphasized in this article, with the consequence that such simple models are necessarily slighted.

separate steps. In the first step, the forces between the atoms in the molecule are determined or modeled, while in the second step one considers the dynamics determined by these forces. This dynamics is done either quantum mechanically, which is correct but difficult, or via classical mechanics, which is often a good approximation to the quantum result. In either case, the forces describing the dynamics are sufficiently complex to necessitate numerical computer solutions. As an introduction to the nature of intramolecular dynamics and to issues of interest in this area, we consider two examples. A. Two Model Calculations As a first example, consider nuclear motion in a fouratom system, A B C D. Two qualitatively different energy ranges are possible. In the first, the system is provided with sufficient energy to induce vibrational and rotational motion but insufficient energy to break any of the bonds. This is bound-state intramolecular dynamics. In the second regime, there is sufficient energy to allow molecular dissociation to one or more of the molecular products (e.g., A B + C D). Typically, each of the interatomic bonds will have a different dissociation energy, the energy required to break the bond. Thus, several dissociation “channels” are possible, such as A B C D → A B+C D

E1

A B C D → A+B C D

E2

A B C D → A+B+C+D

E3

where the lowest energy required for each of the particular processes is arbitrarily labeled E 1 , E 2 , etc. Consider the specific case of NaBrKCl for which theoretical, classical dynamics studies are available at energies where two dissociative channels are energetically accessible. (Questions as to the validity of the classical picture are relegated to later sections.) This system possesses attractive forces between the atoms such that the bound NaBrKCl species lies at an energy of approximately 40 kcal/mol below NaBr + KCl or NaCl + KBr. Specifically, consider the case where energized NaBrKCl is formed by the collision NaBr + KCl → NaBrKCl

II. QUALITATIVE DYNAMICS Information regarding intramolecular dynamics is available from three sources: experimental studies on specific molecular systems, theoretical computations on specific systems or models, and formal studies of typical (“generic”) systems. At present, the latter two provide considerably greater detail than the first and involve two

with sufficient energy to dissociate as NaBrKCl → NaCl + KBr NaBrKCl → NaBr + KCl. The initial collision between NaBr and KCl is here regarded as the preparatory step to the subsequent intramolecular dynamics and decay of the intermediate

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FIGURE 1 Time dependence of each of the six interatomic distances during a trajectory describing the collision of NaBr with KCl. System energy is E = 0.0912 a.u. [From Brumer, P. (1972). Ph.D. dissertation, Harvard University.]

NaBrKCl species. An analogy to the collision, bound molecule dynamics, and subsequent decay is a system of two balls, attached by a spring, colliding with two other such balls on a billiard table containing a deep hole in the center. The springs are breakable and endowed with the ability to exchange between pairs of particles. The basic dynamics in classical mechanics is embodied in trajectories, that is, the dynamics following precise specification of the system initial conditions. Comparison with experiment then involves averaging results over a set of trajectories consistent with the specific experimental conditions. Two such trajectories are considered in Figs. 1 and 2, where we show the time dependence of the distances between all the atoms during the collision. In one case the initial conditions lead to a long-lived intermediate, and in the other case they lead to a short-lived species. Consider the shorter trajectory (Fig. 1) first. A careful examination of the figure indicates the initial oscillations of the bound NaBr and KCl, with the other atomic distances shrinking in time as the two diatomics approach one another. The collision between them occurs at approximately 12,000 atomic time units (denoted atu, where an atomic time unit is 2.4 × 10−17 sec) and is promptly followed by decay to NaCl + KBr. Thus, for these particular initial conditions, the intermediate species NaClKBr is only a fleeting phase in the collision. Results in Fig. 2 are in sharp contrast, showing a long-lived intermediate that displays complex dynamics in a bound energized molecule.

Energy Transfer, Intramolecular

Here, the system forms a bound four-body molecule at approximately 38,000 atu that persists until t = 240,000 atu. During this time the various bond distances go through a variety of values between 4 and 11 Bohr radii, a sort of vibrant dance of four bound particles. Decay follows thereafter to the product diatomics. In a classical picture, knowledge of the distances and momenta of the atoms as a function of time permits complete knowledge of all properties. Figure 3 shows, for example, the calculated energies in each of the alkali–halide bonds during the course of the dynamics for the same collision as seen in Fig. 2. Figure 4 contains an analogous picture of the rotational energy of the four-atom system. The dynamics is clearly marked by extensive energy exchange between the bonds and among rotation and vibration. These examples provide a picture of the complexity of individual trajectories in bound intramolecular dynamics. Such a trajectory emerges from a precise specification of initial particle momenta and coordinates. A comparison with real phenomena requires, however, averaging over all initial conditions not precisely specified in the given experiment. For example, the experiment may only have initially fixed the translational and internal energies of the colliding partners. Figure 5 provides a typical example of the result of averaging over a set of trajectories where the trajectory in Fig. 2 is one participant. Here, we show a typical measurable in the decay of NaClKBr, that is, the probability of independently finding the products NaCl and KBr with particular vibrational energies. Note that the results are simpler than the complex underlying trajectories. This is a result of both the comparatively simpler question being asked and the averaging implicit in the computation. The details of the dynamics ongoing during the course of the collision also often simplify as a result of averaging over a set of trajectories. Consider, for example, Fig. 6, which shows the average vibrational energy in each of the four alkali–halide bonds during the collision; in this case the calculation only shows trajectories prior to dissociation. The results are to be compared to Fig. 3, although the overall energies in the two figures are somewhat different. One sees that each of the bond energies changes rapidly over the initial time period but then tends to level out. This is an example of apparent intramolecular energy relaxation during the course of the dynamics. That is, the system seems to reach, at least with respect to these variables, a relatively invariant state reminiscent of the behavior of macroscopic systems relaxing to equilibrium. The observation of relaxation behavior is, of course, not ubiquitous and is dependent on gross system conditions (e.g., total energy, total angular momentum, etc.). Understanding conditions under which intramolecular dynamics may be approximated by a statistical model is indeed one of the

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FIGURE 2 Similar to Fig. 1 but at a lower energy, E = 0.0227 a.u. [From Brumer, P. (1972). Ph.D. dissertation, Harvard University.]

main themes in intramolecular energy transfer. Such statistical models have both advantages and disadvantages. They are advantageous in that they considerably simplify the description of the dynamics. They are disadvantageous in that they imply that the final state of the system is relatively insensitive to the initial state. That is, the outcome of a chemical event is not readily influenced by altering initial conditions. The example discussed here displays a number of relevant features. Clearly noticeable is the complexity of individual trajectories modeling long-lived dynamics, as well as the possibility of simplifications that result if a statistical description that includes relaxation applies. The participation of rotations as well as vibrations in the dynamics is also evident. In addition, it makes clear the important

role of the relative rates of intramolecular energy transfer and other competitive processes. That is, the degree of intramolecular energy flow within the molecule depends on the length of time the system exists as a bound entity, as well as on some (as yet undefined) “rate of intramolecular energy flow.” In cases where competitive processes such as dissociation (or the ever-present radiative emission) are possible, effective intramolecular energy flow requires a larger rate of energy transfer than of competitive processes. (Note, however, that in the case of competitive dissociative or other nuclear rearrangement processes, both the energy transfer and the competitive process are governed by the same set of underlying dynamical equations for the nuclear motion. This leads to a natural difficulty associated with attempting to partition

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FIGURE 3 Vibrational energy in each of the four alkali–halide bonds, as a function of time, for the trajectory shown in Fig. 2. [From Brumer, P. (1972). Ph.D. dissertation, Harvard University.]

the dynamics into distinct parts labeled intramolecular dynamics and dissociation.) The computation described above is completely classical: the nuclear motion is assumed to be well described by Newton’s equations. The extent to which classical mechanics provides a useful description of intramolecular energy flow is another focus of current research in this area. As one example of the validity of classical mechanics, consider the bound-state dynamics of a three-atom system confined to a line, that is, A B C. Computations on the case where the A B and B C bonds are anharmonic have been performed using both classical and quantum

mechanics, where the initial state is a mixed state (see Section III.C). One useful measure of system dynamics is the probability of the system returning to the state from which it started. Figure 7 shows the time dependence of this probability for one case, where the quantum and classical results are seen to be in excellent agreement. In sharp contrast is the comparison shown in Fig. 8, which corresponds to the probability of return to another, higher energy initial mixed state where classical–quantum disagreement is substantial. The effect has also been seen for the higher energy decay of A B C to AB + C as shown in Figs. 9 and 10. Specifically, the probability of the

FIGURE 4 Total rotational energy associated with the NaClKBr system during the trajectory shown in Fig. 2. [From Brumer, P. (1972). Ph.D. dissertation, Harvard University.]

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FIGURE 5 Probability of producing a specific vibrational state of NaCl or of KBr from collisions of NaBr with KCl at E = 0.0411 a.u. [From Wardlaw, D. (1982). Ph.D. dissertation, University of Toronto.]

system remaining in its initial state is shown in Figs. 9 and 10 for dynamics initiated in two different mixed states. Once again, agreement is achieved between classical and quantum mechanics in one case, but there is substantial

disagreement in the other, with the classical being more statistical. The origin of this difference lies, in this case, in the existence of so-called quantum trapping states, which lead to classical results that behave more statistically than the quantum. Such states occur when there is a very asymmetric distribution of energy among the bonds in the molecule. Although here classical mechanics is more statistical than quantum mechanics, this is not always the case. The computations described above are useful in that they provide some qualitative insight into the nature of intramolecular dynamics and energy flow in particular systems. They are, in essence, theoretical experiments on given systems in that they provide only hints of the general rules that govern rates, nature, and degree of intramolecular energy interchange. In the sections that follow, we describe the current state of understanding on the general principles that underlie these processes. Prior to doing so, we emphasize, in the next section, some of the fundamental issues related to intramolecular energy transfer that have been alluded to above. B. Fundamental Issues: Qualitative Overview The goal of science is to provide a qualitative and quantitative description of natural phenomena. Such a description

FIGURE 6 Average energy in each of the four alkali–halide bonds as a function of time. Here, the average values are obtained from a set of NaBr + KCI trajectories. “Error bars” show the range of values associated with the set of trajectories incorporated in the calculation. [From Brumer, P. (1972). Ph.D. dissertation, Harvard University.]

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FIGURE 7 Classical (dashed) and quantum (solid) probability, as a function of time, of a coupled Morse oscillator system remaining in its original state. [From Kay, K. (1980). J. Chem. Phys. 72, 5955.]

is most useful if it is as simple as possible. For example, there is little reason to invoke relativistic quantum mechanics to describe planetary motion; classical mechanics will normally suffice. Furthermore, such a scientific description is most useful if it is computationally tractable. Thus, for example, thermodynamics often provides a more useful route to disallowing certain processes in a bulk system than does a full dynamics calculation involving the underlying Avogadro number of molecules. It is in this spirit that the utility of two approximate descriptions—classical mechanics in lieu of quantum mechanics, and statistical approaches in lieu of full dynamics calculations—is a central theme in contemporary intramolecular dynamics. Indeed, they intertwine in an interesting fashion. Consider first the essential difference between a dynamics description and a statistical description of a system. By

FIGURE 8 Same as Fig. 7, but for a different initial state. [From Kay, K. (1980). J. Chem. Phys. 72, 5955.]

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the former we mean a description based on Newton’s equations in classical mechanics or Schrodinger’s equation in quantum mechanics. By the latter we mean descriptions characteristic of nonequilibrium statistical mechanics, such as the Boltzmann or Fokker–Planck equation. Dynamics (either quantum or classical) is based on a set of basic equations that are time-reversal invariant. This property means that the final state of a process may be time reversed to recover the initial state. It implies that the final state of dynamical evolution contains all the information associated with the initial state. As a consequence, relaxation to a final equilibrium state, independent of the fine details of the initial state, does not occur, and hence, the final state may be a sensitive function of the initial state. Relaxation to equilibrium is, however, a familiar feature in macroscopic systems, and the equations of statistical mechanics are designed to provide a nonmicroscopic description that encompasses the relaxation process. The link between the underlying time-reversible equations of motion and the macroscopic irreversible equations is not well established and has been the subject of extensive, long-standing discussions on the basic equations of nature. Typical questions include the following. Are time-reversible equations more fundamental than statistical relaxation equations, or do they have equal, but independent, roles as models of nature? Is observed relaxation a consequence of coarse graining associated with macroscopic measurements on intrinsically time-reversible systems? Isolated-molecule dynamics is expected to be a sufficiently elementary process to permit observation of microscopic reversibility in the dynamics and, hence, to display a dependence of the outcome of dynamics on initial conditions. This dependence is desirable since the ability to retain information about initial conditions is necessary in order to achieve the technologically desirable goal of externally influencing chemical reactions. However, a great many experiments, perhaps with insufficiently well-characterized preparation and measurement, have indicated that time-irreversible relaxation is a useful model for many intramolecular processes. Thus, isolatedmolecule intramolecular dynamics serves as a laboratory for the study of the inter-relationship between irreversible relaxation behavior in systems that are fundamentally describable by time-reversible equations of motion. It also presents an experimental challenge to prepare sufficiently well-characterized states to observe time reversibility and sensitivity to initial conditions. Two further issues of fundamental importance, that of quantum–classical correspondence and that of “quantum chaos,” are intimately linked to studies in intramolecular dynamics. Extensive theoretical studies, beginning in the early 1960s, showed that a great many phenomena involving the dynamics of atoms and molecules are well

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FIGURE 9 Classical (dashed) and quantum (solid) probability of a coupled Morse–harmonic oscillator system remaining in its initial state. The energy is sufficient to allow dissociation. [From Kay, K. (1984). J. Chem. Phys. 80, 4973.]

described by classical mechanics. That is, atomic and molecular dynamics are at the borderline between classical and quantum mechanics, in that dynamic phenomena in molecules are often well approximated by classical dynamics. This is not always the case, as shown in the example in Section II.A, and so studies in intramolecular dynamics provide insight into the utility of classical mechanics as an approximation to quantum mechanics. Phrased in this manner, this appears to be a question of relative accuracy, with the expectation that the process is qualitatively similar in both mechanics. Recently, however, major qualitative distinctions between classical and quantum mechanics have been noted. Specifically, it is now known that even small classical mechanical systems (e.g., two

degrees of freedom) can display highly statistical behavior, termed chaos. One can show formally, however (as will be described further below), that such chaotic behavior is not possible in bound-state quantum mechanics. Further, typical semiclassical schemes that base quantization on the classical system motion (e.g., Einstein–Brillouin– Keller quantization) do not hold in this classically chaotic regime. Computational studies have indicated that chaotic behavior is expected in classical mechanical descriptions of the motion of highly excited molecules. As a consequence, intramolecular dynamics relates directly to the fundamental issues of quantum vs classical chaos and semiclassical quantization. Practical implications are also clear: if classical mechanics is a useful description of intramolecular dynamics, it suggests that isolated-molecule dynamics is sufficiently complex to allow a statistical-type description in the chaotic regime, with associated relaxation to equilibrium, and a concomitant loss of controlled reaction selectivity.

III. DYNAMICS: THEORY This section provides an introduction to the theory of classical and quantum intramolecular dynamics, with emphasis on general principles. A. The Hamiltonian A complete description of the dynamics of any molecular system is contained in the Hamiltonian H , which is the energy operator in quantum mechanics or the energy function in classical mechanics. In general, the Hamiltonian is a function of the electronic and nuclear degrees of freedom, as is the description of the system dynamics. This complex problem simplifies through the adoption of the Born– Oppenheimer approximation, which is the assumption that nuclear and electronic motion are independent due to their substantially different time scales and masses. This assumption allows one to first solve for the dynamics of the electrons and then obtain the forces experienced by the nuclei as determined by this fixed-electron configuration. Within the Born–Oppenheimer approximation, the nuclear Hamiltonian may be written in the form H (q, p) = T (p) + V (q),

FIGURE 10 As in Fig. 9, but for a different initial state. [From Kay, K. (1984). J. Chem. Phys. 80, 4973.]

(1)

where T (p) is the nuclear kinetic energy and V (q) is the nuclear potential energy. Classically, p is the momentum of the nuclei, whereas in quantum mechanics it is the momentum operator. In general, q has 3N components for an N -atom system. However, it is always possible to eliminate three coordinates corresponding to the center of mass

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of the system, and we shall assume this reduction to the internal molecular coordinate system has been carried out; thus, q denotes 3(N − 1) coordinates. Central to the nature of dynamics is the notion of coupled vs uncoupled degrees of freedom. Consider, for example, a system whose Hamiltonian is the sum of two terms, such that H (q, p) = h 1 + h 2 ,

(2)

where h 1 depends on a different set of coordinates and momenta than h 2 . Under these circumstances, the dynamics of the degrees of freedom in h 1 are independent of those in h 2 . That is, the total system is composed of two independent subsystems. Introduction of a coupling term, to give H = h 1 + h 2 + V (1, 2),

(3)

where V (1, 2) is a term dependent on the coordinates and momenta of both h 1 and h 2 , induces energy exchange and interrelated dynamics among the entire system. Major qualitative changes in the dynamics can result from small perturbations or couplings. It is important to note that although Eq. (3) is conceptually pleasing, there is no unique division into subsystems and into intersubsystem coupling for a given physical system. Rather, such a division must be motivated by an experimental or theoretical interest in a particular property, such as the energy flow between subsystems h 1 and h 2 . Central to the nature of intramolecular energy transfer is an understanding of the effect of the coupling V (1, 2) on the energy flow among the subsystems represented by h 1 and h 2 .

p(t) = A = const 1 (5)

B. Classical Mechanics

q(t) = q(t, A).

There are a number of formulations of classical mechanics, each providing different insights into its nature. For example, Hamilton’s method, used here, describes dynamics in terms of trajectories in generalized coordinates and momenta. Consider an M degrees of freedom system with system Hamiltonian H (q, p), where (q, p) is a complete set of M conjugate generalized coordinates and momenta. The time evolution of the system is given by Hamilton’s equations, dqi /dt = ∂ H/∂ Pi

phase space, with (q, p) as coordinates. A trajectory is a curve in this space parametrized by the index t. Modeling a realistic system necessitates producing a set of trajectories with varying initial conditions and looking at averages of system properties over this set. It is this trajectory technique, where V (q) is a model or realistic potential and where Hamilton’s equations are solved numerically on a computer, that has been extensively used to study intramolecular dynamics in small molecules. Results typical of those obtained were shown in Section II.A. The flexibility of Hamilton’s approach lies in the appearance of generalized canonical coordinates (p, q). As a consequence of their generality, one may seek out the set of coordinates and momenta within which the dynamics is most easily performed and understood. For example, a conservative Hamiltonian system may have, along any trajectory, a constant value of total angular momentum J. Such a quantity is said to be a constant of the motion and can prove useful as a momentum, since the equation of motion for J is particularly simple, dJ/dt = 0. Indeed, the idea of seeking constants of motion for use as coordinates or momenta is the central goal of the Hamilton–Jacobi approach to classical dynamics. The essential approach is simple. One seeks a set of M constants of the motion A of the system via a systematic procedure. Once found, these momenta are known to be constant along any trajectory, and the time dependence of the conjugate coordinates is generally simple. With the constants of the motion known, along with the procedure for generating them, it is a relatively straightforward algebraic problem to express the desired q(t), p(t) in terms of these constants and their conjugate coordinates. That is, we have

d pi /dt = −∂ H/∂qi

(4)

with i = 1, . . . , M. Specifying the state of the system at t = 0 via the initial conditions q(t = 0) = q0 , p(t = 0) = p0 then leads to a solution, a trajectory p(t), q(t), from which the time dependence of all system properties along that trajectory can be computed. The motion may be best visualized as taking place in a 2M-dimensional space, termed

Although we shall not deal with Hamilton–Jacobi theory here, the concept of a set of constants of the motion is vital to an understanding of the issue of intramolecular energy flow and statistical vs nonstatistical behavior. In essence, the number of global constants of the motion provides a method for grouping systems into general categories. Consider an M-degrees-of-freedom system. How many constants of the motion can we identify for a given trajectory? The answer is clearly 2M, with the simplest set consisting of the initial values of the coordinates and momenta that define the trajectory and that surely constitute a set of 2M equations for q(t), p(t) in terms of constants. That is, q(t) = q(q0 , p0 ; t)

p(t) = p(q0 , p0 ; t).

These constants of the motion, however, are of little interest. Although knowing them does identify the trajectory

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uniquely, it provides no reduction of the complexity of solving the full problem for q(t), p(t). Clearly, what we seek is a set of global constraints that define smooth surfaces in phase space on which the trajectories lie. Members of such a set of functions are called global constants of the motion. The number of global constants of motion that a system possesses provides a means of generally classifying the system behavior. 1. Integrable Systems Consider first the case in which an M-degrees-of-freedom system possesses M independent global constants of the motion, denoted K. Selecting these as the M generalized momenta K, with conjugate momenta Q, allows us to write the Hamiltonian as H (K). Hamilton’s equations then become d K i /dt = −∂ H/∂ Q i = 0 (6) d Q i /dt = ∂ H/∂ K i = ωi (K), where the last equation defines the frequencies ωi (K). Systems allowing this description are termed integrable, or regular, and possess a number of important properties: 1. Trajectories lie on M-dimensional surfaces in phase space whose topology is that of a torus. The tori are labeled by the values of K. For example, in a two-degreesof-freedom system, the motion lies on the surface of a doughnut (see Fig. 11). 2. The frequencies of motion about the independent directions on the torus are given by ω(K). 3. The time dependence of the coordinates and momenta for a given trajectory are given by the Fourier series (where n is a vector of M integers)  q(t) = qn (K)exp[in · ω(K)t] n

p(t) =



(7) pn (K)exp[in · ω(K)t].

n

Note then that trajectories in such systems come arbitrarily close, during the course of their dynamics, to their original starting position. For this reason, such dynamics is termed “quasiperiodic.” The vast majority of “textbook” problems dealt with in elementary and advanced analytical mechanics treatises are of this type. Examples include the hydrogen atom or the small-vibrations Hamiltonian, although these systems tend to be, in addition, separable, that is, of the form H = H (K 1 ) + H (K 2 ) + · · · . Trajectories in integrable systems are stable with respect to small changes in initial conditions. In particular, consider a trajectory [q(t), p(t)] emanating from initial conditions q(0) = q0 , p(0) = p0 and an initially close trajectory [q (t), p (t)] originating from q (0) = q0 + δ Q, p (0) = p0 + δ P with δ Q and δ P very small. Then define d(t) as the time-dependent “distance in phase space” between these two trajectories, d(t) = [ p1 (t) − p1 (t)]2 + [ p2 (t) − p2 (t)]2 + · · · + [q1 (t) − q1 (t)]2 + · · ·]1/2 .

This quantity measures the rate at which two nearby trajectories separate as a function of time. 4. Then, for regular systems, d(t) grows linearly with t, a relatively slow rate of separation characteristic of stability. All these features are, in a qualitative sense, indicative of essentially predictable motion in integrable systems. behavior is repetitious, or at least describable by a simple set of frequencies. The linear growth of d(t) suggests that knowledge of the behavior of a single trajectory allows prediction, over a fair length of time, of the behavior of its initially nearby neighbors. As a simple example of regular dynamics, to be embellished later, consider a system of two uncoupled oscillators, H = H1 ( p1 , q1 ) + H2 ( p2 , q2 ),

(9)

where the individual oscillators, with Hamiltonians Hi (qi , pi ) = T ( pi ) + V (qi ), have potentials terms of the Morse form, V (qi ) = Di [exp(−qi /ai ) − 1]2 .

FIGURE 11 Sample quasiperiodic trajectory in a two-degreesof-freedom system as it moves on the surface of a torus in phase space. The trajectory shown is actually periodic; in general, the trajectory will fill the entire torus surface.

(8)

(10)

Here, Di is the oscillator dissociation energy and ai is a system parameter. Equation (10) might, for example, model three atoms on a line where the coupling potential between the oscillators has been eliminated. The individual Hamiltonians H1 and H2 are then the conserved integrals, and their numerical values remain constant at their initial values throughout the dynamics. Since this is a two-degrees-of-freedom system and since two constants of the motion exist, the system is integrable.

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488 Consider a measurement on the dynamics of this system. If an experiment were able to begin with an initial state consisting of a single trajectory, then the time evolution would be on a torus in phase space and there would be no energy transfer between the subsystems H1 and H2 . Sufficiently long-time observations on the system would reveal repetitive quasiperiodic motion. Two remarks regarding such measurements are, however, important. First, if one is interested in the dynamics of a subcomponent other than H1 and H2—say one interrogates the time dependence of the harmonic oscillator p12 /2m 1 + q12 —then the measurement would reveal energy flow into and out of this subsystem. Quasiperiodicity would still be evident, however, after a suitable time. Second, if the time scale of the experiment is short compared to the relevant system frequencies, then this quasiperiodicity will not be manifest. The essential point then is that the nature of the measurement of interest determines whether the regular system behavior is fundamental to, or observable in, the particular experimental study. There is another feature of integrable systems that is important. Specifically, consider the concept of statistical behavior in dynamics. The fundamental features of such behavior are that a trajectory at energy E fills the entire volume of phase space associated with that energy E, that a set of trajectories relaxes to a long-time limit that no longer varies with time, and that the final state is dependent solely on the energy of the system. It is clear that the first of these properties is not satisfied by a regular system, since a trajectory lies on the surface of a torus of dimensionality M, whereas the constraint to constant energy would confine dynamics to a larger surface of dimension 2M−1. It would also appear, from the list of properties above, that a regular system does not relax. This is, in fact, not the case. That is, properties (1)–(4) constitute features of the trajectories of a regular system. As already remarked, however, typical comparisons with physical systems require information on the average behavior of the time development of a collection, or ensemble, of trajectories. It is therefore important to note that despite the quasiperiodic behavior of integrable systems, an ensemble of trajectories in a regular system can relax to a long-time, stationary distribution. The final relaxed state of the system is, however, intimately related to the initial conditions of the dynamics. This is clear from the simplest of considerations. That is, each of the trajectories in the set of trajectories retains its original values of the conserved quantities. Thus, the final state of the system will depend on more than just the total overall energy of the system. Each of properties (1)–(4) gives rise to useful computational tools for the theoretical identification of integrable behavior in models of molecular motion. Relationships

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to actual experimental techniques and measurements are, however, not well formulated. The regular system constitutes one major category of observable dynamical behavior. Systems that are integrable are well known and have been experimentally observed, as discussed later. A second major category of Hamiltonian systems emerges from formal ergodic theory, which defines a set of increasingly idealized statistical systems. Such systems (in terms of increasing statistical characteristics) are termed ergodic, mixing, K-systems, and Bernoulli systems. Each category imposes additional conditions, leading to requirements difficult to verify for realistic systems. Thus, they are to be regarded as idealized models of statistical motion. 2. Ergodic Consider first the integrable system where each trajectory lies on the surface of a torus. Two conditions are possible. In the first, the trajectory wraps about the torus and closes on itself without covering the torus completely. An example is shown in Fig. 11, where it is clear that this property arises if the frequencies of motion about the torus are related to one another by the relation n 1 ω1 + n 2 ω2 . Such a set of frequencies is said to be rationally related and results in the trajectory returning exactly to its original position. On the other hand, the frequencies on the torus may not be rationally related, in which case the trajectory fills the entire surface of the torus. Under such conditions the dynamics is said to be ergodic on the torus. This formal terminology does not correspond to the historical use of the term ergodic as found in the physics literature. There, ergodic tends to mean ergodic on the energy hypersurface, that is, on the (2M−1)-dimensional surface in 2M-dimensional phase space that results from constraining the system to constant energy. For clarity, we shall term this E-ergodic. Thus, the characteristic of an E-ergodic system is the existence of a single trajectory at each energy E that comes arbitrarily close to all points on the energy hypersurface. It is important to note, however, that this property does not ensure that the system displays irreversible relaxation during the course of the dynamics. A pictorial analog of possible motion of an ergodic system is provided by imagining a speck of carbon in a continuously stirred fluid. The carbon speck, representative of the system in phase space, moves throughout the fluid without constraint, but does not settle down to some long-time stationary state. 3. Mixing A system that is ergodic but has the rudimentary properties associated with statistical irreversible behavior is the mixing system. Such a system displays the following properties.

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Consider the system at energy E. Denote the average over the energy surface by the expression f , where f (q, p) is any dynamical property. Then, 1. lim f [q(t), p(t)] = f  t → ∞. 2. The correlation between any two dynamical properties, that is,

g[q(t), p(t)], f [q(0), p(0)] − g[q(t), p(t)] f [q(0), p(0)], goes to zero as t → ∞. 3. Subdivide the total phase space into regular regions of particular volume. Then the probability of going from region i to region j in the long-time limit depends only on the size of the phase space regions i and j.

tween nearby trajectories that grows exponentially, that is, d(t) = d(0)exp(kt), indicative of trajectory instability. This set of properties gives rise to useful theoretical indicators of irregular motion, but connections with actual experimental observables are not well established. From the viewpoint of measurement, if one were able to prepare a single trajectory as the initial state of an irregular system, then the subsequent measurement of any property, other than energy, would show continual variation with time. The trajectory would, in addition, display no tendency to return to the original state over any finite time. If one prepared an ensemble of trajectories, it would approach a long-time stationary distribution dependent solely on energy.

4. Typical Molecular Systems Thus, a mixing system satisfies a number of simple properties that are in qualitative agreement with statistical relaxation dynamics. A particle of soluble colored material stirred into water provides a pictorial analog of mixing dynamics. Once again, the fluid models the phase space. The system evolves over time to reach a final macroscopically invariant distribution of uniformly colored fluid throughout the container. It is unfortunate that the formal definitions of ergodic, mixing, etc. systems involve the infinite time limit. As a consequence, a system may, for example, still be mixing even if relaxation is not observed in the finite time associated with a realistic measurement. This limitation significantly reduces the practical utility of formal concepts such as mixing behavior. A host of other formal systems with additional, and hence stricter, requirements have been defined. Here, we only mention the C-system, which is ergodic and mixing and which possesses the important characteristic that the distance d(t) between any two initially close trajectories in phase space grows exponentially in time. This trajectory instability leads to the rapid parting of trajectories from one another and, hence, the inability to predict the dynamics of trajectories, even for a relatively short time period, from knowledge of the dynamics of their neighbors. A system that displays characteristics of mixing as well as exponential divergence of adjacent trajectories is termed irregular or chaotic. In contrast with the characteristic properties of a regular system, an irregular system displays (1) trajectories that lie upon the (2M−1)dimensional energy hypersurface in phase space (additional simple constants of the motion such as angular momentum may also be incorporated), (2) and (3) trajectory dynamics that cannot be written in terms of a Fourier series involving a simple set of discrete frequencies and their overtones and combinations, and (4) a distance d(t) be-

Both regular and irregular motions are extremes of behavior, and their relation to the dynamics of realistic systems has principally been established through numerical computer studies. These studies indicate that many, but certainly not all, molecular systems display behavior characterizable as regular at low energies and irregular at higher energies. The example of carbonyl sulfide, OCS, is shown in Fig. 12, where the percentage of phase space not showing exponential divergence is shown. The system is seen to display a transition to chaotic motion at an energy of approximately 14,000 cm−1 . By 20,000 cm−1 , close to dissociation, almost all of the phase space is irregular. To appreciate the origin of the regular behavior at low energies, we note two common approximations

FIGURE 12 Percent regular trajectories as a function of energy for model OCS. The symbol D denotes the molecular dissociation energy. [From Carter, D., and Brumer, P. (1982). J. Chem. Phys. 77, 4208.]

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in low-energy molecular motion. The first, rotation– vibration decoupling, assumes that the rotational and vibrational motions are essentially uncoupled at low energies, that is, that the Hamiltonian is the sum of vibrational and rotational parts: H = Hvib + Hrot .

(11)

Second, we recall the standard normal-mode procedure for small-amplitude vibrational motion wherein, at sufficiently small energy, the vibrational Hamiltonian is of the form Hvib = H0    1  2 Pi + λi Q i2 + V (P, Q), = 2 i

(12)

with V (P, Q) sufficiently small to be negligible. The Q, P are called normal coordinates and momenta. Thus, lowenergy vibration is well approximated by a sum of M harmonic-oscillator Hamiltonians. In some instances an alternative separable Hamiltonian, composed of the sum of bond Hamiltonians, provides a superior separable representation. In either case, the low-energy vibrational motion is regular and separable. The situation changes dramatically with increasing energy as V (P, Q) becomes larger and the system begins to exchange energy between the decoupled harmonic oscillators. The subsequent dynamics, as observed in the measurement of the energy in a normal mode, depends intimately on the nature of the coupling, which is typically expandable in the form  V (P, Q) = Vn (P, Q), (13) n

where Vn (P, Q) denotes polynomial terms of the form Q ik P jm with k + m = n. As a simple example of the effect of coupling, consider a two-degrees-of-freedom system with V (P, Q) = V2 (P, Q) = AQ 1 Q 2 .

(14)

It is convenient to first identify the constants of the motion in the harmonic Hamiltonian and use them as the new momenta. Consider then the momenta   Ii = (4λi )−1/2 Pi2 + λi Q i2 (15) and conjugate coordinates  1/2  θi = cot−1 −λi Q i Pi .

(16)

In these coordinates, H0 assumes the form H0 = ω1 I1 + ω2 I2 , 1/2

(17)

where ωi = λi . We shall assume ω1 and ω2 to be unequal. These specific types of momenta and coordinates I are termed action-angle variables.

The relationships in Eqs. (15) and (16) allow us to rewrite V2 (P, Q) as V2 = A(I1 I2 /ω1 ω2 )1/2 [cos(θ1 −θ2 )−cos(θ1 +θ2 )], (18) where A is a constant. Hamilton’s equations of motion [Eq. (4)] then provide expressions for d Ii /dt that are nonzero due to the coupling V2 . In the event that the coupling is small, one may approximate the solution for the time dependence of the angles as that of the time dependence in the absence of the perturbation. This approach, a classical perturbation theory, gives the following result for the time dependence of Ii (t): 

I1 I2 1/2 I1 (t) = I1 (0) − A ω1 ω2   × cos (ω1 − ω2 )t + θ10 − θ20 (ω1 − ω2 )   − cos (ω1 + ω2 )t + θ10 + θ20 (ω1 + ω2 ), (19) where θ10 , θ20 are the initial values of the angles. The quantity I2 (t) is similar, but out of phase. Thus, the action variables oscillate about their unperturbed values with frequencies (ω1 − ω2 ) and (ω1 + ω2 ). Since ω1 − ω2 is assumed large, the total variation of I1 and I2 as a function of time is small. The result is quite different if the system is resonant, that is, ω1 = ω2 . In this case, the effect of the perturbation is more drastic, and energy can be exchanged completely, albeit periodically, between the two harmonic oscillators. There are several reasons why the example treated above is a gross oversimplification of the situation in molecules. First, the unperturbed system is assumed harmonic, that is, linear in I. Second, the perturbation has been assumed to be composed of a single term. Third, only one type of perturbation has been included. We now qualitatively examine the important effects associated with the breakdown of these simplifying assumptions. a. Anharmonicity of H0 . In general, H0 is not harmonic, but is rather of the general anharmonic form H (I). As a result, the zero-order system frequencies ωi (I) = ∂ H0 /∂ Ii are no longer independent of the actions I; that is, the frequencies depend on the energy content of the oscillators. As a consequence, anharmonic systems will display regions of the I1 , I2 space where ω1 (I) and ω2 (I) are resonant as well as other regions where they are not. Thus, in the course of the dynamics, the zero-order system can go into and out of resonance as the energy of the oscillator varies. Regions of I space where the system is resonant are called resonance zones. Note that despite the coupling, the dynamics within this resonance region is regular.

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A well-known example provides a picture of the resonance zone and the “trapping effect” associated with a nonlinear resonance. Consider a child’s swing being pushed at a fixed frequency. The nonlinear swing, well approximated by a pendulum, will gain energy from the “pusher” until the system is well out of resonance. At this stage the swing loses energy until it once again comes into resonance with the driving frequency. The system is therefore effectively trapped in a range of swing energies determined by the resonance zone associated with this driven pendulum. A similar effect is associated with the single resonance region associated with ω1 (I) = ω2 (I) in the example above, the system being essentially trapped in the region about the resonance center if the dynmics is initiated in that region. b. Other coupling contributions. The above discussion emphasizes the ω1 = ω2 resonance, which results from the assumed form of the coupling in Eq. (14). In general, the coupling is more complicated, but is still expected to be expandable in the form  V (I, θ) = Vm,n (I1 , I2 ) exp(inθ1 + imθ2 ). (20) mn

The V2 coupling term in Eq. (14) is an example of |n| = 1, |m| = 1 contributions to this expansion and leads to the ω1 = ω2 condition for resonance. Similarly, the n, m term in this series leads to an “n, m resonance” at action variables satisfying nω1 (I) = mω2 (I). Once again, within the neighborhood of this single resonance, the system displays regular energy transfer between the zero-order oscillators. We note that the size of the resonance zones tends to decrease with increasing n, m. c. Overlapping resonances. When a few terms in Eq. (20) contribute to the coupling, there is little reason to expect that specific regions of I space are influenced by solely one resonance. Under rather general conditions, the resonance regions in phase space arising from different terms in the coupling expansion [Eq. (20)] may overlap. Numerical studies have shown that energy flow between the zero-order oscillators assumes chaotic characteristics in regions of overlapping resonances. As an example, consider the resonance and resonanceoverlap structure associated with a collinear molecule A B C where H0 is the sum of two Morse oscillators [Eq. (10)] corresponding to the bond potentials and the coupling term is the form Ap1 p2 . Here, p j is the momentum associated with the jth bond. With E i defined as the energy of the ith bond and E the total energy of the system, quantitative application of resonance theory allows for the explicit determination of E i , E regions dominated by either a single resonance or by overlapping resonances. Sample results are shown in Fig. 13, where the solid shad-

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FIGURE 13 Resonance structure of a model system A B C where each atom has a mass equal to that of carbon. The A B bond has frequency 1000 cm−1 and dissociation energy Dc , whereas the B C bond has corresponding parameters 1300 cm−1 , 1.5Dc . Black areas denote single-resonance regions, and cross-hatched areas denote regions of overlapping resonances. [From Oxtoby, D., and Rice, S. A. (1976). J. Chem. Phys. 65, 1676.]

ing indicates regions dominated by a single resonance and the cross-hatched areas are those dominated by overlapping resonances. In the case shown, there is a general trend toward overlapping resonances as the energy increases, consistent with the observation of increasing chaotic behavior with increasing energy. For the particular parameters shown, however, the system, even at energies near dissociation (E = 1.5Dc ), displays regions of regular behavior dominated by a single resonance. Alternate system parameters can result in larger or smaller contributions from overlapping resonances. In summary, the picture that emerges with respect to energy transfer between specified zero-order oscillators is qualitatively straightforward. The coupling between the specified oscillators induces nonresonant energy transfer between the oscillators if the system is initiated, and remains, within a nonresonant region of I values. Resonant energy transfer results if the system begins within a resonance zone, or enters the resonance zone, during the dynamics. In both cases, energy transfer has welldefined pathways: the energy transfer is well described in terms of the time-dependent energy content of the zero-order oscillators. Finally, chaotic energy transfer between the zero-order oscillators is expected in the I-space regime dominated by overlapping resonances. Computational results on bound molecules indicate that the volume of resonance regions increases with increasing system energy. One important aspect of this discussion is worthy of emphasis. Specifically, the subdivision of the system into zero-order oscillators and coupling terms, and the

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492 subsequent expansion of the coupling term, must be motivated by the experimentally measured quantities. Specifically, one may see apparent chaotic motion between particular zero-order oscillators even if the system is regular. This would be the case if the time scale of measurement is short and the observed oscillators are not those directly related to the conserved integrals of motion.This feature also makes clear that overlapping resonances do not necessarily ensure true irregular motion. Detailed studies on the dynamics of realistic molecular systems are just becoming available. As a consequence, it is unclear whether the vast majority of highly excited molecules are weakly coupled with few overlapping resonances or are strongly chaotic. As a specific example of resonant coupling with weak coupling characteristics, and hence a specific energy-transfer pathway, we discuss below the study of overtones in the benzene molecule. As an example of chaotic energy transfer, we call attention to the NaBrKCl example discussed in Section II. Recent experimental studies on benzene have shown that the absorption spectrum contains local mode features, that is, evidence of local isolated bond motions. In the benzene case, the C H bonds, if they contain sufficient energy, appear directly in the spectrum, as if they were decoupled from the remainder of the molecular framework. In particular, one sees evidence of excitation to the overtones of C H stretch, that is, five, six, seven, etc. quanta of energy in the bond. The experimental results further indicate that if energy is deposited in these bonds, it would transfer to the remainder of the benzene nuclear framework within about 10−13 sec. Although apparently rapid, this rate of energy transfer is substantially slower than that expected from a C H bond democratically linked to all the degrees of freedom in the ring. Detailed quantum and classical studies of the dynamics of benzene indicate the following picture. Consider first the immediate local environment of a C H bond attached to the ring (Fig. 14). Two C C bond distances are

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FIGURE 15 A schematic of the coupling scheme linking the C H stretch to the various modes of the benzene ring. Schemes (a) and (b) represent the same coupling schemes described in two different zero-order mode languages. [From Sibert, E. L., Hynes, J. T., and Reinhardt, W. P. (1984). J. Chem. Phys. 81, 1135.]

labeled s1 and s6 , with the C H bond distance being labeled s. Also shown is the angle β associated with the wag motion of the C H relative to the ring. The picture that emerges from numerical studies is that with increasing excitation of the C H bond, the C H bond frequency comes into resonance with the wag, where the resonance is characterized as n = 2, m = 1. Energy transfer from the C H bond first occurs as resonant energy transfer to the wag. Energy is subsequently transferred from the CCH wag to the remainder of the modes of the molecule. This is pictorially shown in Fig. 15. Trajectory calculations of the time dependence of the flow of energy out of the excited C H mode, for various degrees of excitation, are shown in Fig. 16. A complementary picture of the growth of energy in the ring modes of the benzene framework and into the lower lying states of C H on the benzene ring is shown in Figs. 17 and 18. On the time scale shown, the energy flow out of the C H bond is irreversible. Agreement with experiment is good, providing evidence that energy flow in this case occurs through a well-defined pathway of resonances. The comparative quantum calculations are discussed later. Further experimental and theoretical efforts are underway to establish the extent to which energy-transfer mechanisms in molecules are either chaotic or rather specific in their nature. C. Quantum Dynamics

FIGURE 14 Coordinates defining the C H bond distance (s). C C bond distances s6 , s1 ; and wag angle β in benzene. [From Sibert, E. L., Hynes, J. T., and Reinhardt, W. P. (1984). J. Chem. Phys. 81, 1135.]

Molecules are, of course, properly described by quantum mechanics, and classical mechanics is recognized as a particular approximation. Nonetheless, an introductory description of intramolecular energy transfer via classical mechanics has proven useful since it contains few concepts that are truly unfamiliar to the macroscopic world.

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FIGURE 16 Time dependence of the average energy in the C H oscillator for three cases: excitation in the ninth C H vibrational level (v = 9), v = 6, and v = 5. [From Sibert, E. L., Hynes, J. T., and Reinhardt, W. P. (1984). J. Chem. Phys. 81, 1135.]

tions prevent a discussion of these quantum phenomena, and the interested reader is referred to the bibliography for details. We focus rather on the dynamical consequences of energy quantization in quantum mechanics. This property means that a system can only exist at specific energy values, a property shared by other observables as well. Energy is, however, intimately linked to dynamics, since the Hamiltonian determines system time propagation, as discussed later. One important remark is in order. That is, although quantum phenomena have been observed in molecular systems, we possess only the very qualitative “traditional” rules regarding conditions under which quantum effects predominate. Specifically, if the initial state involves large classical actions and the initial state is one that is allowed classically, then quantum effects tend to be small. Considerably more work is necessary, however, before more quantitative, predictive statements can be made and before our understanding of classical/quantum correspondence in bound molecular systems is complete. Considerations of the quantum dynamics of bound molecules shows that, in the absence of the emission of radiation from energized molecules, all dynamics is quasiperiodic and regular. That is, quantum mechanics does not admit the possibility of long-time relaxation to a time-independent stationary state, a property that characterizes a classical mixing system. This property creates

Although it is possible to cast both quantum and classical mechanics in a similar formal language (i.e., distributions in phase space and a Liouville propagator), standard quantum mechanics is based on a mathematical structure that is substantially different from that of classical Hamiltonian mechanics. We first provide a brief qualitative summary of some results of quantum investigations, and then we present details that can be best appreciated by the reader who is well versed in quantum mechanics. First and foremost, we note that classical mechanics does not allow a number of phenomena that occur in nature. A familiar example is tunneling, in which a system has finite probability of being in a region of phase space where it is not permitted classically. The simplest example of tunneling occurs in a system consisting of a particle moving in a potential that has two minima with a potential barrier between them. Classically, a particle initiated in one of the wells with energy below the barrier height is confined to that well forever. In the quantum case, however, the system flows between the two wells: it “tunnels” through the potential barrier. Tunneling effects are most certainly important in the intramolecular dynamics of systems at energies below such potential barriers. Less well known are symmetry effects, resonances, etc. that can play important roles in intramolecular dynamics. Space limita-

FIGURE 17 Growth of energy content of ring modes in benzene associated with the v = 6 case in Fig. 16. Note that only a few ring modes, labeled by their frequency, are shown. [From Sibert, E. L., Hynes, J. T., and Reinhardt, W. P. (1984). J. Chem. Phys. 81, 1135.]

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Quantum mechanics describes system behavior in terms of operators that represent measurable quantities, their eigenfunctions, which describe possible states of the system, and their eigenvalues, which correspond to allowable values of the measurable. If the system is presumed best described in terms of a specification of the system energy, then one looks for states |ψ that are eigenfunctions of the molecular Hamiltonian operator H. That is, one solves the problem H|ψ(t) = −i h ∂|ψ(t)/∂t, ✟

(21)

where t is the time. The nature of this equation is such as to admit the solutions |ψ j (t) = |ψ j  exp(−i E j t/ h ), ✟

(22)

where |ψ j  is the solution to the eigenvalue problem H|ψ j  = E j |ψ j .

FIGURE 18 Probability of finding i quanta in the C H oscillator as a function of time for the case of initial v = 6. [From Sibert, E. L., Hynes, J. T., and Reinhardt, W. P. (1984). J. Chem. Phys. 81, 1135.]

a number of difficulties in understanding the formal relationship between classical and quantum mechanics, particularly for energized molecules that display classically chaotic behavior. Practically, however, one finds that if the system is close to the classical limit, then quantum and classical dynamics agree over a significant time scale. This time scale is expected, in the vast majority of typical chemical experiments, to be in excess of the time of interest for the process. Under these circumstances, the formal discrepancy between classical and quantum mechanics is irrelevant to the specific chemical problem. Nevertheless, since quantum mechanics does not admit anything other than quasiperiodic behavior, attention has recently been focused on other quantities that might provide the qualitative distinction between quantum systems that display, more or less, statistical behavior.

(23)

Here, the Hilbert space vector |ψ j  has a coordinate space representation ψ j (q) = q|ψ j , and |ψ j (q)|2 is the probability of observing a given value of q when the system is in a state defined by energy E j . In general, the system may be degenerate, in which case several E j may have the same numerical value. This treatment and the one that follows provide an idealized picture in which the molecule is entirely isolated from external influences. Such an ideal picture cannot, in fact, apply. Specifically, although one may be able to experimentally isolate the molecule from interactions with other molecules (e.g., via high-vacuum techniques), the molecule will always interact with the background radiation field to radiate energy. In this discussion we regard this emission as a small perturbation that can be introduced as part of the measurement process. Consider time dependence in quantum mechanics, with the experimentally prepared initial state assumed completely specified as the Hilbert space vector |φ(0). That is, the system is initially in a pure state and may be expanded in a linear combination of energy eigenstates |ψ j  as  |φ(t = 0) = c j |ψ j  c j = ψ j |φ(0). (24) j

Then, from Eq. (22), the subsequent time evolution is given by  |φ(t) = c j |ψ j  exp(−i E j t/ h ). (25) ✟

In the alternative case, the so-called mixed state, one does not have a complete specification of the initial state; that is, the initial state cannot be described as a single Hilbert space vector, nor can it be written in terms of a linear combination of |ψ j . It can, however, be written in terms of a density matrix,

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ρ(0) =

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wn |χn  χn |,

(26)

n

where the states |χn  can be written in terms of a linear combination of |ψ j  and where n wn = 1, with wn equal to or greater than 0. The essential feature of such states is that ρ(0) is lacking information on the relative phases of the participating eigenstates |ψ j . We focus on the pure state, although it is the exception, rather than the rule, in experimentally prepared systems. In accordance with quantum mechanics, a measurement of a particular property during the course of the system time evolution consists of evaluating the average value of the corresponding operator. For example, if one measures the property described by the quantity F, then the average value of F as a function of time is given by

F(t) = φ(t)|F|φ(t)  = di, j exp(iωi, j )

(27)

i, j

di, j = ψ j |F|ψi c∗j ci , where ωi, j = (E i − E j )/ h. Thus, F(t) may be written as a linear combination of terms involving a discrete set of frequencies ωi, j . By analogy with the discussion of classical systems, this sum is seen to be quasiperiodic. The number of terms contributing to the sum in Eq. (27) and the relationship between the frequencies ωi, j determine the kind of qualitative behavior observed. In the case where only a few terms contribute, the dynamics is almost periodic. Such behavior is observable as, for example, a periodic modulation of the fluorescence emitted from a molecule prepared in a linear combination of a few states, a phenomenon known as quantum beats. An example of beats in SO2 is shown in Fig. 19, where the fluorescence reflects the interference between two contributing levels.

FIGURE 19 The intensity of fluorescence, as a function of time, from SO2 created in a superposition state composed of two levels. [From Ivanco, M., Hager, J., Sharfin, W., and Wallace, S. C. (1983). J. Chem. Phys. 78, 6531.]

FIGURE 20 Density of states (i.e., number of states per unit energy interval) as a function of energy for a number of molecules. The abcissa is in number of photons, rather than energy, where the type of laser photon used depends on the particular molecule. For example, the state density for CF3 CH2 OH is plotted vs photons, from an HF laser, associated with the P1 (7) line. The energy of each of these photons is 0.01660 a.u. Other photons used are the P1 (6) line with a photon energy of 0.01683 a.u. and the CO2 10P(20) line with energy of 0.00430 a.u. per photon. [From McAlpine, R. D., Evans, D. K., and McClusky, F. K. (1980). J. Chem. Phys. 73, 1153.]

Such simple behavior emerges only when the initially created state is composed of a few levels. This is seldom the case, as seen from Fig. 20, which shows the density of states (i.e., the number of states per unit energy interval) D(E) for some typical molecules. The quantity D(E) is seen to be an increasing function of the size of the molecule and, for even small molecules such as SF6 , can reach very large values (e.g., 103 cm−1 ). As a result, chemical experiments, whose energy resolution is often not sharp, will typically involve an initial state composed of many many levels. The subsequent dynamics emerges through the simultaneous interference of a multitude of terms, and the resultant behavior is difficult to qualitatively extract from

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a formal sum [Eq. (27)] of contributions from individual levels. For example, the system can display short-time behavior reminiscent of relaxation. Consider the case where the coefficients di, j are distributed in a smooth fashion about a particular frequency value. Then one can show that for short times compared to the density of frequencies, F(t) decays smoothly as a function of time, with a time scale governed by the inverse of the frequency width of the coefficients di, j . This behavior is termed dephasing, to distinguish it from irreversible relaxation of the initial state. That is, despite this decay, the system will eventually reassemble to form the initial state, although this time scale may be exceedingly long. Thus, an experiment measuring

F(t) over a time scale short compared to the recurrence time will show apparent relaxation of F(t). Nonetheless, formally, the system is quasiperiodic. The facts that quantum bound-state dynamics is quasiperiodic, classical mechanics can be mixing, and the latter is expected to approximate the former, make up the essence of the important unresolved problem “what is quantum chaos?” The quantum picture of bound-state dynamics calls attention to an important aspect of intramolecular dynamics. Specifically, the state |φ(t) is composed of a linear combination of states |ψ j . The probability of observing the system in an exact eigenstate |ψ j  at time t is given by |c j exp(−E j t/ h )|2 = |c j |2 . Thus, the population of each exact eigenstate does not change as a function of time. If, in fact, one were solely interested in the population of these exact levels, then there is no such thing as time dependence in the dynamics of bound molecules (other than radiative emission)! Clearly, the focus of intramolecular dynamics and energy transfer is on attributes other than exact eigenstates populations. To appreciate the desired description of energy flow in chemistry, recall the historical origin of the interest in intramolecular energy flow. The most prominant case is that of unimolecular decay, in which a molecule, sufficiently energized, breaks into a variety of products (e.g., ABC → A + BC). In this case the focus of attention, and therefore of the measurement, is on the energy content of the A B bond. This is typical of chemical descriptions in which the analysis is in terms of subunits of the molecule that are not, in themselves, naturally distinct subcomponents of the molecule. Such a description results when a zero-order basis set is used. Specifically, consider the Hamiltonian for a two-degrees-of-freedom system written, as in the classical case, in the form ✟

H = H0 + V

H0 = H1 + H2 ,

(28)

where H1 and H2 describe two distinct subcomponents of interest in a particular experiment and the eigenfunctions of Hi are denoted by |χ ij , where

Hi χ ij = εij χ ij

i = 1, 2.

(29)

The perturbation V couples the zero-order states so that exact-energy eigenstates |ψk  are linear combinations of these zero-order states or vice versa. That is,  1 2 χ χ , = bk ψk . (30) i

j

i, j

k

Using this expression and Eq. (22) gives the following form for the time evolution of these zero-order basis-set states;  1 2 χ χ (t) = bk ψk exp(−i E k t/ h ). (31) ✟

i

j

i, j

k

If the initial state consists of a linear combination of the zero-order states, then the populations of the zero-order states are seen to be time dependent. The degree to which the zero-order states enter into the exact eigenstates [i.e., the nature of the sum in Eq. (30)] is a measure of the strength of the coupling and provides an insight into the nature of the exact eigenstates from the view-point of this particular zero-order basis. It essentially provides the time-independent picture of the possible zero-order states that can be coupled during the dynamical evolution of the system. Equation (28), treated quantum mechanically, admits the same kind of perturbation treatment as in the classical case, with a similar emphasis on isolated resonances and overlapping resonances emerging. As in the classical case, the question of the nature of intramolecular energy flow—whether it is statistical or whether it displays a specific pathway—is of interest. Unfortunately, few quantum calculations on realistic molecular systems have been performed. The example of the dynamics of benzene, initially prepared in an excited state of the C H bond, has, however, been treated quantum mechanically. Here, the relevant zero-order Hamiltonian is of the form H = HL + HN + HLN ,

(32)

where HL is the Hamiltonian for the local C H vibrational motion, HN is the Hamiltonian for the remainder of the molecule, and HLN is the coupling between them. Computations have been carried out, including the coupling between the C H vibration and the CCH wag motion, via terms in HLH . In the adopted model, HN is essentially harmonic so that the zero-order states are of the form |vL , kN  = |vL |kN , denoting v quanta in the C H stretch and k quanta in the ring modes. As in the classical treatment in Section II, interest is in the dynamics of benzene in the energy range where the C H bond is prepared with considerable energy. Figure 21 shows some of the many zero-order energy eigenstates in the energy regime associated with the zero-order state |6L , 0N . Not shown are states with less than four quanta in HL , which, although coupled to the |6L , 0N  state, were too

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FIGURE 21 (a) Some of the zero-order states of benzene in the energy neighborhood near that of excitation to the sixth vibrational level of C H. States are organized, for clarity, in ladders associated with the number of quanta in the C H mode. (b) Levels for the analogous monodeuterated benzene case. [From Sibert, E. L., Reinhardt, W. P., and Hynes, J. T. (1984). J. Chem. Phys. 81, 1115.]

numerous to be included in the computation. For convenience, the levels are stacked in ladders, or “tiers,” with the ladder labeled by the quanta of energy in the C H oscillator. Calculations show that the exact system eigenstates are a strongly coupled mixture of the zero-order states. Figure 22 shows the result of a calculation on the quantum dynamics of benzene initially excited to the sixth vibrational level of C H. The population of the sixth level (denoted |cCH (t)|2 ) is seen to decay rapidly in time, with concomitant growth of population in the various tiers. The essential features of this quantum calculation are in accord with the classical results by the same authors and describe the dynamics of benzene, initially energized in the C H vibration, as decay via a specific pathway of resonances between zero-order modes. Although specific examples of this kind are of considerable importance, general rules regarding quantum intramolecular dynamics would be far more useful. Indeed, this was the motivation for investigating ergodic, mixing, etc. systems classically. Similarly, it is the reason for the general interest in the question of “what is quantum chaos?” We comment briefly (and only qualitatively) on recent developments in this area, noting first that an appropriate definition of chaos can only involve properties of the central participants in the time evolution, that is, the energy eigenstates and eigenvalues.

As a starting point, note that chaotic classical systems possess a number of prominent features. First, the system relaxes to a long-time stationary distribution. Second, the long-time limit is sensitive to only a few simple constants of the motion (e.g., the total energy). Third, as a direct consequence, the computation of system properties can be done by replacing the actual dynamics by simplified statistical models. The first property is clearly not satisfied by a quantum bound system, which has been formally shown to be quasiperiodic. Nonetheless, one possibility is that the system approach a long-time value about which fluctuations are small and that this long-time value be sensitive to only a few simple properties. Indeed, if all wave functions in an energy interval are basically the same (i.e., have similar properties), then the system will evolve in a fashion that is relatively insensitive to the nature of the preparation. Further, if there are no integrals of motion other than the total energy, then one might expect the energy eigenvalues to display rather simple properties reflecting this characteristic. These remarks motivate one current view on the nature of quantum chaos. Specifically, in such a system the eigenfunctions are proposed to be similar to one another in character as the energy varies, and the probability of observing a particular level spacing is expected to be of a specific form (the Wigner adjacentlevel distribution). The former requirement, coupled with the condition that all eigenfunctions be orthogonal, that is,

ψi |ψ j  = 0,

(33)

hints at the nature of these wave functions; that is, they display erratic nodal patterns. This is clearly not the case

FIGURE 22 The time dependence of various quantities associated with the quantum dynamics of benzene initially excited to the sixth level of C H bond excitation. Labels are defined in the text, other than Pi (t), which denotes the probability of being in the i th tier of energy levels. Only three tiers were included in this calculation. [From Sibert, E. L., Reinhardt, W. P., and Hynes, J. T. (1984). J. Chem. Phys. 81, 1115.]

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FIGURE 23 Contour plot of ψ(q) associated with a typical state of the particle in a stadium potential. Only positive contours are shown. Note the highly erratic pattern of contour lines proposed to be a characteristic of wave functions associated with chaotic classical systems. [From Brumer, P., and Taylor, R. D. (1983). Faraday Discuss Chem. Soc. 75, 171.]

for the vast majority of typical systems studied in elementary quantum mechanics (e.g., the H atom, the particle in a box), which are, in fact, integrable and separable in classical mechanics as well. Note that the condition of erratic nodal patterns makes sense with respect to the view resulting from an analysis in terms of a zero-order basis set. In particular, expanding such exact wave functions in any arbitrary basis is expected to yield populations spread over all zero-order wave functions in the energy neighborhood. Thus, there is a form of statistical coupling between all zero-order basis functions. Numerical computations on model systems have, in fact, revealed erratic wave functions for some systems that are classically mixing. An example is shown in Fig. 23, where the system is a particle confined by infinite potential walls to a region that is the shape of a racetrack (the so-called stadium system). The nodal patterns are clearly highly disordered and in marked contrast with the simple nodal lines associated with, for example, the separable particle in a rectangle case. Unfortunately, there is no one-to-one correspondence between systems that display chaotic classical behavior and the observation of erratic wave function nodal patterns.

IV. STATISTICAL APPROXIMATIONS AND DYNAMICS Accurate dynamical studies, such as those discussed above, are limited to small molecular systems and relatively short times (e.g., typically 100 vibrational periods). These restrictions stem from a number of sources. First, reliable computations of forces between atoms in large molecular systems are extremely difficult, and few quantitatively accurate models of interatomic forces exist. Sec-

Energy Transfer, Intramolecular

ond, even if such potentials are available, large-molecule motion treated quantum mechanically involves huge numbers of participating energy eigenstates. Techniques for efficiently computing large numbers of eigenstates in systems with significant numbers of degrees of freedom are only now being developed. Alternative techniques, which rely on the direct, numerical, temporal propagation of initial states [i.e., via Eq. (21)], do not utilize eigenstates but suffer inaccuracies in long-time applications. An analogous difficulty exists in attempting numerical studies of dynamics using classical mechanics; in this case, the exponential growth of distances in phase space (see Section III) translates into the rapid growth, with time, of numerical errors. Since the vast majority of interesting molecules have many degrees of freedom, the need for models that simplify the dynamics is evident. The most popular of such models relies on a statistical assumption of relaxation during the course of the dynamics. That is, one assumes that after preparation of the energized molecule, the system relaxes to a well-defined state that is dependent on only gross features (e.g., the total energy) of the preparation. Assumptions of this kind predate detailed dynamical studies of molecular dynamics. To appreciate the simplifications resulting from such models, consider the paradigm case of unimolecular decomposition (A → B + C, where A, B, and C are molecules), where this assumption leads to statistical theories of the rate of unimolecular decay. Here, A, sufficiently energized to dissociate, is prepared by any of a variety of means (e.g., laser excitation, collisions, or as the product of a chemical reaction). A detailed computation of the dynamics of this process for a realistic molecule, an intractable computational feat, would entail the following steps. One first specifies the exact nature of the molecular potential, the nature of the process that prepares the excited molecule, and the state of the molecule prior to preparation. Second, the exact dynamics of the evolution of the molecule, from preparation to decay, is computed. Such a computation must be repeated for each and every type of initial state of the molecule and each and every type of state preparation. In contrast, the formulation of a typical statistical model proceeds as follows. First, define the region of phase space in which the molecule A is to be regarded as being bound and a complementary region in which it is characterizable as B + C. Then assume that the rate of dissociation of the excited molecule depends primarily on the magnitude of simple known constants of the motion, that is, energy E and total angular momentum J. Assume further that any initially prepared state rapidly relaxes to a uniform distribution over the surface in phase space characterized by fixed E, J and within the region where A is regarded as bound. The computation of the

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rate of decay of such a system then entails an analysis of the rate at which the relaxed uniform distribution crosses over into the dissociation region. Such a computation is straightforward in both classical and quantum mechanics, if a number of approximations on the structure of the molecule are made, and leads to statistical theories, such as RRKM theory, discussed in the bibliography. The essence of such theories is, then, that the system undergoes memory erosion during the dynamics so that information on initial conditions is lost; only knowledge of E and J remains. Similar theories have been proposed for a wide variety of processes that involve the participation of long-lived molecular intermediates. These include chemical reactions that proceed via collision complexes, photodissociation where molecular preparation is through controlled laser excitation, molecules adhering to surfaces where detachment is induced via a variety of means, etc. Such theories have the advantage of yielding rather general results, which are amenable to both theoretical and experimental analysis. For example, in the case of unimolecular decay, the rate constant for dissociation is found to increase with increasing energy and decrease with the number of participating degrees of freedom in the system. Experimental studies on the validity of such theories have been ongoing for many years, as discussed briefly in the next section. The theoretical examination of the validity of such approaches is more recent and links directly to the issues discussed in Section III. At present, classical mechanical studies have shown the possibility of both statistical and nonstatistical decays, depending on the degree and extent of exponential divergence of trajectories in phase space. The greater the degree and extent of exponential separation, the closer the agreement with statistical approaches. Similarly, quantum-mechanical studies have shown that model systems can display unimolecular rate constants whose energy dependence is inconsistent with that predicted by simple statistical theories. It is fair to say, however, that a clear understanding of the interrelationship between molecular properties and the validity of statistical theories is in its early stages of development.

V. EXPERIMENTAL STUDIES The ideal experiment on intramolecular energy transfer, as yet unachieved, entails a number of simple features. Specifically, the molecule is prepared in a well-defined and well-characterized state and evolves for a known time interval, after which the state of the system is precisely determined via a high-finesse experimental probe. The importance of each of these components to the resolution of even the most qualitative of questions (e.g., is the energy transfer statistical or not) should be clear. If, for exam-

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499 ple, as is the case in some of the available techniques, the initial state of the molecule is in itself highly randomized energetically, then a subsequent measurement that shows that the energy is statistically distributed among the system modes is of little consequence. That is, this particular observation would be a consequence of the preparation, as distinct from the evolution, of the molecule. The dynamics of molecules with low energy content is usually well described by a Hamiltonian that is the sum of independent degrees of freedom, and absorption spectroscopy has provided considerable information on the nature of such systems. The situation with respect to energized molecules is far more complex. A number of experimental tools have been utilized over the past 30 years to examine the nature of energy transfer in such systems, as discussed next. Many suffer from the inaccurate knowledge of the initial system state. In our brief description of several experimental methods it will become clear that experimental tools have been rapidly developing over the past few years and that an explosion of highly informative experimental data is now underway. First and foremost, note that interest is in the nature of intramolecular dynamics of molecules in isolation. That is, observations must be made over a time scale where the molecule does not collide with others in the reaction vessel. Modern techniques allow very low pressures under which such measurements can be made. Most desirable among these methods are beam techniques in which molecules are studied in a low-density beam produced, for example, by vaporizing molecules in an oven. Experiments prior to this “beam age” (circa 1960) often inferred information about intramolecular dynamics from bulk data, which contained effects due to collisions, with resultant loss in accuracy. Measurement techniques in typical experiments can be subdivided into two categories. The first and most modern probes the bound molecular dynamics directly, for example, by observing radiation emitted or absorbed during the course of the dynamics. The second infers information about the nature of bound molecular dynamics by indirect means, typically by analyzing the outcome of a process that involves the molecule of interest as a longlived intermediate. These latter types of measurement are readily clarified by considering the NaCl + KBr example discussed in Section II. Specifically, this particular reaction proceeds via the bound NaC1KBr intermediate to yield two different sets of products, either NaCl + KBr or NaBr + KCl. The probability of observing these two product “channels” depends on the nature of the NaCIKBr dynamics. That is, the product ratio will provide insight into whether intramolecular energy transfer in NaCIKBr is rapid or not. More extensive measurements will entail analysis of the internal states of the product molecules

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500 (see Fig. 5), as well as of the relative velocities of the products. Thus, these kinds of measurements allow one to infer information on the characteristics of the dynamics of the intermediate without directly observing it. Other examples of this kind include measurements on the products of unimolecular decay, photodissociation, etc. Similarly, most experiments on intramolecular energy transfer fall into one of two categories with respect to preparation of the molecule. In the crudest, the system is prepared by a “coarse” technique where little detail about the initial molecular state is available. Such, experiments include preparation via collision that is, where the molecule of interest, A, collides with, and absorbs energy from, another molecule B and preparation via reaction, where the molecule A is the product of a “precursor” reaction. Although the bulk of early work on intramolecular dynamics was carried out with these techniques, far greater insight emerges from modern experiments in which the molecule A is prepared by the absorption of radiation. The optimum experiment would therefore proceed by preparing the molecules in a precise state using beam methods and laser excitation, followed by measurement of the radiative emission as well as other properties of the bound molecule. Such experiments are, in fact, underway on a variety of molecules in several laboratories around the world. In addition, information on the bound-state dynamics of molecules has emerged from pump-probe techniques in which two lasers are utilized, one to prepare the molecule in the desired state and the second to interrogate the dynamics. Along with these experimental developments, we note a need for reliable theories to understand the interrelationship between the observed features and the nature of the dynamics. Such developments are in progress.

VI. CONTROL OF DYNAMICS We noted, in Section I, that the study of intramolecular energy transfer is linked to the practical goal of controlling the dynamics of molecules. Since the 1980s theoreticians have made giant strides which make control over mollecular dynamics feasible. Space restrictions prevent anything other than a brief comment; the reader is referred to the Accounts of Chemical Research reference for further details. Equation (27) makes clear that the dynamics of a molecular process is intimately linked to the phases of the system preparation, contained in the di, j coefficients. It is clear, then, that if one were able to control the phases and magnitude of these terms, the subsequent system dynamics would also be controlled. Recently, a number of theoreticians have noted that coherent laser sources transfer phase

Energy Transfer, Intramolecular

information to the molecule upon which they impinge. As a consequence, by controlling the phases of these laser sources, one can affect the nature of the molecular dynamics. More careful examination indicates that the experimentalist must control the relative phases of two laser sources, rather than the absolute phase of a single source, a far more feasible prospect. These proposals for the control molecular dynamics and chemical reactions rely heavily on quantum interference phenomena similar to that seen in the famous “doubleslit” experiment. As a consequence they herald a new age in molecular reaction dynamics, one in which quantum aspects of molecular motion are utilized to alter molecular dynamics.

VII. SUMMARY Understanding the nature of intramolecular energy flow in isolated molecules is of great practical and fundamental interest. Early developments, both theoretical and experimental, were hampered by a number of technological problems now being overcome. As a consequence, general features that determine the rate and extent of intramolecular energy transfer are slowly emerging. These include generic features of classical Hamiltonian systems and the way in which coupling terms influence the nature of the dynamics, the dependence of observed energy transfer on the zero-order system, the interaction between state preparation and state measurement on the qualitative interpretation of the dynamics, and the differences and similarities between the quantum and classical views of dynamics. Nonetheless, general rules regarding the rates and extent of intramolecular energy flow have yet to be established. Similarly, a number of fundamental issues arising in the study of intramolecular energy flow have yet to be resolved. Rapid technological developments in computational and experimental tools hold great promise for substantial developments over the next decade.

ACKNOWLEDGMENT We are grateful to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of the research on which this overview is based.

SEE ALSO THE FOLLOWING ARTICLES ATOMIC AND MOLECULAR COLLISIONS • CHAOS • CHEMICAL KINETICS, EXPERIMENTATION • CHEMICAL PHYSICS • COLLISION-INDUCED SPECTROSCOPY • DYNAMICS OF

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ELEMENTARY CHEMICAL REACTIONS • MECHANICS, CLASSICAL • QUANTUM MECHANICS • STATISTICAL MECHANICS

BIBLIOGRAPHY Brumer, P. (1981). Adv. Chem. Phys. 47, 201. Brumer, P., and Shapiro, M. (1988). Adv. Chem. Phys. 70, 365. Brumer, P., and Shapiro, M. (1989). Accts. Chem. Res. 22, 407. Faraday Discuss. Chem. Soc. (1983). 75. Felker, P. M., and Zewail, A. M. (1988). Adv. Chem. Phys. 70, 265. Forst, W. (1973). “Theory of Unimolecular Reactions,” Academic Press, New York.

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501 Gruebele, M., and Bigwood, R. (1997). Int. Rev. Phys. Chem. 17, 91. Gruebele, M. (2000). Adv. Chem. Phys. 114, 193. Lehmann, K. K., Scoles, G., and Pate, B. H. (1994). Annu. Rev. Phys. Chem. 45, 241. Levine, R. D. (1969). “Quantum Mechanics of Molecular Rate Processes,” Oxford Univ. Press, London. Noid, D. W., Koszykowski, M. L., and Marcus, R. A. (1981). Annu. Rev. Phys. Chem. 32, 267. Rice, S. A. (1975). “Excited States” (E. C. Lim, ed.), Vol. 2, Academic Press, New York. Rice, S. A. (1981). Adv. Chem. Phys. 47, 117. Stechel, E. B., and Heller, E. J. (1984). Annu. Rev. Phys. Chem. 34, 563. Wyatt, R. E., Iung, C., and Leforestier, C. (1995). Accts. Chem. Res. 28, 423.

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Hydrogen Bond Krzysztof Szalewicz University of Delaware, Newark, Delaware

I. Historical Perspective II. Quantum Mechanical Description of Hydrogen Bond III. Analytic Representations of Potential Surfaces IV. Nature of Hydrogen Bond V. Properties of Hydrogen Bonds VI. Types of Hydrogen Bonds VII. Hydrogen Bonds in Clusters VIII. Hydrogen Bonds in Solids IX. Hydrogen Bonds in Liquids X. Proton Transfer XI. Hydrogen Bonds in Biological Structures

GLOSSARY Basis functions In the context of solutions of the electronic Schr¨odinger’s equation for hydrogen-bonded system, this term refers to Gaussian functions of the form exp[−αr2 ], where r is the position vector, multiplied by powers of the coordinates x, y, and z. The basis functions are usually located at the nuclear positions and near the midpoint of the hydrogen bond (midbond functions). Linear combinations of such basis functions form molecular orbitals. Coulomb interaction The interaction between two charged particles. According to Coulomb’s law of electrostatics, the energy of such interaction is q1 q2 /R, where qi is the charge of the ith particle and R is the particles’ separation.

Dimer A complex formed by two molecules that do not react chemically with each other. The two constituent molecules of the dimer, called monomers, are disturbed upon the formation of the dimer but preserve their identity. Electron density The probability density of finding any electron of a molecule at a given point in space. Often visualized as the “electron cloud.” Electron density can be obtained by integrating the square of the modulus of an electronic wave function over the coordinates of all electrons but one. Euler angles A set of three angles that uniquely determines the orientation of a solid body in space. Hamiltonian The operator appearing in the Schr¨odinger equation of quantum mechanics. Operators act on wave functions, transforming them into other wave

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506 functions. For an isolated molecule the Hamiltonian contains second partial derivatives with respect to coordinates of all constituent particles—which describe the kinetic energy—and the sum of Coulomb interactions between all the particles—which describes the potential energy. Hartree-Fock method A method of solving the Schr¨odinger equation that assumes that particles’ motions are independent of each other and a given particle interacts only with the averaged charge distribution of other particles. The electronic wave function can then be approximated by an antisymmetrized product of one-electron functions (orbitals). The effects neglected by the Hartree-Fock method are called correlation effects. Molecular simulations Computer modeling of the motion of an assembly of atoms or molecules. In molecular simulations only the motions of nuclei are considered, (i.e., one assumes that the electronic Schr¨odinger equation has been solved providing intermolecular interaction potentials.) In practice empirical interaction potentials are utilized in most cases. Two main approaches are used: the Monte Carlo (MC) method and molecular dynamics (MD). The former relies on statistical sampling of the configuration space of the systems, whereas the latter solves classical mechanics (Newton) equations to find trajectories of molecules. Perturbation expansion A method of solving the Schr¨odinger equation by dividing the Hamiltonian H into an unperturbed part H0 and the perturbation V . H0 is chosen such that an accurate solution of the “unperturbed” Schr¨odinger equation H0 0 = E 0 0 is possible. The wave function  and the energy E which solve H  = E are expanded as power series in V starting from 0 and E 0 , respectively, which leads to a hierarchical set of equations for the coefficients in these power series called the wave function and energy corrections, respectively. The simplest implementation of the perturbation method is called the RayleighSchr¨odinger perturbation theory. Quantum mechanics Theory describing behavior of matter and radiation on the atomic and subatomic scale. Applied to chemical problems, quantum mechanics accounts for the properties of atoms and molecules in terms of the interactions between the constituent particles: electrons and nuclei. The motion of these particles is described in the nonrelativistic quantum mechanics by the Schr¨odinger equation. Schr¨odinger equation Partial differential equation of quantum mechanics used to calculate wave functions and energies of atoms and molecules. In chemistry it is often sufficient to consider only the time-independent version of this equation which can be written as

Hydrogen Bond

H K = E K K , where H is the Hamiltonian,  K is the wave function, and E K is the energy of the system in its K th state. The ground state of the system is denoted by K = 0. Supermolecular approach The method of calculating the interaction energies by subtracting the monomers’ total energies from the dimer total energy. Symmetry-adapted perturbation theory (SAPT) The method describing the intermolecular forces including the hydrogen bonds. Wave function A solution of the Schr¨odinger equation. The square of the absolute value of a one-particle wave function is the probability density of finding the particle at a given point in space. The wave functions replace the trajectories of classical mechanics.

IN SIMPLEST TERMS the hydrogen bond is a type of intermolecular interaction characterized by the equilibrium configuration involving a hydrogen atom located close to the line connecting the two nearest nonhydrogen atoms of the interacting molecules. Intermolecular interactions are interactions between molecules that do not lead to formation of chemical bonds. Such interactions are also called noncovalent interactions or intermolecular forces. To discuss properties of hydrogen bonds, it will be convenient to denote one of the monomers as R1 -X-H and another one as Y-R2 . R1 and R2 are arbitrarily large molecular fragments, X and Y are sufficiently electronegative atoms— in most cases oxygen, nitrogen, or fluorine—and H is a hydrogen atom. Usually there is a lone electron pair on atom Y. The hydrogen-bonded dimer can be then denoted as R1 -X-H· · ·Y-R2 , where R1 -X-H is called the hydrogen donor and Y-R2 is called the hydrogen acceptor. In a typical hydrogen-bonded complex, the atoms X-H· · ·Y form approximately a straight line. The name hydrogen bond— which comes obviously from the central position of the hydrogen atom in the dimer—is misleading because this atom as such has no particular bonding properties (sometimes the term hydrogen bridge is used, which perhaps would be more appropriate). Instead, the bond results from a balance of four fundamental physical interactions taking place between the whole monomers. The definition of the hydrogen bond will be discussed further in sections VI and VII. Examples of hydrogen-bonded systems are presented in Fig. 1. Two monomers can be connected by more than one hydrogen bond, as in the case of the formic acid dimer shown in Fig. 1(b). Sometimes also some interactions within a single molecule resemble hydrogen bonds, as in the case of salicylic acid shown in Fig. 1(d). While hydrogen bonds involving N, O, and F atoms (both as donors and acceptors) are most common, also C and P can

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FIGURE 1 Examples of hydrogen-bonded systems. The hydrogen bonds are indicated by broken lines. (a) Ammonia· · ·hydrogen chloride; (b) formic acid dimer; (c) formamide dimer; (d) intramonomer bond in salicylic acid.

play the role of donors, and S, Cl, and Br can appear in both roles. The hydrogen bond is a special case of intermolecular or interatomic interactions. Consider two monomers: each of them can be an atom or molecule, separated by a distance R. To be precise, let R denote the distance between the centers of masses of each monomer. When R is sufficiently large, the monomers will typically attract each other (exceptions include large-R interactions dominated by dipole–dipole terms that can be of either sign, depending on mutual orientation). When R is shortened, up to some point the attraction will keep increasing. Upon a further shortening of the intermonomer separation there are two possibilities. The monomers may undergo a chemical reaction, forming a new molecule and releasing energy on the order of 100 kcal/mol. This new molecule is bound by chemical forces. In most cases, however, the monomers just begin to repel each other, with the repulsion increasing very quickly with the decrease of R. The forces acting between two such monomers are called intermolecular forces, and a typical shape of the potential energy surface

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507 in this case is shown in Fig. 2. The value of R where the attraction changes to repulsion is sometimes called the σ point, Rσ , of the potential (it is, of course, dependent on the mutual orientation of the two molecules). This distance can also be called “the distance of closest approach” of the two monomers. Although monomers with enough energy can get a little closer, the repulsive wall is so steep that the differences compared to Rσ are minor. The two monomers interacting via intermolecular forces can be bound or not. In a dilute gas, most monomers will have a positive energy of relative motion (i.e., these monomers will be in a scattering state, getting momentarily close to each other and then separating into remote regions of space.) A smaller number of monomers will be in bound states, staying at a finite distance from each other that is close to the equilibrium distance at the minimum of the potential, Re . The energetic location of such a bound state and the vibrational wave function are indicated in Fig. 2. The intermolecular forces are much weaker than the chemical forces because the interaction energy at the point where the attraction is strongest is typically smaller than 20 kcal/mol. For the weakest intermolecular bonds, the depth of the interaction potential can be as small as 0.02 kcal/mol, as is the case for the helium dimer. Intermolecular forces are the most prevalent forces of nature around us. The structure of all liquids and of most solid matter (an exception are metals) is determined by intermolecular forces. These forces also play a major role in shaping the properties and functions of biological systems. The intermolecular forces are often called van der Waals forces, although some authors reserve this name only for the interactions not stronger than a couple of kcal/mol. The boundary between chemical forces and intermolecular forces is, of course, flexible. Systems exhibiting the strongest intermolecular interactions may also be classified as being bound by chemical forces. An additional, although still subjective, criterion in borderline cases can be the extent of the deformation of a monomer’s geometries and electron densities upon the formation of the complex. The physical processes leading to intermolecular forces are well understood. The total interaction energy is composed of four components: electrostatic, induction, dispersion, and exchange energies. The electrostatic component is due to Coulomb interactions of the unperturbed charge distributions of the monomers. The charge distributions do get perturbed during the interaction: the charge distribution on monomer A produces an electric field on monomer B which in turn induces a charge deformation (i.e., polarizes monomer B and vice versa). This process gives rise to the induction energy. The two discussed components are completely defined in the framework of classical electrostatics. The third component, the dispersion

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FIGURE 2 A generic one-dimensional interaction potential. The equilibrium separation Re and the separation Rσ , where the potential changes from repulsive to attractive, are marked. De denotes the potential depth at the equilibrium. The position of the ground vibrational level and the ground vibrational wave function are indicated. D0 is the dissociation energy.

energy, is of quantum mechanical origin. It results from correlations of the fluctuations of the electronic charges on monomers A and B. The exchange energy is also of quantum origin. This component is the consequence of the electrons tunneling through the potential barrier between the monomers. The name follows from the fact that the electrons are exchanged between monomers during this process. Sometimes the exchange energy is interpreted in terms of the “repulsion of the electron clouds.” This picture—which derives from the fact that the exchange energy is proportional to the overlap of electronic charge distributions—does not reflect the physics of the interaction as well as the tunneling picture. Hydrogen bonds result from the same physical forces as all other intermolecular interactions. Thus, from this point of view there would be no need to define hydrogen bonding as a distinct process. Therefore, the phenomenon of hydrogen bonding is related to the structural characteristics of hydrogen-bonded dimers rather than to the physical nature of the interaction. As we will discuss below, if two monomers can be brought together to form a hydrogenbonded structure, this structure will likely be close to the

minimum on the potential energy surface. Thus, the usefulness of the hydrogen bonding concept may simply result from our ability to predict structures of dimers and larger clusters. A further reason for recognizing hydrogen bonding as a phenomenon worth a separate treatment are spectroscopic properties of the X-H stretching motion when the hydrogen atom participates in a hydrogen bond. The fundamental frequency of this vibration is significantly lowered (red shifted) in the dimer compared to that in the isolated monomer. The shift is very sensitive to the molecular environment and therefore provides a major tool to investigate the structure of hydrogen-bonded clusters, liquids, and solids. The hydrogen bond has been the subject of numerous review papers and monographs. It is always treated in works devoted to intermolecular interactions [see, e.g., Hobza and Zahradnik (1988), Stone (1996), Jeziorski and Szalewicz (1998), M¨uller-Dethlefs and Hobza (2000), as well as a special issue of Chemical Reviews (1994)]. Monographs restricted to the hydrogen bond include Schuster (1984), Jeffrey (1997), Scheiner (1997), and Desiraju and Steiner (1999).

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I. HISTORICAL PERSPECTIVE The subject of hydrogen bond (or bonding) is certainly a popular one. The ISI Science Citation database lists more than 26,000 papers that use this phrase in the title, abstract, or as a keyword. In the year 1999 alone there were about 3,000 such papers published. The importance of the hydrogen bond stems mainly from the fact that for certain types of molecules formation of such bonds determines the structure of the dimers or of larger molecular clusters. Thus, hydrogen bonds are characteristic features of many clusters, biological aggregates, and of condensed phases. Chemical structures containing what we now call hydrogen bonds were considered already in the early 1900s. The concept itself appeared around 1920 in works of Huggins and of Latimer and Rodebush. The term hydrogen bond was used for the first time by Pauling in the early 1930s. In the same period it became clear that anomalous properties of bulk water are due to the formation of hydrogen bonds. An important paper on this issue was published in 1933 by Bernal and Fowler (despite their not using the term hydrogen bond). The idea was then extended to other “associated” fluids. The hydrogen bonding concept was popularized by Pauling’s 1939 book Nature of Chemical Bond. While initially the experimental evidence of hydrogen bonding was coming from thermodynamic measurements of anomalous properties and from X-ray measurements of crystal structure, in the 1930s it was realized that the formation of a hydrogen bond has a profound effect on the frequency of the X-H stretch. This started the infrared investigations of hydrogen bonds, which became the most sensitive and the most widely applied experimental method of studying this phenomenon in clusters and in the liquid and solid phases. In the condensed phase the vibrational spectra are often determined with the Raman techniques, which detect lines corresponding to vibrational transitions in scattered radiation of visible frequencies. Since the early 1950s the nuclear magnetic resonance (NMR) spectroscopy has been applied to hydrogen-bonded systems. This technique is, however, utilized less often than the infrared spectroscopy due to complexity of the spectra. In the 1960s the microwave spectra in gas phase for some fairly strongly bound clusters were measured and gave precise information about structures of the clusters. Starting in the 1970s, molecular beam techniques provided a major tool for investigating small clusters. Spectroscopic measurements in molecular beams produced—via the rotational spectra—a wealth of information on geometries of clusters, including even very weakly bound ones. Also in the 1970s, neutron scattering techniques were applied to the condensed phase containing hydrogen bonds (although the earliest such studies

date back to the 1950s). The method of neutron diffraction has an advantage over the X-ray diffraction—it gives information about the positions of hydrogen atoms. In the 1980s high-precision near-infrared spectra of clusters in molecular beams gave reliable information on the frequencies of intermonomer vibrations in hydrogen-bonded clusters. In the 1990s it became possible to measure the same frequencies directly using the techniques of far-infrared spectroscopy. Very recently these methods have enabled a rather complete spectroscopic characterization of small clusters such as the water dimer.

II. QUANTUM MECHANICAL DESCRIPTION OF HYDROGEN BOND The hydrogen bond can be completely described from first principles by solving the Schr¨odinger equation for a set of molecules. In practice such solutions involve several approximations. Despite these approximations, with the current computer capabilities the solutions predict properties of small clusters with accuracy approaching, in some cases, experimental accuracies and provide extremely useful information on hydrogen-bonded systems, including information on systems and properties that cannot be measured. In addition to numerical information, quantum mechanics provides the framework necessary to understand the hydrogen-bond phenomenon. The theoretical approach most useful for an analysis of hydrogen bonding is symmetry-adapted perturbation theory (SAPT). For a more detailed presentation of SAPT and references to the original papers, see Jeziorski and Szalewicz (1998). This approach serves four main purposes: (a) it provides the basic conceptual framework within which intermolecular interactions—including hydrogen bonds—are discussed; in particular it provides the standard division of the intermolecular interaction into four fundamental components: electrostatic, induction, dispersion, and exchange; (b) it is the source of models used in the construction of empirical potentials (empirical force fields); (c) it provides asymptotic constraints on any potential energy surface derived either from experiment or from theory; and, finally, (d) it can accurately predict the complete intermolecular potential energy surfaces for hydrogen-bonded molecular complexes. The last goal can also be obtained using the so-called supermolecular approach [Chalasinski and Szczesniak (1994), van Duijneveldt et al. (1994)].

A. Interaction Energy To study hydrogen bonding it is sufficient to use the time-independent, nonrelativistic Schr¨odinger equation.

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As usual for most of chemistry, the Born-Oppenheimer approximation is assumed. This approximation relies on the observation that the electronic motion is orders of magnitude faster than the nuclear motion. Therefore, the motion of electrons is computed by solving the electronic Schr¨odinger equation with nuclei clamped in space. The solution of this equation provides the potential energy surface governing motion of the nuclei. For investigations of intermolecular interactions, this surface is conveniently divided into the energies of monomers and the interaction energy E int . In the dimer case E AB (R, ω A , ω B ; Ξ A , Ξ B ) = E A (Ξ A ) + E B (Ξ B ) + E int (R, ω A , ω B ; Ξ A , Ξ B ),

(1)

where E AB is the total energy of the dimer, E X is the total energy of monomer X, ω X denotes the Euler angles defining the orientation of monomer X in a dimer-embedded coordinate system, and Ξ X is the set of internal coordinates of each monomer. The energies entering Eq. (1) are solutions to the electronic Schr¨odinger equation. It should be noted that in definition (1) the energies of the dimer and of the monomers correspond to the same values of the internal coordinates Ξ X . In most cases of hydrogen-bonded systems, the monomers are nearly rigid compared to the dimer (i.e., the amplitudes of intramonomer motions are much smaller than those of the intermonomer ones). This is reflected in the monomer vibrational frequencies being much higher than the intermolecular ones. Thus, one can expect that the dependence of the interaction energy on the intramonomer coordinates can be neglected and that a large body of spectroscopic, scattering, and bulk phase experiments can be interpreted in terms of effective potentials depending on intermolecular coordinates only. This approximation leads to a dramatic simplification in studies of hydrogen-bonded complexes. For general complexes consisting of N atoms, there are 3N nuclear degrees of freedom, (i.e., 3N coordinates are needed to describe the position of the system in space). However, different locations of the center of mass of the complex in space as well as rotations of the whole system do not change interaction potentials. Because of these translational and rotational invariances, the energy surface of a system containing N atoms depends on 3N − 6 coordinates (3N − 5, i.e., one coordinate for a diatomic dimer like Ar2 ). Even for relatively small systems like the water dimer 3N − 6 = 12 is a large number of degrees of freedom to treat. The “frozen” monomer approximation reduces the number of degrees of freedom to only 6 for any dimer consisting of two general molecules. The simplest and apparently most natural way of obtaining rigid-monomer potentials is to perform interaction energy calculations assuming equilibrium monomer coor-

dinates. The set of equilibrium coordinates, which will be denoted by re , describe the geometry of the monomer at its potential energy minimum. Another reasonable choice of intramonomer coordinates is the geometry averaged over monomer vibrations, r 0 . The two geometries will be different since the monomer vibrations are always to some extent anharmonic. However, intuitively one can expect that the best effective potential can be obtained by averaging of the complete, monomer geometry-dependent potential over an appropriate vibrational wave function of the monomer. Computation of such an averaged potential, E int 0 , although as expensive as the computation of the complete potential energy surface, represents a useful task because its availability simplifies dramatically the spectroscopic, scattering, and bulk phase computations. For atomdiatom complexes the E int 0 potentials predict spectral quantities that are only about 0.1% different from those obtained from full three-dimensional nuclear dynamics calculations. Recently it has been shown [Jeziorska et al. (2000)] that the r 0 geometry is the optimal choice if only a single monomer geometry can be considered. The spectra of atom-diatom complexes computed using r 0 potentials exhibit deviations from the spectra computed with three-dimensional potential, which are about four times smaller than the analogous deviations produced by the re potentials. The rigid-monomer approximation will not work well if the monomers are too floppy and for complexes involving charged monomers. In the latter case the reason is that the monomers may be fairly significantly distorted upon the formation of the complex. For biopolymers, certain intramolecular coordinates do vary significantly, and these coordinates have to be included in the description of hydrogen bonds in such systems. In the supermolecular method the interaction energy is computed by subtraction of the individual energies E AB , E A , and E B . These energies are in practice calculated using finite basis sets. As a result, the monomer part of the dimer energy is improved by utilizing the basis functions of the interacting partner. This leads to a spurious lowering of the interaction energy, referred to as the basis set superposition error (BSSE). This error can be removed using the counterpoise (CP) technique introduced by Boys and Bernardi. Denoting by E Sσ (G) the electronic energy of system S at geometry G computed with basis set σ , the definition of the CP correction takes the following form β

α∪β

α∪β

δ E CP = E αA (AB) + E B (AB) − E A (AB) − E B (AB), (2) where α and β stand for the basis sets used for monomers A and B, respectively. The use of AB geometry for a monomer means that the nuclei of this monomer are at the same relative positions as in the dimer AB, and the basis

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functions originating from the interacting partner (often called the “ghost” functions) are located at the same spatial positions as in the dimer. The SAPT approach by definition does not include BSSE, since it calculates the interaction energy directly rather than by subtraction of monomer energies from the energy of the dimer. This feature made it possible to prove the correctness of the CP method by comparisons of supermolecular and SAPT calculations [van Duijneveldt et al. (1994)]. B. Perturbation Theory of Hydrogen Bond In the SAPT method, the total Hamiltonian H of the dimer is partitioned as H = H A + H B + V , where H X , X = A or B, is the total Hamiltonian for monomer X , and V is the intermolecular interaction operator. The operator V collects Coulomb interactions of all the particles of monomer A with those of monomer B. This partition means that the unperturbed operator is chosen as H0 = H A + H B and V is the perturbation. The interaction energy is then obtained directly in the form of a perturbation series in V , (1) (1) (2) (2) E int = E pol + E exch + E pol + E exch + ···,

(3)

with each term of the sum having a well-defined physical meaning. The polarization components, denoted by the subscript “pol,” are determined by the standard Rayleigh(1) Schr¨odinger perturbation expansion. The correction E pol is the classical electrostatic (Coulomb) interaction energy of two unperturbed charge distributions and will be written (1) as E elst . The remaining components, denoted by the subscript “exch,” are the exchange corrections accounting for the effect of resonance tunneling of electrons between the interacting systems. These contributions originate from the antisymmetrization of the polarization corrections to the wave function. The second-order corrections can be further divided into induction and dispersion components: (2) (2) (2) E pol = E ind + E disp ,

(2) (2) (2) E exch = E exch−ind + E exch−disp .

(4) The polarization energies through second order have a simple physical interpretation and can be rigorously expressed through monomer properties. 1. Electrostatic Interaction The electrostatic energy, the lowest order polarization component, is defined as    (1) E elst (5) = 0A 0B V 0A 0B , where 0X is the  unperturbed wave function of monomer X and  f | g = f ∗ gdτ with the integration extending over all electron coordinates. The electrostatic energy can be expressed in terms of the total charge distributions ρ A (r )

and ρ B (r ) of the monomers, showing clearly its relation to the Coulomb law  1 (1) E elst = ρ A (r 1 ) (6) ρ B (r 2 ) d 3r 1 d 3r 2 . |r 1 − r 2 | The total electric charge distribution ρ X (r ) for monomer X is the sum of the electronic contribution—which can be obtained from the wave function 0X —and the nuclear contribution. The electrostatic interaction plays a major role in determining the structure of dimers consisting of polar molecules, in particular hydrogen-bonded systems. The evaluation of the electrostatic interaction energy for such systems is often performed by approximating the electrostatic potential of a molecule by that resulting from a set of point charges or from a single-center or multicenter distribution of multipole moments. However, one should emphasize that the electrostatic energy contains also important short-range terms due to the mutual penetration (charge overlap) of monomers’ electron clouds. This short-range part of the electrostatic energy, neglected both in the monocentered and distributed multipole expansions, makes significant contributions to the stabilization energy of hydrogen-bonded systems. The electrostatic interaction can be either attractive or repulsive. This property is best illustrated by large-R interactions of two neutral molecules possessing dipole moments. The electrostatic dipole– dipole term dominates then the interaction energy. This term can be either attractive or repulsive, depending on the mutual orientation, with the maximum and minimum of equal magnitudes. Often the electrostatic interaction energy alone, or rather its asymptotic form, is used to determine the approximate equilibrium orientation (but not the equilibrium separation) of hydrogen-bonded clusters. Due to the importance of this interaction, such predictions are frequently correct, and a few examples will be given in section V. The electrostatic predictions are not correct when other components of the interaction energy have anisotropies significantly different from the anisotropy of the electrostatic component (cf. section V). In fact, those anisotropies are always to a smaller or larger extent different so that the exact minimum structure of a hydrogen-bonded cluster can be found only by taking into account all of them. 2. Induction Interaction (2) The second-order polarization energy E pol is given by the expression

(2) E pol

=

     0A 0B V  KA  LB 2 KL

E 0A + E 0B − E KA − E LB

,

(7)

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512 where K and L label excited states of the monomer A and B, respectively, and the prime over the summation sign indicates that the term with K = 0 and L = 0 is excluded. X X Eigenfunctions  M and eigenenergies E M are solutions of the Schr¨odinger equation, with the Hamiltonian H X for monomer X describing the ground and excited states of this system. The induction energy is obtained when the sum over states in Eq. (7) is restricted to functions excited on only one monomer (i.e., to the functions of the (2) form 0A  LB and  KA 0B ). The corresponding part of E pol , (2) (2) (2) (2) denoted by E ind , is given by E ind = E ind (A) + E ind (B), where    (2) E ind (A) = 0A  B (1) (8) ind (A) (2) and a similar expression holds for E ind (B). The induction (1) function ind (A) is defined by      KA  B 0A (1) ind (A) =  KA . (9) A A E − E K 0 K =0

The operator B = 0B |V 0B  is the electrostatic potential generated by the unperturbed monomer B. Equation (8) has the form of the second-order energy correction arising when monomer A is perturbed by the static electric field generated by the (unperturbed) monomer B. Notice that Eqs. (8) and (9) include only the coordinates of electrons belonging to system A, all the effects of system B entering via the potential B . The second-order induction energy results, thus, from the polarization of the monomers by the static electric fields of unperturbed partners (in older literature the induction contribution is sometimes referred to as the polarization energy). Asymptotically, at large intermolecular distances R, this effect is fully determined by the permanent multipole moments and static multipole polarizabilities of the monomers. At finite R, additional monomer information is needed to account for the short-range, penetration part (2) of E ind . Because monomers that form hydrogen bonds almost always are very polar, the induction energy for hydrogenbonded clusters makes a very significant contribution to the total interaction energy. It is also quite anisotropic although less than the electrostatic energy. Some authors proposed to identify within the induction energy a term called the charge transfer energy. This term was assumed to describe the interactions due to the transfer of a part of the electronic charge from monomer A to monomer B or vice versa. Although such a transfer certainly takes place to some extent, a quantitative determination of this component proved to be not possible thus far. Methods proposed by various authors lead to dramatically different estimates of the size of this effect. Therefore, we will not discuss the hypothetical charge-transfer energy com-

Hydrogen Bond

ponent any further. In contrast, the difference between the dimer electron density and the sum of the densities of the two monomers is perfectly well defined and can be computed to interpret experimental observations related to the charge density, like, for example, NMR spectra. 3. Dispersion Interaction (2) is defined as The second-order dispersion energy E disp the difference between the second-order polarization and (2) (2) (2) induction energies, E disp = E pol − E ind . Therefore it can be written as       0A 0B V  KA  LB 2 (2) E disp = E A + E 0B − E KA − E LB K =0 L =0 0    (10) = 0A 0B V (1) disp ,

where (1) disp

=

 K =0 L =0

   KA  LB V 0A 0B



E 0A + E 0B − E KA − E LB

 KA  LB ,

(11)

is the “dispersion function” representing the leading intermolecular correlation contribution to the dimer wave function. This function is a sum of products of wave functions that are describing the electronic excitations on both the monomer A and B; therefore the dispersion interaction can be viewed as the stabilizing energetic effect of the correlation of instantaneous multipole moments of the monomers. Since the dispersion energy is a correlation effect, it cannot be reproduced at the Hartree–Fock level of theory. The dispersion force results from the dependence of electrons of system B on the position of an electron in molecule A; therefore it goes beyond the averaged charge distributions characteristic of the Hartree–Fock method. The dispersion energy is usually the most isotropic component of the interaction in hydrogen-bonded clusters. Although it does not have a large effect on the cluster’s equilibrium orientation, it contributes significantly to the energy of hydrogen bonds, being of similar size to the induction energy for medium-size systems but playing an increasingly larger role as the system size grows. 4. Exchange Interaction The sum of low-order polarization corrections, even when evaluated exactly without the use of the multipole expansion, is not able to predict the existence of the van der Waals minimum or the repulsive wall at shorter intermolecular separations. The repulsive contributions are due to the electron exchange (i.e., to the physical process of the [resonance] tunneling of electrons between interacting systems). The unperturbed function 0 = 0A 0B as well

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as higher-order polarization functions such as (1) ind (X ) and describe the situation when the electrons stay within (1) disp their monomers, but the permanent and instantaneous polarization is allowed (therefore the name “polarization expansion”). The true wave function contains also components corresponding to tunneling of one, two, or more electron pairs between interacting units. In the case of two hydrogen atoms, both electrons can simultaneously tunnel in opposite directions between the two potential energy wells in the six-dimensional configuration space (one well with electron 1 at nucleus A and electron 2 at nucleus B and the other with the electrons exchanged). Because the two wells are equivalent by symmetry, the tunneling leads to the resonance splitting of the unperturbed energy level into the energies of the singlet and the triplet states. Asymptotically at large R the exact wave function becomes a linear combination of two equally weighted “resonance strucA B A B tures” 0 = φ1s (r 1 )φ1s (r 2 ) and P12 0 ≡ φ1s (r 2 )φ1s (r 1 ), the second of which cannot be recovered by a low-order X polarization theory (φ1s is the 1s orbital of hydrogen centered on atom X). It is clear that A0 ≡ √12 (0 ± P12 0 ) rather than 0 should be used as the unperturbed wave function in the perturbation theory of intermolecular interactions. Unfortunately, the use of A0 in the conventional Rayleigh-Schr¨odinger perturbation procedure (employing the sum of the monomer Hamiltonians as the unperturbed operator) is impossible since A0 is not an eigenfunction of the zeroth-order operator H0 . Therefore, a modification of the perturbation procedure is required such that the function A0 can be utilized in the perturbation development. Such a modification, usually referred to as symmetry adaptation, was first attempted in 1930 by Eisenschitz and London, and there has been continued activity in this field since the late 1960s. The currently most popular variant of symmetry adaptation, called the Symmetrized Rayleigh– Schr¨odinger method, was proposed in 1978 by Jeziorski, Chalasinski, and the present author. Since the exchange effects decay exponentially with distance, they are negligible for large R. In the region of

cases the exchange contribution at the potential minimum is the component that is largest in magnitude. It is also significantly anisotropic. 5. Multipole Expansion As the distance between monomers grows larger, the importance of various components of the interaction energy changes. The exchange effects, dominating at short separations, become negligible due to their fast, exponential decay with R. The polarization components decay much slower. This is easy to understand for the electrostatic energy which—in the case of neutral polar monomers—has to reduce for very large distances to a dipole–dipole interaction proportional to 1/R 3 . One can show in general that at large R the interaction energy E int has the following asymptotic expansion in powers of 1/R: E int (R, ω A , ω B , Ξ A , Ξ B) ∼

∞ 

Cn(ω A , ω B , Ξ A , Ξ B)R −n.

n=1

(12) Although it is only asymptotically convergent in Eq. (12) can approximate the exact interaction energy arbitrarily closely when R is sufficiently large. Therefore, the knowledge of the Cn coefficients is very useful in estimating the interaction energy at large distances, and is necessary to guarantee the correct large-R asymptotic behavior of empirical or theoretically derived potential energy surfaces. The angular dependence of the coefficients Cn (ω A , ω B , Ξ A , Ξ B ) for fixed internal geometries Ξ A , Ξ B of the monomers can be expressed in a closed form. The coefficients Cn can be computed from properties of monomers such as multipole moments and polarizabilities. The relevant formulas are obtained from the polarization series truncated at some finite order by replacing the potential V by its asymptotic expansion in powers of 1/R. For the Coulomb potential 1/|r 1 −r 2 |, such expansion has the form

l< ∞   1 (−1)l B (l A + l B )! = √ |r 1 − r 2 | l A ,l B =0 m=−l< (2l A + 1)(2l B + 1)(l A + m)!(l B + m)!(l A − m)!(l B − m)!

× r1l A r2l B YlmA (θ1 , φ1 )Yl−m (θ2 , φ2 )R −l A −l B −1 , B the potential minimum these effects are, however, always very important. In the case of hydrogen-bonded dimers near the potential minimum, the electrostatic, induction, and dispersion effects are all negative (the latter two are in fact always smaller than zero). The positive exchange energy cancels a large part of the attractive effect. In many

(13)

where ri , θi , φi are the polar coordinates of ith particle and l< denotes the smaller of l A and l B . The coordinates of particle 1 are measured in system A, while those of particle 2 are measured in system B. The two coordinate systems have their z axes along the same line, and the other axes of one system are parallel to the

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514 respective axes of the other system. The functions Ylm (θ, φ) are the standard spherical harmonics. The expansion is valid only when R > r1 + r2 . Combining the equivalents of Eq. (13) for other Coulombic terms in the intermolecular interaction operator V , one gets the asymptotic expansion of V . Individual terms in this expansion are interpreted as arising from the interaction of a 2l A pole moment on monomer A (i.e., an instantaneous dipole, quadrupole, etc., moment formed by an electron) with a 2l B -pole moment on monomer B. Therefore the expansion in Eq. (13) is often called the multipole expansion of the potential. Equation (13) shows that after the multipole expansion of the potential is performed, the coordinates of electrons of molecule A are separated from those of molecule B. Therefore, if this expansion is substituted in Eqs. (5), (8), (10), and (11), the asymptotic interaction energy is expressible in terms of integrals each involving only coordinates of one monomer. Thus, this energy is expressible in terms of monomer properties only. This fact makes the calculations in the asymptotic region much easier than for the finite distances, where this approximation cannot be applied. It is important to realize that the multipole expansion has built-in dependence on the mutual orientation of two monomers. Thus, asymptotically the anisotropy of the polarization part of the interaction energy is precisely predicted by this expansion. Although for finite separations this anisotropy is modified by the penetration effects discussed below, the asymptotic prediction remains useful at finite R. The lowest power of 1/R appearing in the asymptotic expansion determines the behavior of the energy for very large separations. Monomers forming hydrogen bonds almost always have nonzero dipole moments. The electrostatic dipole–dipole interaction becomes then the dominating long-range term at very large R, because it decays as 1/R 3 . The consecutive electrostatic terms, dipole– quadrupole and quadrupole–quadrupole, decay as 1/R 4 and 1/R 5 , respectively. For neutral monomers with nonvanishing dipole moments, the induction energy decays as 1/R 6 , whereas the dispersion energy always decays as 1/R 6 (i.e., both components decay faster than the electrostatic interaction of polar monomers). Thus, the electrostatics strongly dominate the long-range interaction energy for hydrogen-bonded systems. For charged systems the role of electrostatics is even more important: the charge–charge, charge–dipole, and charge-induced dipole terms decay as 1/R, 1/R 2 , and 1/R 4 , respectively. Notice that for polar monomers at some angular orientations the interaction energy will be positive rather than negative for large enough R due to the positive dipole– dipole contribution. When the intermolecular distance gets sufficiently short, the multipole expansion fails to provide a good ap-

Hydrogen Bond

proximation to the interaction energy. The distance where this starts to happen is significantly larger than the equilibrium separation. The quantitative measure of the deviation can be obtained by comparing the results of SAPT calculations and of the multipole expansion for a given configuration. At large enough distances SAPT agrees with the asymptotic expansion arbitrarily well. In addition, SAPT electrostatic, induction, and dispersion energy components can be directly compared with analogous components of the asymptotic expansion, and the agreement is reached for each component separately. As the distance decreases, the exchange effects are becoming important so that the total interaction energies predicted by SAPT and by the asymptotic expansion begin to differ. The individual components begin to differ as well, although these differences are smaller than the exchange effects. The discrepancies in the components appear because the terms neglected when assuming the multipole expansion of the interaction potential are not negligible anymore. These terms are related to the overlap of electron charge distributions and therefore are called the overlap or penetration effects. When analytic forms of the polarization components are developed using the multipole expansion, the penetration effects have to be taken into account. Most often this is done by multiplying the 1/R terms by the so-called damping functions, decreasing the magnitude of these components for smaller R. In addition to damped 1/R terms, the polarization energies contain also purely exponential components that have to be included for an accurate modeling of these energies. The type of analysis of the interaction energy presented here is very useful for obtaining analytic fits to potential energy surfaces.

C. Many-Body Effects The interaction energy of a system consisting of N molecules can be defined similarly to the dimer energy of Eq. (1) as  E i (Ξi ), (14) E int (ξ 1 , . . . , ξ N ) = E tot (ξ 1 , . . . , ξ N ) − i

where E tot is the total energy of the N -mer, E i is the energy of the ith monomer, and ξi = (Ri , ω i , Ξi ) stands for the set of all coordinates needed to specify the spatial position Ri , orientation ω i , and the internal geometry Ξi of the ith monomer. The N -mer interaction energy can be expressed as a sum over 2, 3, . . . , N -body interactions: E int = E int [2, N ] + E int [3, N ] + · · · + E int [N , N ], (15) where K -body contributions to the N -mer energy, E int [K , N ], can be written as the following sums:

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E int [2, N ] =



E int (ξi , ξ j )[2, 2],

(16)

i< j

E int [3, N ] =

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E int (ξi , ξ j , ξ k )[3, 3],

(17)

i< j
and so on. The two-body or pairwise additive interaction energies E int [2, 2] are analogous to the dimer interaction energies defined by Eq. (1). The higher-body terms (i.e., the nonadditive contributions to the N -mer interaction energy) are defined recursively. For example, the three-body nonadditive contribution to a trimer interaction energy, E int [3, 3], is the difference between the total interaction energy of a given trimer and the sum of all pair energies. It should be noted that the many-body expansion of the interaction energy can be defined only when the quantum states of all subsystems are unambiguously defined. For strongly interacting systems such as liquid metals or chemically bound molecules this condition is not fulfilled and the applicability of the many-body expansion can be questioned. The nonadditive effects play an important role in condensed phases of hydrogen-bonded systems. However, it is important to realize that even if a cluster or solid is described using only two-body potentials, the properties are not just a superposition of dimer properties. The simple reason is that a given monomer may interact via twobody potentials with more than one nearest neighbor and that it interacts not only with its nearest neighbors but also with all the other molecules in the cluster, liquid, or solid. Since the other molecules are farther apart, the latter interaction is almost always attractive and therefore stabilizing. An instructive exercise is to consider the minimum structures and binding energies of a linear chain of atoms interacting via a Lenard–Jones potential (this potential will be discussed in section IV). Assuming for simplicity that all the bonds in an N -mer are of the same length, one can easily find the increased binding energies per monomer and decreased interatomic distances as the chain grows larger. Similar summations over a threedimensional lattice mimic the lowest order trends in rare gas crystals. One may use the phrase pairwise-additive cooperative effects to describe the properties of N -mers resulting from two-body forces only, while the true nonadditivity would define pairwise-nonadditive cooperative effects. The three- and higher-body nonadditive forces may introduce both a stabilizing or a destabilizing effect. In the case of rare gas crystals, the three-body nonadditive contribution to the interaction energy is positive, whereas for hydrogen-bonded systems the net effect is usually stabilizing. The long-range asymptotic effects have been investigated for more than 50 years, starting from the derivation of the familiar Axilrod-Teller-Muto triple-dipole disper-

sion nonadditivity in the third order of perturbation theory. Only recently, however, have the many-body effects (particularly in hydrogen-bonded systems) become the subject of quantitative investigations also for finite intermolecular separations, and most of the available information concerns the three-body nonadditivities. At the minimum configuration of the water trimer, this term contributes about 15% to the trimer’s interaction energy. Simulations of liquid water performed with two-body only and two-body plus three-body potentials give internal energies differing by 22%. Clearly, the nonadditive effects are very important for water and most other hydrogen-bonded clusters and the condensed phase. Very little is known about the fourbody and higher nonadditivities; however, the available results indicate that these are smaller than the three-body ones, contributing only a few percent to the interaction energy. Figure 3 presents the leading components of the threebody nonadditivity for hydrogen-bonded systems (or for any significantly polar systems). The electrostatic energy is always purely additive. The dispersion energy is nonadditive, but this nonadditivity plays a minor role in hydrogen-bonded systems. Thus, the nonadditivity reduces in practice to the exchange and induction components. The S n components of the first-order exchange nonadditivity are proportional to the nth power of overlap integrals between the orbitals of the monomers. The phys(1) ical mechanism of E exch [3, 3](S 2 ) interaction is as follows. The two-body exchange interaction between a pair of monomers, A and B in Fig. 3, leads to a deformation of the electronic charges, which can be described mainly by a quadrupole moment on this pair. This quadrupole moment is represented in the figure by a set of charges on AB. This set of charges can also be replaced by two dipole moments on monomers A and B, symbolized by the dotted arrows. The exchange quadrupole interacts electrostatically with the permanent dipole moment µ of molecule C. The (1) next term, E exch [3, 3](S 3 ), is of purely exchange origin, as it results from cyclic permutations involving three electrons from three monomers. This is the minimal number of electrons needed to perform exchange between all three monomers. Terms with higher powers of S result from exchanges involving more electrons. The next term, the non(2) additive induction energy E ind [3, 3], is the most important nonadditive component for hydrogen-bonded systems. It is the only term used—in the asymptotic approximation— in the so-called polarizable intermolecular potentials. The mechanism leading to this term results from the permanent dipole moment of one of the monomers, A in Fig. 3, polarizing another monomer and therefore producing on it an induced dipole moment µind , shown as the broken arrow on monomer B. This induced moment interacts in turn with the permanent dipole moment of molecule C.

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FIGURE 3 Major components of three-body nonadditivity for hydrogen-bonded systems.

III. ANALYTIC REPRESENTATIONS OF POTENTIAL SURFACES Knowledge of the interaction potential for a hydrogenbonded system on a grid of points would not be convenient for most applications. Therefore, analytic representations of this potential are needed (i.e., reasonably simple mathematical expressions depending on the intermolecular and possibly intramolecular coordinates). Parameters in such an analytic potential can be fitted to the computed interaction energies. Alternatively, the parameters can be adjusted by comparing predictions of a trial potential to the observed quantities. The existing potentials for hydrogenbonded systems range from very simple functions containing only a few terms to rather elaborate functionals with dozens of parameters. The simplest forms of the intermolecular potentials are the Lennard–Jones potential A B − 6 12 R R and the so-called “exp-6” potential E int (R) =

(18)

C , (19) R6 where A, A , B, C, and α are adjustable parameters. The 1/R 12 and the exponential terms model the repulsive wall, while the 1/R 6 term models the dispersion energy. There is nothing special in the choice of power 12—any other E int (R) = A e−α R −

large exponent would do—except that it speeds up calculations since (R 6 )2 = R 12 . The potentials of Eq. (18) or (19) can represent isotropic interactions of apolar systems. For modeling the hydrogen-bonded interactions, the potentials have to be supplemented by at least the multipole interactions, dipole–dipole, dipole–quadrupole, etc. Alternatively, one may assign to chosen sites on each monomer— typically atoms but other choices are possible—charges simulating the multipoles of the monomer. Most of the popular potentials fitted to thermodynamic properties of liquids are a combination of an isotropic term of the form of Eq. (18) or (19) and of terms describing the Coulombic interactions of a set of distributed charges. Examples include the SPC potentials of Berendsen et al. and the TIP potentials of Jorgensen et al. developed in the 1980s to describe liquid water. More accurate representations of the interaction potentials, in particular those fitted to ab initio calculations or to spectroscopic data, have to assume more complicated forms. An example of such a potential can be the SAPT-5s potential for water [Groenenboom et al. (2000)] which is of the form:  

qa q b E int = f 1 δ1ab , Rab + Aab g ab (Rab )e−βab Rab R ab a∈A,b∈B   ab

Cnab , (20) + f n δn , Rab (Rab )n n=6,8,10

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where the sum extends over eight sites (five of them symmetry distinct) a(b) in molecule A (B). Three of the sites are on the atomic nuclei, two are in the region of lone pairs, and the three remaining are on the bisector plane between the hydrogens. Not all sites are involved in all interactions. The first term represents the Coulombic interactions between charged sites. The exponential terms are multiplied by polynomials

3  m g(R) = 1 + am R

better than the effective two-body potentials. Applications of the former potentials in condensed phase simulations is a few times more time consuming than calculations with the latter ones. Obviously, the many-body effects have also to be taken into account if ab initio two-body potentials or the potentials fitted to dimer spectra are to be used in condensed phase simulations.

IV. NATURE OF HYDROGEN BOND

m=1

for increased flexibility. In addition to the leading induction and dispersion 1/R 6 terms, also the terms with inverse powers of 8 and 10 are included (odd powers are nonzero in an exact expansion but are very small). All 1/R terms are damped using the Tang-Toennies function: f n (δ, R) = 1 − e−δ R

n 

(δ R)m /m!

(21)

m=0

The potential of this form is capable of fitting the computed points to within about 0.1 kcal/mol. The potentials become still more complicated if the rigid monomer approximation is abandoned. Unfortunately, at the present time very little reliable information is available about flexible-monomer potentials. What is often done is to assume that the flexible potential can be obtained by simply letting the sites centered on atoms to move with atoms, but this assumption has not been truly tested so far. The potentials discussed above are pairwise or twobody potentials (i.e., potentials describing dimers). Yet, in many cases such potentials are fitted to thermodynamics data for liquids and solids. In such media the pairwise nonadditive effects are usually quite important. Therefore, potentials of this type are called “effective” two-body potentials since they approximate the many-body effects by an unphysical distortion of the two-body potential relative to the exact two-body potential. As a consequence, the effective two-body potentials perform poorly in predicting pure dimer properties such as dimer spectra or second virial coefficients. In fact, the effective two-body potentials perform poorly also in predicting trimer properties (although the three-body component dominates the nonadditive effects, cf. section III.C). A partial solution to the problem is offered by the socalled polarizable potentials. As discussed in section III.C, the major component of the three-body nonadditivity for hydrogen-bonded systems are the induction effects. These effects can be modeled by computing the electric field on a given monomer due to other monomers and finding the dipole moments induced by this field. The polarizable water potentials are able to predict the dimer properties much

To see how the interplay between the four fundamental physical forces creates the hydrogen bond, we will consider a number of hydrogen-bonded systems: N2 · · ·HF, hydrogen fluoride dimer, HF· · ·ClF, water dimer, dimethylnitramine (DMNA)· · ·methyl alcohol complex, and ammonia dimer. The first molecule represents a relatively weak hydrogen bond, with the interaction energy at the potential minimum of only −2.3 kcal/mol. The water dimer, certainly the most important hydrogenbonded system, is bound twice as strongly, by 5.0 kcal/mol, and the bond in the HF dimer is of similar strength. The complex of DMNA with methyl alcohol, bound by about 6 kcal/mol, is discussed to show a system for which a very plausible hydrogen-bonded configuration is not the global minimum structure. The components of the interaction energy of N2 · · ·HF as functions of the orientation of HF denoted by the angle θ2 are shown in Fig. 4. For θ2 = 0◦ (180◦ ) the configuration is linear, of the form N-N· · ·F-H (N-N· · ·H-F). The preference for the hydrogen-bonded minimum at θ = 180◦ is clearly due to the electrostatic interaction. The negative dipole–quadrupole term dominates this component (the magnitude of this term is maximized at the N-N· · ·H-F geometry), but the next term in the multipole expansion, the quadrupole–quadrupole interaction, is not negligible, contributing close to 50% of the electrostatic energy at some configurations. At the minimum this term is attractive and contributes about 28%. A rotation of N2 can only make both terms less attractive, so that it is clear that we do not need to consider other positions of this monomer. As Fig. 4 shows, the minimum configuration is also preferred by the induction energy and (slightly) by dispersion. Even HF the rather small term denoted by δ E int —which contains the induction and exchange-induction effects of the order higher than second computed at the Hartree–Fock level of theory (these effects cannot at present be separated, HF therefore δ E int is not included in the four fundamental interaction energy components)—favors the same configuration. On the other hand, the N-N· · ·H-F configuration is strongly disfavored by the exchange effects. This is to be expected, since the size of these effects is proportional to the overlap of electronic clouds and when HF rotates from

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FIGURE 4 Components of the interaction energy for N2 · · ·HF shown as functions of the angle θ2 formed by HF with ˚ close to the minimum separation, and the the intermolecular axis (θ2 = 0◦ for the N–N· · ·F–H geometry) at R = 3.6 A, N2 molecule lying on the intermolecular axis. SAPT, symmetry-adapted perturbation theory. (Data from Jankowski et al., 2000.)

0 to 180◦ around its center of mass, close to the F atom, the overlap strongly increases. The increase of the exchange repulsion is in fact larger than the absolute increase of the magnitude of electrostatic interaction. Only the inclusion of the induction and dispersion effects leads to the hydrogen-bonded minimum. N2 · · ·HF is a good example of quite a typical situation in hydrogen-bonded complexes. The prediction based on electrostatics alone gives the correct minimum angular configuration, which is then confirmed by a more detailed analysis, despite the fact that the other physical components (and their anisotropies) may be of similar size or larger than in the case of the electrostatic component. However, there are also cases where predictions based on electrostatics fail (see below). Of course, the electrostatic model cannot predict the radial position of the minimum since the electrostatic interaction only increases in magnitude as the separation decreases (in most cases this is true, but for some combinations of dipole and quadrupole moments the electrostatic energy may change sign at short R). Therefore this model assumes a hard sphere repulsion at a distance determined from the empir-

ical van der Waals radii of atoms. Since hydrogen bonds are shorter than other van der Waals bonds, the radius of ˚ is conventionally neglected the hydrogen equal to 1.2 A when adding van der Waals radii. For example, for water ˚ reasonthe sum of oxygen’s van der Waals radii is 2.8 A, ˚ ably close to the accurate value of Re = 2.9 A. A similar calculation including also the hydrogen atom gives too ˚ large a distance of 3.6 A. Some authors describe an even simpler model of the hydrogen bond than the purely electrostatic model. This model is based on the observation that the X-H bond is polarized such that there is a net positive charge on the hydrogen and negative on the atom X. The positive charge on H is attracted to a negative charge on the electronegative atom Y. Since both X and Y are negatively charged, the hydrogen bond has to be linear to attain the minimum repulsion of these charges. This model is too simple to completely predict the minimum geometry in most cases. Let us see how this model works for N2 · · ·HF. Although due to symmetry there is obviously no charge flow from one nitrogen atom to another in N2 , there is an excess

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electron density on the molecular axis outside the nuclei, where the lone pairs are located. Thus, the positive charge on the hydrogen is attracted to this place and the negative charge on F is repelled, leading to a linear hydrogen bond. However, this model is not able to determine the angle between the N-N and N· · ·H-F axes. The linear structure of the whole dimer cannot be explained by repulsion of the negative charge on F by the negative charge on the other side of the N2 molecule, as there are regions of either sign between the nitrogen nuclei. Our next example will be the hydrogen fluoride dimer. It is a good example to show that the electrostatics restricted to the dipole–dipole interaction (or the equivalent charge interaction model) is not sufficient to determine the minimum orientation even for interactions of diatomic monomers. Such a model predicts the linear F-H· · ·F-H structure dictated by the optimal orientation of the dipoles, which disagrees with the observed structure in which the hydrogen acceptor is tilted away from the linear geometry by about 60◦ H

F H· · · ·F

#

The electrostatic model gets the right geometry if one includes the dipole–quadrupole and quadrupole-quadrupole interactions. The former interaction is zero for linear geometry, as the dipoleA –quadrupoleB term cancels with the quadrupoleA – dipoleB term, but the sum of the two becomes negative for a tilted acceptor. The quadrupole– quadrupole interaction also prefers nonlinear structures. In fact, a simple electrostatic calculation involving only the dipole and quadrupole moments predicts the tilting angle of 68◦ , in reasonable agreement with the observed value. The overall stabilizing electrostatic effect between the linear and the tilted configurations is about 2 kcal/mol. As in the case of N2 · · ·HF, the exchange interaction acts opposite to the electrostatics, but the destabilizing exchange effect when going from the linear F–H· · ·F–H to the nonlinear configuration is in this case smaller than electrostatic stabilization. The induction and dispersion energies both favor the nonlinear configuration but only by a fraction of kcal/mol. Experiment and ab initio calculations also predict that the bonding hydrogen is somewhat tilted away from the F–F axis, by about 10◦ down on the diagram above. An example where electrostatics does fail to predict the correct minimum structure is the complex formed by HF and ClF. This complex is experimentally found to exist in the “anti-hydrogen bonded” HF· · ·ClF configuration rather than in the hydrogen-bonded structure ClF· · ·HF predicted by electrostatic arguments analogous to those discussed for the HF dimer.

Consider now the water dimer. The minimum structure, shown in Fig. 5, can be rationalized in a simplistic way as follows. The water molecule has two lone electron pairs in a tetrahedral arrangement with the hydrogen atoms. Since there is an excess negative charge on the lone pair, the O-H· · ·O line should lie along the position of one of the pairs. If one uses an additional argument that the acceptor’s hydrogens should maximize the distance from the free hydrogen of the donor to minimize the repulsion of positive charges, an almost correct dimer structure is obtained (cf. Fig. 5). Now let us see what structure would be predicted taking into account the electrostatic interaction only. Although for interactions involving diatoms the dipole–dipole model predicting a linear dimer may sometimes be correct, it is rather clear that it will rarely work for general monomers. Indeed, for the water dimer the electrostatic picture limited to the dipole–dipole interaction would predict a head-to-tail geometry without specifying the dihedral angle between the planes of the monomers. If we use an additional argument taking into account the repulsion of the charges on the hydrogens, a bifurcated structure is predicted, far from the true minimum (this structure, however, is a saddle point on the potential energy surface, see section VIII.A). That this approach cannot be sufficient can be seen by recalling the charge distribution of the water monomer. The simplest representation of this distribution is two positive charges on the hydrogens and a twice larger negative charge on the oxygen—quite a different distribution than a dipole along the symmetry axis. The description of the water electrostatics limited to the dipole–dipole term does work at very large intermonomer separations, as this term decays slower than other electrostatic terms. At distances near the minimum, the dipole– quadrupole and the quadrupole–quadrupole interactions are, however, very substantial. Taking them into account ˚ gives a geometry very close the true for a fixed R of 3 A minimum shown in Fig. 5: the hydrogen bond nonlinearity angle α is only 8◦ too large and the angle β between the O-O axis and the bisector of the hydrogen acceptor is 9◦ too large compared to the values computed using the SAPT method. The complete, ab initio computed electrostatic energy (including the charge penetration effects) predicts the minimum parameters even more accurately: both α and β are only 2◦ too small. The minimum structures of the DMNA complex with methyl alcohol are shown in Figs. 6 and 7. The first structure is the global minimum and as one can see it is not hydrogen bonded. The local minimum shown in Fig. 7 is hydrogen bonded but it is located 1.1 kcal/mol above the ˚ larger global minimum of 5.8 kcal/mol. It is also at 1.1 A separation than the global minimum. Sure enough, the electrostatic prediction favors the hydrogen-bonded structure. The peculiar shape of the global minimum structure

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FIGURE 5 Water dimer equilibrium geometry. The system has the Cs symmetry. The angles are computed from the symmmetry-adapted perturbation theory (SAPT)-5s potential (Mas et al., 2000) and the angles measured by Odutola and Dyke (1980) are given in parentheses. The equilibrium separation is a mixed theoretical/experimental value.

results from competing tendencies to maintain a reasonably favorable electrostatic interaction and to maximize the dispersion attraction while at the same time keeping the methyl groups of both monomers as far apart as possible to reduce the exchange repulsion. It is interesting that

all the components of the interaction energy at the global minimum are fairly close to the corresponding components at the local minimum, except for the dispersion energy, which is more than 2 kcal/mol larger in magnitude in the former configuration. Thus, the dispersion component plays here a decisive role. The dispersion energy is not particularly anisotropic for this complex, and the increase in

FIGURE 6 The global minimum structure of the dimethylnitramine complex with methyl alcohol. The carbon atoms are gray, hydrogens—white, nitrogens—blue, and oxygens—red. (From Bukowski et al., 1999.)

FIGURE 7 The secondary (local) minimum structure of the dimethylnitramine complex with methyl alcohol. The carbon atoms are gray, hydrogens—white, nitrogens—blue, and oxygen—red. (From Bukowski et al., 1999.)

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FIGURE 8 The nonhydrogen bonded (a) and hydrogen-bonded (b) structures of the ammonia dimer.

its magnitude results mainly from the shortening of R. For the electrostatic component the increase in the magnitude due to the decrease in separation is offset by the decrease due to less favorable orientation at the global minimum. The final example is the ammonia dimer. This system is unusual in this respect that the question whether it is hydrogen bonded does not have a yes or no answer. Ammonia forms hydrogen-bonded complexes with many molecules, for example, the H3 N· · ·HCl complex shown in Fig. 1. Therefore, one would expect the ammonia dimer to be hydrogen bonded. Indeed, numerous ab initio calculations predicted the hydrogen-bonded structure shown in Fig. 8(b). However, spectroscopic measurements by Nelson et al. (1987) clearly indicated that the dimer has the nonhydrogen-bonded structure similar to that depicted in Fig. 8(a) (the positions of the hydrogens were determined up to a rotation around monomers’ symmetry axes). This controversy was explained by Olthof et al. (1994) based on accurate quantum solutions for the dimer rovibrational states. They have shown that while the equilibrium structure is hydrogen bonded, the large amplitude of intermolecular motions leads to vibrationally averaged structure similar to that shown in Fig. 8(a). Table I lists physical components of the interaction energies at the minima for the systems discussed in this section. TABLE I Components of Interaction Energies (in kcal/mol) at the Minima of Selected Dimersa,b N2 · · ·HF Re E elst E ind E disp

(HF)2

(H2 O)2

DMNA· · ·HOCH3

3.56

2.73

2.95

−2.74 −1.68

−5.87 −2.00

−7.50 −2.88

−6.2 −2.9

3.22

−5.3

−1.59

−1.87

−2.77

E exch

4.25

4.79

8.91

9.1

HF δ E int

−0.42 −2.18

0.10 −4.85

−0.82 −5.05

−0.4 −5.7

E int

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a Data from Jankowski et al. (2000), Rybak et al. (1991), Mas and Szalewicz (1996), and Bukowski et al. (1999). b The equilibrium separations are given in A. ˚ E elst , E ind , E disp , and E exch are the electrostatic, induction, dispersion, and exchange compoHF contains both nents, respectively, of the interaction energy while δ E int induction and exchange-induction components of the order higher than second.

Perhaps the most important observation based on these data is how important all the components are in determining the depths at the minima. In all cases the magnitudes of the electrostatic and exchange energies are similar to or larger than the magnitude of the total interaction energy. The magnitudes of the induction and dispersion energies are typically at least half of the magnitude of the interaction energy. Clearly, the total strength of interaction results from a subtle balance between these components. One can observe various trends in the physical components. Since all the systems are similarly polar and the equilibrium separations are not much different, the electrostatic energies are of similar size except for N2 · · ·HF. In the latter case the electrostatic interaction is about twice smaller than for the other dimers due to the lack of a dipole moment on N2 . The dispersion energy is larger in magnitude for DMNA· · ·HOCH3 than for other systems, which reflects the general increase of the importance of this term with the increase in the number of electrons. It is worth mentioning that in the literature the dispersion energy is often considered to be a negligible component of interactions in hydrogen-bonded systems. Table I clearly shows that this is not the case. The reason for this misconception could be that reliable values of dispersion energy for hydrogenbonded systems were not known until 1970s. In fact, the first ab initio calculation of the dispersion energy for such a system was the work of Jeziorski and van Hemert published in 1976. The exchange energies are larger for the water dimer and for DMNA· · ·HOCH3 than for the other two systems. This is a consequence of the large magnitude of the sum of attractive contributions for the two former dimers. A stronger attraction results in a shorter intermonomer separation and, consequently, in a larger charge overlap. A question sometimes discussed in the literature is whether the hydrogen bond is a purely electrostatic phenomenon or does it have a covalent character. From the examples presented previously, it is prefectly clear that the former interpretation is not valid. The electrostatic interaction plays an important role in shaping the hydrogen bond, but it is only one of the four fundamental effects, the other three interactions being of comparable importance. However, this statement does not imply that the hydrogen bond should be called a covalent bond. The name covalent bond is used primarily in the context of chemical bonds (except for those in highly ionic compounds like inorganic salts). The covalent bond is formed between open-shell systems (i.e., systems with partially occupied electronic shells). The internal coordinates of the interacting monomers undergo significant changes upon formation of a chemical bond. The chemical forces are distinctly different from the intermolecular forces, which arise between closed-shell systems. The distinction can be further

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522 related to the physical picture of hydrogen bonding involving the electrostatic, induction, dispersion, and exchange forces. Such a picture does not apply to covalently bound compounds. Only the exchange interactions (related to electron tunneling) can be identified in covalently bonded compounds. However, the contribution of these interactions to the chemical bond energy is stabilizing, in contrast to intermolecular interactions where it is always repulsive. Therefore, it is better to reserve the descriptor “covalent” to the chemical bonds. In fact, Muller-Dethlefs ¨ and Hobza (2000) use the term noncovalent interactions for intermolecular interactions (including hydrogen bonds).

V. PROPERTIES OF HYDROGEN BONDS The hydrogen-bonded complexes are characterized by several structural and spectroscopic properties that provide means of identifying the presence of such bonds. The most typical hydrogen bonds would exhibit all those properties. Some authors require that all the characteristics listed below should be present to classify dimers as hydrogen-bonded ones, but recently the more “relaxed” definition of the hydrogen bond given in section I, which does not include this requirement, has become generally accepted. (a) The H· · ·Y distances in hydrogen-bonded systems are shorter than generally expected of intermolecular bonds. This fact is a simple consequence of the strength of hydrogen bonds and has been historically important for the formation of the concept of hydrogen bonding. The distances between atoms in molecular crystals have been known for a long time. Based on these data, Pauling defined the van der Waals radii of elements such that the sum of such radii gives the interatomic distance. For example, the radii for oxygen, nitrogen, and fluorine are 1.4, 1.5, ˚ respectively. Also, somewhat more arbitrarand 1.35 A, ˚ was assigned to hydrogen ily, a radius of about 1.1–1.3 A ˚ (other authors use even 1.0 A). The hydrogen bonds are much shorter than the sum of van der Waals radii of hydrogen and atom Y. In the case of the water dimer discussed ˚ whereas the sum above, the H· · ·O distance is about 2.0 A, ˚ of H and O van der Waals radii gives 2.5 A. (b) The fundamental frequency of the X-H bond stretch is lowered upon the formation of a complex resulting in a red shift, which is a very sensitive probe of hydrogen bonds in clusters and in the condensed phase. The origin of this effect can be understood as follows. In an isolated monomer the fundamental frequency of the X-H vibration is proportional to the energy difference between the ground (v = 0) and the first excited (v = 1) vibrational states of the monomer. When the transition takes place in a dimer, it involves also the ground van der Waals intermolecular vibrations and therefore its frequency will be

Hydrogen Bond

different. As is shown in Fig. 9, the difference in transition energies is equal to the difference of the dimer dissociation energies D0 in the interactions potentials corresponding to the ground and excited X-H vibrations E monomer − E dimer = D0 (v = 1) − D0 (v = 0). Depending on the sign, this difference is called either the red or blue shift. For a hydrogen-bonded system the excited-state potential is almost always deeper; therefore the frequency shift is to the red. This deepening of the potential is usually attributed to an increase in magnitude of the electrostatic and induction components, although recent ab initio calculations have shown that the change of the dispersion component also plays a major role. The increase of the former two components can be related to the lengthening of the X-H distance upon vibrational excitation. The longer distance results in larger magnitudes of multipole moments, which increases both the electrostatic and induction energies. This fact explains also why the magnitudes of the red shift are correlated with strengths of hydrogen bonds. The bond strength is measured by its dissociation energy D0 . Generally one may expect that if D0 is getting larger, also the difference D0 (v = 1) − D0 (v = 0) (i.e., the red shift) will get larger. There exist, in fact, empirical formulas relating the red shift to the strength of hydrogen bond. The red shift is accompanied by an increase in the intensity of the X-H stretch due to the appearance of the induced dipole moments, as discussed next. Upon formation of a hydrogen-bonded dimer, most of the other monomer vibrational frequencies are also shifted, either to red or to blue, but these shifts are typically a few times smaller in magnitude than the red shift of the hydrogenbonded X-H stretch. (c) The dipole moment of a hydrogen-bonded complex is larger than the vector sum of the monomer’s moments. This effect is due to the presence of the induced dipoles. Although a dipole moment of a cluster is well defined, attempts to determine the dipole moments of the individual monomers within the cluster encounter difficulties in the partition of the total quantity. For large water clusters and for ice, the literature values vary from about 2.3 to 3.1 Debye, depending on the method of partition, compared to the monomer’s value of 1.86 Debye. (d) The length of the X-H bond is often increased upon the formation of the cluster. This increase is almost negligible for the dimers. In the case of the water dimer, recent ab initio calculations have found that the O-H distance ˚ compared to its equilibrium increases by only 0.006 A ˚ value of 0.958 A [Klopper et al. (2000)]. However, for ice some experiments indicated that this bond can be length˚ a factor of seven more than for ened to about 1.000 A, the dimer. These experimental observations remain controversial and can be alternatively explained by disordering of oxygen atoms off the hexagonal axis. Another often

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FIGURE 9 An illustration of the origin of the red shift in vibrational frequencies upon formation of hydrogen bond.

˚ [Kuhs and Lehmann quoted experimental result is 0.973 A ˚ more than the monomer equilibrium (1986)], only 0.015 A value. An increase of this order is predicted by ab initio calculations for clusters of size of about six molecules.

VI. TYPES OF HYDROGEN BONDS The hydrogen bonds are classified based mainly on the strength of interaction as measured by the depth of the interaction potential De at the minimum of the complex. Usually three classes are distinguished: weak, moderate, and strong bonds, with energetic boundaries at about 2 and 15 kcal/mol. The weak hydrogen bonds involve less polar X-H groups in proton donors, like C-H or P-H groups, or less polar acceptors, like the N2 molecule in the N2 · · ·HF complex discussed above. Also, the hydrogen bonds where X-H attaches to a π bond on the acceptor belong to this group (examples of such bonds are given in Fig. 10). The weakest hydrogen bonds considered in the literature are about 0.5 kcal/mol. Most of hydrogenbonded complexes of interest form the group of moderate hydrogen bonds. Water dimer or hydrogen fluoride dimer are typical examples for this group. Other well-known dimers in this group involve carboxylic acids, base pairs of

nucleic acids, and typical hydrogen bonds forming within or between proteins. The strong hydrogen bonds involve ionic species. Examples include Cl− · · ·H2 O, F− · · ·H2 O, H3 O+ · · ·H2 O, and F− · · ·HF with interaction energies of about 15, 30, 35, and 40 kcal/mol, respectively. In the case of the last complex, the bond is so strong that this complex could also be classified as a chemically bonded ˚ one. Furthermore, the H· · ·F− distance of about 1.3 A is closer to distances typical for chemical bonds (about ˚ than to H· · ·Y hydrogen bond distances (about 2 A). ˚ 1 A) Strong hydrogen bonds result in a significant distortion of the monomer’s structure. The H-F bond in F− · · ·HF is of the same length as the F− · · ·H hydrogen bond (i.e., the complex is centrosymmetric). Thus, the F-H distance is

FIGURE 10 Examples of hydrogen-bonded compounds where an X-H group attaches to a π bond. (a) Complex of chloroform with benzene. (b) Complex of hydrogen chloride with ethylene.

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524 ˚ to 1.30 A ˚ upon the dimer formaincreased from 0.93 A tion. This is, however, an extreme case. For F− · · ·H2 O the ˚ hydrogen-bonded O-H distance increases by about 0.1 A ˚ from its equilibrium-isolated monomer value of 0.96 A, and this size of elongation is typical for ionic systems. The hydrogen bonding between ions and water is the principal mechanism of the hydration processes. Most of hydrogen-bonded molecules exhibit to a larger or lesser extent all of the properties described in section VI. There exist cases, however, when some of these features are not present or even the trend is opposite. An example are complexes in which the X-H vibrations are blue shifted (see Hobza et al., 1999, and Gu et al., 1999). Some authors questioned if such complexes should be considered hydrogen bonded. According to the definition adopted in this article, the hydrogen-bonded classification will be appropriate if the equilibrium geometry has a nearly linear X-H· · ·Y shape, and the bond is reasonably strong (i.e., not much weaker than about 1 kcal/mol).

VII. HYDROGEN BONDS IN CLUSTERS

Hydrogen Bond

of the two monomers agrees very well with results of modern ab initio calculations. The equilibrium O-O distance cannot be directly measured, but spectral data provide the value of R −2 0 . This value together with the difference between R −2 0 and Re from dynamical calculations in the SAPT-5s potential (Groenenboom et al. 2000) gave a mixed experimental/theoretical estimate of 2.91 ± 0.005 ˚ (Mas et al., 2000), which agrees to all digits with the A value obtained in large-scale ab initio calculations [Klopper et al., 2000]. The depth of the water dimer potential has been accurately determined by ab initio calculations to be 5.0 kcal/mol (Klopper et al., 2000). Knowledge of the limit value of De allowed scaling of the dissociation energy computed from the SAPT-5s potential (Groenenboom et al., 2000) to obtain an estimate of D0 equal to 1165 ± 54 cm−1 , in excellent agreement with the value derived from experiments equal to 1168 ± 245 cm−1 (Curtiss et al., 1975). The water dimer comparison represents an unprecedented level of agreement between theory and experiment on the equilibrium parameters for a hydrogenbonded system. As an example, Fig. 11 presents the intermonomer vibrational-rotational-tunneling (VRT) energy levels of

Small clusters offer substantial advantages in investigations of hydrogen bonds compared to the condensed phase, where one can observe only statistically averaged properties. Clusters can be detected in bulk gas phase or in molecular beams, the latter method being presently the method of choice. Various spectroscopic techniques applied to investigations of hydrogen bonds have been discussed in section II. Most information about clusters comes from infrared techniques, both near and far infrared. The former method operates in the range of several thousand inverse centimeters and excites one of the vibrational modes in a monomer with a simultaneous transition between various van der Waals vibrations. The van der Waals transitions have much lower frequencies because the intermonomer motion is much slower than the intramonomer one. The far infrared method operates in the range from a few to a few hundred inverse centimeters and therefore can be used to observe pure transitions between the intermolecular vibrational modes. Another technique used to investigate hydrogen-bonded clusters are matrix isolation methods, where clusters are included in solid matrices made mostly of rare gases. Such spectra, however, give much less precise information due to relatively strong interactions of polar molecules with the matrix. A. Dimers Rotational spectroscopy measurements predicted the structure of the water dimer shown in Fig. 5. A similar structure had been predicted even earlier by ab initio calculations. The experimental equilibrium angular orientation

FIGURE 11 Vibrational-rotational-tunneling (VRT) spectra of the water dimer. AT, acceptor twist (A ); AW, acceptor wag (A ); DT, donor torsion (A ); GS, ground state (A ). Broken lines A symmetry and solid lines, A symmetry with respect to the equilibrium structure. K = 0, rotational ground state, K = 1, first excited state. The calculated results are from Groenenboom et al. (2000), while the experimental data are from Braly et al. (2000).

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Hydrogen Bond TABLE II Vibrational Frequencies of Water (in cm−1 ) Monomer a

Bend Symm. stretch Asymm. stretch

Dimer inter b

Donor torsion Acceptor wag Acceptor twist Stretch In-plane bend Out-of-plane bend Acceptor H-O-H bend

86 112 121 187 370 565 1593c

Donor H-O-H bend Donor bridge O-H stretch

1611c 3530d

Donor free O-H stretch

3730d

Acceptor symm. stretch

3600d

Dimer intra

Liquid

Ice I f

watere

Acceptor asymm. stretch Intermonomer stretch Librational H-O-H bend O-H stretch Intermonomer stretch Librational H-O-H bend O-H stretch

1595 3657 3756

3745d 65, 162 430, 650, 795 1581, 1641 3051, 3233, 3393 3511, 3628 50, 164, 229 840 1650 3220

a

Fland et al. (1976). Groenenboom et al. (2000). The three lower frequencies are from exact calculations, whereas the three higher ones are harmonic. c Bentwood et al. (1980), in argon matrix. d Huang and Miller (1989). e Carey and Korenowski (1998), measured at temperature of 24◦ C and pressure of 256 bar. f Franks (1972), p. 133–135. b

the water dimer. The numerical values of the intermonomer vibrational frequencies are also presented in Table II. The water dimer potential has eight equivalent minimum structures differing only by a permutation of hydrogen atoms. The monomers can tunnel between these minima through the separating barriers (this tunneling should not be confused with the electron tunneling leading to the exchange forces). The major tunneling processes are shown in Fig. 12. The tunneling results in a splitting of the vibrational energy levels. The presence of tunneling splittings explains the name VRT for these spectra. The donor-acceptor interchange splittings are of the order of 1 cm−1 and would not be visible on the scale of Fig. 11. The levels shown, denoted as 1 and 2, are averages of sets of three levels split by the donor-acceptor interchange tunneling. The distance between the levels 1 and 2 is the measure of the acceptor tunneling splitting. The bifurcation tunneling does not lead to additional splittings but to small shifts of levels. The abbreviations GS,

DT, AW, and AT in Fig. 11 denote the ground state (A ), donor torsion (A ), acceptor wag (A ), and acceptor twist (A ) modes. The solid lines refer to A symmetry and the dashed lines to A symmetry with respect to the point group Cs of the equilibrium structure. Experimental levels not shown have not been measured to date. The K = 0 and 1 columns describe the rotational ground and first excited states, respectively. The agreement between the spectra measured and calculated from the SAPT-5s potential is very good for most levels. Some of the existing discrepancies may result from misassignments of the measured lines. The intramonomer vibrational frequencies change significantly upon the formation of a hydrogen-bonded dimer compared to the isolated monomers’ frequencies. Table II presents the fundamental vibrational frequencies for the water monomer, dimer, and condensed states of water. The monomer frequencies are related to the characteristic vibrational types of motion called normal modes. There are three normal modes in the water monomer: a bending mode and the symmetric and asymmetric stretches. In the water dimer there are six intramonomer modes. We expect that the vibrational modes of the acceptor will be closer to those in the monomer than those of the donor. Since the dimer bending modes are available only from measurements in matrices, we should use monomer’s frequency from matrix measurements, which is equal to 1590 cm−1 in argon matrix. Thus, as Table II shows, the acceptor’s bend is blue shifted by 3 cm−1 , while the donor’s bend is blue shifted by 21 cm−1 . We also expect that the stretch of the O-H bond containing the bridge hydrogen will be affected most. This motion cannot be anymore classified as a symmetric or asymmetric stretch, and instead we distinguish just the bridge O-H and donor’s free O-H stretches. Compared to the frequency of the monomer symmetric stretch, the bridge O-H stretch is red shifted by as much as 127 cm−1 . The free O-H stretch is shifted 26 cm−1 to the red compared to the monomer’s asymmetric stretch. For the acceptor molecule we still can recognize the symmetric and antisymmetric stretches, which are red shifted by 57 and 11 cm−1 , respectively.

B. Trimers and Larger Clusters The structures of hydrogen-bonded clusters have been the subject of extensive research in the past 20 years [see, e.g., Xantheas and Dunning (1998); Buck et al. (1998); Lee et al. (2000)]. The hydrogen-bonded N -mers with N = 3, 4, and 5 tend to be of cyclic form. The preference for the cyclic form can be understood from simple arguments. Consider water clusters as an example. Various isomers of water clusters from trimer to hexamer are displayed in Fig. 13. A chain of three water molecules can be formed

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FIGURE 12 Water dimer tunneling paths. The numbers below transition states are barrier heights computed from the symmetry-adapted perturbation theory (SAPT)-5s potential.

such that the two hydrogen bonds are in dimer equilibria configurations. Neglecting the three-body effects this configuration would be bound by about 11 kcal/mol, including about 1 kcal/mol energy from the “second-neighbor” interaction. If instead a cyclic trimer is formed, three hydrogen bonds are present. Of course, the energy of each bond is smaller than dimer’s minimum energy of 5 kcal/mol; however, this decrease turns out to be small and the two-body binding effect at the trimer minimum is about 14 kcal/mol. Three-body nonadditive energy adds about 2 kcal/mol,

resulting in a trimer bound by about 16 kcal/mol. Thus, the trimer is more strongly bound than the dimer in the sense that the separation of the equilibrium dimer into monomers requires about 5 kcal/mol of energy (neglecting zero-point energy) while separation of one monomer from the trimer’s equilibrium structure requires 11 kcal/mol. Often the binding energy of a cluster is divided by the number of monomers, defining the average binding energy as  N = −E int /N ,

FIGURE 13 Water clusters from trimer to hexamer. The top row shows the global minima isomers, the structures in the bottom row are local minima isomers. The global minimum structure of the hexamer is called “cage,” whereas the noncyclic local minima hexamers in the bottom row are known under names “prism,” “book,” and “bag,” respectively.

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where E int is the interaction energy at the minimum of the cluster (global or local). For the smallest clusters this quantity changes rapidly (it is equal to 2.5 kcal/mol for the water dimer and 5.3 kcal/mol for the cyclic trimer). This rapid change is mainly due to two-body cooperative effects. Even for the chain-like trimer the number of hydrogen bonds increases by a factor of two, whereas the number of monomers increases by a factor of 1.5 compared to the dimer, and for the cyclic trimer the number of hydrogen bonds increases by a factor of three. In the increase of  N between the dimer and the cyclic trimer about 0.7 kcal/mol is due to nonadditive effects. The number of hydrogen bonds per monomer grows further in larger clusters, reaching the maximum value of four bonds per water molecule in ice. Similar considerations to those for the trimer explain the greater stability of the cyclic tetramer and pentamer over the possible chain structures. In these two cases the number of hydrogen bonds is four and five, respectively. Unlike in the step from the dimer to trimer, there is no increase of  N resulting from increase of the number of hydrogen bonds per monomer. However,  N does increase quite substantially, by nearly 2 kcal/mol, between the trimer and the tetramer. This increase is due to the “second-neighbor” two-body interactions and to the increased number of three-body interactions: there are four trimers in a cyclic tetramer. However, the increase in  N between tetramer and pentamer is less than a half kcal/mol despite the number of three-body interactions growing to 10. One of the reasons is the “frustrated” structure of the pentamer with two free hydrogens on the same side of the ring, which leads to an increased repulsive contribution. For the pentamer one can construct various pyramidal structures like the one shown in Fig. 13 and some of them are local minima; however, all of them are energetically significantly above the cyclic pentamer despite having more hydrogen bonds. This is because the dimer configurations in those structures are far from the dimer’s minimum structure, which reduces the magnitude of the two-body component. In contrast to smaller clusters, there exist several energetically close isomers for the water hexamer. The six hydrogen bonds in the cyclic structure may provide less stabilization than eight bonds that are possible in a cagelike structure. Indeed, the latter structure has been shown both by ab initio calculations and by experiments to be the lowest one. However, the cyclic hexamer as well as several other isomers shown in Fig. 13 are energetically very close, with only about 1% or smaller differences in the binding energies. This process of forming more and more hydrogen bonds per molecule continues as the size of the cluster increases and at the limit of infinite cluster (i.e., in ice, each molecule participates in four hydro-

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527 gen bonds and the interaction energy per molecule, ∞ , is about 14 kcal/mol). An interesting question is how large a cluster one has to consider to get close to the ice-binding energy. Ab initio calculations for water decamers gave 10 of only about 9 kcal/mol [Lee et al. (2000)]. Unfortunately, for clusters with N larger than 10 reliable ab initio results are very difficult to obtain. In addition, the potential energy surfaces for clusters of this size contain enormous numbers of local minima and therefore finding the global minimum is a highly nontrivial task. The infrared spectra of water clusters provide information which in conjunction with ab initio calculations can be used to determine clusters’ structures. As an example, spectra of water clusters with N = 8, 9, and 10 measured by Buck et al. (1998) are shown in Fig. 14. The octamer and decamer are of particular interest due to their possible importance in liquid water. Let us consider the octamer first. One of possible octamer structures is displayed in Fig. 14. One can see that this structure includes exactly two types of water molecules: those that are double-proton donors but accept only a single proton (and will be denoted as DDA monomers) and those that are single-proton donors and double acceptors (DAA). All hydrogen bonds are between a DDA and a DAA molecule but are of two kinds, depending whether the O-H group is on DDA or on DAA monomer. Therefore, one may expect that the spectrum will consist of three groups of lines: two connected with the two discussed types of hydrogen bonds and the third one due to the vibrations of free hydrogens. Indeed, Fig. 14 shows that this is the case. Based on calculations using an empirical model, the lower frequency was assigned to bonds with the OH group belonging to DAA. This frequency is red shifted by as much as about 550 cm−1 from the monomer’s symmetric stretch, even more than the largest red shifts in ice or liquid water (comparing peak values). The peak corresponding to the O-H stretch in a DDA monomer is red shifted by only about 110 cm−1 (i.e., even less than in the water dimer). The free O-H stretch is at 3700 cm−1 , midway between the two stretches in the monomer. The splittings of the peaks are related to the symmetric and asymmetric types of motion. Although the octamer structure of D2d symmetry shown in Fig. 14 agrees with the spectrum, it is not known at this point whether it is indeed the minimum structure since a similar isomer of S4 symmetry provides an equally satisfactory interpretation of the spectrum. Quantum mechanical calculations [Lee et al. (2000)] predict that the two isomers have energies close to within 0.1 kcal/mol. Buck et al. assumed that both isomers might be present in the beam. Calculations of Lee et al. show that (harmonic approximation) spectra of the two isomers do differ significantly enough to distinguish between them, provided that higher resolution spectra are

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same types of hydrogen bondings. However, each spectrum has also distinctly different features. In the nanomer spectrum one can clearly see that the lowest frequency band is much broader than in the octamer. This can be related to the fact that one of the water monomers in the structure presented in Fig. 14 is of DA type. Calculations of Lee et al. (2000) indeed predict a smaller red shift for this monomer so the O-H vibration connected with the DA monomer is responsible for the features around 3150 cm−1 . A further reason for the broadening of the bands in the nanomer compared to the octamer is that the DAA monomers are no longer equivalent. Notice also that the position of the strongest peak in the hydrogen-bonded OH band moves to lower frequencies for nanomer compared to octamer and to still lower in the case of decamer, indicating the increased strength of hydrogen bonding. For decamer none of the theoretically predicted spectra agrees well with the experimental spectrum. However, the “butterfly” structure (c) in Fig. 14 agrees much better than the “cage” structure (a).

VIII. HYDROGEN BONDS IN SOLIDS

FIGURE 14 Spectra of water clusters from octamer to decamer. The horizontal axis is frequency in cm−1 . The vertical axis is the dissociation (Diss.) fraction of clusters. (From Buck et al., 1998.)

measured. This will also require, however, more accurate ab initio calculations since although the overall agreement of the ab initio spectrum generated by Lee et al. with the experimental spectrum is reasonable, some details are significantly different. For example, the splitting of the band at 3550 cm−1 is measured to be only 30 cm−1 , while the computed splitting is about 80 cm−1 . Spectra of the two other clusters shown in Fig. 14 are similar to the octamer spectrum, indicating generally the

X-ray and neutron diffraction measurements on hydrogenbonded solids provide precise data on their structure, including information about internuclear distances. These distances can be compared to those found for dimers from spectroscopic measurements. It should be realized, however, that the distances to be compared are in fact somewhat different quantities. The spectral information about internuclear distances is obtained from rotational con−1/2 stants (i.e., the distances are defined as R −2 0 , where the brackets denote averaging over the ground vibrational wave function). The diffraction data give just R0 . Both quantities are different from the equilibrium distances Re −1/2 calculated ab initio. For the water dimer, the R −2 0 and ˚ respectively. The Re O-O separations are 2.98 and 2.91 A, ˚ measured value of the O-O separation for ice is 2.75 A. ˚ The shortening of about 0.2 A in ice compared to the ˚ between dimer is twice as large as the shortening of 0.1 A the dimer and the trimer. The decrease of intermolecular distances in solids compared to dimers is related to the increase in the strengths of binding due to both pair-wise additive and nonadditive effects. For example, about 30% of the trimer shortening is due to two-body forces. Solids of hydrogen halides allow one to relate the observed trends in vibrational spectra to the differences in their pairwise additive and nonadditive effects. The different hydrogen halide crystals show fairly varying properties. At low temperatures HF, HCl, and HBr have the same type of crystal structure consisting of zig-zag

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Hydrogen Bond TABLE III Intramonomer Vibrational Frequencies of Hydrogen Halides (in cm−1 ). HF

HCl

HBr

Monomer

3961a

2886a

2556b

Dimer

3868a

2857a

2496b

3931a 3065 3275 3404 3585

2880a 2699 2715 2744 2754

2550b 2395 2406 2431 2440

Solidc

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a

Pine and Howard (1986). Maillard et al. (1979), in argon matrix. c Kittelberger and Horning (1967) for HF and Savoie and Anderson (1966) for HCl and HBr. b

hydrogen-bonded chains, while the structure of HI is not precisely known. The chains are loosely packed, the distance between chains being larger than the distance between the nearest halide atoms in a chain. The triples of nearest heavy atoms form the angle of 116◦ for HF and 90◦ for HCl and HBr, close to the halide-halide-free hydrogen angles in the corresponding dimers. The most striking differences are in the red shifts of X-H vibrations displayed in Table III. The HF crystal exhibits an extremely large shift of 896 cm−1 , whereas for HCl and HBr the shifts are only 187 and 161 cm−1 , respectively. The differences are much smaller in the dimers, where the corresponding shifts are 93, 29, and 60 cm−1 . The main reason for these differences appears to be the significantly larger X-Y separations in HCl (and HBr) than in HF clusters, which lead to much smaller nonadditive effects in the former case. In ˚ The the dimers the relevant separations are 3.8 and 2.7 A. bond is longer for HCl simply because the chlorine atom is larger than fluorine and charge-overlap repulsive effects are larger for a given separation. Also the dipole moment of HCl is about twice as small as that of HF—therefore the electrostatic interaction is about a factor of four smaller in magnitude at a given distance—but this loss of attractive interaction is partly made up by the dispersion interaction, which is larger for HCl. As a result, the strength of the hydrogen bond is only twice as small in HCl than in HF. The increase in the heavy atom separation has, however, a more significant impact on nonadditive effects. The leading nonadditive term is proportional to the product of the square of the dipole moment and polarizability. Because the polarizability of HCl is about twice that of HF, this product is roughly twice as small for HCl than for HF at a given separation. However, because the intermonomer ˚ larger than in HF, the 1/R 6 distance in HCl is almost 1 A decay of the induction nonadditivity means that there will be an additional sixfold decrease of the magnitude of the

nonadditivity in HCl compared to HF (assuming monomer separations in crystal unchanged from the dimer separations). Overall, one can expect that the stabilization due to the cooperative effects will be an order of magnitude smaller for HCl than for HF crystals. Let us now discuss the solid phase of water. Ice exists in several polymorphic phases. The one most commonly investigated is the hexagonal form (ice Ih). The structure of ice Ih obtained from simulations with a SAPT potential is shown in Fig. 15. The oxygen atoms form a hexagonal lattice, and each water molecule is in the center of a tetrahedron, participating in four hydrogen bonds. There are several possible arrangements of the hydrogen atoms in such a structure, and these are realized stochastically. There are four different nearest neighbor water dimer configurations in ice Ih. An interesting property of ice is its response to an external electric field. The dielectric constant of any polar medium is a measure of the ability of the molecules to reorient in the electric field. It is believed that in ice the

FIGURE 15 The structure of ice Ih. The top view is along the hexagonal axis, and the bottom one is from a perpendicular direction.

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530 reorientation is achieved by a collective proton transfer rather than by a true rotation of molecules. This mechanism is supported by the properties of the dielectric constant, as the time of the dielectric orientation in ice has been found to increase rapidly with decreasing temperature. Such strong temperature dependence is characteristic of processes that require activation energy, as the collective proton transfer does. ˚ As already discussed, the O-O distance in ice is 2.75 A, significantly smaller than in the dimer, showing the increased strength of binding due to cooperative effects. This increased binding is also demonstrated by the increased value of the red shift. The shift which in the dimer case is 127 cm−1 , becomes equal to 437 cm−1 for ice Ih (cf. Table II). The sharp transition line in the case of the water monomer becomes a very diffuse band in ice, with a bandwidth of about 500 cm−1 . This broadening reflects the fact that different environments of the O-H bond in the lattice exhibit different shifts, as well as temperature broadening. The H-O-H bend is seen as another broad band centered at 1650 cm−1 (i.e., 55 cm−1 blue shifted compared to the monomer). The librational bend with peak at 840 cm−1 extends from 400 to 1050 cm−1 . This range includes the dimer’s out-of-plane bend and is close to dimer’s in-plane bend. However, the librational motions in ice are collective motions of the lattice and may be very different from the dimer’s dynamics. The same is true about the intermonomer stretch band, which occupies the region from 50 to 360 cm−1 . Still, one of the peaks, at 164 cm−1 , is very close to the dimer’s stretch. Some hydrogen-bonded crystals exhibit phenomena of (anti)ferroelectricity, which are analogues of (anti) ferromagneticity. At sufficiently low temperature a ferroelectric crystal may show electric polarization, resulting in one side of the crystal charged positively and the opposite one negatively. This polarization may exist in zero external electric field. In a ferroelectric crystal molecular dipole, moments spontaneously orient themselves within small but macroscopic regions of space called domains in such a way that the domains are electrically polarized. Ferroelectricity ceases in a given material above a characteristic temperature called the Curie temperature for this crystal, because above this temperature the thermal motion overcomes the tendency of the dipoles to align. At the Curie temperature a ferroelectric crystal has an order–disorder phase transition. The best known ferroelectric hydrogenbonded crystal is KH2 PO4 . Also one of the crystal forms of water, ice IX, is ferroelectric with Curie temperature of 72◦ K. Below this temperature, the stochastic distribution of protons around oxygens characteristic of ice Ih changes into an ordered distribution, leading to an alignment of dipole moments.

Hydrogen Bond

IX. HYDROGEN BONDS IN LIQUIDS The hydrogen-bonded liquids are much more difficult to describe than either the gas phase (where the ideal gas approximation provides a good zeroth-order description) or the solid phase (with its lattice symmetries and long-range order). The hydrogen-bonded liquids, often called associated liquids, are so different from simple liquids that their properties are called anomalous. How profound the differences are can be illustrated by the fact that water would be expected, based on its molecular mass, to be a gas at room temperatures. Similarly, many organic compounds would not exist as solids were they not hydrogen bonded (e.g., the sugars glucose and sucrose). The anomalous behavior of the density of water as a function of temperature, with the maximum at 4◦ C, is another manifestation of hydrogen bonding. The fact that ice floats on water—the consequence of this density dependence—is quite important for life on earth, since organisms survive winter in water under a layer of ice. Structural information about liquids can be obtained from neutron or X-ray diffraction experiments. This information is usually presented in the form of radial atom– atom pair correlation (or distribution) functions g AB (R) giving the probability of finding an atom of type B at the distance R from atom A and vice versa. It is worth mentioning that the functions g AB (R) are not directly measured and result from a fairly complex processing of experimental data involving several assumptions. Different methods of processing of the same set of data can give significantly different pair correlation functions [Soper (2000)]. Figure 16 shows the oxygen–oxygen pair correlation function for water at ambient conditions. This function is ˚ zero up to the distance of closest approach, about 2.6 A, and then increases rapidly, showing the strongest maximum at a value close to the equilibrium separation in the ˚ This peak represents the first coordination dimer of 2.9 A. shell of the liquid and is followed by a minimum resulting from geometrical restrictions. The integral of gOO (R) up to R = Rmin1 , the position of the first minimum, defines the coordination number (i.e., the number of water molecules in the first shell). The experiments show that this value is slightly larger than five molecules, while it is four for ice. Thus, in liquid the water molecules can get into the interstitial spaces (i.e., into the cavities formed by the ice lattice, see Fig. 15). This explains why the water density increases when ice melts. As we go to still larger R, the second peak arises—indicating the second coordination shell—and then the pair correlation function oscillates with a decreasing amplitude around the limiting value of gOO (R) = 1—indicating a more and more random distribution.

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FIGURE 16 The oxygen–oxygen pair correlation function for liquid water at ambient conditions.

The vibrational spectrum spectrum of liquid water is presented in Fig. 17 and the centers of the peaks are listed in Table II. The most striking feature of this spectrum is that it consists of very broad bands. The band in the region between 3000 and 3800 cm−1 corresponding to the O-H stretching excitations is about 400 cm−1 wide at half height and shows little structure. This lack of structure is not due to insufficient spectral resolution (which is about 0.5 cm−1 ) but is an inherent feature of this spectrum. The major reason for the large width of the band is the broad distribution of environments of the O–H bonds in liquid water. Some O–H bonds are in ice-like clusters and therefore are strongly red shifted. The strongest peak at 3233 cm−1 , red shifted by 424 cm−1 , is usually attributed to this type of environment. It is quite close to the strongest peak at ice I at 3220 cm−1 . The Gaussian deconvoluted peak at about 3628 cm−1 , red shifted by 79 cm−1 compared to the average of the two stretching modes in the monomer, corresponds likely to a water molecule playing the role of a donor to only one other molecule or being only an acceptor. In both cases such a molecule possesses a “free” O-H bond whose frequency can be only little shifted. One may also mention that the peak at 3051 cm−1

may actually not be due to an O-H stretch vibration but may originate from the overtone of the H-O-H bend. In the dimer this overtone is around 3180 cm−1 . In addition to the broad range of red-shifts of O-H stretching, there are a number of other factors contributing to the smoothness of the spectral band. One of them is the fact that an excitation of the intramonomer transition is in most cases coupled to some excitation of the intermonomer modes. The smallest of the latter excitations are of the order of 100 cm−1 , which is comparable to the width of the band. Finally, the lines are broadened by thermal effects (i.e., by collisions with other water molecules). In addition to the O-H stretch band, Fig. 17 shows also the H-O-H bend region and the intramonomer vibration region. The frequency ranges not shown in Fig. 17 are relatively featureless. In the H-O-H bend region one may distinguish two frequencies likely corresponding to bendings in molecules with the prevailing donor versus acceptor character. Compared to the monomer, one of the frequencies is blue and another red shifted. The blue-shifted frequency is very close to the H-O-H bend frequency in ice I. The intramonomer frequencies have been classified as either corresponding to stretches (translations) or to

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FIGURE 17 Raman vibrational spectrum of liquid water at temperature of 24◦ C and pressure of 128 bar. (From Carey and Korenowski, 1998.) The figure shows also the Gaussian deconvoluted peaks.

librations. Comparison with the dimer frequencies may suggest that two of the librational peaks, at 430 and at 650 cm−1 , may indeed be related to dimer’s in-plane and out-of-plane bends. The frequency at 795 cm−1 may be an overtone or may result from collective motions like in ice. One of the stretches, at 162 cm−1 , is close to the dimer stretch at 187 cm−1 . The other frequency, at 65 cm−1 , classified usually as a stretch, may also result from torsional motions of donors, which have 86 cm−1 frequency in a dimer. Several models for the structure of liquid water have been proposed in the past. Some of them, like the bentbond model with all water molecules four-coordinated as in ice, can be discarded based on the molecular simula-

tions, which show a significant disorder of the hydrogenbond network. The picture based on such simulations defines the current model of liquid water. While ice-like structures prevail, some molecules form less than four and some do not form any hydrogen bonds at a given instant. Neutron scattering measurements coupled with Monte Carlo simulations indicate that on the average each water molecule participates in about 3.5 hydrogen bonds. During the thermal motion the hydrogen bonds are broken and made; thus a given molecule can move back and forth from an ice-like cluster environment with four hydrogen bonds, through intermediate hydrogen bond coordinations, to a monomer not forming any hydrogen bonds. This model explains the essential features of the pair correlation

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functions, the vibrational spectra of water, the decrease of volume on melting, and several other anomalous properties of water. For example, the time constant of the dielectric relaxation in water has been measured to be about six orders of magnitude shorter than in ice and much less temperature dependent. This indicates that the response to the electric field is achieved in liquid water by partial rotations of “free” water monomers and of small clusters— processes that do not involve activation barriers. An open question about the structure of liquid water is whether it contains one or a few predominant cluster sizes and structures. If there are a few such clusters, water should exhibit some characteristic properties of a multicomponent liquid, and there is some experimental evidence for this. Among other structures, octamers and decamers shown in Fig. 14 have been postulated. Other authors believe that more likely structures are cyclic hexamers, as these naturally result from ice Ih and have been the predominant structures observed in some molecular simulations. A snapshot from a Monte Carlo simulation with a SAPT potential at ambient conditions is shown in Fig. 18. This figure suggests that liquid water is a collection of a variety of cluster structures. Structures and properties of other strongly associated liquids like hydrogen fluoride or carboxylic acids share many features with liquid water. Therefore, a separate discussion of other pure liquids is not needed here. Spectroscopic techniques can also be applied to solutions of hydrogen-bonded species in solvents that do not form hydrogen bonds. This method is particularly useful for investigations of solutions, with two solutes forming a strong

FIGURE 18 A snap shot from a Monte Carlo simulation of liquid water at ambient conditions using a symmetry-adapted perturbation theory (SAPT) potential.

hydrogen bond. At sufficiently low dilutions, spectral line shifts characteristic of dimers can be observed. NMR techniques are often used in such measurements because the small number of dimers in fairly well-defined geometries produce spectra that are easy to interpret. Molecules of all nonpolar and certain polar compounds form aggregates in water. Because most of these molecules do not form aggregates in other solvents, the process is clearly related to the specific structure of water. Aggregation of molecules in water is commonly called hydrophobic interaction. This aggregation is a process with interesting thermodynamic properties. In some cases the change of energy due to the aggregation is positive; therefore, from this point of view the aggregation should not take place. Thus, the entropic change has to be the driving force of the process. The entropy that is the measure of the disorder of a system is increased upon aggregation compared to dispersion. If nonpolar solute particles are dispersed, water molecules tend to orient themselves around the solute molecules in an orderly fashion. Formation of aggregates leads then to a loss of order.

X. PROTON TRANSFER Proton transfer from one molecule to another is one of the most studied phenomena in chemistry. It is the essence of acid–base chemistry. This process is related to hydrogen bonding, as a hydrogen-bonded complex can be an intermediate step: + A-H + B   A-H · · · B   A− · · · H-B+   A− + H-B ,

(22) where A-H· · ·B is a hydrogen-bonded complex and A− · · ·H-B+ is an ionic pair, bound mainly by the Coulombic attraction between the charges. Proton transfer requires breaking of the chemical bond, a highly energetic transition compared to energies of hydrogen bonds. Therefore, the proton transfer processes are very distinct from the processes considered in the rest of this article, which include breaking and making of hydrogen bonds only. Almost all neutral dimers form only one stable complex in the gas phase, the neutral complex A-H· · ·B. The energy of the two ions is much higher than that of the two neutral molecules, and the barrier along the proton transfer path between the two minima may also be high. The proton transfer will therefore be a rare event. The situation changes in solution, where both complexes are stabilized by solvation. Some solvents will stabilize the ionic complex more than the neutral one, up to the point that it becomes the global minimum on the proton transfer path. The extent of proton transfer is determined then by the acidity of A-H and the basicity of B. Experimentally

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534 the transfer from the neutral to the ionic form can be observed by a very significant increase of the dipole moment of the complex. The neutral forms have dipole moments in the range of 1–3 Debye, whereas the ionic ones have dipole moments around 10 Debye. The ionic complex can also be dissociated into free solvated ions. The whole proton transfer process is a complicated many-step reaction. It involves more degrees of freedom than just the proton motion because the reorientation of the solvent molecules is critical. The rate of the proton transfer process depends on the barrier height on the reaction path. There is little reliable information available on the barrier heights for the proton transfer in solutions. In some solvents (e.g., in water), the process is simplified since the solvent effects are so strong that neither the neutral complex nor the ionic complex are formed, except as very short-lived intermediates. Then the process can be well described as a single-step reaction. There are some charged dimers that exhibit an intrinsic symmetry on the proton transfer coordinate. All those systems are strongly bound, typically by more than 10 kcal/mol. The best known example is F− · · ·HF (bound by about 40 kcal/mol). Clearly, the structure FH· · ·F− must possess, the same energy. Early electronic structure calculations were predicting that the intermolecular potential for the proton motion along the transfer path is a symmetric double-well potential. This could give rise to interesting tunneling phenomena. However, more accurate calculations have found that there is only a single minimum on this path. Thus, the structure of the system is better described as [F· · ·H· · ·F]− . The potential is similar for [H3 O2 ]− , [HCl2 ]− , and [HBr2 ]− . Only for weak proton donors like CH4 there probably is a double minimum well with a small barrier. Also the cationic complexes like [H5 O2 ]+ usually have near-centrosymmetric potentials, with barriers smaller than 1 kcal/mol. An exception could be [H7 N2 ]+ , where the barrier could be around 1 kcal/mol.

XI. HYDROGEN BONDS IN BIOLOGICAL STRUCTURES Hydrogen bonds are of great significance in determining the structure of the biologically most important compounds like proteins and nucleic acids. Since virtually all biological processes take place in aqueous solution, equally important to the hydrogen bonds between biopolymers are hydrogen bonds between biopolymers and water and between water molecules surrounding biopolymers. Water stabilizes biopolymers and, as discussed above, is often the medium stimulating the formation of various kinds of aggregates essential in living matter. Most biological compounds simply do not exist in their biologically functional form in nonaqueous solutions. The strength of

Hydrogen Bond

hydrogen bonds is just right for biological applications. The energy needed to break such a bond is somewhat larger than the thermal energy at ambient temperatures; therefore, hydrogen-bonded structures are well-defined. At the same time, in proper environments these bonds can dissociate and associate quickly, allowing dynamical biological processes. The chemical bonds could not have been used for this purpose because the energies required to break them are much larger. Proteins are polypeptides (i.e., polymers formed from amino acids). About 20 different amino acids appear in proteins. The sequence of amino acids determines the primary structure of a protein. The peptide link between the adjacent amino acids; in other words, the group O # C# ❝ N # H can act both as proton donor and acceptor (at the oxygen atom). The polypetide chains are connected by hydrogen bonds formed between different peptide link groups as shown in Fig. 19. A single chain can form a helical structure, the so-called α helix, where peptide link groups that are above each other form hydrogen bonds. Another chain arrangement are the pleated sheets (β sheets), where chains of polypeptides placed next to each other are connected by hydrogen bonds. These spatial arrangements are called secondary structures. All polypeptides have one structure or the other and often have alternating regions of either one. The spatial arrangement of fragments consisting of α helices and β sheets is called the tertiary structure (see Fig. 19), and hydrogen bonds contribute to its stability as well. Enzyme molecules are a class of polypeptides whose function in an organism is chiefly determined by their shape. In particular, the shape is critical near the catalytic center of the enzyme (key and lock mechanism). Because these shapes are determined by hydrogen bonds, it follows that hydrogen bonds are centrally important to the functions of life. In some cases hydrogen bonds can also directly participate in a catalytic reaction of an enzyme. Of similar importance are hydrogen bonds in nucleic acids, although they play a different role there than in proteins. These bonds are responsible for the specific pairing of the nucleic bases. There are only a few bases that are the building blocks of nucleic acids, and their specific sequence contains all the genetic information. Only certain pairs of bases, called complementary pairs, form strong hydrogen-bonded complexes binding the double helix structure of nucleic acids. For example, cytosine connects with guanine and thymine with adenine as shown in Fig. 20. The double helix is stabilized by two backbone

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FIGURE 19 The structure of proteins. (a) The primary structure is the determined by the sequence of amino acids in the polypeptide chain. Four of such acids shown are valine (Val), glycine (Gly), serine (Ser), and leucine (Leu). (b) The secondary structures: α helix and pleated β sheet. Hydrogens or side chains connected to the carbon atoms situated between the peptide link groups (Cα carbons) are not shown. The hydrogen bonds are indicated by dotted lines. Most of the hydrogen bonds in proteins are N-H· · ·O = C bonds, but C-H· · ·O = C bonds are also present, and one such bond is marked on the α helix. (c) Fragments of proteins built of pleated sheets and α helices connected by additional hydrogen bonds (not shown) are creating the tertiary structure. (d) Well-determined units such as the one shown in (c) may repeat a number of times in a large protein. Mutual arrangements of such units determine the quaternary structure of a protein. The units can be linked to each other by hydrogen bonds.

strands on the outside consisting of sugar molecules to which the bases are attached and of phosphate groups linking the sugar molecules together, as shown in Fig. 21. Two such strands form a double helix displayed in Fig. 22. The strands are connected by hydrogen bonds between the nucleic bases. There are two classes of nucleic acids: ribonucleic acid (RNA) and deoxyribonucleic acid (DNA). RNA plays a function in the cellular synthesis of proteins, whereas DNA is the carrier of the genetic information, except for some viruses that use RNA for that purpose. There are two main differences between RNA and DNA: the sugar of the backbone is ribose and 2-deoxyribose, respectively, and one of the bases is different. There are four major bases appearing in either RNA or DNA. In the former case these are guanine (G), adenine (A), cytosine (C),

and uracil (U). DNA uses thymine (T) instead of uracil. Thus, a given nucleic acid alphabet contains to a very good approximation just four letters. Only very exceptionally do some other bases appear, mainly methylated forms of major bases found in ribosomal RNA. The hydrogenbonded pairs GC and AT (AU in RNA) are more stable than the other combinations and better fit into the geometry of the backbone, and for this reason they are called complementary pairs. Their hydrogen-bonded configurations shown in Fig. 20 were proposed by Watson and Crick. The complementarity is the basis of the molecular mechanism of nucleic acid replication as well as transcription from DNA to RNA and translation from RNA to proteins. The genetic information coded by the sequence of the nucleic bases is replicated when DNA separates into two

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Hydrogen Bond TABLE IV Interaction Energies of Base Pairsa Base pair

E int + E def

T

∆H(T)

GC CC AU AT (Hoogsteen) AT UU TT

−23.4 −17.5

381 381

−20.5 −14.6

−12.7 −11.8

323 298

−10.3 −9.5

−10.0

323

−7.8

∆H(T) experimentalb −21.0 −16.0 −14.5 −13.0 −9.5 −9.0

a Ab initio values from Hobza and Sponer (1996). Experimental enthalpies from Yanson et al. (1979). All energies in kcal/mol, temperature in degrees Kelvin. b The experiment in not sensitive to the geometry of the complex. Assignments have been made assuming that the observed complexes correspond to the minimum structures determined in ab initio calculations.

FIGURE 20 The adenine–thymine and guanine–cytosine complementary pairs. Notice two types of hydrogen bonds: N-H· · ·O=C and N-H· · ·N.

single strands, each of which serves as a template for a new strand. The copying process producing a new strand is guided by the same pairs of hydrogen-bonded nucleic bases that appeared in the original double-stranded DNA.

Interaction energies and enthalpies of formation for a few base pairs are listed in Table IV. The enthalpies have to be computed in order to compare with measured values. Enthalpies of formation are related to the dissociation energies—under assumption of the complete separation of the rotational and vibrational motion—by the formula H (T ) = −D0 + E vib (T ) + 4RT,

FIGURE 21 A single strand of DNA. The sugar molecules are connected to each other by the phosphate groups forming the DNA backbone. A nucleic base is attached to the other end of each sugar molecule.

(23)

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monomers to those in the complex. The ab initio enthalpies of formation listed in Table IV agree well with the measured ones. One should point out, however, that due to the size of the system, the ab initio calculations had to be performed at a relatively low level of theory and used small basis sets. Also the accuracy of the experimental data is uncertain, and the measurement of Yanson et al. is the only source of data available. The complementary pairs GC and AT listed in Table IV are in configurations shown in Fig. 20. Many other hydrogen-bonded dimers are observed in crystal structures of nucleic acid bases, involving both homo and hetero pairs, and some of them are included in Table IV. The Hoogsteen AT pairing can be obtained from the AT configuration in Fig. 20 by rotating the adenine around the N-C-N axis in the six-membered ring. The GC pair is the strongest bond of all nucleic acid pairs. This is clearly due to the fact that this is the only pair with three hydrogen bonds. The Watson-Crick AT pair is bound only half as strong. The bond in the Hoogsteen AT pair is in fact little stronger, but this pair cannot be used in DNA due to steric reasons of connecting to the sugars of the backbone.

SEE ALSO THE FOLLOWING ARTICLES BIOPOLYMERS • BORON HYDRIDES • COHESION PARAMETERS • ENERGY TRANSFER, INTRAMOLECULAR • HALOGEN CHEMISTRY • PERTURBATION THEORY • PROTEIN STRUCTURE • QUANTUM MECHANICS FIGURE 22 The DNA α helix consisting of two strands such as those shown in Fig. 21. The sugar–phosphate backbones are shown only schematically as blue ribbons. The two strands are connected by the hydrogen bonds between the nucleic bases.

where R is the gas constant, T is temperature in Kelvin, and E vib (T ) is the vibrational thermal energy defined in the harmonic approximation as  ωi E vib (T ) = , (24) exp(ωi /RT ) − 1 i where ωi are the complex’s fundamental excitation energies. Enthalpy of formation accounts for the fact that in a given temperature the dimers can be created at various excited vibrational levels as well as for changes in the rotational energies during the formation of the complex. The ab initio data in Table IV include also the monomer deformation energies E def . These relatively small (1–2 kcal/mol) destabilizing contributions describe the change of monomers’ energies upon the deformation of monomers’ geometries from the values in isolated

BIBLIOGRAPHY Bentwood, R. M., Barbes, A. J., and Orville-Thomas, W. J. (1980). J. Mol. Spectrosc. 84, 391. Braly, L. B., Liu, K., Brown, M. G., Keutsch, F. N., Fellers, R. S., and Saykally, R. J. (2000). J. Chem. Phys. 112, 10314. Buck, U., Ettischer, I., Melzer, M., Buch, V., and Sadlej, J. (1998). Phys. Rev. Lett. 80, 2578 (1998). Bukowski, R., Szalewicz, K., and Chabalowski, C. (1999). J. Phys. Chem. A 103, 7322. Camy-Peyret, C., Fland, J. M., Maillard, J. P., Guelachvili, G. (1977). Mol. Phys. 33, 1641. Carey, D. M., and Korenowski, G. M. (1998). J. Chem. Phys. 108, 2669. Chalasinski, G., and Szczesniak, M. M. (1994). Chem. Rev. 94, 1723. Chemical Reviews (1994), 94, Issue 7. The American Chemical Society. Curtiss, L. A., Frurip, D. J., and Blander, M. (1975). J. Chem. Phys. 71, 2703. Desiraju, G. R., and Steiner, T. (1999). “The Weak Hydrogen Bond in Structural Chemistry and Biology,” Oxford Univ. Press, Oxford and New York. Fland, J. M., Camy-Peyret, C., and Maillard, J. P. (1976). Mol. Phys. 32, 499. Franks, F. (ed.) (1972). “Water, a Comprehensive Treatment,” Vol. 1.

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538 Groenenboom, G., Mas, E. M., Bukowski, R., Szalewicz, K., Wormer, P. E. S., and van der Avoird, A. (2000). Phys. Rev. Lett. 84, p. 4072. Groenenboom, G., Wormer, P. E. S., van der Avoird, A., Mas, E. M., Bukowski, R., and Szalewicz, K. (2000). J. Chem. Phys. 113, 0000. Gu, Y., Kar, T., and Scheiner, S. (1999). J. Am. Phys. Soc. 121, 9411. Hobza, P., Spirko, V., Havlas, Z., Buchold, K., Reinmann, B., Barth, H.-D., and Brutschy, B. (1999). Chem. Phys. Lett. 299, 180. Hobza, P., and Sponer, J. (1996). Chem. Phys. Lett. 261, 379. Sponer, J., Leszczynski, J., and Hobza, P. (1996). J. Phys. Chem. 100, 1965, 5590. Hobza, P., and Zahradnik, R. (1988). “Intermolecular Complexes—The Role of Van der Waals systems in Physical Chemistry and Biodisciplines,” Elsevier, Amsterdam. Huang, Z. S., and Miller, R. E. (1989). J. Chem. Phys. 91, 6613. Jankowski, P., Tsang, S., Klemperer, W., and Szalewicz, K. (2000). J. Chem. Phys., submitted. Jeffrey, G. A. (1997). “An Introduction to Hydrogen Bonding,” Oxford Univ. Press, Oxford and New York. Jeziorski, B., and Szalewicz, K. (1998). “Intermolecular Interactions by Perturbation Theory”, in “Encyclopedia of Computational Chemistry,” edited by P. von Ragu´e Schleyer, N. L. Allinger, T. Clark, J. Gasteiger, P. A. Kollman, H. F. Schaefer III, and P. R. Schreiner, Wiley, Chichester, UK, 1998, vol. 2, pp. 1376–1398. Jeziorska, M., Jankowski, P., Jeziorski, B., and Szalewicz, K. (2000). J. Chem. Phys. 113, 2957. Kittelberger, J. S., and Horning, D. F. (1967). J. Chem. Phys. 46, 3099. Klopper, W., van Duijneveldt-van de Rijdt, J. G. C. M., and van Duijneveldt, F. B. (2000). Phys. Chem. Chem. Phys. 2, 2227. Kuhs, W. F., and Lehmann, M. S. (1986). Water Sci. Rev. 2, 1.

Hydrogen Bond Lee, H. M., Suh, S. B., Lee, J. Y., Tarakeshwar, P., and Kim, K. S. (2000). J. Chem. Phys. 112, 9759. Mas, E. M., and Szalewicz, K. (1996). J. Chem. Phys. 104, 7606. Mas, E. M., Bukowski, R., Szalewicz, K., Groenenboom, G., Wormer, P. E. S., and van der Avoird, A. (2000). J. Chem. Phys. 113, 0000. Maillard, D., Schriver, A., Perchard, J. P., and Girardet, C. (1979). J. Chem. Phys. 71, 505. M¨uller-Dethlefs, K., and Hobza, P. (2000). Chem. Rev. 100, 143. Nelson, D. D., Fraser, G. T., and Klemperer, W. (1987). Science 238, 1670. Odutola, J. A., and Dyke, T. R. (1980). J. Chem. Phys. 72, 5062. Olthof, E. H. T., van der Avoird, A., and Wormer, P. E. S. (1994). J. Chem. Phys. 101, 8430, 8443. Pine, A. S., and Howard, B. J. (1986). J. Chem. Phys. 84, 590. Rybak, S., Jeziorski, B., Szalewicz, K. (1991). J. Chem. Phys. 95, 6576. Savoie, R., and Anderson, A. (1966). J. Chem. Phys. 44, 548. Scheiner, S. (1997). “Hydrogen Bonding. A Theoretical Perspective,” Oxford Univ. Press, Oxford and New York. Schuster, P. (ed.) (1984). “Hydrogen Bonds,” Top. Curr. Chem. 120, Springer, Berlin. Soper, A. K. (2000). Chem. Phys. 258, 121. Stone, A. J. (1996). “The Theory of Intermolecular Forces,” Clarendon Press, Oxford. van Duijneveldt, F. B., van Duijneveldt-van de Rijdt, J. G. C. M., and van Lenthe, J. H. (1994). Chem. Rev. 94, 1873. Xantheas, S. S, and Dunning, T. H., Jr. (1998). In “Advances in Molecular Vibrations and Collision Dynamics,” edited by J. M. Bowman and Z. Bacic, p. 365, JAI Press, Stamford. Yanson, I. K., Teplitsky, A. B., and Sukhodub, L. F. (1979). Biopolymers 18, 1149.

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Ion Kinetics and Energetics James M. Farrar University of Rochester

I. History II. Ion Energetics III. Ion Kinetics

GLOSSARY

Gas-phase basicity Gibbs free energy change associated with the protonation reaction M,

Appearance energy (AE) Minimum energy required to form a particular fragment ion A+ from a precursor neutral molecule: +

−

AB −→ A + B + e ,

Hr×n = AE.

Breakdown diagram Graph of the relative abundance of parent ion and fragment ion as a function of the energy above the ionization threshold of the precursor neutral molecule. Electron affinity (EA) For a neutral atom or molecule, equal to the energy difference between the enthalpy of formation of a neutral species and the enthalpy of formation of the negative ion of the same structure. The EA is defined as the negative of the 0 K enthalpy change for the electron attachment reaction: M + e− −→ M− ,

Hr×n = EA.

Franck–Condon principle Principle that an electronic transition, such as that effected in ionization, takes place such that the positions and momenta of the nuclei remain fixed. Gas-phase acidity For a molecule AH, the Gibbs free energy change (at 298 K) for the reaction AH −→ A− + H+ .

M + H+ −→ MH+ . Ionization energy (IE) Energy required to remove an electron from a molecule or atom: M −→ M+ + e− ,

Hr×n = IE.

IEs are classified as adiabatic or vertical: The adiabatic IE is the lowest energy required to remove an electron from a molecule or atom and equals the transition energy from the lowest electronic, vibrational, and rotational level of the isolated molecule to the corresponding level of the isolated ion. The vertical IE is the energy change corresponding to ionization leading to formation of the ion in a configuration with the same equilibrium geometry as the ground-state neutral molecule. The vertical IE must always be greater than or equal to the adiabatic IE. KERD Kinetic energy release distribution—the probability distribution for observing product fragments in a dissociative process with a particular value of the relative kinetic energy. MATI Mass-analyzed threshold ionization—the mass spectrometry of ions produced in coincidence with zero-kinetic energy (ZEKE) electrons.

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PEPICO Form of photoionization mass spectrometry in which ions produced by photoionization are detected in coincidence with the accompanying energy-selected photoelectrons. PFI Pulsed field ionization—the application of a short electric field pulse that induces long-lived Rydberg states of atoms or molecules just below the ionization continuum to decay by tunneling through the electric field-induced barrier. Photoionization Production of atomic and molecular ions from precursor neutrals by absorption of photons. Proton affinity (PA) Negative of the enthalpy change of the protonation reaction: M + H+ −→ MH+ ,

Hr×n = −PA.

The PA is a quantity defined at a finite temperature, usually 298 K. Quasi-Equilibrium theory (QET) Theory of mass spectra based on rapid internal conversion of excited electronic states produced by electron impact to the ground electronic state. Statistical redistribution of energy on the-ground state surface, followed by vibrational predissociation, determines the rate of fragmentation. REMPI Resonance-enhanced multiphoton ionization— the ionization process whereby an atomic or molecular species absorbs two or more photons through a transition from an intermediate resonant bound state to a bound state. R2PI Resonance two-photon ionization—a variant of REMPI in which two photons are required to produce ionization, with the first photon accessing a resonant bound intermediate state. ZEKE Zero-kinetic energy electrons, produced by field ionization of excited Rydberg states just below the ionization limit to a particular internal energy state of the resultant ion.

I. HISTORY At the beginning of the twentieth century, groundbreaking experiments by leading scientists of the day, most notably J. J. Thomson and Ernest Rutherford, determined that the electrical nature of matter is fundamentally a property associated discrete particles. Thomson’s discovery of the “electron” as the particle that carries negative charge in matter and Rutherford’s determination that the positive charge in atoms is localized in a very small volume called the “nucleus” provided the foundations of the modern atomic theory. In an electrically neutral atom or molecule, the number of electrons balances the positive charge of the constituent nuclei. Positively charged species have a deficiency of electron charge relative to the

nuclei and are called cations, while negatively charged species are referred to as anions. One of the early triumphs of the theory of atomic structure was the correct description of the quantized energy level structure of atoms formed by the Coulomb attraction of a single electron to a positively charged nucleus. The 1/n 2 dependence of the energy levels on integer quantum number n in one-electron systems yielded an exact fit to the emission spectra of such atoms that had been cataloged at the end of the nineteenth century by Balmer, Rydberg, Bohr, and Ritz. The convergence of such Rydberg series at large n to energy levels of the corresponding ion provided a sharp definition of the ionization energy of an atom. A. Ion Energetics Once the fundamental principle was established that matter exists as a collection of positively charged nuclei surrounded by negatively charged electrons, many experimental observations confirmed that ionized matter is present throughout the Universe. In addition to “cathode rays” produced by thermionic emission from metals or by the photoelectric effect, “positive rays” were also generated in gases. The identification of the latter species as positive ions from the fact that their deflections in electric and magnetic fields were opposite to those of cathode rays occurred quickly. The same experimental techniques of focusing and deflection of charged particles in electric and magnetic fields, in conjunction with improved methods for producing a high vacuum, led to the development of mass spectrometers, devices that sorted these charged particles according to mass. The early history of mass spectrometry focused on the determination of atomic and molecular masses of ions formed by electron impactinduced removal of one (or more) electrons from a stable molecule. The use of incident electrons with reasonably well-characterized energies also led to systematic information on the threshold ionization energies of atoms and molecules, forming the basis for the determination of enthalpies of formation of gaseous ions. In the case of simple electron removal to form the parent ion, the resultant energy is the ionization energy, while bond cleavage occurring in conjunction with ionization, so-called dissociative ionization, yields the appearance energy of the corresponding ionic fragment. The systematic study of ionization and appearance energies for parent and fragment ion formation to yield enthalpies of formation for gaseous ions forms the heart of the subject of “ion energetics.” The electron volt (eV) is the most frequently employed unit of energy in studies of ionization phenomena and corresponds to the energy acquired by a single fundamental charge when accelerated by an electrostatic

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potential of one volt. The definition of the electron volt is as follows: 1 eV = 96.4845 kJ/mol or 23.06036 kcal/mol. B. Ion Kinetics Concurrently with these studies of primary atomic and molecular ionization mechanisms in atoms and molecules under collision-free conditions, mass spectrometrists began to observe pressure-dependent phenomena in the ion sources of their instruments. One of the first such observations goes back to the beginning of the twentieth century, when J. J. Thomson discovered that operating his positive ray parabola apparatus in a hydrogen atmosphere produced signals at a mass-to-charge ratio of 3. Although he correctly attributed this signal to the species H3 , it was only in later high-pressure ion source mass spectrometer experiments by Hogness and Lunn that it was demonstrated unequivocally that this species was produced by a reaction between the primary ionization product H+ 2 and molecular hydrogen. This point of view concerning the rapidity of reactions between gaseous ions and molecules in the gas phase was supported by other pioneers in the emerging field of mass spectrometry, including Aston and Dempster. Although theoretical work by Eyring, Hirschfielder, and Taylor in the mid-1930s predicted that the rate of the ion– molecule reaction + H+ 2 + H2 −→ H3 + H

(1)

was limited only by the frequency with which the reactants collide, a picture that was claimed to describe the rates of many such ion–molecule reactions, systematic studies of such rates were not carried out for another 15 years. The pioneering work of Tal’rose in Moscow on ion formation in methane gas irradiated by electrons initiated the study of ion–molecule reactions as a distinct discipline within the field of chemical kinetics. In those studies, electron impact ionization in methane produced CH+ 4 as a parent ion and CH+ by dissociative ionization. The bimolecular 3 reaction + CH+ 4 + CH4 −→ CH5 + CH3 ,

(2)

occurring by a mechanism in which proton transfer from + CH+ 4 to CH4 and hydrogen transfer from CH4 to CH4 are implicated, was observed to occur on every collision. In addition, the carbon–carbon bond formation reaction + CH+ 4 + CH4 −→ C2 H5 + H2

(3)

was also observed to occur at the collision frequency. Reactions (2) and (3) occurring in methane were also observed by Stevenson and co-workers in 1955 in the ion source of a laboratory mass spectrometer and launched an important era that connected elementary ion–molecule

reactions with ionic products observed in the radiation chemistry of hydrocarbons. Pioneering work by Hamill at the Notre Dame Radiation Laboratory suggested that the ionic products formed in hydrocarbon mixtures irradiated by ionizing radiation from the γ -emission of radioactive nuclei could be explained on the basis of simple ion-neutral collisions. It was the observation in 1958 by Field and Lampe that hydrocarbon cations were capable of abstracting hydride (H− ) ions from neutral hydrocarbon collision partners that led Futrell to connect chemistry occurring in γ -radiolysis with simple ion–molecule reactions. The determination of ionization potentials and appearance energies with instrumentation that developed rapidly in the 1950s and 1960s led to additional studies of the reactivity and stability of gas-phase ions by pioneers such as Franklin and Ausloos. The 1960s brought two major advances in the discipline of ion energetics and structure. The development of highpressure mass spectrometer ion sources led by Kebarle and co-workers allowed the measurement of equilibrium constants for gas-phase ionic systems at well-specified temperatures and, from temperature-dependent studies, the enthalpy and entropy changes associated with gas phase equilibria. These concepts provided the foundation for current studies of molecular recognition in the gas phase. The development of the discipline of elementary reaction dynamics as the foundation of chemical kinetics also occurred in the 1960s. This subject addresses the fundamental question of energy disposal in a chemical reaction and the microscopic details of the collisions that lead to reaction product formation by probing the features of the potential energy surface (or surfaces) that governs the conversion of reactants into products. The application of the methods of collision kinematics and dynamics exemplified in the molecular beam experiments of Datz and Taylor was a natural development, as the techniques of mass spectrometry form the ionic reagents into a directed beam. The measurement of product energy and angular distributions that such an experimental geometry affords yielded a microscopic view of the kinds of collisions that lead to chemical reaction. Some of the pioneering applications of these techniques were accomplished by Mahan and co-workers at Berkeley, Herman in Prague, and Wolfgang at Yale, on collisions of atomic and diatomic ions with hydrogen molecules. By exploiting the state selectivity that ionization of small molecules by ultraviolet radiation provided, ionic collision dynamics made a central contribution to the emerging field of chemical dynamics. A key question in chemical dynamics is how energy that is placed in specific modes of reactant motion affects the outcome of a chemical reaction. The first example of a chemical reactant prepared in a quantized excited vibrational state and

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the observation that energy in this form allowed the system to surmount a potential energy barrier occurred in an ionic system. By using a monochromatic light source in the ultraviolet region of the spectrum, Chupka and co-workers were able to produce H+ 2 ions in selected vibrational states by photoionization of molecular hydrogen. The use of photoionization methods to produce vibrationally stateselected reagents both for bimolecular collision studies and for unimolecular decay of energy-selected ions played a central role in contemporary developments in ionic reaction dynamics. Along with more sophisticated methods for reagent preparation and product state-resolved detection came advances in the production of chemically more complex reagents. The field of ion kinetics has provided benchmark data for testing theories of chemical reaction dynamics in elementary atom–diatom systems. The field has also provided a wealth of studies on more complex processes at the microscopic level, including organometallic chemistry related to catalysis, cluster ion studies relevant to atmospheric and materials chemistry, the ion chemistry of electrical discharges and plasmas (relevant to the microelectronics industry), and reactions of biological species that provide useful information on the nature of conformation as a function of solvation. The majority of ionic species are formed by the removal (or the addition) of an electron from (or to) a stable atom or molecule. As a result, ionic species are highly reactive. Because the environment in which ionic species are created is often chemically complex, special techniques for the preparation and handling of such transients are required for reliable determination of ionization and appearance energies, energetic thresholds for chemical reactions, and unambiguous measurements of chemical reaction cross sections and rates. The general techniques of mass spectrometry form the basis for experimental methods that provide information on ion energetics and kinetics.

II. ION ENERGETICS A. Spectroscopic and Ionization Methods in Isolated Atoms and Molecules 1. Threshold Determination The ionization threshold energy for the process M −→ M+ + e−

(4)

is the minimum internal energy change accompanying the removal of an electron from a stable neutral precursor M. The term ionization energy generally refers to the adiabatic ionization energy, denoted IEa . The species M+ is called the parent ion. When M is a molecular species, the ionization process may impart sufficient energy in excess

of the ionization threshold to cause fragmentation of the parent ion. The threshold energies associated with fragments A+ formed through the reaction AB −→ A+ + B + e−

(5)

are the appearance energies, AE, for a given fragment. Appearance energy values result in thermodynamic information only when the reaction coordinate has no barrier in excess of the endothermicity for dissociative ionization, and the products appear promptly on the time scale of product detection in a particular instrument, i.e., there is no “kinetic shift.” The terminology introduced above for the formation of cations may also be applied to anions, which may be formed either through heterolytic bond cleavage in AB, AB −→ A+ + B− ,

(6)

or by electron attachment to a suitable neutral precursor A having a positive electron affinity, A + e− −→ A− .

(7)

Some classes of ionic decomposition processes occur via pathways requiring that the system surmount a barrier on the potential surface in excess of the endothermicity. Simple bond cleavage processes generally occur without such a barrier, but rearrangements often occur with such a barrier. The determination of accurate thermochemical information from fragmentation threshold energies is always subject to the uncertainty that the enthalpy of reaction may differ from the observed threshold energy. The presence of a potential barrier for a fragmentation process may result in excess product translational energy release, and measurement of the kinetic energy release distribution (KERD) often allows an estimate of the barrier height. The term “kinetic shift” refers to experimentally observed ionization thresholds that are higher than true thermodynamic threshold energies because of the finite lifetime of the decaying species. Most experimental apparatuses sample ions at a specific, instrument-dependent time after ionization has occurred, typically about 10−5 sec, so that ions decaying on a time scale long compared to this characteristic observation time will not be detected at threshold. This situation results in an apparent threshold for dissociation that is shifted to higher values of the internal energy of the decaying parent ion by the magnitude of the “kinetic shift.” The kinetic shift thus depends on the time scale for observation and the sensitivity of the instrument for detecting low signal levels for ion formation. Measuring the decay rate as a function of the parent ion internal energy provides a way to correct experimental data for this kinetic shift. Time-resolved ionization methods employing pulsed ion production methods also provide a way to estimate the kinetic shift.

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The most widely used technique for determination of ionization and appearance energies measures the minimum energy required to form a parent or fragment ion from its precursor neutral or to detach an electron from a negative ion. Ionization of a given species may be effected by absorption of a photon, by collision with electrons having kinetic energy in excess of the ionization threshold, or by interaction with excited atoms or molecules having excitation energies in excess of the ionization energy of the neutral. Such measurements can be carried out by determining the coincidence between the ions and the electrons formed. The venerable technique of electron impact is the most widely employed in mass spectrometric determination of ionization and appearance energies. The development in the 1960s of electron monochromators, in which electron beams with kinetic energy spreads of as little as 0.005 eV are generated by passing the beam through deflectiontype electron energy selectors, led to important advances in measuring ionization and appearance potentials. The incident electron excites the molecular precursor through an energy transfer collision in a vertical Franck–Condon transition to excited electronic states. The Franck–Condon principle states that the most probable ionizing transition will be the one in which the positions and momenta of the nuclei do not change. When the equilibrium geometries of a neutral precursor and its corresponding parent ion are similar, the energy dependence of the ionization threshold will be a sharp step function, forming the ground vibrational state of the ion. However, when the equilibrium geometry of the ion is significantly different from that of the neutral, e.g., is accompanied by a significant change in one or more bond lengths or angles, the transition to the lowest vibrational level of the ion is no longer the most probable one. The vertical ionization energy corresponding to the maximum transition probability will populate excited vibrational levels of the ion preferentially. The geometry change upon ionization can be sufficiently large that ionization to produce the lowest vibrational level of the ion at the adiabatic ionization energy will not be observed. In the mid-1960s, the availability of tunable ultraviolet radiation laboratory sources led to the replacement of electron beams with photon sources. Rare gas resonance lamps producing continuum sources of radiation that could then be passed through monochromators allowed the field of photoionization mass spectrometry to develop. The determination of photoion yield as a function of ionization energy, the photoionization efficiency curve, led to determination of ionization potentials with accuracies exceeding those of electron impact methods. Modern photoionization experiments often utilize laser or synchrotron light sources with narrow bandwidths and may employ collimated molecular beam sources that reduce the effects of

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69 Doppler broadening. These refinements make the determination of ionization energies quite precise, in the favorable case where the neutral and ionic ground states do not involve significant geometric rearrangements. As with electron impact ionization, the Franck–Condon principle predicts that a large change in geometry upon singlephoton absorption will result in an ionization threshold that is gradual, making the exact ionization threshold difficult to locate accurately. In the case where ionization is direct, the narrow bandwidth of modern photoionization sources often provides a means to achieve the production of molecular ions with vibrational state selection. At the lowest ionization threshold, ions are formed in the ground vibrational state. As the frequency of the photoionization radiation is increased, thresholds for the formation of molecular ions with vibrational excitation are encountered, and the photoionization efficiency curve shows stepfunction increases in ion production as excited vibrational states become accessible. In the favorable case of photoionization of H2 , rotational state selection of the resultant molecular ions can also be achieved. At the threshold for the formation of a molecular ion in a given excited vibrational state, the product ion distribution will contain contributions from all lower vibrational states that are energetically accessible, degrading the quantum state purity of the parent ions. In addition, the simple step-function behavior that accompanies direct ionization may be complicated by autoionization processes, in which the initial optical excitation is to an excited Rydberg state of the neutral molecule. As discussed below, energy analysis of the electrons that accompany the ionization process can sort out the details of such complex ionization processes. The rapid development of intense laser sources has led to the development of ionization methods based on the absorption of multiple photons. Ionization energies for molecules have been measured using resonant twophoton ionization (R2PI) or resonance-enhanced multiphoton ionization (REMPI) of rotationally cooled species in a molecular beam. In such studies, the internally cooled beam of molecules is excited to a specific quantum state with a tunable laser; while holding this excitation energy constant, a second, independently tunable laser ionizes the beam of excited molecules. By tuning the first excitation laser to different transitions, the entire Franck–Condon accessible region of the intermediate electronic state is probed, yielding the adiabatic ionization energy. The experiment thus contains an internal consistency check that leads to more reliable values of ionization threshold. The production of ions by multiple photon absorption is complicated by the loss of information on the Franck–Condon factors for the initial photon absorption steps. However, in cases where the ionization is an (n + 1)-photon process, with the first n photons providing access to a Rydberg

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state of the precursor neutral, the final photon absorption process often occurs from a specific vibrational state of the neutral to an ion with the same vibrational quantum number. Under such conditions, vibrational state selection can be achieved by multiphoton absorption. The method has been applied successfully in the production of NH+ 3 cations with specific excitation in the ν2 umbrella-bending mode. The photoionization process is accompanied by the emission of electrons. The technique of photoelectron spectroscopy is based on a measurement of the kinetic energies of those electrons, either independently of the ion formation process or in coincidence with it. In photoelectron spectroscopy, the precursor atom or molecule is irradiated with light of a known energy sufficient to eject an electron from the material through the process M + hν −→ M+ + e− (KE),

(8)

KE(e− ) = hν − I − E ∗ (vib,rot).

(9)

where

In this expression, I is the binding energy of the electron and E ∗ (vib,rot) is the internal energy of the product ion M+ . The technique may also be employed to determine the electron affinities of atomic or molecular anions. Conventional photoelectron spectroscopy, invented by Turner and Siegbahn, developed through the application of the helium or neon resonance lines (21.218 and 40.813 or 16.848 and 16.671 eV, respectively) from discharge lamp photon sources. The development of synchrotron radiation sources as well as tunable vacuum ultraviolet laser sources has also enhanced the utility of photoelectron spectroscopic methods. In an experiment, the ejected electrons will be produced with a distribution of kinetic energies degermined by the distribution of energy levels in the M+ ion product. The energy distribution of the emitted electrons is called the photoelectron spectrum. The photoelectron band shapes reflect not only the vibrational energy level structures in the ground and excited electronic states of the ion, but also the M+ ← M transition probabilities dictated by Franck–Condon factors. When the equilibrium geometries of the ion and the corresponding neutral are similar, the observed threshold of the first photoelectron band generally yields the adiabatic ionization energy. The ionization energies are thus determined with an energy resolution characteristic of the resolving power of deflection-type energy analyzers, or as low as 0.005 eV. On an energy scale appropriate to optical measurements, this resolution corresponds to 40 cm−1 , significantly poorer than the subcentimeter value afforded by narrow-bandwidth optical sources. Threshold photoelectron spectroscopy (TPES) was developed to overcome the inherent limitations in the

spectral resolution of conventional photoelectron spectroscopy. When the full energy of the photon is absorbed in the optical transition leading to ionization, the ejected electron has zero kinetic energy. As the photon energy is increased beyond an ionization threshold, the atom or molecule continues to absorb light and electrons are ejected with nonzero kinetic energy. The resulting “hot” electrons can be discriminated from the zero kinetic energy electrons by the fact that they are moving in the center of mass coordinates, and the electrons with zero kinetic energy are stationary in this frame of reference. The zero-kinetic energy-threshold electrons remain in the interaction region, and a small electric field extracts them through a small aperture or a restricted channel. An important variant of the threshold photoelectron spectroscopy technique arises from a consideration of the nature of the Rydberg states of the precursor neutral that converge to the ground state of the ion, defining the ionization energy. Additionally, each electronic, vibrational, and rotational (rovibronic) state of the ion has a unique Rydberg series converging to that state of the ion. While conventional TPES focuses on the zero-kinetic energy electrons in the ionization continuum, i.e., KE(e− ) → 0+ , the technique of ZEKE spectroscopy is concerned with the detection of Rydberg states in a very narrow band just below the ionization limit of each robronic state of the ion. These long-lived states are separated from the fastdecaying low-n Rydberg states by a simple time delay in the detection process. The usual method for carrying out this delayed ionization is to apply a pulsed electric field of magniture ∼1 V/cm approximately 10 µsec after the primary excitation step and after the “prompt” electrons from short-lived states have moved away. The small applied field then ionizes the long-lived Rydberg states by field ionization. The lifetimes of such high-n Rydberg states are observed to be nearly two orders of magnitude longer than expected and arise from external perturbations that operate only on the Rydberg states lying extremely close to the ionization threshold. The acronym “ZEKE,” referring to the detection of these zero-kinetic energy electrons, is commonplace, although the term “ZKE” has also been used in the literature, more correctly referring to the zero-field production of threshold electrons. The use of a pulsed electric field is referred to as “pulsed field ionization” (PFI). ZEKE spectroscopy differs from other methods of photoelectron spectroscopy because it is a multistep process. The successive steps are required to isolate the narrow band of long-lived high-n Rydberg neutrals from the background hot electrons and their counterions from the low-n Rydberg states. The ions produced in coincidence with ZEKE electrons can also be detected, forming the basis for mass-analyzed threshold ionization (MATI) spectroscopy.

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Owing to the kinematic restriction that results from detecting the ion, this method has an intrinsically lower energy resolution than detecting the corresponding ZEKE electron. However, that disadvantage is offset by the desirability of producing mass-selected ions with a known vibrational excitation. For the purpose of producing ions with wellcharacterized levels of vibrational excitation, as well as for studying the thermochemistry of ionic fragmentation processes, the coincident detection of photoions and their corresponding electrons yields a powerful combination. The technique known as photoelectron–photoion coincidence (PEPICO) correlates ejected electrons formed with zero kinetic energy with their corresponding positive ions. Below the dissociation threshold, this method provides a way to produce parent ions with vibrational state selection. At energies where parent ions undergo dissociation to form one or more fragment ions, one measures the relative probabilities for the formation of the daughter ions from parent ions with a well-specified internal energy, a quantity called the “breakdown diagram.” The ions can be detected at differing times after the ionization event for determination of the time dependence of the dissociation process. By measuring the time-of-flight distributions for state-selected ions, a detailed analysis of the lineshapes can yield the kinetic energy release distributions for stateselected fragmentation processes occurring on the time scale of transit through the instrument. Photoelectron spectroscopy can also be carried out by measuring the distribution of flight times of photoemitted electrons. This method is useful in systems in which the absorbing species are produced by a pulsed source and the photolysis radiation is also pulsed. These electron time of flight methods have been used to elucidate structures of transient species such as free radicals and clusters produced in pulsed photolysis sources and in assessing the vibrational-state purity of ions produced in multiphoton ionization processes, particularly those in which the final photon absorption process is from a Rydberg state whose geometry is similar to that of the ion. 2. Ion Enthalpies of Formation from Threshold Energies The methods for determining ionization and appearance energies under collision-free conditions as described above must be employed according to well-documented procedures of chemical thermodynamics to yield the appropriate enthalpies. The conversion of the internal energy changes for these primary ionization processes to enthalpies of formation requires that attention be paid to a number of important principles and conventions. The enthalpy of formation of a molecular cation is obtained

by summing the enthalpy of formation of the precursor molecule to the adiabatic ionization energy (IEa ) and subtracting the enthalpy of formation of the electron, f H 0 (M+ ) = f H 0 (M) + IEa − f H 0 (e− ).

(10)

The analogous expression for an anion requires the value for the electron affinity: f H 0 (M− ) = f H 0 (M) − EA + f H 0 (e− ).

(11)

Similarly, the enthalpy of formation of a positive fragment ion A+ with appearance energy AE formed by dissociative ionization of precursor molecule AB is given by f H 0 (A+ ) = f H 0 (AB) − f H 0 (B) − f H 0 (e− ) + AE. (12) As we have already noted, this expression assumes that the dissociation reaction coordinate has no barrier in excess of the dissociation energy and that the dissociation lifetime is rapid compared with the time scale of measurement in the specific instrument. The molecular quantities of ionization energy, electron affinity, and appearance energy refer to differences in zero-point energy levels of the appropriate potential energy surfaces of reactants or products and are, thus, internal energies. Their conversion to the corresponding enthalpies at finite, nonzero absolute temperatures requires correction factors that depend on how the free electron enthalpy is accounted for. The enthalpy of formation of a chemical species is defined as the difference between the enthalpy of the compound and the sum of the enthalpies of the elements comprising it. In the case of a cation or anion, the balanced chemical reaction corresponding to its formation from the elements will also contain the electrons that are liberated or captured in the ionization process. The enthalpy of formation of the free electron must therefore also be accounted for in the ionic enthalpy of formation. Although the numerical value of the enthalpy of formation of a given ion should be independent of the convention whereby one accounts for the free electron enthalpy, two distinct procedures for dealing with this issue have evolved in the literature. The computation of an enthalpy change at a finite temperature relative to the 0 K value requires the evaluation of the following expression:  T H 0T (K) = H00K + CP (T  ) dT  . (13) 0

The evaluation of this expression for an ionization process, in which an electron appears explicitly as a reactant or product, requires a consistent approach. In the “Electron Convention,” used predominantly in the thermodynamics community, and the “Ion Convention,” employed in the ion physics/chemistry communities, important differences

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arise in the manner by which the integrated heat capacities of the elements are used to compute enthalpies at nonzero absolute temperatures. The more widely used Electron Convention defines the enthalpies of formation of elements in their standard states to be zero at all temperatures. Under such a convention, the integrated heat capacities of the elements must be accounted for implicitly in other parts of the calculation. The Ion Convention is based on the definition of the enthalpy of formation of a given element at temperatures above absolute zero to be equal to the integrated heat capacity of that species. At the zero of absolute temperature, the enthalpies of formation of a given ion are identical in both conventions. Because standard tabulations of thermochemical quantities have historically used both conventions, there is a need to define the conventions with clarity and caution users of such data to be aware of the conventions. In the Electron and Ion conventions, at 0 K the enthalpy of formation of the electron is zero and the enthalpies of formation of the ions are exactly equal to the 0 K enthalpy of formation of the precursor molecule plus the ionization energy of M. The following equations treat anions and cations consistently: f H 0 (M+ ) 0 K = f H 0 (M) 0 K + IEa , −

f H (M ) 0 K = f H (M) 0 K − EA. 0

0

(14) (15)

At temperatures above 0 K, e.g., 298 K, where many standard thermochemical data are reported, the Electron and Ion conventions differ in the following manner: in the Electron Convention, the electron is treated like a standard chemical element, and its enthalpy of formation is constrained to be zero at all temperatures. However, the integrated heat capacity is not taken to be zero. Under this condition, the expressions for the enthalpies of formation of positive and negative ions reduce to f H 0 (M+ )298 K = f H 0 (M)298 K + IEa + C, (16) f H 0 (M− )298 K = f H 0 (M)298 K − EA − C. (17) In these expressions, C is the integrated heat capacity for the electron. The integrated heat capacities for the ion M+ and its precursor neutral M effectively cancel in the above expressions. The Electron Convention has been used historically in the ion thermochemistry literature despite the fact that the additive correction to the enthalpy of ionization from the integrated heat capacity of the electron is small and temperature dependent, thereby causing confusion. In contrast, the standard Ion Convention treatment of ion enthalpies of formation that appears in much of the literature on ion physics/chemistry assumes that the integrated heat capacity of the electron, C, equals the enthalpy of formation of the electron at temperature T . In the Ion

Convention, the enthalpies of formation of cations and anions reduce to f H 0 (M+ )298 K = f H 0 (M)298 K − f H 0 (e− )298 K + IEa + C,

(18)

= f H 0 (M)298 K + IEa ,

(19)

and f H 0 (M− )298 K = f H 0 (M)298 K + f H 0 (e− )298 K − EA + C,

(20)

= f H 0 (M)298 K − EA.

(21)

Note that in both expressions, the equality of C and the enthalpy of formation of the electron at temperature T (=298 K) results in expressions that have no dependence on the integrated heat capacity of the electron. The calculation of C, the integrated heat capacity of the electron, is based on treating the free electrons as an ideal gas and computing the heat capacity according to Boltzmann statistics. At 298 K, the heat capacity at constant pressure is that of an ideal gas, C = 5/2 RT, or 6.197 kJ/mol. This computation allows data tabulated under the Electron Convention to be rationalized with Ion Convention data as follows: f H 0 (M+ )298 K (IC) = f H 0 (M+ )298 K (EC) − 6.197 kJ/mol, −

(22)

−

f H (M )298 K (IC) = f H (M )298 K (EC) 0

0

+ 6.197 kJ/mol.

(23)

For the most precise work, however, the evaluation of the heat capacity of the free electron gas should be carried out with Fermi–Dirac statistics, resulting in a value of C = 3.145 kJ/mol. This value is recommended in all cases where the electron is explicitly a reactant or product in an ionization reaction. While many earlier compilations of data have used the Electron Convention, the most recent tabulation of standard enthalpies described in the NIST Webbook (http://www.nist.gov) employs the Ion Convention. It does not introduce any temperature dependence, however small, to ion enthalpies of formation and is, therefore, considered a simpler and less confusing representation of data. This convention is also consistent with the literature of the ion physics and chemistry community over the past 50 years. In any application of tabulated thermochemical data, investigators are cautioned to be exceedingly clear about the conventions of the data they employ.

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B. Measurement of Equilibrium Constant Methods, Keq The preceding methods may be characterized as collisionless, i.e., they determine the properties of isolated atoms or molecules in the gas phase. In contrast, ion/molecule equilibrium processes maintained by collision and characterized by a specific temperature −−−→ A± + B ←−−− C± + D

(24)

may be established in a high-pressure mass spectrometer, flow tube, or ion cyclotron resonance spectrometer. By determining the mass action equilibrium constant through measurement of the relative abundances of the two ions A± and C± after equilibrium has been established by a large number of collisions, the thermodynamic state variables for chemical reaction may be obtained. K eq is defined by the following expression: K eq = [C± ][D]/[A± ][B].

(25)

The neutral reactants B and D are present in great excess compared to the ionic reactants, yielding a ratio [D]/[B] that does not change as equilibrium is established. A single measurement of K eq yields a value for the Gibbs free energy change of reaction at the temperature of the measurement, while a series of measurements at different temperatures permits an experimental evaluation of the entropy and enthalpy changes associated with the reaction through the relation −RT ln K eq = G 0 = H 0 − T S 0 .

(26)

Accurate knowledge of the temperature is essential for a correct application of equilibrium methods. Equilibrium measurements carried out in widely different pressure regimes have been able to reproduce relative spectroscopic ionization energies quite reliably, demonstrating that equilibrium may be achieved readily in such experiments. Such measurements produce only relative thermochemical data; consequently, the resulting thermochemical “ladders” must be calibrated to reliable comparison standards to yield absolute values. Extracting the more useful H 0 values from G 0 measurements requires determination of Third Law entropies, through either temperature-dependent methods, i.e., van’t Hoff plots, d ln K eq −H 0 = , d(1/T ) R

(27)

or applications of statistical mechanical methods. Systematic errors in such estimates arise from inaccurate determination of the reactive neutral pressure in pulsed high-pressure mass spectrometers. When such an instrument operates in the transition regime between molecular

and viscous flow, diffusion of neutral species leads to mass discrimination effects. Other systematic errors arise from clustering of neutral molecules to the ions at low temperatures and pyrolysis of the ions at high temperatures. The high level of development of ab initio quantum chemistry calculations has resulted in reliable estimates of equilibrium geometries of polyatomic molecular ions, from which vibrational frequencies and rotational constants yield absolute entropies that rival or surpass the accuracy of experimental determinations or provide valuable consistency checks. Thermochemical scales derived from equilibrium constant determinations are relative values, and absolute assignments for appropriate quantities require reliable calibration values of ionization energy, electron affinity, or proton affinity, for example. Moreover, because of the interrelationships among the thermochemical data for structurally similar molecules, the scale must be evaluated as a whole, not just for individual molecules, and must demonstrate internal consistency among the variables G, H , and S. A particularly important application of equilibrium methods in mass spectrometry arises in the phenomenon of “chemical ionization.” In analytical mass spectrometry, the extensive fragmentation that accompanies the ionization of large molecules in electron impact ionization is a serious problem. The distribution of ions in a high-pressure ion source at thermal equilibrium is illustrative of the so-called “soft” ionization techniques that have been developed to circumvent the problem of fragmentation when a Franck–Condon distribution of species is created by electron impact or photoionization. The chemical ionization (CI) method employs a chemical reaction between ions generated from a reactant gas and the species to be ionized to generate primary ions. The reactant gas in CI is usually chosen to be a species such as CH4 or H2 which may be ionized by electron impact followed by reactions with the neutral gas that lead to + protonated species such as CH+ 5 and H3 . These species may then undergo proton transfer reactions with the species M to be ionized. Such reactions generally lead to MH+ ions in thermodynamic equilibrium at the ion source temperature with very little fragmentation. 1. Proton and Electron Transfer Equilibria Three important classes of gas-phase ion–molecule reaction schemes demonstrate the value of thermochemical data deduced from equilibrium systems. The first of these concerns the derivation of extensive scales of relative proton affinities, gas-phase acidities, and electron affinities. These results derive primarily from measurements of enthalpy changes for proton or electron transfer reactions:

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AH+ + B −→ BH+ + A

(28)

AH + B− −→ BH + A− .

(29)

or

The proton affinity of substance A is defined as the negative of the enthalpy change for attachment of a proton at a particular temperature, usually 298 K; the gas-phase basicity of A is the negative of the Gibbs free energy change for that process. The negative of the Gibbs free energy change for the dissociation of a species HA into H+ and A− defines the gas phase acidity of HA. Another valuable system is that of charge transfer processes A± + B −→ B± + A,

(30)

which provide information on relative enthalpies of ionization or electron affinity at finite temperatures. Such equilibrium determinations of relative ionization energies closely reproduce the equivalent scale of spectroscopic ionization energies. The most useful application of this equivalence is the determination of ionization energies for species that undergo a large change of geometry upon electron loss. The small Franck–Condon factors for vertical transitions therefore result in poorly defined ionization thresholds. The n-alkanes provide examples of such a system. The conventional view of thermal energy ion– molecule equilibria is that electron transfer occurs in a long-lived A± · B collision complex that survives for many vibrational periods, allowing the ionic configuration corresponding to the equilibrium geometry of the ion, i.e., the geometry corresponding to the adiabatic transition, to be accessed. 2. Solvation Under high-pressure conditions in the ion source of a mass spectrometer, it is possible to measure the equilibrium constants for formation of solvated ionic species. Such information provides insight into the thermochemistry of stepwise attachment of solvent molecules to selected ion cores. The high-pressure mass spectrometer technique allows the attainment of equilibrium cluster ion distributions at specified partial pressures of a clustering ligand at well-specified source temperatures. The method depends on creating and measuring an equilibrium distribution for ligands, B, clustered to ions, Aq+ , of charge q, as follows: Aq+ · Bn + B + M = Aq+ · Bn+1 + M.

(31)

n is the number of solvent molecules, and M is a background gas species that provides collisional stabilization of the clusters. The measured intensities Cn of the species

Aq+ · Bn yield equilibrium constants and thermodynamic parameters as follows:    0 ln K n,n+1 = ln(Cn+1 /Cn ) = −Hn,n+1 RT  0   + Sn,n+1 R . (32) Many of the early studies focused on the thermochemistry of hydration and ammoniation of alkali metal cations, Ag+ , H3 O+ , and NH+ 4 . These studies focused initially on the value of such data in understanding nucleation and gasphase ion solvation phenomena but were quickly appreciated as probes of solvation energies in the bulk solvent. Experimental data show that the ratio of the solvation energy in the liquid phase to the sum of the stepwise enthalpies of solvation up to a given cluster size converges with as few as five or six solvent molecules for many different cations clustered both to water and to ammonia. These observations end support to the very simple Born concept that to first order, solvation can be modeled by the immersion of a sphere of fixed radius and charge in a structureless dielectric continuum. Higher-order corrections to this simple picture come from consideration of surface tension effects. Convergence of these ratios to approximately the same value is indicative of the fact that, beyond the first solvation shell, the majority of the contribution to solvation is from electrostatic interactions between the central ion cavity and the surrounding medium. 3. Molecular Recognition The study of ion energetics and equilibria in the gas phase has made valuable contributions to the subject of molecular recognition. One of the fundamental characteristics of systems that exhibit molecular recognition is the presence of significant noncovalent interactions. Such interactions involve an interplay of entropic and enthalpic effects that are difficult to separate in the condensed phase but that can be sorted out in careful gas-phase studies using methods described in preceding sections. The insight that gas phase energetics studies can provide to this important field of inquiry is especially important in the development of new analytical separations technology that can be applied to important environmental remediation issues such as nuclear waste disposal. Gas-phase experiments provide the benchmark data that allow the development of computational methods capable of accurate predictions of host–guest interactions in relevant condensed-phase environments. A particularly important model system for understanding the microscopic details of molecular recognition is the interaction of alkali metal ions with macrocyclic ligands such as crown ethers, c-(C2 H4 O)n . The n = 6 case, specifically 18-crown-6, denoted 18c6, has generated much

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interest because of the fact that it binds K+ selectively over Rb+ , Cs+ , and Na+ (in that order) in an aqueous environment. In contrast, the gas-phase bond energies reported by Armentrout, using the thermokinetic method of collisioninduced dissociation, for the various alkali metal cations with 18c6 clearly indicate that Na+ is most strongly bound, as one expects from charge density models of electrostatic binding. A consideration of the competition between gasphase solvation of the cations by water and complexation by the crown ether, M+ (H2 O)x + 18c6 −→ M+ (18c6) + xH2 O,

(33)

+

shows that complexation of Na by 18c6 is favored until the number of displaced water molecules corresponds to a complete solvent shell. At this point, K+ selectively binds to the crown ether, as observed in solution. In addition, for all metal cations, the displacement reaction that liberates six water molecules but binds a single 18c6 species is entropically driven. Overall, this comparison demonstrates that the selective binding of K+ by 18c6 in aqueous media is a subtle balance of the solvation and complexation enthalpies, entropic effects, and the solvation of the resultant complex. The results are consistent with a picture in which such selectivity is not an intrinsic property of the metal cation–crown ether complex but depends critically on solvation. This point of view makes it clear that a proper understanding of molecular recognition in such systems must include details of the solvent itself.

1. Ion-Molecule Bracketing Experiments Equilibrium in the ion source of a mass spectrometer cannot be achieved in certain ion–molecule systems because of rapid competitive reactions or because one of the species in the equilibrium of interest is an unstable species such as a free radical. In such cases, it is sometimes possible to obtain an experimental estimate of the enthalpy change of a particular reaction (charge transfer, proton transfer, etc.) by use of a technique known as “bracketing,” in which the ion of interest is reacted with a series of molecules selected to provide a range of values for the relevant thermochemical parameter of interest, e.g., ionization energy, electron affinity, gas-phase basicity, or acidity. Reaction is presumed to occur for exothermic processes and not to occur for endothermic processes. For example, the proton affinity of a species B can be “bracketed” between the know proton affinity of reference compound A and that of reference compound C if the following observations are made: no reaction

BH+ + A −−−−→ AH+ + B, fast reaction

BH+ + C −−−−−→ CH+ + B.

(34) (35)

The notation “no reaction” means either that the reaction products are not observed or that the reaction efficiency, i.e., the rate relative to the Langevin collision rate, is low. The observations lead to the relative ordering PA(A) < PA(B) < PA(C).

C. Thermokinetic Methods In addition to the two general methods described above for determination of thermochemical parameters, i.e., ionization under collision-free conditions and free energy measurements in systems at equilibrium at a well-specified temperature, a third set of methods exists for determinations of reaction enthalpies and enthalpies of formation. These methods are collectively referred to as “thermokinetic” measurements and are based on the relative rates or cross sections for the formation of reaction products under single-collision conditions. The “ion bracketing method” puts constraints on the enthalpy of formation for a specific species by observing the occurrence or nonoccurrence of reactions of that species with known collision partners. The collision-induced dissociation method determines bond energies in polyatomic ions by observing the threshold energy at which collisions with a rare gas atom fragments the parent ion. The “kinetic method” evaluates the difference in enthalpies of reaction of two competitive reaction channels arising from decay of a common precursor, usually a cluster ion. Each of these classes of thermokinetic methods is based on underlying assumptions that restrict the validity of a given method.

2. The Kinetic Method The kinetic method is an approximate scheme to determine relative thermochemical properties based on the rates of competitive dissociation of mass-selected cluster ions. As an example, consider the proton-bound dimer system: k1

B1−H+−B2 −→ B1 H+ , k2

−→ B2 H+ ,

(36) (37)

with rate constants for competitive dissociation as indicated. In its simplest form, the kinetic method deals with systems in which the structures of B1 and B2 are sufficiently similar that it is reasonable to assume that the entropic factors in both decay processes are equal. With the additional assumption that the dissociation processes occur with no barriers in excess of the endothermicities, the following relationship holds:     k1 [B1 H+ ] (H ) ln = ln , (38) ≈ + k2 [B2 H ] RTeff where Teff is the effective temperature of the activated proton-bound dimer. Perhaps the most difficult issue

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associated with the application of this method is the estimation of the effective temperature, which describes a Boltzmann distribution of activated dimer ions which fragments to give the same fragment ion abundance ratio as observed for the non-Boltzmann population sampled in the experiment. The assignment of Teff is made especially difficult because the distribution it describes is time dependent, a function of the observation time characteristic of a particular instrument. Since the method determines differences in enthalpies of reaction, calibration data based on the ion ratios of several species with energetics known from other methods, such as ion/molecule equilibria, are necessary for absolute data determinations. The method may be applied to reactions other than relative proton affinity determinations, including electron affinities, ionization energies, gas-phase acidities, and cation affinities. In favorable cases, entropy changes associated with attachment of an ion to a molecule may be determined from kinetic method data, yielding information on bonding and fragmentation mechanisms in cluster ions, especially those of biological origin. 3. Collision-Induced Dissociation This collisional method obtains bond energies by measuring the kinetic energy threshold for the dissociation of AB+ ions, represented schematically as follows: AB+ + Rg −→ A+ + B + Rg.

(39)

Formally, this process may be thought of as a superposition of bimolecular activation and unimolecular decay. The bimolecular step occurs in the collisional activation of an AB+ ion above one (or more) of its dissociation thresholds; A+ fragment ions appear as products of the unimolecular decay of the activated ion. In the process noted above, Rg is often a rare gas atom. A consideration of collisional dissociation near threshold requires an understanding of the amount of energy transferred in a single encounter, the role of angular momentum in the decay of activated molecules, and the lifetimes for decay of such species. Accurate determination of apparent thresholds has been achieved by modeling the threshold behavior of the system with a cross-sectional form as follows:   σ (E) = σ0 (40) gi (E − E i − E 0 )n E, where the summation extends over internal states labeled by the index i and having degeneracy gi , σ0 and n are adjustable parameters, E is the relative kinetic energy, and E 0 is the 0 K reaction threshold. The dissociation rates of activated molecules close to threshold may be long in comparison with the observation time window, leading to large kinetic shifts. The ac-

cessibility of methods for performing accurate ab initio quantum chemical calculations has led to reliable estimates for the vibrational frequencies and rotational constants of decaying molecules, as required for evaluation of rate constants computed according to statistical Rice– Ramsperger–Kassel–Marcus (RRKM) theory. Threshold CID measurements have proven to be valuable in determining thermodynamic information. By varying the kinetic energy of the AB+ ions in reaction (39) and observing A+ products as a function of incident energy, the threshold energy for CID can be determined, yielding the bond energy of AB+ . The accuracy of this method requires that the probability for transferring all translational energy into the internal energy of the AB+ molecule is finite and nonzero and that there is no activation energy in excess of the bond energy along the reaction coordinate for dissociation. Accurate threshold determinations require attention to the role of multiple collisions with the Rg collision gas, the effect of the internal energy content of the AB+ reactant, and the metastability of the activated AB+ species on the timescale for detection. The generality of the methods has been demonstrated through its application to systems such as small organic molecules, solvated hydronium ions and metal ions, organometallic complexes, atomic clusters, and metal–crown ether complexes. D. Sources of Evaluated Data The foregoing discussions of methods for determining thermochemical properties of gas-phase ions have focused on specific experiments that yield a particular observable parameter. New experimental methods that improve the accuracy and precision of thermochemical determinations, as well as the fact that conventions for expressing such information change with time, make up-to-date critical data compilations of prime importance for consumers. The National Institutes of Standards and Technology (NIST) has continued its historical tradition of maintaining a critical compilation of data, estimating error limits in appropriate cases. The NIST compilations focus on experimental determinations, but the importance of highquality ab initio calculation of enthalpies of formation of ionic compounds should not be ignored. As an example of this last point, the understanding of the structure of the seven-membered cyclic tropyl ion, c-C7 H+ 7 , is particularly illustrative of the value of computations. Neither the tropyl ion, nor the isomeric ben+ zyl, C6 H5 CH+ 2 , and tolyl, C6 H4 CH3 , ions are stable in the neutral precursor forms, forming in the gas phases only through ionic dissociation reactions. Numerous reactions produce the c-C7 H+ 7 ions, among them hydrogen atom loss from toluene and halogen atom loss from benzyl halides and halotoluenes. The tolyl and benzyl ion

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structures can be produced at their thermochemical thresholds, yielding well-established heats of formation. However, the tropyl ion is formed only through a process with a substantial barrier. Thus, the only means for obtaining its energy directly is through ab initio quantum chemical calculations, which have been carried out with the Gaussian G2 methodology. The most current online source of thermochemical data may be found in the NIST Webbook (http://webbook.nist.gov). In particular, the web site http://webbook.nist.gov/chemistry/ion-ser.htm provides a searchable database for enthalpies of formation, electron affinities, ionization energies, gas-phase acidities and basicities, and appearance energies. The NIST data collection includes only data derived from experimental determinations, although the evaluation of the reliability of the data has been enhanced by using computational results, particularly for the proton affinity tabulation.

III. ION KINETICS A. Unimolecular Reactions Decay of molecular ions activated above one or more dissociation thresholds by photon absorption or electron impact and decomposition of an activated species prepared by a collisional excitation, are examples of unimolecular reactions. Such reactions have played a central role in the subject of ion energetics and kinetics and have served as important testing grounds for statistical theories of unimolecular decay. The fundamental assumption of all statistical theories of chemical reaction dynamics is that at some specified location on the reaction coordinate, all energy microstates accessible to the system are populated with equal probability. The manner in which this microcanonical equilibrium is achieved is not specified in the theory, but the implication of the assumption is that intramolecular vibrational redistribution (IVR) is rapid on the time scale of reaction. All statistical theories have this assumption in common but differ in the point on the reaction coordinate at which the statistical approximation is applied and in the manner in which angular momentum conservation is handled. Theories in which the statistical approximation is applied at a critical configuration are called transition-state theories, while those in which the product states accessible by energy and angular momentum conservation are populated with equal probability are called phase-space theories. One of the earliest statistical theories of unimolecular decay of ionic species, developed explicitly to interpret mass spectra, is the quasi-equilibrium theory (QET). In the QET, the electronically excited molecular states that are accessed by the primary ionization event decay rapidly by internal conversion to the ground state of the ion, where

microcanonical equilibrium is achieved, and subsequent vibrational predissociation leads to fragmentation. The breakdown diagram, the graph of fragment ion abundances as a function of the internal energy of the ion, is computed from QET rate constants, and the mass spectrum is the convolution of the energy deposition function for the ionization process, e.g., the Franck–Condon factors for the primary ionization, with the breakdown diagram. QET has enjoyed considerable success in interpreting mass spectra. The QET is formally identical to the Rice–Ramsperger– Kassel–Marcus (RRKM) theory of unimolecular decay, in which the rate constant for dissociation to reaction products of an energized species with total angular momentum J and internal energy E ∗ over a barrier of E 0 is given by the following relation: k(E ∗ ) =

σ N + (E ∗ − E 0 , J ) . hρ(E ∗ , J )

(41)

In this expression, N + (E ∗ − E 0 , J ) is the total number of energy levels associated with the 3N − 7 bound internal degrees of freedom of the critical configuration of total angular momentum J and energy E ∗ − E 0 . The quantity ρ(E ∗ , J ) is the density of internal energy states of the activated ion at energy E ∗ , σ is the symmetry number of the reaction coordinate, and h is Planck’s constant. The evaluation of rate constants by RRKM theory requires estimates of vibrational frequencies for activated molecules and critical configurations, all of which have been facilitated by ab initio calculations of the relevant potential energy surfaces. The barrier height, E 0 , is generally the least well-determined parameter in an RRKM calculation and is often the single parameter varied until the calculated rate constant is in agreement with the observed rate. The validity of the statistical approximation in QET– RRKM–phase-space theories has been tested extensively both experimentally and theoretically. Some of the most incisive tests of statistical theory have been performed with energy-selected ions, prepared as described previously, using methods such as photoelectron–photoion coincidence (PEPICO). Particularly incisive tests of these statistical models have been performed on isomeric C4 H+ 6 systems by Baer and Bowers and their co-workers and on C4 H+ 8 by Bowers and collaborators. The latter study provided validation of an important variant of statistical theory called the “transition state switching” model. B. Bimolecular Reactions 1. Ion-Neutral Collision Cross Sections and Rates Langevin (1905) provided the first theoretical description of the collision frequency between an ion and a neutral species in terms of the long-range potential between these

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species. The sum of the long-range attractive electrostatic potential and the repulsive 1/r 2 centrifugal potential results in a barrier as the reactants approach. For every collision for which the relative kinetic energy exceeds this centrifugal barrier, reaction is assumed to occur. For an ion of charge e approaching a neutral collision partner with polarizability α, the long-range potential behaves as 1/r 4 and the Langevin capture cross section is given by

2α 1/2 σLangevin = πq , (42) µE T where the relative collision energy is E T and the reduced mass of the approaching reactants is µ. The thermal average of this cross section over the relative speed distribution at a fixed temperature yields the Langevin–Gioumousis– Stevenson (LGS) rate constant expression, 1/2 α kLangevin = 2πe , (43) µ a result that is independent of temperature. For neutral collision partners with permanent dipole moments, the longrange potential also has a charge-dipole term that behaves as r −2 cos θ, where θ is the angle between the relative velocity vector and the axis of the dipole. Several models have been proposed to account for interaction of the charge with the dipole, the most successful of which is the average dipole orientation (ADO) model. This model postulates that the electric field of the approaching ion does not fully quench the rotational motion of the dipole and results in a partial locking of the dipole orientation. The ADO cross section is given by the following expression: σ (ν) = πrc2 +

πq 2 α 2πqµD + cos θ. rc2 µν 2 µν 2

(44)

The quantity rc is the distance at the top of the centrifugal barrier for the full effective potential, and the bar denotes that the cross section is computed as an average over the orientation the permanent dipole µD makes with respect to the relative velocity vector of magnitude v. Both the Langevin and the ADO models predict thermal energy cross sections of magnitude 10−16 −10−15 cm2 , corresponding to rates in excess of 10−9 cm3 molecule−1 sec−1 . These predictions were tested in pioneering experiments by Laudenslager and Bowers in the early 1970s that estimated the extent of dipole locking, demonstrating the general validity of such capture theories at thermal energy. The capture hypothesis forms the basis for the LGS model for ion-neutral reactions and many exothermic reactions appear to proceed at the collision frequency. Despite the appeal and widespread validity of the LGS model, a number of ion-neutral reactions have rates significantly

below the LGS limit. The interpretation of such slow reactions has been the subject of numerous investigations. Brauman and co-workers have proposed a model of ionic reactions that proceed on potential energy surfaces with two minima separated by a barrier lower than the entrance channel energy. A schematic surface with these features is shown in Fig. 1. As the reagents approach under the influence of the long range attraction, they enter the first well, the electrostatically bound encounter complex. For reaction to occur, this complex must undergo an atom or functional group transfer over a barrier to a second bound species that is a precursor to products. When this barrier height is below the energy of the approaching reagents, the barrier is not an energetic bottleneck to reaction, but the low density of states for crossing the barrier may reduce the rate for this process below that for decomposition of the initial encounter complex back to reagents. When the branching ratio for decay of this encounter complex favors production of the reactants, the reaction efficiency drops significantly below the LGS prediction. By modeling complex formation and decomposition reaction rates using statistical RRKM theory and computing reaction efficiencies as a parametric function of the intermediate barrier height, one can determine barrier energies for many potential surfaces, correlating them with structural features of the reactants. A particularly important reaction that has been examined in the context of the double minimum potential model is the gas-phase nucleophilic substitution reaction, written symbolically as X− + CH3 Y −→ Y− + CH3 X,

(45)

where X and Y are halogen atoms. Experimental work from the research groups of Bowers, Johnson, and Brauman has provided critical confirmation of this double minimum potential model by trapping the electrostatic complexes, activating them, and observing their decomposition to reactants or products. Experiments by Bowers and collaborators have focused attention on the Cl− + CH3 Br system, with overall reaction as follows: − − −− → −− → Cl− + CH3 Br ← − − [Cl · CH3 Br] ← − − [Br · CH3 Cl] A −− → ← − − Br + CH3 Cl −

B (46)

This reaction is known to occur with approximately 1% efficiency at thermal energies. In addition, the height of the isomerization barrier separating intermediates A and B, as estimated from RRKM calculations, lies approximately 0.085 eV below the energy of the approaching reagents. In the experimental work, the electrostatically bound complex corresponding to the entrance channel is produced in a high-pressure ion source with a narrow range

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FIGURE 1 Double-minimum reaction coordinate showing formation of complex A with rate k1 . The sparse level density for crossing the intermediate isomerization barrier to B with rate constant k2 makes this rate low in comparison with nonreactive decay to reactants, with rate constant k−1 .

of internal energies and is injected into a reverse geometry sector mass spectrometer. To be detected in the time window of these experiments, the metastable ion Cl− ·CH3 Br must be formed with internal energies between about 0.4 and 0.5 eV. The lower limit corresponds to the height of the isomerization barrier, while ions produced with internal energies of more than 0.5 eV decay on a time scale very fast in comparison with the 10-µsec transit time to the second field-free region of the mass spectrometer. Ions with this distribution of internal energies yield the products Br− + CH3 Cl. The data also allow a determination of the kinetic energy release distribution (KERD) for the reaction, thus providing a test of the statistical assumption inherent in RRKM or phase-space treatments of the dynamics. 2. Low-Temperature Dynamics At very low temperatures, quantum mechanical effects that classical electrostatic capture models fail to account for play a critical role in determining reaction rates. Ion chemistry at temperatures below 80 K is especially important in elucidating the role of such quantum effects; molecule formation in the interstellar medium is a particul-

arly important testing ground for such concepts. Experimental methods employing liquid helium-cooled drift tubes, Penning traps, and supersonic expansions such as the CRESU (Cinetique de Reaction en Ecoulements Supersoniques Uniformes) wind tunnel instrument have allowed studies of ion-neutral interactions at temperatures as low as 3 K. At these temperatures, radiative association and three-body association reactions become important, as well as bimolecular processes. Recent low-temperature studies of such collisional processes include the reaction of N+ with molecular hydrogen and its isotopomers. Although the magnitudes of the rotational energies of the neutral reactants, the spin-orbit electronic energies of N+ , and the zero-point energy differences among H2 , HD, and D2 are fairly small, they are significant compared to the magnitude of the collision energy at low temperatures and provide sensitive tests of quantum theories of capture processes. Such quantum capture theories in three dimensions have been developed to solve the Schr¨odinger equation for the long-range attractive entrance channel potential in a coupled rotational-states formalism. A rotationally adiabatic approximation to this theory has been developed by constructing potential curves that describe the evolution of

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isolated rotational states of molecular reagents into hindered rotor states of the ion–molecule complex. The reaction probability of that reagent rotational state is unity when the collision energy lies above a particular adiabatic curve and zero when the curve rises above the collision energy. The cross section is computed by summing over capture probabilities for the rotationally adiabatic curves. These theories yield improved agreement with experimental data on ion–dipole collision processes, at low temperatures. 3. Vibrational-State Resolution in Ion–Molecule Reactions Since the pioneering experiments of Chupka and coworkers, which employed single-photon VUV ionization of molecular hydrogen to prepare H+ 2 with a specific amount of vibrational excitation, the field of study of ion–molecule reaction dynamics with vibrationally state-selected reactants has experienced a number of advances. Another important system in the history of ion–molecule reactions is the hydrogen atom transfer process (47): + N+ 2 + H2 −→ N2 H + H.

(47) N+ 2

A recent study of this system, in which the reactant is produced in the ground electronic state in a single rotational state (J = 2) and with vibrational quantum number v + varied between 0 and 4 has been performed by the group of Kompa. The reactant ions are produced by (2 + 1) REMPI on nitrogen gas in a pulsed supersonic expansion. The spectroscopic method for ion production is a one-color scheme that employs high vibrational levels, v  =10−14, of the two-photon allowed a 1 g state of N2 having a lifetime of 120 µsec, followed by absorption of a third photon to produce the desired N+ 2 reactant ions. This state-preparation scheme requires photons in the wavelength range near 237 nm and is employed in conjunction with octupole ion guides and a scattering gas cell to determine total cross sections for chemical reaction of + N+ 2 formed in vibrational states v = 0–4 over the translational energy range from 0.01 to 2 eV in the center of mass coordinates. An unexpected result of this study was the observation of a small activation barrier to reaction of magnitude 0.01 eV. The data show a rise from this threshold and follow an E −1/2 dependence over the collision energy range from 0.02 to 0.09 eV. The cross sections appear to be larger than the Langevin prediction, calculated with the polarizability of H2 , particularly for reactant ions with vibrational excitation v + ≤ 2, and in fact are in much better agreement with the cross section computed for the attraction of the + charge-transferred reactants H+ 2 + N2 . For N2 ions with

v + = 3 and 4, agreement with the Langevin result is better. At these levels of vibrational excitation, Franck–Condon factors make the charge transfer process less favorable, and the total cross section is more closely described by the N+ 2 + H2 long-range potential. Above a collision energy of 0.09 eV, the reactive cross section rises above the Langevin value as endothermic charge transfer channels open. The most detailed probe of an ion–molecule reaction is the state-to-state cross section. The measurement of angularly resolved state-to-state cross sections has been achieved only for reactants in their ground state, but true state-to-state total cross sections have now been measured in a few favorable cases. A particularly novel method for product state determination with reactant state selection has been developed by Ng. In the differential reactivity method, product vibrational states are distinguished by their differing cross sections for charge transfer with selected molecules. The charge transfer reaction +  + H+ 2 (v ) + H2 −→ H2 + H2 (v ),

(48)

+ in which H+ 2 is prepared in v = 0 and 1 by VUV photoionization provides a particularly good example of its power.  The H+ 2 charge transfer products, formed in v = 0–3, are first mass analyzed, accelerated to 10 eV, and then passed through a collision cell containing N2 , Ar, or CO, where + + charge transfer occurs once again. The N+ 2 , Ar , or CO products are mass analyzed and the cross sections σm for forming these ions by charge transfer from the H+ 2 reaction products are measured. The charge transfer cross sections  for H+ 2 with these gases have different dependences on v , and therefore the gases can be used to probe the product states of H+ 2 formed in the symmetric charge exchange reaction. Letting X v denote the fraction of H+ 2 formed in the vibrational state v  , the following set of simultaneous equations can be solved for X v , since the cross sections σv are known and the σm  s are measured:

X 0 + X 1 + X 2 + X 3 = 1,

(49)

+ + X 0 σ0 (N+ 2 ) + X 1 σ1 (N2 ) + X 2 σ2 (N2 ) + + X 3 σ3 (N+ 2 ) = σm (N2 ),

(50)

X 0 σ0 (Ar+ ) + X 1 σ1 (Ar+ ) + X 2 σ2 (Ar+ ) + X 3 σ3 (Ar+ ) = σm (Ar+ ),

(51)

X 0 σ0 (CO+ ) + X 1 σ1 (CO+ ) + X 2 σ2 (CO+ ) + X 3 σ3 (CO+ ) = σm (CO+ ).

(52)

An extensive set of data at collision energies from 2 up to 16 eV has been measured. When the reactant ions are in the vibrational ground state, at low kinetic energies the

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charge transfer product is also in its ground state, indicating that resonant charge transfer is the dominant process. At increasing collision energies, X 1 increases from 0.0 to 0.17. For vibrationally excited H+ 2 reactants, inelastic re laxation to form H+ in v = 0 is important at all collision 2 energies, increasing in magnitude with increasing collision energy. Of particular interest is the fact that inelastic relaxation forming v  = 0 is substantially more important than inelastic excitation producing v  = 1 or 2. This trend is predicted by theory but underestimated at lower collision energies. More recently, methods based on laser preparation of reagent ions and laser-based detection of product ions have been accomplished for the thermal energy charge transfer reaction DBr+ + HBr → HBr+ + DBr. REMPI was used to prepare DBr+ in specific vibrational and spin-orbit states, and the HBr+ product state was measured by laserinduced fluorescence. The largest rate constants were observed for near-resonant processes in which both the vibrational and the spin-orbit quantum numbers remained unchanged. 4. Mode-Selective Chemistry One of the objectives of quantum state selection is to enhance the rates of chemical transformation by exciting molecular motions correlated with the formation of particularly desirable (and perhaps non-statistically favored) reaction products. In the H+ 2 + He system, the activation barrier occurs late, after the collision system has “turned the corner” of the potential energy surface. Thus, H+ 2 vibrational motion is directed along the reaction coordinate at the saddle point and is effective in promoting reaction. In polyatomic species, the issue of enhancement becomes more complex because rapid intramolecular vibrational relaxation may dissipate selective excitation on a time scale fast compared to reaction. Vibrational energy may also be expected to play specific roles that depend on the nature of the atomic motions in a given normal or local mode. Recent research by Anderson and co-workers on C2 H+ 2 reactant ions produced by REMPI methods has provided another example of mode-selective enhancement of chemical reactions. REMPI methods have been demonstrated to prepare C2 H+ 2 in the ground electronic and vibrational states, as well as with excitation in the ν2 C–C stretching mode and with two quanta in the bending excitation. This state selection is accomplished with (2 + 1) ionization through Rydberg states of gerade symmetry; bending modes promote vibronic coupling of neighboring Rydberg states and, therefore, become optically accessible in the ionization process. State-selected reactant ions produced in this way have been studied in reaction with CH4 as shown below:

+ C 2 H+ 2 + CH4 −→ C3 H5 + H

(H = −0.92 eV), (53)

−→ C3 H+ 4 + H2

(H = −1.33 eV), (54)

−→ C2 H+ 3 + CH3

(H = +0.02 eV). (55)

The condensation reactions (53) and (54) forming new C–C bonds are exothermic, proceeding through a collision complex, and, as expected, show very little effect of vibrational energy. The hydrogen atom transfer reaction (55), is mildly endothermic, however, and shows a significant vibrational effect. With bending excitation in the reactants, the cross section for C2 H+ 3 formation shows a significant enhancement, with the data showing that this form of excitation increases the cross section at the lowest collision energies by a factor of 30 relative to ground-state ions. Moreover, excitation of the bending mode increases the selectivity of formation of the C2 H+ 3 product relative to the C3 condensation products dramatically near threshold. Anderson and co-workers hypothesize that the selective enhancement of C2 H+ 3 production arises from the fact that formation of C2 H+ from C 2 H+ 3 2 requires rehybridization from linear sp to trigonal sp2 , of the carbon atoms in C2 H+ 2 and bending excitation facilitates that motion. This work represents one of the clearest demonstrations of the role of selective vibrational excitation in promoting a specific reaction to date. Another recent example of mode-selective chemistry comes from the laboratory of Zare and co-workers on collisions of state-selected NH+ 3 with ND3 , the products of which are shown below: + NH+ 3 + ND3 −→ NH3 D + ND2 ,

−→

NH+ 3

+

ND+ 3,

−→ NH2 + ND3 H+ .

(56) (57) (58)

The ammonia cation, NH+ 3 , is readily prepared by ˜ and C ˜  Rydberg states of (2 + 1) REMPI through the B NH3 . Because the ammonia molecule is pyramidal in the ˜  Rydberg states and ground state and planar in the B˜ and C in the ion, (2 + 1) REMPI produces NH+ 3 in a progression in the ν2 umbrella bending mode. The ν1 breathing mode corresponds to symmetric elongation and compression of all three N–H bonds. By control of the excitation wavelength, isoenergetic states of NH+ 3 with pure ν2 umbrella excitation or combinations of ν1 breathing and ν2 umbrella motion can be prepared. Two particular combinations of states have been examined in detail: the state with one quantum of breathing excitation and two quanta

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82 of umbrella excitation (denoted ν1 = 1, ν2 = 2, or 11 22 ), with 0.63 eV of internal energy, and the state with no breathing excitation and five umbrella quanta, denoted 10 25 , with a total internal energy of 0.60 eV. Because the two state preparations have essentially the same internal energy, their differential reactivities can address the question of whether the magnitude the internal energy is sufficient to effect a particular reaction or if the form of the internal motion in the reactants plays an important role. All three of the reactive channels of NH+ 3 with ND3 are direct processes that do not involve the intermediacy of a transient complex and, thus, represent good opportunities for demonstrating mode-specific dynamics. A comparison of the reactive cross section for the isoenergetic states 11 22 and 10 25 shows that deuterium abstraction and charge transfer are enhanced, and proton transfer is suppressed by the 10 25 excitation, i.e., the state with enhanced umbrella excitation. In addition to comparing the reactivity of different isoenergetic vibrational state preparations, the study also compares the effect of vibration with equivalent amounts of translational energy, demonstrating that ν2 umbrella excitation is four times more effective in promoting deuterium transfer than an equivalent amount of translational energy.

Ion Kinetics and Energetics

attenuation of ions with a particular mobility and therefore in a particular electronic state. The reaction of Fe+ with propane, C3 H8 , is a particularly interesting system with rich chemistry. In ion chromatography experiments, the Fe+ ions are prepared by electron impact on Fe(CO)5 , and the electronic state distribution is controlled with the ionizing electron beam energy. The 6 D (4s3d6 ) ground state of Fe+ exhibits a high rate for formation of the adduct Fe+ · C3 H8 without additional reaction, while the 4 D (4s3d6 ) and 4 F (3d7 ) excited states exhibit subsequent elimination of H2 and CH4 . The system is particularly interesting in that it is one of a very few that have been studied with all three methods of electronic state preparation, and agreement among the differing studies is excellent.

SEE ALSO THE FOLLOWING ARTICLES CHEMICAL KINETICS, EXPERIMENTATION • DIELECTRIC GASES • ION TRANSPORT ACROSS BIOLOGICAL MEMBRANES • KINETICS (CHEMISTRY) • MASS SPECTROMETRY • POTENTIAL ENERGY SURFACES

BIBLIOGRAPHY 5. Electronic State Selection Electronic excitation of atomic ions provides numerous opportunities to observe dramatic effects in both the overall rates of reaction of such species and the distributions of products in these reactions. Three fundamental methods have been applied to the production of electronically stateselected ions. The first method compares the reactivities of different distributions of electronic states resulting from ions produced in different ways, e.g., electron impact ionization and thermal ionization. The second method uses REMPI for atoms whose Rydberg states are unperturbed by configuration interaction and autoionization. The final method, referred to as “ion chromatography,” uses the differing mobilities of metal ions with different electronic configurations in He. Particularly large differences are observed when vacancies are created in valence s orbitals. By injecting a short pulse of ions into a helium-filled cell with a uniform drift field, the ions separate spatially and temporally as they diffuse through the cell. Electronic state-specific chemistry can be examined by introducing small amounts of a reagent gas and observing the selective

Armentrout, P. B. (1990). “Electronic state-specific transition metal ion chemistry,” Annu. Rev. Phys. Chem. 41, 313. Armentrout, P. B., and Baer, T. (1996). “Gas phase ion dynamics and chemistry,” J. Phys. Chem. 100, 12866. Baer, T., Ng, C. Y., and Powis, I. (eds.) (1996). “The Structure, Energetics, and Dynamics of Organic Ions,” Wiley, New York. Bowers, M. T. (ed.) (1979, 1980). “Gas Phase Ion Chemistry,” Vols. 1–3, Academic, Press, New York. Cooks, R. G., Koskinen, J. T., and Thomas, P. D. (1999). “The kinetic methods of making thermochemical determinations,” J. Mass Spectrom. 34, 85. Farrar, J. M. (1995). “Ion reaction dynamics,” Annu. Rev. Phys. Chem. 46, 525. Farrar, J. M., and Saunders, W. H., Jr. (eds.) (1988). “Techniques for the Study of Ion-Molecule Reactions,” Wiley–Interscience, New York. Lias, S. G., and Bartmess, J. E. (1997). “Gas-Phase Ion Thermochemistry,” National Institute of Standards and Technology, Washington, DC. http://webbook.nist.gov/chemistry/ion/. Ng, C. Y., and Baer, M. (eds.) (1992). “State-Selected and State-to-State Ion-Molecule Reaction Dynamics, Part 1: Experiment,” Vol. LXXXII in Advances in Chemical Physics, Wiley–Interscience, New York. Ng, C. Y., Baer, T., and Powis, T. (eds.) (1994). “Unimolecular and Bimolecular Ion-Molecule Reaction Dynamics,” Wiley, New York. Thomson, J. J. (1913). “Rays of Positive Electricity,” Longmans Green, New York.

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Kinetics(Chemistry) Keith J. Laidler University of Ottawa

I. II. III. IV. V. VI. VII. VIII. IX. X.

Basic Kinetic Principles Molecularity Temperature and Reaction Rates Statistical Theories of Rates Reaction Dynamics Reactions in the Gas Phase Reactions in Solution Reactions on Surfaces Composite Reaction Mechanisms Photochemical and Radiation-Chemical Reactions XI. Homogeneous Catalysis

GLOSSARY Activated complex Configuration of atoms corresponding to an arbitrarily small region near to the col in a potential energy surface. Activation energy Energy defined by the equation E = −R

d ln k , d(1/T )

where k is the rate constant, T the absolute temperature, and R the gas constant. This energy is related to the heights of energy barriers to reaction. Catalysis Substance that increases the rate of a chemical reaction without itself being consumed, and without affecting the energetics of the overall reaction, is called a catalyst. The process is called catalysis.

Encyclopedia of Physical Science and Technology, Third Edition, Volume 8 C 2002 by Academic Press. All rights of reproduction in any form reserved. Copyright 

Composite (complex or stepwise) reaction A reaction that involves more than one elementary reaction. Elementary reaction Reaction in which no reaction intermediates (other than transition species) have been detected or need to be postulated in order to explain the behavior. Half-life Time required for a reactant concentration to reach a value that is the arithmetic mean of its initial and final values. Inhibition Process by which a substance (called an inhibitor) reduces the rate of a chemical reaction. Molecularity Number of reactant particles (atoms, molecules, free radicals, or ions) that are involved in the microscopic event occurring in an elementary reaction. Order of reaction For some reactions the rate of reaction is expressible as

177

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v = k[A]α [B]β where k, α, and β are independent of concentration and time. The exponent α is the order with respect to A, and β is the order with respect to B. These are partial orders; the sum of the partial orders, often written as n, is the overall order. Photochemical reaction Chemical reaction brought about by electromagnetic radiation having insufficient energy to bring about ionization. Potential energy surface Surface resulting from a plot of potential energy against two parameters, such as two interatomic distances. If more than two parameters are used the term hypersurface is employed. Pre-exponential factor The factor A when the rate constant is expressed as k = Ae−E/RT , where E is the activation energy. Rate of reaction Rate of formation of a product, d[Z ]/dt, or the rate of consumption of a reactant, −d[A]/dt, divided by the corresponding coefficient in the stoichiometric equation. Rate constant For a reaction having an order, the rate constant k is the constant appearing in the rate equation. Radiation-chemical reaction Chemical reaction brought about by electromagnetic or particle radiation having sufficient energy to bring about ionization. Reaction cross-section Quantity that is used in collision theory to give a measure of the rate of a bimolecular reaction; it is the probability of reaction multiplied by π d 2 , where d is the collision diameter (i.e., the distance between the centers of two colliding reactant molecules). Reaction (or molecular) dynamics A field that deals with the intermolecular and intramolecular motion that occurs during the course of a chemical reaction and the quantum states of the reactant and product molecules. State-to-state kinetics Study of the transformation of reacting molecules in specified quantum states into product molecules in specified quantum states. Steady-state treatment Application of the hypothesis that for an intermediate X present at very low concentrations, d[X]/dt = 0. Transition-state theory Theory of rates that focuses attention on activated complexes and assumes them to have a concentration corresponding to equilibrium with the reactant molecules. Transition species Species having a state intermediate between reactants and products.

CHEMICAL KINETICS deals with the rates of chemical processes and how the rates depend on factors such as

concentrations, temperature, and pressure. The ultimate objective of a chemical-kinetic investigation is to gain information about the mechanisms of chemical reactions. Such information is also provided by certain nonkinetic studies, but little can be known about the mechanism of a reaction until its kinetics have been investigated. Even then, some doubt must remain about a reaction mechanism; an investigation, kinetic or otherwise, can disprove a proposed mechanism but cannot establish a mechanism with absolute certainty. Chemical kinetics has very farreaching implications in that it relates to many branches of biology, geology, engineering, and even psychology. Theories of chemical kinetics are also applicable to purely physical processes, such as flow processes.

I. BASIC KINETIC PRINCIPLES Several rates can be defined with reference to a chemical reaction. Substances whose concentrations decrease with time are known as reactants, and the rate of decrease of a reactant concentration is known as the rate of consumption (removal or disappearance) of that reactant. Concentration is amount of substance (S.I. unit: mol) divided by volume (S.I. unit: m3 ), and the concentration of a substance A is conveniently written as cA or [A]. The rate of consumption of A is thus defined as vA = −d[A]/dt. −3 −1

(1) −3 −1

Its S.I. unit is mol m s , but mol dm s (also written as mol L−1 s−1 or as M s−1 ) is more commonly used. Rates of consumption of two reactants A and B are only the same if 1 mol of A reacts with 1 mol of B. If, for example, the stoichiometric equation is of the type A + 2B → Y + Z, the rate of consumption of B is twice the rate of consumption of A. The rate of formation of a product of reaction is defined as the rate of increase in its concentration: thus for a product Z vZ = d[Z]/dt.

(2)

Again, the rates of formation of different products are not always the same, nor are they necessarily equal to the rates of consumption of reactants. Thus if the stoichiometry of the reaction is 3A + B → 2Y + Z, and this stoichiometry is preserved throughout the course of reaction, vA vY (3) = vB = = vZ 3 2 The S.I. unit is the same for all of these rate expressions, and mol dm−3 s−1 is most commonly used.

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Sometimes, but not always, it is possible to define a quantity, known as the rate of reaction, which is independent of the reactants and products. This can only be done if the reaction is of known stoichiometry; for some reactions there are numerous minor products and the stoichiometry is uncertain. Another condition for defining a rate of reaction is that the stoichiometric equation must remain the same throughout the course of reaction: for some reactions intermediates are formed in significant amounts, and the stoichiometry varies as the reaction proceeds. If these two conditions are satisfied (i.e., if the stoichiometry is known and is time independent), the rate of reaction is given by any of the expressions that appear in Eq. (3). In other words, the rate of reaction is the rate of consumption or formation divided by the appropriate coefficient that appears in the stoichiometric equation. In the case of products these coefficients are called the stoichiometric coefficients; in the case of reactants the stoichiometric coefficients are the negatives of the coefficients in the rate equation. As seen in Eq. (3), this division has made the four rates equal to one another, so that the rate of reaction is unique for the reaction under the particular conditions of the experiment. It is important to note that rate of reaction, unlike rates of consumption and formation, only has meaning with reference to a specified stoichiometric equation for the reaction. Thus, if the stoichiometric equation had been written as 3 2

A+ B→Y+ Z 1 2

1 2

the rate of reaction would be v = 23 vA = 2vB = vY = 2vZ

(4)

and would therefore be twice the rate specified with reference to the first form of the equation. Note also that some kineticists define rates with reference to the rates of change of amounts of substances rather than their concentrations; the S.I. unit is then mol s−1 , there being no division by volume. In general, rates depend on concentrations of reactants and sometimes on concentrations of products and other substances. For some reactions the rate of reaction v can be expressed by an equation of the form v = k[A]α [B]β ,

(5)

where k, α, and β, are independent of concentration and time. Similar equations apply to rates of consumption and formation. The exponent α is known as the order of reaction with respect to A. The term order was introduced in 1887 by F. W. Ostwald (1853–1932), but the concept of order had been used in 1884 by J. H. van’t Hoff (1852–1911). Similarly the exponent β is the order with respect to B. When there is more than one such order, as in this example, each one is called a partial order, and their sum,

α + β + · · · , is known as the overall order and is often given the symbol n. A simple case is when the rate equation is given by v = k[A]

(6)

An example of such a first-order reaction is the conversion of cyclopropane into propylene

A second-order reaction can involve a single reactant; v = k[A]2 ,

(7)

v = k[A][B].

(8)

or two reactants,

In the latter case the partial orders are unity, and the overall order is two. The process H2 + I2

2HI is an example, being second order in both directions. Reaction orders are not necessarily integral; in the case of the acetaldehyde decomposition, for example, CH3 CHO → CH4 + CO the order is : 3 2

v = k[CH3 CHO]3/2 .

(9)

This failure of the kinetics to correspond to the stoichiometry indicates that the reaction occurs by a composite mechanism, the nature of which is considered in Section IX. The constant k that appears in the preceding equations is known as the rate constant. Its units depend on the order of reaction. Thus, if a reaction is first order [Eq. (6)] and v is expressed as mol dm−3 s−1 and [A] as mol dm−3 , the unit of k is s−1 . Similarly, for the second-order reaction the unit of k is usually dm3 mol−1 s−1 . Just as rates of consumption and formation depend, in general, on the reactant or product under consideration, so do the corresponding rate constants. The rate constant that derives from the rate of reaction, for a specified stoichiometric equation, is unique. By no means do all equations have rate equations of the type shown in Eq. (5). Sometimes reactant concentrations appear in the denominator of the rate equation, and this is indicative of a composite mechanism. If a chemical reaction is sufficiently slow it is possible to mix reactants together or to raise the temperature of the reaction system very rapidly so as to start the reaction, and then to measure concentrations of reactants or products at various times. Such concentrations may be measured by chemical methods or by physical methods such as spectrophotometry. Two procedures are then available

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180 for determining orders of reactions and rate constants. One method is the method of integration, which involves integrating the differential equation for the rate and obtaining expressions for concentrations of reactants and products as a function of time. This integrated equation, which describes the time course of the reaction, is then fitted to the experimental results. A second method, the differential method, involves determining rates from the slopes of concentration–time curves. An analysis is made of the way in which the rates depend on the concentrations of reactants, and the order and rate constant can then be deduced. For rapid chemical reactions special methods must be used. For reactions that are not extremely rapid, flow methods are useful. Reactants may be mixed together in specially designed vessels and then passed along a tube, concentrations being measured at various positions by physical techniques such as spectrophotometry which allow instantaneous determinations to be made. A commonly used technique is the stopped-flow method, in which reactants are mixed and the flow suddenly stopped, so that the mixture is trapped in a vessel in which, by physical methods, concentrations can be studied as a function of time. The results can be analyzed by suitable adaptations of either the method of integration or the differential method. Some reactions, however, are so rapid that mixing cannot be achieved sufficiently rapidly. For such reactions the relaxation methods, such as the temperature-jump (T-jump) method may be used. They were developed in 1954 by M. Eigen (b. 1927). In this technique a reaction system at equilibrium is subjected to a very rapid rise in temperature which causes the equilibrium to shift (relax) to a new position of equilibrium. Physical methods are available for following the concentration changes during this relaxation, and the results can be analyzed to give the rate constants and the order of reaction. A quantity that has proved useful for comparing rates of different reactions is the half-life, or half-period. The half-life of a given reactant is the time that it takes for half of it to be consumed during the reaction. The way in which the half-life depends on reactant concentrations varies with the order of the reaction; for the special case of a first-order reaction there is no dependence of half-life on reactant concentration. For a reaction of any order the halflife is inversely proportional to the rate constant; half-lives are therefore useful in giving an inverse measure of the rate of a reaction and can be used for comparing reactions of different orders. Sometimes the rate of a chemical reaction is affected by the addition of a substance that is not consumed in the process and does not affect the equilibrium constant for the reaction. When the rate is increased in this way the added substance is known as a catalyst, and the effect is

Kinetics (Chemistry)

called catalysis, this name having been coined in 1836 by J. J. Berzelius (1779–1848). Catalysis can be classified as homogeneous catalysis (Section XI) in which only one phase is involved, and as heterogeneous or surface catalysis (Section VIII) in which the reaction occurs at an interface between phases. In biological systems catalysis is brought about by enzymes, which are proteins: the action is sometimes homogeneous and sometimes heterogeneous. A special type of catalysis, brought about by the product of a reaction, is known as autocatalysis. For example, ester hydrolysis is catalyzed by acids, which are products of the reaction; the process may therefore first accelerate. The term catalyst is sometimes applied to a substance that is consumed but which in other ways acts like a catalyst: such substances, however, are better called pseudocatalysts or activators. Mechanisms of catalysis are considered in Sections VIII and XI. An inhibitor is a substance that reduces the rate of a chemical reaction. Such substances were formerly called negative catalysts, but this terminology is not recommended since their action is quite different from that of a true catalyst. Inhibitors, in fact, act either by interacting with a catalyst and rendering it less effective or by removing active intermediates such as free radicals. The term inhibitor is often applied to substances that are consumed during the course of reaction. If a reaction in the absence of an inhibitor proceeds with a rate v0 , and in the presence of inhibitor with rate v, the degree of inhibition is defined as v0 − v v εi = =1− . (10) v0 v0

II. MOLECULARITY Chemical reactions can be classified as either elementary or composite. An elementary reaction is one which, as far as can be determined, goes in a single stage; the reactants pass smoothly through an intermediate state and then become products. If a reaction is elementary no specific intermediates can be detected or need to be postulated in order to explain the kinetic behavior. Composite reactions, also known as complex or stepwise reactions, occur in more than one stage, and therefore involve two or more elementary reactions; they are considered in more detail in Section IX. For an elementary reaction, but not for a composite reaction, the term molecularity can be employed. The molecularity of an elementary reaction is the number of reactant particles (atoms, molecules, free radicals, or ions) that are involved in each individual chemical event. For example, the cyclopropane isomerization appears to be elementary, in that each chemical act involves a single cyclopropane molecule: the molecularity is unity, and the

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reaction is said to be unimolecular. For the elementary reaction Br + H2 → HBr + H, the molecularity is two, and the reaction is said to be bimolecular. A reaction having molecularity of three, such as 2NO + Cl2 → 2NOCl is said to be trimolecular. Reactions of higher molecularities are unknown: it is unlikely for more than three molecules to come together in a single chemical event, and instead the mechanism is composite. It is important to distinguish clearly between the molecularity and the order. The latter is a purely experimental quantity, which is concerned with how the rate depends on reactant concentrations; the concept of order applies to some composite reactions. The molecularity of an elementary reaction, on the other hand, is arrived at by inference from all of the evidence available about the reaction. One such piece of evidence is the order. If a reaction in the gas phase appears to be elementary and has an order of one, it is reasonable to conclude that it is unimolecular. However, as will be seen in Section VI, unimolecular gas reactions become second order at low pressures, and it is therefore unsafe to conclude that a second-order gas reaction is bimolecular: it may be a unimolecular reaction in its second-order region. With reactions in solution the question arises as to whether the solvent should be included in the molecularity. It is usually considered that it counts in the molecularity if it enters into the overall reaction but not if it exerts only an environmental effect. Thus, a process in aqueous solution in which a compound was being hydrolyzed, with the reaction of a water molecule with each solute molecule, would be regarded as bimolecular. The isomerization of cyclopropane in solution, however, is unimolecular. Similar conventions are applied to reactions on surfaces. If individual molecules undergo decomposition or isomerization on a surface the reaction is usually described as unimolecular, even though the surface atoms are also involved. This, however, would not be done if the material of the surface entered into the final products.

III. TEMPERATURE AND REACTION RATES A considerable number of empirical equations have been proposed to express the dependence of reaction rates on temperature, and for over 60 years—from about 1850 to about 1910—there was much uncertainty and confusion. The difficulty was that several of the proposed equations fitted the data equally well, and the problem was finally

resolved in favor of equations for which there was a useful physical interpretation. One of the earliest equations relating the rate constant k to the absolute temperature T was k = Ae DT

or

ln k = ln A + DT.

(11)

This equation was proposed in 1862 by M. Berthelot (1827–1907). The parameters A and D are empirical quantities, and this equation requires that a plot of ln k against the absolute temperature T will be linear, as is approximately true in many cases. It is of interest that recently (1982–1985) C. M. Hurd and co-workers have shown that this equation is obeyed when a process occurs by quantummechanical tunneling (see Section IV), and they have provided an interpretation in terms of atomic vibrations. In 1884 J. H. van’t Hoff presented the equation, derived from thermodynamics, for the temperature dependence of an equilibrium constant K : d ln K H ◦ , (12) = dT RT 2 where R is the gas constant and H ◦ the standard change in heat constant (enthalpy) during the overall process. He then noted that an equilibrium constant is the ratio of rate constants for the reaction in forward and reverse directions, his argument being as follows: If a reaction A+B

Y+Z is elementary in both directions, the rate from left to right is k1 [A][B] and that from right to left is k−1 [Y][Z]. At equilibrium the two rates are equal so that   [Y][Z] k1 = = K. (13) [A][B] eq k−1 This being so, van’t Hoff argued, k1 and k−1 must show the same kind of temperature dependence as K , and therefore for a rate constant k d ln k E , (14) = dT RT 2 where E is some energy term. van’t Hoff did not assume that E itself is temperature independent but considered several possibilities. If E is temperature independent Eq. (14) integrates to ln k = ln A −

E RT

or k = Ae−E/RT ,

(15)

where A is a constant. Another possibility considered by van’t Hoff is that E is linear in temperature; in that case the equation reduces to the form k = AT m e−E/RT ,

(16)

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where m is a constant. Most kinetic data fit this equation very satisfactorily, and it is often used in modern compilations of kinetic data. The reason that equations of such different forms fit data reasonably well is that in kinetic work the temperature range is usually quite limited. In 1889 S. A. Arrhenius (1859–1927) showed that van’t Hoff’s simplest equation [Eq. (15)] applied satisfactorily to a number of results, and he also suggested a very simple interpretation for the equation; consequently this equation, although first given by van’t Hoff, is now always called the Arrhenius equation. Arrhenius pointed out that increasing the temperature by 10◦ C often approximately doubles the rate of reaction, but that the average molecular energies do not increase to anything like that extent. He concluded that reaction rates cannot depend on the average molecular energies, and he postulated that a pre-equilibrium is first established between reactant molecules and some highly energized intermediate. Thus, for a bimolecular reaction between A and B the process would be represented as A+B

AB∗ → Y + Z The intermediate complex AB∗ , being of high energy, is formed only in very small amounts, and its concentration increases strongly with increase in temperature. The rate is proportional to the concentration of AB∗ , and the large temperature coefficients are therefore explained. Arrhenius’s interpretation is essentially correct, and it was later made more precise by the application of statistical procedures to reaction systems. Figure 1 shows the distribution of energy in a system, as given by the treatments of J. Clark Maxwell (1831–1879) and L. E. Boltzmann (1844–1906). The fraction of molecules having energy in excess of a specified value E (per mole) is e−E/RT , which is the fraction that appears in the Arrhenius equation [Eq. (15)]. The interpretation of the equation is

FIGURE 2 Schematic Arrhenius plot of In k against 1/T. The slope is −E/R, and the activation energy E is thus defined by E ≡ −R [d ln k/d (1/T)].

that only those colliding molecules having joint energy in excess of E are able to undergo reaction; other collisions are ineffective, the reactant molecules merely separating unchanged. The energy E that appears in Eq. (15) is known as the activation energy or the energy of activation. The parameter A, which has the same units as the rate constant, is called the pre-exponential factor (formerly the frequency factor). The activation energy is obtained experimentally by plotting ln k against 1/T and measuring the slope, as shown schematically in Fig. 2; the slope is equal to −E/R. On the basis of a statistical argument it was shown in 1939 by R. H. Fowler and E. A. Guggenheim that the activation energy obtained in this way is the average energy of the molecules actually undergoing reaction minus the average energy of all of the reactant molecules. The activation energy is conveniently considered with reference to a potential energy surface, in which potential energy is plotted against bond distances and angles. A simple type of surface is obtained for a reaction A + B—C → A—B + C

FIGURE 1 Distribution of energy in a gas, according to the treatments of Maxwell and Boltzmann.

in which A, B, and C are atoms. The system A . . . B . . . C requires three parameters to describe it; these might be the A–B, B–C, and A–C distances, or two of the distances and an angle. The potential energy would therefore have to be plotted against three parameters, which would require a four-dimensional diagram. Since such a diagram cannot be constructed or visualized it is necessary to use a series of three-dimensional diagrams in which one parameter has been fixed at a particular value. For example, the A–B–C angle might be fixed at 180◦ , and for some systems this is the most probable configuration.

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FIGURE 3 Schematic potential energy surface for a reaction showing the minimal reaction path (dotted line) and a typical trajectory (dashed line).

represented as X‡ . The state of an activated complex is known as the transition state. Activated complexes play a very important role in theories of reaction rates, as will be seen in Sections IV and V. The preceding discussion has been confined to a system of three atoms, with the angle A–B–C held constant. If more atoms are involved, it is necessary to consider potential energy hypersurfaces for which there are more than three dimensions. These can be dealt with by extensions of the methods outlined; the data for the hypersurfaces can be stored in a computer, and activated complexes can again be identified. Sometimes much more complicated reaction systems are treated in an approximate way in terms of three-dimensional surfaces such as that shown in Fig. 3, and such treatments, although not precise, are very helpful in considering the course of reactions.

IV. STATISTICAL THEORIES OF RATES This type of three-dimensional diagram is conveniently represented on a two-dimensional surface as a contour diagram, and an example of such a diagram is shown in Fig. 3. The course of the reaction is then represented by motion of the system from the top left-hand region to the bottom right-hand region. In the top left-hand region the A–B distance is large, and the configuration therefore corresponds to A + B—C; in the bottom righthand region the B–C distance is large and the system is A—B + C. Many calculations have been made to determine the general shapes of potential energy surfaces, some of them involving pure quantum-mechanical theory and others introducing a certain amount of empiricism. All of the calculations have shown that the surfaces for A . . . B . . . C systems are of the general form shown in Fig. 3. The course of an individual reaction involves the motion of the system along the lower part of a rising valley until it reaches a col or saddle-point; then the system descends into another valley and finally reaches the state A—B + C. The dotted line in the diagram shows what is called the minimal reaction path, which corresponds to the path of steepest descent from the col into the two valleys. If it were to follow this minimal reaction path the system would make the least expenditure of energy. In reality, however, systems will follow a variety of reaction paths, or trajectories, depending on the initial conditions of the reactants A and B–C. A typical trajectory is illustrated by the dashed line in Fig. 3. This particular trajectory leads to considerable vibrational energy in the product molecule B–C, a matter that is referred to further in Section V. Those configurations of atoms that correspond to an arbitrary small region near the col of a potential energy surface are known as activated complexes, and are often

Most measurements of rates of chemical reactions are made on bulk systems in which the reacting molecules are distributed over a range of energy states (cf. Fig. 1). Some theories of reaction rates therefore focus on the bulk, or macroscopic, systems. This requires them to be statistical treatments. It is useful to consider theories of reaction rates in terms of the Arrhenius equation [Eq. (15)] which has two parameters, the pre-exponential factor A and the activation energy E. If each one of these can be calculated from first principles a complete theory of rates has been attained. If a potential energy surface has been calculated the height of the barrier to reaction is known, and the activation energy can be deduced from this height (although it is somewhat difficult to do this precisely). Progress is being made towards the reliable calculation of potential energy surfaces by quantum-mechanical methods, but success has so far only been achieved for systems involving few nuclei and electrons; no doubt the problems for more complicated systems will be overcome in the future. For the treatment of the pre-exponential factor A there have been several useful treatments, and there has been much more success in calculating values that agree with experiment. The first of these was simple collision theory, in which reacting molecules were treated as if they were hard spheres, and the frequencies of their collisions were calculated on the basis of kinetic theory. This treatment leads to a collision frequency factor z, and the rate constant is then obtained by multiplying z by e−E/RT , which is the fraction of collisions in which there is sufficient energy for reaction to occur:

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k = ze−E/RT .

(17)

A particularly clear formulation of this kind was presented in 1918 by W. C. McC. Lewis (1885–1956), and on its basis he calculated a value of the pre-exponential factor for the reaction 2HI → H2 + I2 that was in excellent agreement with experiment. However, later work showed that rather large deviations from experiment are obtained for reactions in which the reacting molecules are more complicated. This collision theory is evidently too simple and unlikely to be generally reliable. One weakness is the assumption that molecules are hard spheres, which implies that any collision with sufficient energy will lead to reaction; if the molecules are more complicated, this is not the case. A more fundamental objection to the treatment is that when applied to forward and reverse reactions it cannot lead to an expression for the equilibrium constant that involves the correct thermodynamic parameters. More recent work has involved a similar approach but has treated molecular collisions in a more realistic and detailed way. The most successful of the later statistical treatments has been transition-state theory, first formulated simultaneously and independently in 1935 by H. Eyring (1901– 1981) and by M. G. Evans (1904–1952) and M. Polanyi (1891–1976). Transition-state theory treats the rates of elementary reactions as if there were a special type of equilibrium, having an equilibrium constant K ‡ , between reactants and activated complexes. The rate constant is then given by k = (kT / h)K ‡ ,

(18)

where k is the Boltzmann constant (the gas constant per molecule) and h is the Planck constant. This assumption of equilibrium can be justified in most cases, but it breaks down for certain types of potential energy surfaces, and particularly when the barrier is low so that the reaction is rapid. An important and simplifying feature of transitionstate theory is that it focuses attention on the activated complex. How the system reaches the transition state and how it behaves after reaching the transition state, is not considered to affect the rate of reaction. The equilibrium constant K ‡ can be treated in two different but equivalent ways. In the first procedure, the methods of statistical mechanics are used. Equilibrium constants can be calculated reliably in terms of partition functions for the molecules involved, and the same can be done for the particular equilibrium constant K ‡ . What are required are the masses of the reactant molecules and the activated complex, their moments of inertia, and their vibrational frequencies. For the reactant molecules this is usually straightforward. These parameters are also known for an activated complex if a reliable potential energy sur-

face has been calculated. If it has not, the parameters can usually be estimated sufficiently well to lead to useful approximate rate constants. The second procedure is to employ a thermodynamic formulation. The equilibrium constant K ‡ can be ex‡ pressed as e− G/RT , where ‡ G, known as the Gibbs energy of activation, is the change in Gibbs energy when the reactants become activated complexes. The Gibbs energy of activation can, in turn, be expressed as ‡ H − T ‡ S, where ‡ H is the enthalpy of activation and ‡ S the entropy of activation. The rate constant k can thus be formulated as kT − ‡ G/RT k= e h KT ‡ S/R − ‡ H/RT = e . (19) e h This equation is of the same form as Eq. (16); the factors ‡ e− H/RT corresponds to e−E/RT , while the pre-exponential ‡ factor (kT / h)e S/R shows dependence on temperature. ‡ However, H is not quite the same as the activation energy E, but adjustments can easily be made for different types of reactions. Much effort has gone into testing the validity of the original formulation of transition-state theory (now often referred to as conventional transition-state theory) and to improving it or proposing alternative theories. Useful extensions of the theory have been suggested, some of them involving locating the activated complex at a position other than at the col in the potential energy surface. Improvement has been achieved, but only at the cost of making very extensive calculations and making assumptions the justification for which is not always entirely clear. In addition to these extensions of conventional transition-state theory there have been some entirely different approaches, such as the dynamical treatments to be considered in Section V. Imperfect as it is—as are all scientific theories— conventional transition-state theory is of importance in providing a conceptual framework with the aid of which much insight is gained into how chemical reactions occur. It is possible without even making any numerical calculations to make qualitative predictions of many important kinetic effects. So far no alternative treatment has provided any such insight. Sometimes chemical reactions occur to some extent by quantum-mechanical tunneling. Usually systems must surmount the col in the potential energy surface (Fig. 3), but quantum-mechanical theory allows the possibility of going from the initial state A + BC to the final state AB + C without passage over the col. This arises when very light species, such as electrons and hydrogen atoms, are involved in the reaction. For example, in a reaction such as,

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Tl2+ + Fe2+ → Fe3+ + Tl+ , there is simply the transfer of an electron, which tunnels through the potential energy barrier without going over it. Tunneling is also important in a reaction such as D + H2 → DH + H, where D is a deuterium atom. Here there is a transfer of a hydrogen atom H, and the reaction involves some passage over the energy barrier but also some tunneling. Since the rate of tunneling increases much less with temperature than the passage over the barrier, quantum-mechanical tunneling is relatively more important at low temperatures, and it is at such temperatures that much of the evidence for tunneling has been obtained. Quantum-mechanical tunneling is an important factor that must be considered in connection with kinetic-isotope effects. Suppose, for example, that the reactions D + H2 → DH + H and D + D2 → D2 + D are compared. The ratio of rate constants depends on certain factors that enter into the partition functions for the reactants and activated complexes but also on the fact that quantum-mechanical tunneling is important for the first reaction but not for the second. The deuterium atom D is twice as heavy as the H atom, and tunneling is quite unimportant when D, or any heavier atom, is transferred in a chemical process.

V. REACTION DYNAMICS It is important to have knowledge, on the molecular or microscopic level, of the elementary act that occurs during the course of chemical change. Work in this field, referred to as-reaction dynamics or molecular dynamics, deals with the intermolecular and intramolecular motions that occur in a chemical reaction and the quantum states of the reactant and product molecules. There are two main reasons for studying chemical dynamics. One is to test the validity of the statistical theories that were outlined in the preceding section. The other is that there are important applications (e.g., lasers) in which it is necessary to have information about the energy states of products of reaction, information that is not provided by the statistical theories. The results of dynamical calculations and experiments are frequently expressed in terms of reaction cross sections rather than rate constants. The reaction cross section for a bimolecular reaction is defined as Pr πd 2 , where Pr is

the probability of reaction and d, the collision diameter, is the distance between the centers of the reactant molecules when they collide. In practice the reaction cross section, given the symbol σ , is the quantity that has to be postulated in using collision theory to interpret calculated or experimental rates. Much of the theoretical work in molecular dynamics has been based on potential energy surfaces that have been calculated for individual reactions. Dynamical calculations are then carried out for various initial states of the reacting molecules. For a reaction A + B—C, for example, one chooses a particular vibrational and rotational state for the molecule B–C, and particular translational states for A and B–C. Other details of the collision between the two are also selected. One then calculates, on the basis of dynamics, the path that the system takes on the potential energy surface. A diagram or mathematical description that describes the motion of a reaction system over a potential energy surface is known as a trajectory. Ideally the dynamical calculations are based on quantum mechanics, but this presents difficulty and more often classical calculations are made; there is good reason to conclude that not much error is then introduced. Even when the trajectories are obtained classically, the initial states of the reactants are usually selected on the basis of quantum theory. A large number of trajectories have been calculated, and the results have provided considerable insight into how reactions proceed. Calculations by Karplus and coworkers on the reaction H + H 2 → H2 + H have revealed a number of interesting features. One is that not all of the vibrational energy of H2 can contribute to allowing the system to surmount the potential energy barrier. Another is that, as assumed in transition-state theory, the system passes directly through the col and does not remain there and perform a number of vibrations. On the basis of calculations made for a variety of initial conditions it was possible to compare calculated rates with those obtained from transition-state theory, and the agreement was very good. For the reaction Br + H2 → HBr + H, the agreement is not as good. Other trajectory calculations have been made for potential energy surfaces that do not have the shape shown in Fig. 3 but instead have a basin at the col. It is then found that the activated complexes are trapped for a short period in the basin, and perform several vibrations before they become products. A number of trajectory calculations have been made, particularly by J. C. Polanyi (b. 1929) and co-workers, with the object of exploring how the slopes of potential energy surfaces affect the transfer of energy during the

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course of a chemical reaction. Three questions of particular importance are: 1. If energy is released in a chemical reaction (i.e., if the reaction is exothermic), what is the distribution of the energy among the translational, vibrational, and rotational states of the product molecules? 2. What type of energy is particularly effective in leading to a successful chemical reaction? 3. If a collision occurs with substantially more than enough energy to surmount the barrier, in what form is the excess energy released? The answers to all three questions depend on the form of the potential energy surface. In the schematic potential energy surface shown in Fig. 3 the energy level of the products, A—B + C, is lower than that of the reactants (i.e., the reaction is exothermic). Also, the surface was drawn in such a way that at the activated state there is not much change in the B–C distance, but that the A–B distance is considerably greater than it becomes in the product A–B. This type of surface is called an attractive or an early downhill surface. In the trajectory shown in Fig. 3 there is not much vibrational energy in the reactant B–C, but much more in the product A–B, and this is typical of this type of surface. If, on the other hand, an exothermic reaction has a late downhill surface, or a repulsive surface, the calculations showed that not so much of the energy released passed into vibrational energy of A–B, most of it becoming translational energy of A–B and C. The answer to the second question can be appreciated by considering the reverse processes. Suppose that the surface is as shown in Fig. 3 but that the reaction is A—B + C → A + B—C. The reverse of the trajectory shown in Fig. 3 therefore applies, and the reaction absorbs energy (i.e., it is endothermic). Vibrational energy is now much more important than translational energy in surmounting the barrier. If there is only translational energy the system is likely to bounce against the inner wall of the surface and return to the entrance valley. With regard to the third question, the disposition of excess energy, there is a tendency for excess translational energy to appear as translational energy in the products and for excess vibrational energy to appear as vibrational energy. This effect has been referred to as adiabaticity, and theoretical reasons for it have been proposed. These conclusions have been amply confirmed by many experimental results. Broadly speaking the experimental investigations fall into two main classes: 1. Experiments in which narrow beams of reactant molecules, in preselected states, are brought into contact

with each other. The primary purpose of these molecularbeam studies is to determine the fate of the reactants and products after the beams have come together. 2. Experiments carried out in bulk systems in which the energy states of the reaction products are determined by spectroscopic techniques. The reaction itself may occur in a static or flow system, and often the reactant molecules have been put into particular energy states. The term chemiluminescence is applied to such studies, since they are concerned with radiation emitted by the products. Ideally, in both types of investigation, the reactant molecules are put into selected translational, vibrational, and rotational states, and the corresponding states of the reaction products are determined. When this objective is more or less achieved the expression state-to-state kinetics is applied. Molecular-beam studies have only been carried out to any extent since the 1950s, and the techniques have developed considerably since that time. A particular advance has been the use of mass spectrometers for determining the nature, speeds, and direction of the molecules after the collision has occurred. Analysis of the results yields detailed information about the distribution of angular momentum among the reaction products, the reaction cross sections, the quantum states of the product molecules, and the lifetime of the collision complex. Many reactions fall into one of two main classes: 1. Reactions having high reaction cross sections (i.e., occurring at relatively high rates) and in which the reaction products are scattered forward with respect to the center of mass of the system. This type of process is said to occur by a stripping mechanism. 2. Reactions having small cross sections, in which there is backward scattering of the reaction products. The mechanism is then said to be a rebound mechanism. It has been found that reactions for which the potential energy surface is attractive (Fig. 3) tend to occur by a stripping mechanism, while when it is repulsive there is a rebound mechanism. The reasons for this have been worked out. When a reaction occurs either by a stripping or a rebound mechanism the lifetime of the activated complex must be short; if it were long enough for rotation to occur the products would separate in random directions and the reaction is then said to occur by an indirect or complexmode mechanism. When the lifetime of the complex is short, as in the stripping and rebound mechanisms, the expression direct or impulsive is used. The occurrence of a reaction by a complex-mode mechanism appears to be associated with the existence of a basin in the potential energy surface.

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The chemiluminescence investigations of chemical reactions have as their origin some pioneering investigations carried out in the 1920s and 1930s by M. Polanyi. In the course of these studies it was found, for example, that in the reaction Cl + Na2 → NaCl + Na a good deal of the energy liberated resides in the product NaCl molecule in the form of vibrational energy, as indicated by the prime. These investigations have been extended, particularly by J. C. Polanyi and co-workers, who have introduced new experimental techniques and have studied reactions of a variety of types. It was these investigations, together with parallel theoretical studies, that led to the classification of different types of potential energy surfaces. Both the molecular-beam and chemiluminescence techniques have been applied to the investigation of state-tostate kinetics. In a molecular-beam experiment the translational energies of the reactant molecules can be well controlled, and by laser excitation the molecules can be put into particular vibrational and rotational states. In the chemiluminescence experiments the reactant molecules are sometimes put into particular vibrational and rotational states by forming them in prereactions. State-to-state studies have been carried out only since the early 1970s, but they have already revealed much important information. By and large they have confirmed and amplified the previous theoretical and experimental studies in reaction dynamics. One general result that has been obtained is that when reactions are substantially endothermic, their barriers are usually late ones, so that they are enhanced more by vibrational than by translational energy. On the other hand, when reactions are only slightly endothermic or are exothermic, the barriers are generally earlier, and translational energy then plays a more important role in leading to reaction. State-to-state studies have also led to some clarification of the influence of rotational energy on reactivity, although the matter is still not entirely clear. When the translational and vibrational energies are held constant and the rotational quantum state of a reactant molecule is steadily increased, the rates sometimes first decrease and then increase. As rotational speeds increase, the time that a molecule spends in a favorable orientation decreases, and this factor leads to a decrease in reaction probability. At higher rotational speeds, however, the preferred orientations are obtained again at short intervals during the approach of the reactants, and this can give rise to higher rates as the rotational energies increase. Also, at the higher rotational speeds there is a greater proportion of energy in the form of rotational energy, and this again can contribute to the effects observed. An important development has been the spectroscopic detection of transition species, which are defined as molec-

ular entities having configurations between those of the reactants and products of a chemical reaction. The term transition species covers a much broader range of configurations than does the term activated complex, which is defined as existing in an arbitrarily small region of space. It has long been known that spectral lines are broadened when a gas is at higher pressures, and this was interpreted in 1915 by H. A. Lorentz (1853–1928) as being due to collisions between the molecules. It was only in 1980, however, that J. C. Polanyi and co-workers were successful in detecting similar effects from molecules that were undergoing reactive collisions. For example, they studied the reaction F + Na2 → F . . . Na . . . Na → NaF + Na∗ The product Na∗ is in an electronically excited state and emits the familiar yellow D-line. On both sides of this line there was “wing” emission, and the evidence indicated that this was due to the transition species F . . . Na . . . Na. Techniques have now been developed for studying the course of chemical change in great detail. In particular, femtosecond flash photolysis has been used to observe transition species, notably by Philip R. Brooks and by Ahmed H. Zewail.

VI. REACTIONS IN THE GAS PHASE Reactions in the gas phase are in some respects easier to understand than those in solution or on surfaces, and what is learned about gas reactions is valuable in leading to an understanding of the other types of processes. The most straightforward of gas reactions are those that are elementary and bimolecular. There are few bimolecular reactions involving molecules, as opposed to atoms and free radicals, since most second-order reactions involving molecules occur in more than one stage, i.e., they are composite. Such composite reactions are dealt with in Section IX. The prototype of all bimolecular reactions is the process H + H 2 → H2 + H The measurement of the rate of this reaction is made possible by the fact that H2 exists in two forms, designated ortho (o) and para ( p). To explain the kinetics of this reaction a large number of theoretical studies have been made. It is now possible to calculate, by pure quantum mechanics, an accurate potential energy surface for this reaction. For this particular reaction, which has a symmetrical potential energy surface, conventional transition-state theory is in good agreement with the best of the dynamical treatments. One matter of considerable interest is the kinetic isotope effect observed when H atoms are replaced by deuterium

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(D) or tritium (T) atoms in this system. These kinetic isotope effects still cannot be treated completely satisfactorily, largely because the theory of quantum-mechanical tunneling is by no means satisfactory. The effects are, however, broadly understood in a semiquantitative way. Among other bimolecular gas reactions that have been extensively investigated are so-called abstraction or metathetical reactions involving atoms and free radicals. An example is

suggestion made in 1921 by F. A. Lindemann, later Lord Cherwell (1886–1957). Lindemann’s hypothesis, as later interpreted and developed by C. N. Hinshelwood (1897–1967) and others, is as follows. In a unimolecular process an energized molecule A∗ is first formed by a collision between two molecules of the reactant A:

CH3 + C2 H6 → CH4 + C2 H5 .

The energization process is second order, the rate being k1 [A]2 , and the rate of the reverse de-energization process is k−1 [A∗ ][A]. The energized molecule A∗ may undergo de-energization, but it may also undergo a process in which the reaction products are formed:

In this process the free methyl radical CH3 abstracts a hydrogen atom from the ethane molecule. There has also been much study, by mass spectrometric techniques, of bimolecular reactions involving ions, such as the reaction +

+

O + N2 → NO + N. For most such reactions conventional transition-state theory provides a useful, but not always precise, interpretation of the results. A few trimolecular reactions in the gas phase have also been investigated; examples are 2NO + Cl2 → 2NOCl and 2NO + O2 → 2NO2 . The interpretation of these reactions was a considerable triumph for conventional transition-state theory. Simple collision theory proved unsatisfactory for trimolecular reactions, owing to the difficulty of defining a collision between three molecules, and usually led to very serious overestimations (by several powers of ten) of the rate constants. Similar difficulties are encountered with dynamical treatments, and these have still not been satisfactorily resolved. Conventional transition-state theory, by regarding the activated complex as being in equilibrium with the reactants, leads to a very simple formulation of the rate constant and to values in good agreement with experiment. It also very neatly explains the rather marked negative temperature dependence of the pre-exponential factors for these reactions. The elucidation of unimolecular gas reactions proved to be much more difficult than that of bimolecular and trimolecular reactions. Whereas the rates of the latter reactions can be interpreted in terms of collisions between two and three molecules respectively, no such treatment appears at first sight to be possible for unimolecular reactions. At one time it was widely held that collisions are not at all involved in unimolecular reactions, and that instead the reactions occur as a result of the absorption of radiation emitted by the vessel walls. However, this proved to be incorrect, and the problem was resolved as a result of a

A+A

A∗ + A

A∗ → products The rate of this first-order process is k2 [A∗ ]. If the pressure is high enough the de-energization process will be more rapid than the reaction to form products. This being so, the energized molecule A∗ is essentially in equilibrium with normal molecules, and its concentration [A∗ ] is proportional to the first power of the concentration [A]. Since the rate is proportional to [A∗ ] it is also proportional to [A]; i.e., the kinetics are first order. At sufficiently low pressures, on the other hand, the time between successive collisions between A∗ and A becomes longer than the time that elapses before A∗ can become products. The removal of A∗ molecules by chemical change then seriously diminishes the concentration of activated molecules. At extremely low pressures the energized molecules A∗ almost inevitably become products, as there are few collisions to inactivate them. The rate of reaction is then equal to the rate of energization and is thus proportional to the square of the concentration. If this Lindemann–Hinshelwood hypothesis is correct, unimolecular gas reactions should be first order at high pressures and should become second order at low pressures. This behavior has now been confirmed for a large number of reactions. In its original form the hypothesis had some difficulty in interpreting results quantitatively, but a number of extensions of the original hypothesis have been made, notably by R. A. Marcus whose treatment is consistent with transition-state theory.

VII. REACTIONS IN SOLUTION Reactions in solution or in the liquid phase are intrinsically more difficult to understand than reactions in the gas phase because of the complications arising from the effect of the liquid. When a solvent is present there are two possibilities: the solvent may exert purely an environmental effect on the reaction, or it may also enter into the reaction.

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The latter case is exemplified by the hydrolysis of an organic compound in aqueous solution; water molecules are chemically involved in the reaction. Some understanding of solvent effects has been provided by comparisons of the same reaction in the gas phase and in solution. Some reactions, however, do not occur at all in the gas phase, and one must then be content with comparing their rates in different solvents. When such comparisons are made it is sometimes found that the solvent does not have much effect on the rate. When, on the other hand, ions are involved as reactants or products, solvents usually have a much greater effect on rates, because of the rather strong electrostatic interactions between ions and solvent molecules. Theoretical studies have also contributed to an understanding of reactions in solution. For reactions between neutral species the frequencies of collisions are of the same order of magnitude in solution and in the gas phase. However, there are important differences between the distribution of collisions. When two molecules collide in solution they are “caged” in by surrounding molecules and within a very short period of time they are likely to undergo one or more additional collisions before they separate. Such a set of collisions, occurring in rapid succession, is known as an encounter. This tendency for collisions in solution to occur in sets has no effect on reactions involving an activation energy, because reaction may occur on any collision within the set. However, atomic and free-radical combinations do not involve an activation energy and occur on every collision. They therefore occur at the first collision in the set, so that the remaining collisions in the encounter do not contribute to the rate. The pre-exponential factor is therefore not related to the frequency of collisions but rather to the frequency of encounters. This cage effect, also known as the Franck– Rabinowitch effect, has other important consequences. In a photochemical reaction in solution, for example, a pair of free radicals produced initially may, owing to their being caged in by the surrounding molecules, recombine before they can separate from each other. This effect is known as primary recombination. When reaction occurs between ions in solution some rather special effects become important. These can be understood in an approximate but very useful way by focusing attention on the dielectric constant of the solvent. The forces between charges, whether they are attractive or repulsive, are inversely proportional to the dielectric constant of the medium. The dielectric constant of water, for example, is about 78 at room temperature, and the force of repulsion between two ions A+ and B+ in wa1 of that in the gas phase. Even in aqueous ter is only 78 solution the forces are still strong enough to have a substantial effect on the collision frequencies and therefore on

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189 the pre-exponential factors of such reactions. In water at 25◦ C, for example, the pre-exponential factor of a reaction of the type A+ + B+ is apt to be about two orders of magnitude lower than for the corresponding reaction between neutral molecules. By contrast, for a reaction in aqueous solution between singly charged ions of opposite signs, A+ + B− , the pre-exponential factor at 25◦ C is usually about 102 greater than that between neutral molecules. Similar considerations allow useful predictions to be made about the effects or different solvents on reaction rates. If a reaction is of the type A+ + B+ the forces are repulsive and will be smaller the higher the dielectric constant of the solvent. The reaction therefore proceeds more rapidly in aqueous solution than in alcohol, since water has a higher dielectric constant than alcohol and reduces the repulsions to a greater extent. When on the other hand the reaction is of the type A+ + B− , the rate will be greater in alcohol than in water, since water greatly reduces the attractions between the reactants. Quantitative formulations of these effects have been worked out. Another matter of considerable importance in connection with ionic reactions in solution is the effect on rates of the ionic strength. This property, introduced in 1921 by G. N. Lewis (1875–1946), is defined as 1 2 I = z ci , (20) 2 i i where ci is the concentration of each ion in solution and z i is its charge number (e.g., +1 for Na+ , −2 for SO−2 4 ): the summation is made over all of the ions present in the solution. Qualitatively the ionic strength has the same kind of effect as the dielectric constant; increasing I reduces the forces between ions. Thus for a reaction of the type A+ + B+ , increasing the ionic strength increases the rate by reducing the repulsion. For a reaction A+ + B− , increasing the ionic strength decreases the rate by reducing the attraction. These ideas were put into quantitative form in 1922–1924 by J. N. Brønsted (1879–1947), N. J. Bierrum (1879–1958), and J. A. Christiansen (1888– 1969). Special effects arise for a solution reaction that is extremely rapid, in which case the rate may depend on the rate with which the reactant molecules diffuse through the solvent. Two effects are to be distinguished, macroscopic diffusion control and microscopic diffusion control. If a rapid bimolecular reaction in solution is initiated by mixing solutions of the two reactants, the observed rate may depend on the rate with which the solutions mix, and one then speaks of mixing control or macroscopic diffusion control. Even if this effect has been eliminated the rate of a reaction may be influenced by the rate with which the reactant molecules diffuse towards each other. This effect

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190 is known as microscopic diffusion control or encounter control. If the measured rate is almost exactly equal to the rate of diffusion one speaks of full diffusion control. An example of this is provided by the combination of H+ and OH− ions in solution, a reaction that is so fast that the rate is almost entirely controlled by the diffusion of the ions towards each other. For some reactions the rates of chemical reaction and diffusion are similar to each other, and one then speaks of partial diffusion control. Quantitative formulations have been worked out for the rate constants for both full and partial diffusion control. Diffusion rates do not vary greatly from one system to another and for uncharged reactants in aqueous solution at 25◦ C diffusional rate constants are approximately 7 × 109 dm3 mol−1 s−1 . If the chemical rate constant is substantially greater than this there is therefore appreciable diffusion control. Most reactions, however, have much smaller rate constants because of an energy barrier to reaction, and are therefore not affected by diffusion. For reactions between ions the diffusion rates and chemical rates are increased if the ions are of opposite signs, and they are decreased if they are of the same sign.

Kinetics (Chemistry)

sorbed molecules, so that 1 − θ is the fraction that is not covered. If [A] is the concentration of gas molecules, then the rate of adsorption, which can only occur on bare surface, is ka [A](1 − θ ), where ka is a constant. The reverse desorption process is a unimolecular process and its rate is kd θ. At equilibrium the two rates are the same so that ka [A](1 − θ ) = kd θ from which it follows that K [A] , (22) 1 + K [A] where K is equal to ka /kd . Equation (22) leads at once to kinetic equations that apply to unimolecular processes on surfaces. The rate is proportional to θ and is therefore given by θ=

k K [A] (23) 1 + K [A] If [A] is sufficiently small that k[A] 1, this reduces to v=

v = k K [A]

Many reactions are affected by the surface of the vessel in which they occur, with an increase in rate. The effect is often a catalytic one, the surface remaining unchanged, and substances are often deliberately introduced into reaction systems with the object of increasing the rates. Surface catalysis is of very great importance in technical work. The fact that the enhanced rates are due to adsorption at the surface has long been known. An important advance was made in 1916 by I. Langmuir (1881–1957) who showed that in many cases of adsorption the gas molecules are held to the surface by bonds of the same character as covalent chemical bonds, and the term chemisorption has been applied to this type of adsorption. Langmuir also developed adsorption isotherms that relate the fraction of surface covered to the pressure or concentration of a gas. Later H. S. Taylor (1890–1974) emphasized the fact that surfaces are never smooth on the atomic scale, and that surface sites are therefore of variable activity. Certain sites, which he called active centers, are particularly active and it is on these that catalysis occurs for the most part. Taylor also showed that the process of chemisorption itself is accompanied by an activation energy, and he referred to this type of adsorption as activated adsorption. The kinetic equations that apply to reactions on surfaces are given to a good approximation on the basis of the Langmuir isotherm, which can be derived as follows. Let θ be the fraction of a surface that is covered by ad-

(24)

and the reaction is first order. If [A] is sufficiently large that K [A] 1 the rate is v=k

VIII. REACTIONS ON SURFACES

(21)

(25)

and is now independent of concentration; the order of the reaction is thus zero. One example of a reaction that changes from zero-order kinetics at high concentrations to first-order kinetics at low concentrations is the decomposition of ammonia on the surface of a metal such as tungsten or iron. When two substances are undergoing reaction at a surface there are two possibilities. The reaction may be an interaction between two molecules that are adsorbed sideby-side on a surface, or it may be an interaction between an adsorbed molecule of one kind and a gas molecule of the other. The kinetics are satisfactorily interpreted by an extension of Langmuir’s isotherm [Eq. (22)], which for two gases A and B gives the following expressions for the fractions of surface covered by A and B respectively: θA =

K A [A] 1 + K A [A] + K B [B]

(26)

θB =

K B [B] 1 + K A [A] + K B [B]

(27)

Reactions involving interaction between two adsorbed molecules are said to occur by Langmuir–Hinshelwood mechanisms, and their rates are proportional to the product θA θB : k K A K B [A][B] v = kθA θB = (28) (1 + K A [A] + K B [B])2 If either concentration is held constant and the other one increased, the rate passes through a maximum. The

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most favorable situation is for equal numbers of A and B molecules to be on the surface, since then the number of A–B pairs is a maximum. If the concentration of either is then increased, the other is displaced from the surface, and the number of A–B pairs diminishes. Mechanisms in which a gas molecule reacts with an adsorbed molecule of the other kind are known as Langmuir–Rideal mechanisms. If, for example, the interaction is between gaseous A and adsorbed B the rate is proportional to [A]θB and is therefore v=

k K B [A][B] 1 + K A [A] + K B [B]

(29)

There is therefore no longer a maximum in the rate, which now approaches a limiting value if the concentration of one reactant is increased with the other held constant. This difference in kinetic behavior allows a discrimination between the two mechanisms, both of which sometimes occur simultaneously. Since about 1970 there have been many important advances in experimental techniques for the study of surfaces and adsorbed layers of molecules on surfaces. Techniques that have been particularly valuable in the investigation of solid surfaces are field-ion microscopy (FIM) and low-energy electron diffraction (LEED). These techniques have shown that surfaces have different types of surface sites, such as atoms at terraces, atoms at steps, atoms at kinks, and adatoms which project out of the surfaces. A technique that has provided valuable information about adsorbed films is infrared spectroscopy. As an example of its use we mention some work on the adsorption of ethylene (C2 H4 ) on various surfaces. It has been found that the manner in which ethylene is adsorbed depends on the availability of hydrogen. If no hydrogen is available the adsorption tends to be of the dissociative type; the ethylene splits into C2 H2 and 2H, all of which are adsorbed separately. With hydrogen present, however, adsorbed H atoms add on to C2 H4 and ethyl radicals (C2 H5 ) become attached to the surface. Evidence of this kind is of great importance in understanding the kinetics and mechanisms of the hydrogenation of ethylene and many other processes. Reactions on surfaces frequently undergo inhibition by added substances, which themselves become adsorbed on the surface and reduce its catalytic activity. This effect is commonly referred to as poisoning of the surface.

IX. COMPOSITE REACTION MECHANISMS Many of the reactions familiar to the chemist occur in more than one stage, and their mechanisms are therefore

described as composite, complex, or stepwise. One indication that a reaction is composite is that the kinetic equation does not correspond to the stoichiometric equation. A simple example is the gas-phase reaction between nitric oxide and hydrogen, the stoichiometric equation for which is 2NO + 2H2 → N2 + 2H2 O If the process were elementary it would be second order in NO and second order in H2 , with an overall order of four. In fact, the rate is proportional to [NO]2 [H2 ]. The reason is that the reaction occurs in two steps, 2NO + H2 → N2 + H2 O2 H2 O2 + H2 → 2H2 O The second reaction is rapid compared with the first, so that the overall rate is that of the first reaction and is proportional to [NO]2 [H2 ]. Because reactions can occur in steps, the order of a reaction is often less than corresponds to the stoichiometry. A reactive collision between three molecules is much less likely than one between two, and one between four molecules is exceedingly unlikely; indeed it is doubtful whether elementary reactions between four molecules ever occur. A reaction whose stoichiometric equation involves more than three molecules always proceeds more rapidly by two or more processes of lower molecularity. Even if the kinetics of a reaction does correspond to the stoichiometry it may still be the case that the mechanism is composite. For example, the reaction H2 + I2 → 2HI and the reverse decomposition are both second order, and the processes were formerly thought to occur in a single stage. Later work, however, has shown that reactions involving I atoms are also involved. The decomposition of ethane, C2 H6 → C2 H4 + H2 , is first order and was long thought to be elementary; however it occurs entirely by a free-radical mechanism, to be considered later. Composite mechanisms can be classified in a number of different ways. First, there is the usual classification according to whether the process occurs in the gas phase, the liquid phase, or on a surface. Another classification is according to whether the process is thermal, photochemical, or radiation chemical. A thermal reaction occurs simply by virtue of the heat energy present in the system; molecules can pick up more energy on colliding with other molecules, and eventually an energetic collision will lead to reaction. Photochemical and radiation-chemical reactions are brought about as a result of radiation absorbed in the system.

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Another characterization of composite mechanisms is in terms of the types of elementary reactions occurring. Sometimes reactions occur in parallel, such as A→Y A → Z, and are called simultaneous reactions. When there are simultaneous reactions, there is sometimes competition, as in the scheme A+B→Y A + C → Z, where B and C compete with one another for A. Reactions occurring in forward and reverse directions are called opposing: A+B

Z

FIGURE 4 Variations with time of the concentrations of A, X, and Z for the mechanism A → X → Z.

Reactions occurring in sequence, such as A→X→Y→Z are called consecutive reactions. Reactions are said to exhibit feedback if a substance formed in one step affects the rate of a previous step. For example, in the scheme A→X→Y→Z the intermediate Y may catalyze the first reaction (positive feedback) or inhibit it (negative feedback). More than one of the above features may of course occur in a composite mechanism. Sometimes a composite reaction mechanism involves a cycle of reactions such that certain reaction intermediates consumed in one step are regenerated in another. The intermediates may be atoms, free radicals, or ions. For example, the reaction between hydrogen and bromine, to be considered later, includes the steps Br + H2 → HBr + H H + Br2 → HBr + Br. When this feature exists, and the cycle is on the average repeated more than once (as it is in this reaction under usual conditions), the reaction is called a chain reaction, and the active intermediates (here H and Br) are referred to as chain carriers. The preceeding two reactions are referred to as chain-propagating steps. Substances that are formed during the course of a reaction but do not remain to any extent as final products are known as transient intermediates, or transients. A simple reaction scheme is A→X→Z and if the reaction goes to completion the way in which the concentrations of A, X, and Z vary with time is illustrated

in Fig. 4. The rate of formation of Z is proportional to [X], and since [X] starts at zero and passes through a maximum, the initial rate of formation of Z is zero. This rate, however, increases, and passes through a maximum when [X] is at its maximum. In a composite mechanism it is sometimes possible to identify a rate-determining or rate-controlling step. An example is provided by the mechanism previously given for the 2NO + 2H2 reaction. Since the first step is slow and the second rapid, the first step determines the rate. The rate may also be controlled by a step that is not the first; in that case the overall rate is not equal to the rate of the slow step but is proportional to it. Only in the case of very simple reaction mechanisms is it possible to obtain explicit expressions for concentration changes as a function of time. In some situations a reliable solution to the problem may be obtained by application of the steady-state treatment. The condition for this to be valid is that the intermediates are present only at concentrations that are very low compared to those of the reactants. If this condition is satisfied by an intermediate X, the rate of change of its concentration is always much less than those of the reactants, and to a good approximation can be set equal to zero: d[X]/dt = 0

(30)

This approximation can be safely applied to intermediates such as atoms and free radicals. The reaction between hydrogen and bromine provides a good example of a composite mechanism to which the steady-state treatment can be applied, since the intermediates are the atoms Br and H. If the back reaction is prevented from occurring (by removal of the product HBr as it is formed) the mechanism is

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(1)

Br2 → 2Br

(2)

Br + H2 → HBr + H

(3)

H + Br2 → HBr + Br

(−1)

The solution for [CH3 ] is  1/2 k1 [CH3 ] = [CH3 CHO]1/2 . k4

2Br → Br2 .

Reactions (2) and (3) constitute chain-propagating reactions. Reaction (1) is referred to as chain-initiation process and reaction (−1) as a chain-ending step. Application of the steady-state treatment to the Br atoms gives k1 [Br2 ] − k2 [BF][H2 ] + k3 [H][Br2 ] − k−1 [Br] = 0. (31) For H atoms the steady-state equation is 2

k2 [Br][H2 ] − k3 [H][Br2 ] = 0.

(32)

(37)

The rate of formation of the product CH4 is k2 [CH3 ] [CH3 CHO] and is therefore  1/2 k1 vCH4 = k2 [CH3 CHO]3/2 . (38) k4 The three-halves-order kinetics actually observed for this reaction is thus explained. Ethane decomposition is also a chain reaction, and the main features have now been elucidated. The initiation process is the dissociation of the ethane molecule into two methyl radicals: C2 H6 → 2CH3 .

There are thus two equations in the two unknowns [Br] and [H], and the solution for [Br] is   k1 1/2 [Br] = [Br2 ]1/2 . (33) k−1

These, however are not chain carriers; they abstract a hydrogen atom from a C2 H6 molecule

The rate of consumption of H2 is k2 [Br][H2 ], and therefore   k1 1/2 vH2 = k2 [H2 ][Br2 ]1/2 . (34) k−1

Methane (CH4 ) is, in fact, found as a minor product of the reaction. The C2 H5 radical is a chain carrier, since it decomposes into C2 H4 and H, which then abstracts a hydrogen atom from an ethane molecule:

This is also the rate of consumption of Br2 . Experimentally the reaction is, in fact, first order in H2 and one-half order in Br2 . Much work has shown this mechanism to be almost certainly close to the truth. Many organic decompositions in the gas phase occur by composite mechanisms involving the participation of atoms and free radicals, and in 1934 F. O. Rice (b. 1890) and K. F. Herzfeld (1892–1978) proposed several types of chain mechanisms to explain the kinetic behavior observed. The acetaldehyde decomposition appears to occur largely by a mechanism that can be simplified to CH3 CHO → CH3 + CHO

(1) (2)

CH3 + CH3 CHO → CH4 + CH3 CO

(3) (4)

H + C2 H6 → H2 + C2 H5 . In this pair of chain-propagating steps the major products of the reaction, ethylene (C2 H4 ) and hydrogen, are formed without any loss of chain carriers. There is a special class of reactions in which there is chain branching, a concept suggested in 1927–1928 by N. N. Semenov (1896–1946) and by C. N. Hinshelwood. When a pair of ordinary chain-propagating steps occurs there is no change in the number of chain carries. When there is chain branching, however, there is an increase in the number of carriers. An example is the pair of reactions O + H2 → OH + H.

2CH3 → C2 H6 .

The radical CHO produced in reaction (1), the initiation reaction, breaks down into CO + H, and H atoms combine to form H2 which is found as a minor product. The C2 H6 formed in the termination reaction is also a minor product. Reactions (2) and (3) are chain-propagating steps. The steady-state equations are now, for CH3 : k1 [CH3 CHO] − k2 [CH3 ][CH3 CHO] + k3 [CH3 CO] (35)

and for CH3 CO: k2 [CH3 ][CH3 CHO] − k3 [CH3 CO] = 0.

C2 H5 → C2 H4 + H

H + O2 → OH + O

CH3 CO → CH3 + CO

−k4 [CH3 ]2 = 0

CH3 + C2 H6 → CH4 + C2 H5 .

(36)

In each of these reactions two carriers have been formed from one; if the two reactions are added together the result is H + O2 + H2 → 2OH + H, so that the H atom is regenerated but has produced two OH radicals which can undergo further reactions. When such chain branching occurs the number of chain carriers in the reaction can increase extremely rapidly. The rate of reaction thus increases, and the result may be an explosion. Mixtures of oxygen with substances such as hydrogen, carbon monoxide, and various organic substances do in

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194 fact explode under certain conditions, and the reason is that chain branching is involved. Reactions in which substances react together with the formation of much larger molecules known as polymers are frequently chain reactions. Thus, a substance containing a carbon–carbon double bond may be caused to polymerize by introduction of a free radical R which adds on to the double-bonded compound to form another free radical:

The resulting free radical adds on to another monomer molecule and the process continues with the eventual formation of a much larger molecule.

X. PHOTOCHEMICAL AND RADIATION-CHEMICAL REACTIONS There are two kinds of radiation, electromagnetic and particle. The former behaves in some experiments as if it were a beam of photons, but these have no mass and can be regarded as packets of energy. Particle radiation, on the other hand, consists of beams of particles having mass. Cathode rays and β radiation, for example, consist of beams of electrons, and α radiation consists of a beam of helium nuclei. Particle-generating machines such as cyclotrons produce beams of protons, deuterons, and other particles. All of these types of radiation are capable of bringing about chemical reaction provided that the energies are sufficiently high. Reactions induced in this way are referred to as either photo-chemical or radiation-chemical reactions. The distinction between the two types is not sharp and is sometimes made on the basis of whether ions are produced in the reaction. With radiation of lower energies, such as electromagnetic radiation in the visible and near-ultraviolet regions of the spectrum, there is no ion formation, and the resulting reaction is referred to as photo-chemical. With ultraviolet radiation of higher frequencies, with X-rays and γ -rays, and with high-energy particle radiation, ions are usually formed, and the process is then described as radiation-chemical.1 Sometimes a distinction is based on specificity. With radiation of lower energy the resulting reaction is often of simple stoichiometry and can then be called photochemical. Higher-energy radiation, however, breaks up molecules into a number of fragments which lead to a variety of products; the term radiation-chemical can then be applied. 1 They should not be called “radiochemical,” which would cause confusion with radiochemistry, which is concerned with radioactive substances.

Kinetics (Chemistry)

In order for a photochemical reaction to occur the radiation must be absorbed, and with the advent of the quantum theory it became possible to understand the relationship between the amount of radiation absorbed and the extent of the chemical change that occurs. It was first realized by A. Einstein (1879–1955) that electromagnetic radiation can be regarded as a beam of particles, which G. N. Lewis (1875–1940) later called photons: each of these particles has an energy equal to hν, where ν is the frequency of the radiation and h is the Planck constant. In 1911 J. Stark (1874–1957) and independently in 1912 Einstein proposed that one photon of radiation is absorbed by one molecule. This relationship, usually referred to as Einstein’s Law of Photochemical Equivalence, applies satisfactorily to electromagnetic radiation of ordinary intensities but fails for lasers of very high intensity. The lifetime of a molecule that has absorbed a photon is usually less than about 10−8 sec, and with ordinary radiation it is unlikely for a molecule that has absorbed one photon to absorb another before it has become deactivated. In these circumstances there is therefore a one-to-one relationship between the number of photons absorbed and the number of excited molecules produced. Because of the high intensity of lasers, however, a molecule sometimes absorbs two or more photons, and one then speaks of multiphoton excitation. Even with ordinary radiation it may appear that the law of photochemical equivalence is not obeyed, and this arises because of two factors. One is that a molecule that has absorbed a photon may become inactivated before it has had time to enter into reaction. When this alone is the case the ratio of the number of molecules undergoing reaction to the number of photons absorbed (a quantity known as quantum yield or photon yield) is less than unity. The second factor is that the reaction may occur by a composite mechanism. For example, in the photochemical decomposition of hydrogen iodide (HI) into hydrogen and iodine, the quantum yield is 2; that is, one photon brings about the decomposition of two molecules of hydrogen iodide. The reason is that the mechanism is (1)

HI + hν → H + I

(2)

H + HI → H2 + I

(3)

I + I → I2 .

In the first step one photon, designated by its energy hν, breaks apart an HI molecule, and the H atom produced interacts with a second HI molecule [reaction (2)]. The sum of reactions (1), (2), and (3) is 2HI + hν → H2 + I2 , which explains the quantum yield of 2. This mechanism is not a chain reaction, since no cycle of reactions is repeated. When chain processes are

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involved the quantum yields may be very large. In the photochemical reaction between hydrogen and chlorine, quantum yields of over 106 have been reported, and it was this fact that led W. Nernst (1864–1941) to propose, in 1918, the following chain mechanism for the reaction: (1)

Cl2 + hν → 2Cl

(2)

Cl + H2 → HCl + H

(3)

H + Cl2 → HCl + Cl

(4)

2Cl → Cl2 .

Reactions (2) and (3) are chain-propagating steps, and since they occur rapidly they are repeated many times, leading to high quantum yields. One important photochemical technique is flash photolysis, in which an intense flash of radiation of very short duration initiates a reaction by producing excited molecules, atoms, and free radicals. The method, first used in 1950 by G. Porter and R. G. W. Norrish (1887–1978), has been applied extensively to the study of fast reactions, both in the gas phase and in solution. Extremely rapid processes can be studied by the use of pulsed lasers, which can have a duration of less than 1 psec (10−12 sec). This is sufficiently short to allow the study of the fastest of chemical reactions, and even molecular relaxation process in which molecules are changing their vibrational and rotational states. Some molecules do not absorb radiation at convenient wavelengths, and the technique of photosensitization is then useful. This term applies to the absorption of radiation by a substance known as a photosensitizer, which then transfers energy to a colliding molecule, causing it to undergo chemical change. For example, hydrogen does not absorb in the near ultraviolet, but mercury vapor does so at a wavelength of 253.7 nm which corresponds to an energy of 469.4 kJ mol−1 . If hydrogen saturated with mercury vapor is irradiated at this wavelength, the mercury atoms, normally in a 61 S0 state, are excited to the 63 P1 state:     Hg 61 S0 + hν → Hg 63 P1 . On collision with a hydrogen molecule an excited Hg atom can bring about dissociation into atoms:     Hg 63 P1 + H2 → Hg 61 S0 + 2H. The dissociation of a mole of H2 atoms requires 432 kJ of energy, so that the excited Hg atoms have more than enough energy for this process to occur. Other molecules, such as hydrocarbons, can be decomposed by similar photosensitization processes. Radiation-chemical reactions have mechanisms similar to photochemical reactions, the difference being in the nature of the radiation-chemical primary process. Unlike photochemical primary processes, these processes are usually composite, leading in a series of steps to atoms,

ions, and free radicals which undergo further reactions. For example, if hydrogen is irradiated with α particles the overall primary process is largely the production of hydrogen atoms, and the process is usually written as α

H2
2H. Several elementary processes lead to this dissociation. One is the ejection of an electron by the α particle, the hydrogen molecule becoming an H+ 2 ion which dissociates into H + H+ : − α + H 2 → H+ 2 +α+e + H+ 2 → H + H.

The H+ ion produced is then likely to pick up an electron with the formation of a hydrogen atom. In radiation chemistry the analog of flash photolysis is pulse radiolysis. For example, linear electron accelerators have been used to give pulses of very high energy with durations of a few microseconds, and with special techniques much shorter pulses, of the order of a nanosecond (10−9 sec), have been achieved. The term chemiluminescence is used to describe the radiation emitted as a result of a chemical reaction. Some of the radiation emitted by flames is produced in this way, although some of it is blackbody radiation resulting from the high temperature of the burnt gases. The radiation emitted by fireflies and by certain tropical fish is chemiluminescence.

XI. HOMOGENEOUS CATALYSIS Catalysis at surfaces, or heterogeneous catalysis, has been outlined in Section VIII, and the present section deals with homogeneous catalysis, where only one phase is involved. Substances whose reactions undergo homogeneous catalysis are commonly known as substrates. There are several different types of homogeneous catalysis. Examples in the gas phase sometimes involve chain mechanisms. Reactions in solution are commonly catalyzed by acids and bases, and many reactions in biological systems are catalyzed by enzymes, which are proteins. Reactions in aqueous solution are often catalyzed by ions of variable valency. No single mechanistic pattern applies to all cases of catalysis, but several different types of catalysis occur according to the following scheme: C+S

X+Y X + W → P + Z. Here C is the catalyst and S the substrate, while X is a reaction intermediate which forms the product P in the second step. The species Y, W, and Z undergo further

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processes that do not affect the kinetics. In the case of a unimolecular surface-catalyzed reaction the scheme is simplified to: C+S

X

The protonated acetone then gives up a different proton to a basic species B present in solution:

X → P. The intermediate X is now an addition compound formed from C and S, and in the second step it forms products. This scheme also applies to many reactions catalyzed by enzymes. For reactions catalyzed by acids and bases, however, it is necessary to include the additional species Y, W, and Z. Suppose, for example, that the reaction of a substrate S is being catalyzed by an acid HA. A typical mechanism is HA + S

SH+ + A− SH+ + B → P + BH+ . In the first step a proton H+ is transferred to the substrate, with the formation of the intermediate SH+ . In the second step this intermediate SH+ transfers a proton to a basic species B (which may be identical with A− or may be a water molecule), and in this step the product P is formed. For example, the reaction between acetone and iodine, CH3 COCH3 + I3 → CH3 COCH2 I + HI, is catalyzed by acids, and the rate is proportional to the acetone concentration and also to the concentration of acid. The rate is, however, independent of the concentration of iodine, and indeed the rate remains the same if iodine is replaced by bromine. This suggests that the slow and ratedetermining step does not involve iodine at all, but is the conversion of the ordinary form of acetone (known as the keto form) into another form (the enol form) which can react with iodine or bromine very rapidly:

The way in which an acid HA catalyzes the formation of the enol form is believed to be as follows. First the acid transfers a proton to the oxygen atom on the acetone molecule:

This is a very common pattern in acid catalysis. In the first step the acid transfers a proton to the substrate, which in the second step transfers another proton, the product being formed either simultaneously or in a subsequent step. In basic catalysis (which also occurs with the iodination of acetone), the substrate molecule first transfers a proton to the basic catalyst, and in a second step accepts a proton at another position. Sometimes when reactions are catalyzed by acids and bases there is little sign of any effect other than catalysis by hydrated hydrogen ions (usually written as H3 O+ ) or by hydroxide ions. One then speaks of specific acid–base catalysis, and the rate of reaction might be of the form   v = k0 [S] + kH+ [S] H3 O+ + kOH− [S][OH− ]. (39) The coefficient k0 relates to any uncatalyzed reaction, and kH+ and kOH− are catalytic coefficients. It is sometimes found that acidic species other than hydrogen ions and basic species other than hydroxide ions are capable of catalyzing reactions. One then speaks of general acid–base catalysis. For example, in an aqueous solution containing acetic acid and sodium acetate the rate coefficient k(=v/[S]) can be expressed as   k = k0 + kH+ H3 O+ + kOH− [OH− ] + kHA [HA] + kA− [A− ],

(40)

where HA is acetic acid and kHA the corresponding catalytic coefficient; A− is the acetate ion and kA− its catalytic coefficient. If an acid has a large dissociation constant, for its dissociation into ions, it tends to have a large catalytic coefficient; similarly the catalytic strength of a base is greater the larger the base dissociation constant. In 1924 J. N. Brønsted proposed the following relationship between ka , the catalytic coefficient of an acid, and K a , its acid dissociation constant: ka = G a K aα ,

(41)

where G a and α are constants. A similar relationship applies to a base catalyst. Catalysis by enzymes, the biological catalysts, is much more specific than that by acids and bases. Some enzymes show absolute specificity, in that they are only known to be able to catalyze a single chemical reaction. Others show a lower specificity, in that they are able to catalyze only a

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certain type of reaction, such as ester hydrolysis. Enzymes commonly show stereochemical specificity, being capable of catalyzing a reaction of one stereochemical form of a substance and not the other. All enzymes are proteins, and they are often associated with nonprotein substances, known as co-enzymes or prosthetic groups. Some enzymes require the presence of certain metal ions. The catalytic action of an enzyme appears to involve only a small region of the enzyme surface, known as the active center. The mechanism of action of enzymes is similar in some respects to that of acids and bases but shows some important differences. One difference relates to the influence of the pH of the solution. If there is catalysis by acid and base the rate passes through a minimum as the pH is varied, because there is acid catalysis at low pH and base catalysis at high pH. With enzymes the rate passes through a maximum as the pH is varied. This is explained as due to the presence of at least two ionizing groups at the active center. For catalysis to be possible one of these groups must be in its acidic form and the other in its basic form, and this combination can only occur at intermediate pH values. The rates of enzyme-catalyzed reactions also pass through a maximum as the temperature is varied. The increase at lower temperatures is the normal effect, due to there being an activation energy for the enzyme-catalyzed reaction. Enzymes, however, like all proteins, become denatured at higher temperatures and lose their catalytic activity. Many patterns of behavior are found for the effect of substrate concentration on the rate of an enzyme-catalyzed reaction. For a reaction involving a single substrate the simplest behavior, described as Michaelis–Menten kinetics, is when there is a hyperbolic relationship between the rate v and the substrate concentration [S]: V [S] v= K m + [S]

(42)

Here K m is called the Michaelis constant and V is the limiting rate at high substrate concentrations. Figure 5 illustrates this behavior and notes that K m is equal to the concentration at which v is equal to V /2. This type of kinetics was explained by L. Michaelis (1875–1949) and M. L. Menten (1878–1960) in terms of the formation of an addition complex between the enzyme and the substrate, the complex breaking down in a second step into products with release of the enzyme: E+S

ES ES → E + P Under usual conditions the substrate is greatly in excess of the enzyme, so that the concentration of free enzyme [E]

FIGURE 5 Variation of rate v with substrate concentration [S] for a reaction obeying Michaelis–Menten kinetics [Eq. (42)].

may be less than the total concentration [E]0 , since some enzyme has been converted into the enzyme–substrate complex ES; thus [E]0 = [E] + [ES]

(43)

Application of the steady-state treatment to the reaction scheme gives k1 [E][S] − k−1 [ES] − k2 [ES] = 0 and elimination of [E] through Eq. (43) gives   k−1 + k2 [E]0 = [ES] +1 k1 [S]

(44)

(45)

The rate is k2 [ES], and thus v=

k2 [E]0 k−1 +k2 + k1 [S]

1

(46)

or v=

k2 [E]0 [S] k−1 +k2 + [S] k1

(47)

This is equivalent to Eq. (42) with V = k2 [E]0 and K m = (k−1 + k2 )/k1 . This is the simplest mechanism leading to Michaelis– Menten kinetics, but many other mechanisms also do so. To investigate mechanisms in more detail it is necessary to use special techniques for studying very rapid reactions, such as the stopped-flow and temperature-jump methods (Section I). Various factors give deviations from Michaelis–Menten kinetics. For example, sometimes the complex ES adds on an additional substrate molecule to give ES2 ; if this does not react as rapidly as ES there is a falling off of the rate at higher substrate concentrations. Many enzyme-catalyzed reactions involve two substrates, one of which for example may oxidize the other. In such cases several mechanisms have been identified. Sometimes one substrate A first forms a complex EA with

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198 the enzyme, and EA then reacts with the other substrate B to form a ternary complex EAB which then gives products: this is the ordered ternary complex mechanism. Alternatively, in the random ternary complex mechanism the ternary complex EAB may be formed either via EA or EB. In another mechanism the reaction of enzyme with one substrate A leads to one product of reaction, together with an intermediate which reacts with B to give another product. All of these mechanisms lead to similar types of kinetic behavior, and discrimination between them requires careful investigation. In recent years there has been much interest in immobilized enzymes, where the enzyme has been attached to a solid support. This has the advantage in technical work of allowing an enzyme preparation to be used many times. The study of the kinetics of immobilized enzymes is also valuable in leading to an understanding of how enzymes behave in living systems, where they are often immobilized. Kinetic studies have shown that in such systems the rate is sometimes diffusion controlled (Section VII). Other types of homogeneous catalysis do not follow a general pattern. Sometimes, for example, substances bring about catalysis in gas reactions by bringing about chain-initiation processes, or by becoming involved in chain-propagating processes. Ions of variable valency are good catalysts for certain solution reactions. For example, singly charged silver ions (Ag+ ) can bring about a reduction with the formation of Ag2+ , which then can bring about an oxidation; the Ag+ − Ag2+ system can therefore act as a mediator in oxidation–reduction systems.

Kinetics (Chemistry)

It is almost always the case that a catalyst exerts its action by reducing the activation energy, which means that it introduces the possibility of an alternative reaction path involving less energy and therefore a higher rate of reaction.

SEE ALSO THE FOLLOWING ARTICLES ATOMIC AND MOLECULAR COLLISIONS • CATALYSIS, INDUSTRIAL • CHEMICAL KINETICS, EXPERIMENTATION • ION KINETICS AND ENERGETICS • LASERS • MULTIPHOTON SPECTROSCOPY • PHARMACOKINETICS • PHOTOCHEMISTRY, MOLECULAR • POTENTIAL ENERGY SURFACES • RADIATION PHYSICS • STATISTICAL MECHANICS

BIBLIOGRAPHY Bamford, C. H., and Tipper, C. F. H., eds. (1969). “Comprehensive Chemical Kinetics,” Elsevier Publishing Co., Amsterdam. This series of many volumes, begun in 1969 and still appearing, is a valuable source of reference to all aspects of chemical kinetics. Laidler, K. J. (1987). “Chemical Kinetics,” 3rd ed., Harper & Row, New York. Laidler, K. J. (1996). “Glossary of Terms used in Chemical Kinetics, Including Reaction Dynamics,” Pure Appli. Chem., 68, 149. Pilling, M. J., and Seakins, P. W. (1995). “Reaction Kinetics,” 2nd edi., Oxford University Press, Oxford, UK. Zewail, A. H. (1990). “The birth of molecules,” Scientific American, November, p. 76.

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Membranes, Synthetic (Chemistry) Michael E. Starzak State University of New York, Binghamton

I. II. III. IV.

Membranes Membrane Modifiers Membrane Characterization Membrane Phenomena

GLOSSARY Amphipathic Having both hydrophilic and hydrophobic regions in distinct regions of the same molecule. Ionophore An organic molecule that binds a polar ion to create a membrane soluble complex. Casting Solvent is evaporated from solvated membrane material on a support to produce a membrane. Permeability coefficient The proportionality factor with units of velocity that relates concentration gradient to flux through a membrane. Selectivity Membrane ability to pass certain species while rejecting others. Ultrafiltration Water permeation through a membrane— reverse osmosis.

A MEMBRANE is a special structure or phase that separates two macroscopic phases. The membrane normally forms a thin barrier between the macroscopic phases so that its total volume is small relative to the separated phases. Although the membrane is generally insoluble in

either of the two bulk phases, its utility often arises from its selectivity, i.e., its ability to facilitate the transport of specific materials between the two bulk phases. The net flux through the membrane is characterized by a permeability coefficient P with units of velocity. The net flux of a material through the membrane is often proportional to this permeability coefficient and the concentration gradient of permeant between the two bulk phases. Naturally occurring materials like β-alumina can conduct ions like Na+ , Ag+ and H+ , and water. Sections of this material can then be used as membranes to regulate the flow of these ions without modification of the basic solid structure. More often, the goal is the preparation of synthetic membranes with specific properties that permit the membrane to be used in a particular application. For example, a synthetic membrane permeable only to water would be useful for reverse osmosis where pure water is separated from an aqueous solution. The state of the art in membrane research is often defined by methods of preparation that give a membrane with the qualities desired. A membrane with few desirable permeability properties can also be modified by adding molecules that impart

345

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346 the proper selectivity or permeability. The membrane surface might be modified to produce a useful membrane. Special molecules might be added to or formed with the membrane to produce the desired properties. For example, special protein channels can be inserted into membranes to produce a membrane selective to a specific ion. The membrane is then permeable only to this species. In most cases, the ultimate goal is the development of a selective membrane optimal permeability or adsorption. Membranes are available for a large number of research and industrial applications including gas separation, fuel cells, reverse osmosis, dialysis, sensors, and purification. They also serve as the support for special processes like the oxidation reduction processes in photosynthesis or cellcell communication. The synthesis and production of membranes with structures that control material flow are presented in Section I. Section II surveys special molecules incorporated into membranes to impart specific properties to these membranes. Section III describes some techniques and instrumentation used to characterize membranes and pore structure. Section IV illustrates some special equilibrium and kinetic physical phenomena most conveniently observed with membrane systems.

I. MEMBRANES A. Monolayer Membranes Biological systems are protected by membranes with molecular structures that serve as models to illustrate the structure and behavior of more complicated membranes. Monolayer membranes are made from amphipathic molecules that have a polar, water-soluble, or hydrophilic terminal region and a hydrocarbon, nonpolar or hydrophobic, region that usually contains long hydrocarbon chains whose length dictates membrane thickness and stability. Hydrocarbon chains of 16–20 carbons generate a self-stabilizing hydrocarbon phase that is insoluble in water. Soap molecules with a weak acid terminal group or phospholipids with two hydrocarbon chains and a polar phosphate-centered head group united on a glycerol base can spontaneously form membranes. If these amphipathic molecules are spread on a water surface, the polar groups dissolve in water to “root” the molecules. Since the polar heads are insoluble in hydrocarbon, the molecules normally spread to form a monolayer membrane on the surface of the water. In this state, the long hydrocarbon chains can be parallel to the water surface to create an extended, random monolayer. A more-ordered, liquid crystalline monolayer is generated by applying surface pressure to the monolayer. A surface balance has floating barriers that move to compress the

Membranes, Synthetic (Chemistry)

surface layer toward a high pressure limit where the hydrocarbon chains are forced parallel and normal to the surface. This more ordered monolayer has some of the properties of a liquid-crystal. The hydrocarbon chains are parallel but observation of the top surface of the monolayer would reveal a random liquidlike structure. This ordered monolayer is used as an air/solution barrier or as a medium for selective extraction of organic material from the aqueous phase. The ordered monolayer also serves as the building block multilayer membranes. A glass slide or other support is passed through the monolayer. With each pass, a monolayer membrane is transferred from the surface layer to the support. If done properly, this monolayer is stable and is not disturbed as additional monolayers are added with additional passes through the ordered surface monolayer. A biological membrane separates two aqueous phases. The outer surfaces of the membrane must be hydrophilic to stabilize the two membrane/solution interfaces. This is accomplished with a bilayer membrane, in which the ordered hydrocarbon chains from two monolayers merge to form a stable central hydrocarbon region approximately twice the length of the hydrocarbon chains. The opposed polar head groups form outer hydrophilic surfaces to stabilize the structure and the solution/membrane interfaces (Fig. 1). The bilayer structure, which is the most common membrane in biological systems, was originally revealed by experiments that first determined the total membrane area in a specific number of red blood cells. The cells were then destroyed, phosopholipid was separated and spread on a surface balance. The area occupied by the lipid on the surface balance was exactly twice the total surface area of the red blood cells. A lipid or soap membrane of large area is inherently unstable. A soap bubble is a good example. Research membranes are often formed over very small pinholes by applying solutions of the membrane material using a

FIGURE 1 Ion channels in the plasma membrane. The membrane phospholipids are arranged in a bimolecular layers with their polar heads on the outside and their hydrophobic tails inside. The bilayer is about 30 A˚ thick. Sitting in it are various intrinsic proteins, including the channels shown here.

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pipette or brush. Bilayers are also created as hemispherical bubbles at the end of a Teflon™ tube. Pressure is applied to the solution phase within the tube to expand membrane material painted across the tube outlet into a hemispherical bubble. In each case, solvent moves toward the support walls to form a Gibbs plateau, a ring of solvent and membrane surrounding the bilayer membrane. With the proper concentrations, purity, and overall cleanliness, such membranes are quite stable. However, a biological cell must remain stable for the life of a cell. Biological lipid membranes are believed to act as stabilizing filler between the proteins scattered throughout the membrane. The proteins provide the support while restricting the area needed for an integral bilayer. Larger synthetic bilayers might be stabilized using membrane molecules that crosslink to form a more durable structure. Solvent-free bilayer membranes are formed by forming monolayers on water surfaces on either side of a Teflon™ sheet containing a pinhole. As the water (and monolayer) level is raised slowly past the pinhole, monolayers from opposite sides merge to form the solvent-free bilayer across the pinhole. A vesicle or liposome is a spherical bilayer that encloses solution and separates it from an external solution. They might be considered synthetic cells with integral bilayer membranes. Vesicles form spontaneously when a solution is agitated with ultrasound. Vesicles formed in this manner are called small, unilamellar vesicles, (SUV). Larger vesicles (large unilamellar vesicles, LUV) are formed by repeated freeze thaw cycles of a lipid solution. Vesicles of a specific diameter are formed by high-pressure extrusion of the lipid solution through polycarbonate membranes. The membrane pores must be homogeneous of relatively constant diameter. A vesicle can fuse with another vesicle or a planar bilayer to create a new homogeneous membrane. Lipid bilayer fusion is often mediated by a polyvalent ion like Ca2+ which reduces the Debye lengths of the charged double layer to permit the two membranes to reach a separation distance where attractive forces can mediate the fusion. Membrane fusion might facilitate the transfer of drugs in the vesicle interior to biological cell. It might also permit drug delivery to specific cells or act as a time-release transfer mechanism. The molecules within a bilayer can aggregate into different phases with different degrees of order. Phase formation has been studied using fluorescence imaging.

reagents, often enhances reaction rates. Zeolites with parallel pores suitable for transport are formed by allowing the zeolite structure to develop around regularly spaced organic molecules that function as templates. The organic material is then purged to produce membrane with regular, parallel pores. Pore diameters can be controlled accurately by the choice of template molecule. C. Silica Membranes Many glass, i.e., silicon dioxide, membranes have some intrinsic porosity and selectivity. The pH “glass” electrode selectively absorbs H+ ions to give a potential proportional to the logarithm of the H+ concentration. Other membranes have been developed as ion-specific electrodes where the membrane is selectively permeable to ions to give a potential difference proportional to the concentration of that ion. Sensors function in a similar manner. A membrane is formed directly on an electrode support so ions or solute reaching the sensor can induce changes at the support electrode that can be detected electrically. The basic principle remains the same; the membrane facilitates transport of a selected material between two bulk phases and the transport is reduced to some observable signal that can be measured to establish the presence and concentration of the species selected. Glass membranes can be formed with small pores that permit separation of hydrogen from other gases. For example, the naturally occurring pores in Corning’s Vycor glass are filled with additional silicon dioxide to produce pores of a size suitable for gas separation using chemical vapor deposition. SiH4 and oxygen gas, flowing across opposite surfaces, diffuse into the pores and react to create SiO2 with a structure containing pores suitable for the gas separation. Colloidal silica can also be cast, i.e., aggregated on a suitable surface, to form silica membranes. The colloids aggregate into sols on the substrate to form protomembranes that can be calcined to stable, gas-permeable glass membranes. D. Capillary Membranes Silica membranes with homogeneous pores are also made by heating and drawing bundles of capillaries. As the capillary length increases, the inner capillary diameter decreases. The process is repeated to create capillaries with the appropriate inside diameter. The bundles are then cut laterally to produce planar membranes.

B. Zeolite Membranes Zeolites are aluminum-silicon compounds that form structures with random pores and cavities. The environment within the cavities, which can be selective for specific

E. Chemical Etching Homogeneous channels with a narrow size distribution are created by irradiating 10–20 µm polymer films such

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348 as polyethylene terephthalate with heavy ions like Ar+9 produced by a cyclotron. The ions create damage tracks that can be expanded by chemical etching to pores of the proper diameter. The density of pores is dictated by the time and intensity or radiation while the pore diameter is determined by the etching time. F. X-Ray Lithography A surface covered with a photoresist and a gold mask can be irradiated with high-intensity, well-collimated X-rays that degrade the exposed regions of the polymer. The degraded polymer is removed to create a primary template. Galvanoforming produces a secondary metallic template that is used as a mold for a polymer membrane with homogenous pores (Fig. 2). G. Cast Dense Membranes Polymer membranes are also formed by casting. The organic polymer that will constitute the membrane is dissolved in a solvent and spread on a support. The solvent evaporates leaving an integral membrane with the requisite properties. Membranes are also formed without solvent by spreading polymer melts on the surface to leave a membrane when the system cools to ambient temperature. The final membrane structure is controlled by factors like

Membranes, Synthetic (Chemistry)

the cooling rate, polymer solution concentration, and the composition of the polymer or polymers selected. Hollow fiber (tubular) membranes are made by spinning or extruding the polymers. The hollow fiber membranes provide a large surface area for transfer and are ideal for separations in flow systems. Membrane polymers include polypropylene, poly (vinylidene difluoride), polysulfone, poly(ether sulfone), poly(ether ether ketone), polyvinyl alcohol, polyacrylonitrile, polycarbonate, and poly(ethylene terephthalate). These “tortuous pore” membranes resemble porous sponges where permeant moves through a convoluted path as it passes between bulk phases. Membranes are considered dense when they have less than 50% void volume. However, even dense membranes can be relatively porous and might require additional treatment to produce a more selective membrane. The membrane characteristics are often altered by treating the membrane surface to produce a region with the requisite properties. The bulk of the membrane then functions simply as a porous support. The properties of the membrane skin might depend on the composition of the casting mixture. The surface might be modified by high-temperature annealing or by plasma treatment, where a discharge in an oxygen atmosphere produces the desired changes in membrane properties. The surface might also be modified by covalently grafting new molecules with the desired properties to the surface. All such treatments attack the surface rather than the interior support region of the dense membranes. H. Porous Membranes A membrane is classified as porous when the void volumes within the membrane exceed 50% of the membrane volume. The porous membranes can be generated from the dense membranes by addition of a solvent that induces swelling. The solvent can be evaporated from some membranes to leave large, stable voids within the porous membrane. The surfaces of these porous membranes can also be modified to produce a skin with the desired selectivity and permeability properties. The fibrous membranes are porous membranes based on materials derived from cellulose. These fibrous membranes often have pores that permit small biological entities like small cells or viruses from aqueous solution. Cellulose-based materials include cellulose acetate and cellophane. I. Liquid Membranes

FIGURE 2 Schematic diagram showing the preparation of a metal structure by galvanoforming.

A membrane can be formed by inserting a thin layer of insoluble liquid between two phases. This liquid membrane might be flowed onto one aqueous phase. The second

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aqueous phase is then carefully flowed onto the membrane phase to complete the system. A porous membrane might also be impregnated with the membrane liquid to function as a support for this liquid membrane. Solutes that dissolve in the membrane are absorbed and transported between the bulk phases if there is a concentration gradient. J. Ion Exchange Membranes The beads of an ion exchange resin have fixed ionized sites of common charge. The sites retain ions of opposite charge for variable times to permit a separation in the eluent. The classical example is the separation of the lanthanide series of ions with very similar chemical properties. An ion exchange membrane must have pores that contain fixed surface charge. This fixed charge attracts counterions into the pores for transport while rejecting co-ions, ions with the same charge as those in the surface. The ion exchange membranes are made for modifying polymers to have ionizable groups that can function as the fixed charge. Anionic fixed charge is normally introduced with sulfonic acid or carboxylic acid groups. Cationic fixed charge for an anion exchange membrane uses positive groups like alkyl ammonium. For example, Nafion 117 has a polytetrafluoroethylene backbone with regularly spaced fluorovinyl ether pendant side chains that terminate in a sulfonic acid. The concentration of fixed charge is high and, when the pores and channels within the membrane are filled with water, the system functions as a high concentration electrolytic system. The ion exchange membranes are used in systems where ions of specific charge must be transported. They are used as separators in fuel cells and for electrodialysis where ions must move between phases.

and solute with equal facility, i.e., the solution flows through the membrane. Semipermeable membranes are used for reverse osmosis where an applied pressure selectively forces water from a solution through the membrane to separate it from solute. Strong synthetic membranes are required for effective, i.e., high pressure, purification. Membranes with pores that permit a flow of small solute molecules and ions as well as water can separate these species from much larger protein or polymer molecules. Dialysis membranes with this property are used to purge small ions from a protein solution, leaving a protein with neutralized charge sites. The neutralized semipermeable membrane can be dialyzed with a semipermeable membrane where water is transported from the protein solution to a concentrated solution leaving a more concentrated protein solution. L. Gold-Sulfur Bonded Membranes Metal surfaces are difficult to clean to produce surfaces as supports for homogeneous membranes. Patches of oxidized surface can reject membrane binding to produce gaps in the membrane barrier to spoil the effectiveness of the membrane. Membrane monolayers can be bound to intrinsically clean gold surfaces using a gold-sulfur linkage. Long hydrocarbon chains with a terminal sulfur bind densely and homogeneously on the fold surface to produce an integral membrane. Since the gold substrate functions as an electrode, the combination is particularly suited to the development of chemical sensors. The monolayer is impregnated with molecules that permit selected species to react to the electrode for oxidation or reduction and, through those chemical changes, detection.

K. Semipermeable Membranes Semipermeable membranes that transport only water while blocking solute flow were discovered by Nollet 200 years ago. They are relatively common in biological systems. For example, frog skin is often used as a semipermeable membrane. Synthetic membranes such as cellophane and membranes made with polyvinyl alcohol, polyurethane, and polytrifluorochloroethylene selectively transmit water. Their ability to transport water is quantified with their ultrafiltration coefficient, the proportionality constant that relates the concentration gradient to the net water flux through the membrane. The reflection coefficient measures the relative selectivity of the membrane for water and solute. A reflection coefficient of one represents the ideal semipermeable membrane where all solute is rejected and only water flows through the membrane. A reflection coefficient of zero signifies a membrane with pores that permit the flow of both solvent

II. MEMBRANE MODIFIERS While both selectivity and high permeability are characteristic of an ideal membrane, it might be worthwhile to sacrifice high permeability for excellent selectivity. This condition can be reached by impregnating an impermeable membrane with molecules that provide highly selective pathways through the membrane. Biological evolution has produced a large number of protein molecules that are highly selective to specific chemicals. Although their densities in natural cells are often quite low, molecular biology provides the means to produce large samples of such proteins. If a large number of such highly selective proteins are incorporated into a synthetic membrane, both high selectivity and permeability are possible. The proteins transport material across the membrane by several distinct mechanisms.

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350 A. Carrier Molecules A hydrophobic membrane like a lipid bilayer blocks the passage of both metal ions and water since they are insoluble in the nonpolar central region formed from the hydrocarbon chains of the lipids. Ionic solubility is increased, however, if the metal ion is encased in organic ligands which mitigate the effects of the ion charge in the membrane. The tetraphenylborate anion, with the boron charge center surrounded by four bulky, nonpolar phenyl groups is soluble in the membrane and enters with a single negative charge. The tetraphenylarsenate cation dissolves in the membrane with a net positive charge. The ions are stabilized in sites near the membrane/solution interfaces. Because the bilayer membrane is thin, a small electrical potential in the millivolt range can produce extremely large electric fields (ca. 107 V/m) in the membrane. These fields are sufficient to induce a transition in which the ion passes through the hydrophobic central region to a site at the opposite membrane/solution interface. The motion of the ions produces an observable current in an external circuit. The current decay is exponential indicating independent transitions of the ions. Membrane-soluble carrier molecules can bind an ion from aqueous solution to create a charged ion/carrier complex within the membrane. This soluble ion then moves down a concentration or electrical gradient to the opposite interface where the ion is released to that solution. Carrier transport is modeled in four kinetic steps: (1) the carrier absorbs an ion from solution 1; (2) the ion/carrier complex moves down its gradient to the opposite interface; (3) the ion dissociates from the carrier or ionophore; and (4) the carrier (without ion) diffuses back to the first interface to bind another ion and repeat the four step cycle. For each cycle, one carrier transports one ion across the hydrophobic membrane. A carrier molecule can have structure or binding sites that make it suitable to transport a specific ion or class of ions. Valinomycin, for example, binds univalent K+ ion to initiate the four-step carrier/ion cycle. A class of macrocyclic molecules, the crown ethers, absorbs ions into the central cavity of their crown-like structure to produce the membrane-soluble ion complexes. Special ionophores like lasalocid and A23187 with judiciously placed carboxyl groups are selective for polyvalent ions such as Ca (II) or lanthanide ions like Eu (III). The ionophore wraps around the ion as the carboxyl groups bind to produce the membrane soluble complex ion.

B. Channel Molecules Ions can be transported across a hydrophobic membrane if this membrane is traversed by molecules with a hollow

Membranes, Synthetic (Chemistry)

central core lined with polar groups. An ion, approaching this molecular channel, replaces some of its waters of hydration with channel binding groups so that the channel interior becomes a continuation of the aqueous solution. Transport across the membrane is then a series of transitions between the charged intrachannel binding sites until the ion is released on the opposite side of the membrane. Biological protein channels can be extremely selective. A variety of special channels exist to transport sodium, potassium, and calcium ions, the most common biological cations. Other channels exist to transport anions like carbonate and chloride. The number of known ion-selective channels has grown exponentially with the development of the patch-clamp technique. A specially polished pipette tip is brought carefully into contact with the membrane of a biological cell. Suction is applied to the pipette to hold the cell to the pipette tip. Binding occurs between the glass of the tip and the cell membrane to produce a high-resistance “gigohm” seal. The seal is extremely strong and the pipette can be retracted carrying a small section of the cell membrane across the tip. This patch membrane can be inserted into a second solution to produce a tiny area of membrane separating two aqueous phases. The small membrane area contains very few channels and the system is studied using single channel techniques which can monitor the small currents through a single open channel. The properties of the channel are deduced from the kinetic behavior of the channel. Early studies on channels were restricted to a limited number of biological systems that could be probed experimentally with internal electrodes. For example, the squid axon, the portion of the nerve cell which transmits a nervous impulse to the muscles that contract the animal’s mantle for flight, is roughly ten times larger than a human axon. Piggy-back electrodes with voltage-sensing and current-carrying electrodes glued together were inserted axially into the axon. A voltage clamp was then used to maintain a constant transmembrane potential while current through channels in the axon membrane was monitored. The patch-clamp technique has largely supplanted these more restricted membrane systems since a patch of membrane can be excised from almost any cell. The gramicidin channel, formed by dimerization of two molecules in a loose helical conformation (Fig. 3), has been studied extensively as a model system. The helical dimer spans the membrane to produce a channel through the hollow center of the helix with anionic sites or regions to facilitate cation transport. The channel transmits only univalent cations. Divalent calcium ion blocks ion flow through the channel. The channel exhibits the anomalous mole fraction effect. Both univalent thallous ion and sodium ion are very permeable when gramicidin dimers

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FIGURE 3 Molecular model of the gramicidin A channel. The two monomers are placed head-to-head with their N termini in near contact. (From Ury, D. W., Jing, T. L., Luan, C. H., and Waller, M. (1988). “Current Topics in Membranes and Transport,” 33, 51–90.)

are present. However, in mixtures of the two ions with a small Tl+ cation mole fraction, channel currents fall significantly. Depending on conditions, the thallous ion functions either as a permeant cation or a blocking ion. Channels like alamethicin form when several molecules aggregate to form a structure with a central pore. Alamethicin is a simple example of a gated channel where an applied transmembrane potential induces aggregation of the molecules to form the conducting channel. Channel formation might be viewed as the aggregation of several long rods that span the membrane. The aggregated “rods” leave a central void space that functions as a channel. Aggregation of a larger number of rods produces a larger central channel. The rod analogy has proved to be an accurate description of the more complicated biological channels. Molecular biology studies of the primary protein structure, i.e., the amino acid sequence, reveals sequences that are stable in an α-helix conformation. A channel-forming protein then consists of regions of α-helix of length roughly equal to the membrane thickness punctuated by regions of random protein chain. The protein inserts through the membrane as a series of parallel rods where one rod ultimately forms part of the pore wall (Fig. 4). The protein sections that connect the helical rods organize for different channel functions. The operational channel might consist of 4–6 of these parallel rod proteins which aggregate to form the central pore. The need for four separate channel molecules to produce an operational channel is consistent with the earliest kinetic models for a channel. Hodgkin and Huxley, in 1952, postulated a model for the potassium channel with four independent channel proteins that could change conformation on the millisecond time scale. All four channels

FIGURE 4 The arrangement of helical regions of a single protein molecule embedded in a membrane. Several such units aggregate to create an operational channel.

had to undergo the conformational change to produce an open channel. The sodium channel is transient. On excitation, three proteins change conformation to produce an open channel while a fourth protein changes conformation to initiate a subsequent closing of the channel.

III. MEMBRANE CHARACTERIZATION The effectiveness of a membrane is often determined by how well it performs the function for which it is developed. However, any membrane can be improved if the structure which results from its preparation is known. The structure can suggest changes in the mode of preparation or conditioning that enhance the overall quality. Some physical techniques used to characterize the size, shape, size distribution, and density of pores and the molecular structure of the membrane surface follow. A. Atomic Force Microscopy An atomic force microscope uses the deflection produced in a fine tip by interactions with atoms in the surface of a membrane to reconstruct the surface structure of that membrane. A piezoelectric drive moves the sample surface under the tip. The motions of the tip are converted into a three-dimensional image of the surface. Atomic force microscopy requires no surface preparation so that the membrane can be observed in its normal environment. B. Electron Microscopy Moving electrons demonstrate wave behavior. With the proper electron velocity, produced by moving the electron in an electric field, wavelengths in the nanometer range

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352 are possible. These wavelengths permit resolution of structures on the nanometer scale. Transmission electron microscopy (TEM), where the electrons penetrate through thin cross-sectional samples (50 nm) that have been freezedried and sectioned, uses the wave interference patterns to establish the structure of the sample. Scanning electron microscopy (SEM) utilizes scattered electrons to generate the image and has higher resolution than TEM. However, both techniques require sample preparation which might alter the structure of the sample under observation. In addition, the high voltages required to accelerate the electrons to the proper wavelength can also cause damage to the surfaces.

Membranes, Synthetic (Chemistry)

brane capacitance. The capacitance then determines the membrane thickness if the area of the membrane is known. If the membrane under study also contains channels or pores that transmit ions, the current also contains an ohmic component and the electrical properties of the membrane are modeled as a parallel resistor-capacitor combination. Liposomes or vesicles in solution can be modeled as nonconducting spheres embedded in a conductive medium. The measured capacity of the ensemble of vesicles is converted to a capacitance per unit area of membrane surface which, in turn, determines the membrane thickness.

IV. MEMBRANE PHENOMENA C. Computer-Aided Image Analysis An optical microscope can be used to observe surfaces if it is interfaced with a computer system that can be used to enhance images of the surface. The computer corrects for the image degradation that occurs when the light wavelength is comparable to the dimensions of the object being measured. The computer system resolves the light image by computationally removing the interference effects. D. The Bubble Point Method The size of pores in a membrane can be established by forcing liquid through the membrane under pressure. The time for the appearance of the bubble determines the largest pore size in the membrane. The pressure necessary to produce a bubble can be reduced by using liquid mixtures that minimize the surface tension within the pore. The surface tension is lower if mercury is used as the bubble-forming solvent. In mercury porosimetry, however, some mercury might be retained within the pores to cancel some advantages of this liquid. E. Other Techniques A number of related techniques use adsorbance of a gas or solid at a membrane surface to determine the total surface area. Adsorption techniques can be used to determine the density and size of the pores within the membrane as well. Planar bilayer membranes are characterized by their electrical response since the insulating bilayer membrane and the two conducting ionic solutions are electrically equivalent to a capacitor with the membrane as the dielectric. The current through a capacitor is directly proportional to the rate of change of the voltage on the capacitor, i = C d V /dt. The capacitance, in turn, is related to the thickness of the membrane and its dielectric constant. The membrane capacitance is determined by applying a ramp potential with a constant d V /dt across the membrane to give a constant current that can be converted to the mem-

Membranes display some physical phenomena that might not normally be observed in bulk phases. A. Equilibrium Phenomena 1. Osmotic Pressure A semipermeable membrane that permits only water flow facilitates the flow of water from a higher to lower concentration, e.g., from pure water to a saline solution. The process could continue indefinitely unless the driving force for this flow is counterbalanced with an equal and opposite force. For osmotic pressure equilibrium, the hydrostatic pressure head resulting from the water flow will exactly balance the concentration gradient driving force of the water. To a good approximation, the osmotic pressure is directly proportional to the molar concentration of the solution when opposed by a bath with pure water and can be used to determine the molar concentration of the solution. If the solute weight and solution volume are known, the molecular weight of the solute can be determined. The hydrostatic pressure can be increased to reverse the flow of water so that it flows from the solution phase to the pure water phase. This reverse osmosis or ultrafiltration provides a convenient way to purify water. 2. The Donnan Equilibrium If a membrane is permeable to small cations and anions, the presence of a large membrane-impermeable molecule with counterions can produce a salt concentration gradient across the membrane. For a salt C+ A− , the Donnan equilibrium requires that the product of cation and anion concentrations in each bulk phase be equal [C+ ]1 [A− ]1 = [C+ ]2 [A− ]2 If an impermeable anion, P z− with z cations C+ is present on side 1, the total concentration of C+ in that bath increases and CA must move through the membrane to the

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opposite bulk phase to reestablish the equilibrium condition. The net effect is a reduction of salt CA in Bath 1, the solution with the protein, and a CA concentration gradient across the membrane. 3. The Electrochemical Potential An ion exchange membrane that can pass ions of one polarity while rejecting those of opposite polarity will produce an electrical potential to balance the permeable ion concentration gradient. For example, channels with anion charge will encourage the passage of cation while repelling anion to produce a charge separation and electrical potential V = V2 − V1 = −(RT )/(z F) ln[c2 /c1 ] where RT /F ∼ 25 mV and z is the charge on the permeant ion. A strong, cation permeable membrane might tap the the Na+ concentration gradient as a fresh water river reaches the ocean to produce electrical power.

because the fixed negative charge in membrane channels excludes anions from the pores so that only cations move with the electrial potential gradient; the flow of anions in the opposite direction is reduced significantly by the fixed charge. The waters of hydration on the moving cations are carried through the pores as well to produce the observed water flow that characterizes electroosmosis. 3. Streaming Potential and Current In electroosmosis, an applied voltage produces a flow of neutral water, i.e., the ion flow and water flow are coupled. The reciprocal process occurs when solution is forced through the membrane with charged pores under a hydrostatic pressure. Ions of charge opposite to that fixed in the pore walls are carried along with the water while the co-ions are repelled from the pores. The flow of one type of charge produces a net current through the channels. This streaming current would appear in a closed electrical circuit. If the electrical circuit is open, a streaming potential appears across the pore or channel. 4. Oscillations

B. Dynamic Phenomena 1. Ultrafiltration A water concentration gradient or an externally applied hydrostatic pressure can force water through a semipermeable membrane. The flux of water through the membrane is directly proportional to the concentration gradient. This is an example of Fick’s first law for a flux J J = P(c1 − c2 ) where P is the permeability coefficient. A membrane permeable to both water and solute requires flux equations for both the water and the solute. The fluxes of water and solute are directly proportional to their respective concentration gradients. In addition, the flows can be coupled so that a hydrostatic pressure also produces a solute flux while a concentration gradient, proportional to osmotic pressure π , produces a water flux. The linear equations Jw = L 11 P + L 12 π Js = L 21 P + L 22 π obey the Onsager reciprocal relations L 12 = L 21 2. Electroosmosis For a membrane with pores containing fixed negative charge, the flow of ions through these pores produces a concomitant flow of water. This electroosmosis results

Oscillatory behavior is the exception in homogeneous phases although there are some special chemical reactions that do produce oscillatory behavior. A membrane system is a multiphase system and oscillations can be generated with a judicious choice of external forces on the membrane. One basic oscillatory system uses the kinetic behavior of protein channels for sodium and potassium ion in a nerve membrane. Before electrical excitation, the sodium channels are closed to sodium ion flow and the electrochemical gradient of 110 mV possible with the sodium ion gradient does not develop. The more permeable potassium channels tap the potassium ion gradient to produce an internal electrical potential of −60 mV relative to the external solution as ground. On excitation, the transient channels open and then close. As the sodium channels open, the sodium electrochemical potential of 110 mV appears to produce a peak internal potential of −60 + 110 = + 40 mV, The one-shot oscillation is completed as the sodium channel closes, the sodium-ion induced potential is lost, and the internal potential returns to the potassium channel dominated potential of −60 mV. The entire oscillation is controlled only by the sodium and potassium channels and the sodium and potassium ion concentration gradients across the membrane.

SEE ALSO THE FOLLOWING ARTICLES BATTERIES • CRYSTALLIZATION PROCESSES • DISTILLATION • ELECTROCHEMICAL ENGINEERING • ION

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354 TRANSPORT ACROSS BIOLOGICAL MEMBRANES • MEMBRANES, SYNTHETIC, APPLICATIONS • SOLVENT EXTRACTION • SURFACE CHEMISTRY • WASTEWATER TREATMENT AND WATER RECLAMATION

BIBLIOGRAPHY Aideley, D., and Stanfield, R. (1996). “Ion Channels,” Cambridge Univ. Press, Cambridge.

Membranes, Synthetic (Chemistry) Hiemenz, P. (1986). “Principles of Surface and Colloid Chemistry,” Marcel Dekker, New York. Kesting, R. (1971). “Synthetic Polymer Membranes,” McGraw-Hill, New York. Lloyd, D. R., ed. (1985). Materials Science of Synthetic Membranes, ACS Symposium Series No. 269, Am. Chem. Soc., Washington, DC. Sorensen, T. S. (1999). “Surface Chemistry and Electrochemistry of Membranes,” Marcel Dekker, New York. Starzak, M. (1984). “The Physical Chemistry of Membranes,” Academic Press, New York. Starzak, M. (1994). “Kinetics of Permeation and Gating in Membrane Channels,” 46: 61.

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Micelles Srinivas Manne

L. K. Patterson

University of Arizona

University of Notre Dame

I. II. III. IV. V. VI. VII. VIII.

Brief History Micelle Formation Micellar Structure Micellar Kinetics Solubilization Micellar Catalysis Micelles at Interfaces Surfactant Aggregates at Solid-Liquid Interfaces IX. Conclusion

GLOSSARY Aggregation number Number of monomeric surfactant units per micelle. Amphiphilic Term denoting the dual nature of a molecule that has both a water-insoluble hydrocarbon group and a polar headgroup. For examples, see Fig. 2. Critical micelle concentration (cmc) Concentration of a surfactant in solution above which the formation of micelles occurs. It is generally the concentration above which a physical property associated with surfactant aggregation changes markedly (e.g., surface tension, light scattering, or conductivity). The concept of a critical micelle concentration carries the implication of a phase separation, which is not rigorously true; however, the cmc remains a good working measure of micelle formation. Degree of ionization (α) Fraction of dissociated ionic headgroups at the micelle surface. A value of 1.0

indicates a completely ionized Stern layer while a value of 0 denotes that all headgroups are bound to counter ions. Hydration number Number of water molecules moving with a micelle as a kinetic entity. Generally expressed in terms of water molecules per surfactant molecule. Solubilization Inclusion of normally hydrophobic or only slightly polar materials (solubilizates or substrates) into micelles in aqueous solution. Surfactant Amphiphilic molecule generally perceived as having a polar headgroup, either charged or neutral, and a long-chain hydrocarbon tail.

MICELLE is derived from the Latin term micella meaning “small bit.” In aqueous solution, micelles are aggregates of surfactant molecules that form spontaneously when appropriate conditions, such as concentration, temperature, and ionic strength, are met. Depending both on such

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FIGURE 1 Highly simplified pictorial model of a spherical ionic micelle. Symbols indicate counterions (x), headgroups (), and hydrocarbon chains (vvv). [From Fendler, J. H., and Fendler, E. J. (1975). “Catalysis in Micellar and Macromolecular Systems,” Academic Press, New York.]

conditions and on surfactant structure, these aggregates may be composed of only a few or up to more than 1000 monomeric units. A naive representation of the aggregate is given in Fig. 1. This very simplistic approach is nevertheless useful for orientation to the discussion that follows. The thermodynamic constraints of water exclude the hydrophobic moieties of the surfactant into a principally hydrocarbon microregion while leaving the charged or polar headgroups in contact with the aqueous phase. This ordering of the micelle constituents produces an interfacial region of high polarity or, in the case of charged surfactants, high surface potential. The interfacial region may involve water–hydrocarbon interaction. Although it is discussed in terms of distinct regions, the micelle is viewed as a highly dynamic structure with rapid exchange of surfactant material between the aqueous bulk solution and the hydrocarbon pseudo-phase. The presence of micelles in solution can dramatically alter the solubility of otherwise water-insoluble materials, and their electrostatic properties may strongly mediate the distribution and kinetics of polar species. There has emerged a literature concerning associated catalytic effects for specific types of reactions. Additionally, the overall structure has evoked comparison with certain biological entities for which micelles have been used as simple models. While there are other molecular assemblies that bear the term micelle (bile salts and surfactants in nonpolar media), these systems are not included in this discussion.

I. BRIEF HISTORY Many of the properties of soap in water, used by humans for centuries, may be ascribed to the presence of micelles,

Micelles

although evidence for systematic study of such properties appears only in the nineteenth century. In 1846, Peroz reported that soap solutions have the power to increase the solubility of various substances; W. Kuehne noted in 1868 that cholesterol, which is essentially insoluble in water, aqueous mineral acids, and alkali solutions, is soluble in soap and solutions of bile salts. Oil-soluble dyes were reported by E. Pfluger in 1899 to be solubilized in aqueous soap solutions and bile salts. As early as 1892 C. Engler and E. Dieckhoff noted that the effectiveness of a soap in inducing solubility was related to the length of the paraffin chain in the soap. The concept of an aggregate as responsible for these unusual properties was not suggested until the twentieth century. With the early work of investigators such as J. W. McBain (1913), whose conductivity measurements on stearate stimulated the investigation of aggregation in soap solution, and who demonstrated the reversible character of micelle formation, numerous studies related the unusual physical properties of surfactant-bearing solutions to the characteristics of micelles. It was McBain who introduced the term micelle in 1913 to describe colloidal particles of detergents and soaps. C. R. Bury and E. R. Jones in 1927 related the marked change in activity coefficient for butyric acid solutions to micelle formation and suggested that the dual nature of the fatty acid (hydrophobic and hydrophylic) leads to micelle formation. Bury and Davies in 1930 proposed the concept of the critical micelle concentration (the surfactant concentration above which micelles form); their work noted that this number is a function of the length of the hydrocarbon moiety. The correct interpretation of the role of water in forces responsible for micelle formation (i.e., the hydrophobic effect) was proposed by G. S. Hartley in 1936. He was also responsible for suggesting the roughly spherical model for the micelle, a suggestion that gained general favor later. The development of readily available instrumentation for optical spectroscopy, light scattering, NMR, ESR, and so on, has provided possibilities for investigation of many micelle properties at the molecular level. Especially during the decades of the 1960s and 1970s, application of such techniques, along with a wide variety of theoretical studies, enhanced manyfold the information on micellar aggregates in solution.

II. MICELLE FORMATION The micellar behavior of any surfactant is determined by the molecular structure of the surfactant. All surfactants have in common both a hydrophilic headgroup, which may or may not be charged, and a hydrophobic moiety that would exhibit little water solubility alone. Formulas for

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FIGURE 2 Examples of several micellar surfactants.

typical micellar surfactants are given in Fig. 2, in which the dual, or amphiphilic, nature of these molecules may be easily seen. For example, in the case of sodium dodecylsulfate (SDS), one of the most widely used surfactants in micelle studies, the molecule consists of a charged sulfate headgroup (hydrophilic) and a 12-carbon aliphatic chain (hydrophobic). Such chains are commonly termed hydrocarbon or hydrophobic tails. Also among the most extensively studied groups of surfactants are the tertiary amines (e.g., hexadecyltrimethylammonium bromide, CTAB). For uncharged surfactants, the molecule may have a hydroxyl headgroup, but normally it includes a region of linked ethoxy groups that may undergo some hydration to compensate for the lack of charge on the OH. See the polyoxyethylene alcohol (or ethyleneglycolmonoether) in the figure. It was established early that surfactants dissolve as single molecules in water up to a given concentration (dependent on temperature and ionic strength), beyond which they are no longer soluble in water as monomeric units but form aggregates. The monomeric surfactant concentration beyond which added surfactant forms micelles is termed the critical micelle concentration (cmc). Micelle formation is not a matter of simple phase separation. There is a region over which both monomer concentration increases and aggregation takes place. Experimentally, this region is determined by plotting measurements of some physical or spectroscopic property that changes markedly with micelle formation as a function of surfactant concentration. Data from low and high concentrations are extrapolated to an intersection point that is taken as the cmc. Some of the properties that exhibit such behavior are illustrated in Fig. 3. Any understanding of effects produced in aqueous solution by the presence of micelles must be preceded by a

consideration of the driving forces for molecular organization in this type of aggregate. Two principal factors govern such organization: (1) tailgroup–tailgroup interactions, in which the hydrophobic effect (see Section II.A) causes the nonpolar entities to coalesce; (2) headgroup–headgroup and headgroup–water interactions which oppose the total separation of surfactant into a nonpolar phase. The interplay of these two factors determines the concentration onset of micelle formation, the micelle size distribution, and subsequent physical and chemical properties. A. The Hydrophobic Effect The marked insolubility of nonpolar molecules in water is essentially an entropic effect and arises, not from energy of association between hydrocarbon molecules, but rather from the unique hydrogen-bond character of water structure. Simply stated, the hydrogen-bonded structure of bulk water must be distorted to accommodate hydrocarbon molecules. H2 O forms an ordered, cagelike

FIGURE 3 Concentration-dependent behavior of some physical properties for a micelle-forming surfactant in solution. [From Lindman, B., and Wennerstrom, ¨ H. (1980). Top. Curr. Chem. 87, 1–83.]

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structure around nonpolar solutes (an effect known as hydrophobic hydration) in an effort to give up as few H-bonds as possible. This constraint therefore leads to significantly restricted mobility of H2 O in the hydration layer. The thermodynamic barrier to solubility in water is therefore largely dependent on the surface area of the cavity created in the water for water–hydrocarbon contact, which is proportional to the length of the hydrocarbon. For n-alkanes, it is known that the chemical potential for transfer from pure liquid hydrocarbon to aqueous solution is linearly dependent on the number of CH2 groups in the molecule (n C ) and has been written as µ0HC − µ0aq = (−2436 − 884)n C cal/mol.

(1)

Introduction of a double bond causes the first term to drop from 2436 to 1503. For dialkenes it falls further to 903, indicating that the double bonds increase the overall solubility but do not perturb the essentially linear dependence on the number of CH2 groups present. These considerations, then, provide a measure of the hydrophobic character of the surfactant hydrocarbon chain in isolation from the headgroup. The coalescence of hydrocarbon chains allows the ordered hydration layers to be expelled into the bulk phase, resulting in a considerable net gain in entropy. Indeed, micelle formation is primarily an entropy-driven process; the enthalpy of hydrocarbon association is comparatively weak and can even be endothermic (opposing association). As an example, dimethyl-n-dodecylamine oxide (illustrated in Fig. 2) undergoes or free energy change of micellization of G = −6.2 kcal/mol (a fairly typical value), of which the enthalpic contribution H = +1.7 kcal/mol and the entropic contribution −T  S = −7.9 kcal/mol. Because hydrocarbon structure plays a marked role in alkane and alkene solubility, it also influences strongly the solubility of surfactants. To express aggregation data in thermodynamic terms by analogy with the hydrocarbon data discussed earlier, it is necessary to consider ln cmc as the measure of free energy (the difference in free energy for a surfactant in the micelle and in water may be shown proportional to ln cmc). Indeed, for a group of surfactants with a common headgroup, a linear relationship between hydrocarbon chain length and ln cmc can be observed (Fig. 4). The addition of a double bond in the chain has an effect equivalent to a reduction in chain length of 1–1.5 carbons. Typical examples are the fatty acid soaps, (e.g., salts of stearic, oleic, and linoleic acids). B. Electrostatic Interactions The counterbalancing influence to the hydrophobic character of the surfactant is the polar headgroup, which ex-

FIGURE 4 Plots of ln cmc versus hydrocarbon chain length at 25◦ C, unless otherwise stated: A, alkyl hexaoxyethylene glycol monoethers; B, alkyl sulfinyl alcohols; C, alkyl glucosides; D, alkyl trimethylammonium bromides in 0.5 M NaBr; E, N-alkyl betaines; and F, alkyl sulfates in the absence of added salt at 40◦ C. [Composite from Tanford, C. (1980). “The Hydrophobic Effect: Formation of Micelles and Biological Membranes,” Wiley, New York.]

hibits several features contributing to micelle character. While the headgroups provide water solubility to an otherwise insoluble entity, their localization in the surface region of the micelle (see Fig. 1) introduces large charge– charge repulsions that limit the number of surfactant monomers that may come together. The influence of the headgroups may be seen in the effects of counterion concentration on both cmc and micellar size. By reducing the repulsive forces, the cmc is markedly reduced as may be seen in Fig. 5, which provides data for cmc versus ionic strength. Further, reduction in headgroup–headgroup repulsions also raises the limit to the number of monomers per micellar unit and the aggregation number. This behavior also is illustrated in Fig. 5. The effects upon cmc and micelle size, produced by changing various of the parameters that alter hydrophobic and headgroup

FIGURE 5 Dependence of the cmc and average micelle size on ionic strength and counterion at 25◦ C. Curve 1, Na dodecyl sulfate with added NaCl; curve 2, dodecyltrimethylammonium bromide with added NaBr; and curve 3, dodecyltrimethylammonium chloride with added NaCl. [Composite from Tanford, C. (1980). “The Hydrophobic Effect: Formation of Micelles and Biological Membranes,” Wiley, New York.]

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Micelles TABLE I Effect of Physical and Structural Properties on the Critical Micelle Concentration (cmc), Aggregation Number (n), and Degree of Ionization (α)a Surfactant Chain length C10 N(CH3 )3 Br

cmc (mM)

α

52

0.18

15 3.6

73 107

0.16 0.15

1.7

—

—

3.3 5.8

— —

— —

16 20

1100 670

0.07 0.07

12

69

0.08

18

48

0.13

3.5

71

0.14

2.1

48

0.11

Counterion type C12 N(CH3 )3 Br (0.5 M NaF) C12 N(CH3 )3 Br (0.5 M NaCl)

8.4

59

0.21

3.8

62

0.20

C12 N(CH3 )3 Br (0.5 M NaBr)

1.9

84

0.17

C12 N(CH3 )3 Br (0.5 M NalO3 )

5.1

38

0.24

C12 N(CH3 )3 Br C14 N(CH3 )3 Br Surfactant branching C14 OSO3 Na C12 CH(CH3 )OSO3 Na C11 CH(CH2 CH3 )OSO3 Na Surfactant head size C10 NH3 Br C10 NH2 CH3 Br C10 NH(CH3 )2 Br C10 (CH3 )3 Br C14 N(CH3 )3 Br C14 N(n-Pr)3 Br

63

n

15

43

0.17

15

34

0.18

C12 N(CH3 )3 Br (45◦ C)

17

28

0.19

C12 N(CH3 )3 Br (55◦ C)

17

24

0.20

120 144 153

0.12 0.11 0.11 0.26

C14 OSO3 Na (0.050 M NaCl) C14 OSO3 Na (0.10 M NaCl) Surfactant concentration C12 OSO3 Na (0.012 M)

III. MICELLAR STRUCTURE As a consequence of the interactions discussed previously, the micelle can be visualized as being composed of three regions: (1) the hydrocarbon core, (2) the polar surface layer, and (3) the diffuse double layer. These may be considered separately along with the tools that have been used to give us our present perception of the overall structure. A. The Hydrocarbon Core

Temperature C12 N(CH3 )3 Br (25◦ C) C12 N(CH3 )3 Br (35◦ C)

Salt concentration C14 OSO3 Na (0.0125 M NaCl)

as well. The absence of highly repulsive headgroups in nonionic systems is expected to affect both cmc and micellar size. And as expected, it is found that the cmc is about two orders of magnitude lower for a nonionic surfactant than for an ionic one having the same-length aliphatic group. With the nonionic polyoxyethylene surfactants, micelle size exhibits a marked dependence on the size of the polyoxyethylene chain. With C12 H25 (OEt)8 OH, only small micelles with aggregation numbers up to about 120 are seen at room temperature, even at high concentration, while with the (OEt)6 analog, size can grow to quite large values.

1.7 1.1 0.32 —

83

C12 OSO3 Na (0.020 M)

—

87

0.26

C12 OSO3 Na (0.129 M)

—

110

0.26

— —

— —

Nonelectrolyte additive C12 OSO3 Na (pure H2 O) C12 OSO3 Na (6 M urea) C12 OSO3 Na (0.06 mole fraction MeOH)

5.7 9.5 7.9

64

0.19

C12 OSO3 Na (0.12 mole fraction MeOH)

9.0

23

0.22

a From Menger, F. M. (1977). In “Bioorganic Chemistry III. Macroand Multimolecular Systems” (E. E. van Tamelen, ed.). Academic Press, New York.

interactions, are summarized in Table I for various ionic systems. With nonionic surfactants, for example, the polyoxyethylene alcohol in Fig. 2, the solubility of the headgroup comes from hydration, and the headgroup repulsions are much smaller. Steric factors may play a role

In early work, Hartley suggested that the hydrocarbon interior of the micelle must be liquidlike if one is to understand its extensive solubilization properties. Very little has happened that would alter that view. Since alkanes with structures analogous to the hydrocarbon tails found in surfactants exhibit freezing points generally much lower than the temperatures at which micelles are studied, the liquidlike nature of the core is to be expected. Many studies have been directed at reaching some quantitative view of the micellar interior. Thermodynamic data taken from partial molal properties indicate that these properties in micelles are comparable to those found in liquid hydrocarbons. Several spectroscopic approaches have been used including fluorescence depolarization and ESR measurements involving nitroxide spin labels. Both of these measure the mobility of the probe within the micelle. A somewhat lower mobility than in pure hydrocarbon is found for the fluorescence probes, and a lower mobility has been observed for the nitroxide than in aqueous solution. Such measurements have been criticized by suggesting that the probes themselves significantly perturb the micelle and, in monitoring probe motion, do not give a true picture of the liquid character of the core. Further, some of these probes are, indeed, of a size significant on the scale of a micelle. Additionally, there is evidence that some such probes tend to localize near the micelle surface where fluidity can differ from that found near the micellar center.

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666 An approach that involves no such perturbations is the use of 13 C NMR, from which one may gather information about the conformation of the chains and chain motion. It has been suggested that, on average, the chains tend to be somewhat more extended in the micelle than in pure liquid; the measurements of motion also give indication of a liquidlike structure. B. The Micellar Surface Region It is normal to think of the surface of the micelle as the region in which one finds not only the surfactant headgroup but also bound counterions (in the case of ionic headgroups) and water molecules. This region is also referred to as the Stern layer. The outer edge of this layer defines the shear surface of the micelle as it moves and is only a few angstroms thick, roughly the diameter of the headgroup (see Fig. 1). As stated above, the headgroups are always partially associated with counterions in the Stern layer, which diminishes the surface potential. A number given for concentration local to the Stern layer is about 3 M. These counterions are subject to both coulombic and thermal forces, one serving to attract them to the headgroup and the second to disperse them. The extent to which counterions are unbound from headgroups in the Stern layer is defined as the degree of ionization α and has a profound effect not only on micelle structure but on the whole range of properties that depend on electrostatic interactions. The region beyond the shear surface in which the dissociated counterions lie is called the Gouy–Chapman double layer and may be several hundred angstroms thick. Here the counterions may exchange with the bulk phase. The perception of bound versus dissociated counterions, of course, depends on the method used to measure it. Spectroscopic techniques, such as fluorescence quenching or NMR, measure changes in properties affected by the presence of ions in the surface region; thermodynamic measurements are sensitive to ion concentrations well away from the Stern layer. Transport techniques, on the other hand, measure the quantity of ions that move through solution with the micelle. These approaches give results that agree only qualitatively; however, α generally falls in the region 0.1–0.4. By and large, it is little changed by temperature, total surfactant concentration, or alkyl chain length for micelles of the same shape. (Exceptions are fatty acid sodium salts, for which α decreases from 0.43 for C7 to only 0.26 for C12 .) For systems with amine and sulfate headgroups, a compilation of dependences on various surfactant parameters is included in Table I. The effectiveness of binding has been shown to be dirrerent for different counterions. For alkyl ammonium headgroups, anions bind more effectively in the sequence − F− < Cl−  Br− < No− 3  I . It is to be expected, and

Micelles

is found, that the cmc is lowest for I− , increasing as one goes to the left in the series. For sulfates and sulfonates, the counter-cations bind in the order Li+ < Na+ < K+ < Cs+ . In both cases, smaller counterion size correlates with weaker binding to headgroups. This is because small and energetic ions such as Li+ and F− strongly bind dipolar H2 O molecules, greatly inflating the (now hydrated) ion size and shielding the charge within the hydration shell. If the counterion is made hydrophobic, one has, in addition to electrostatic effects, hydrophobic interactions between surfactant and counterion. An example is the salicylate anion, which binds very strongly to cationic surfactants, decreasing the cmc manyfold and α to near zero. C. Hydration in the Micelle Because there is effectively a hydrocarbon–water interface at the micelle surface, several possibilities must be considered for hydration in the micelle. These involve counterions, headgroups, and the portion of the hydrocarbon chain near the Stern layer. The bound counterions themselves may differ considerably in extent of hydration; the cations noted above maintain a higher degree of hydration than do the anions. There has been considerable evidence that much of the water of hydration is maintained upon binding. From viscosity and diffusion studies, hydration numbers for micelles have been deduced for a variety of charged surfactants; these fall into the range of 5–10 water molecules per surfactant molecule. For uncharged micelles with variable numbers of oxyethylene groups, the hydration number can vary considerably, as seen from Fig. 6. The extent to which water interacts with the hydrocarbon chains of the surfactant is a question yet unsettled; various workers have suggested complete, partial, or no water penetration into the micellar core. For aliphatic

FIGURE 6 Dependence of the number of water molecules per polyoxyethylene chain on the number of oxyethylene units for micelles of sodium dodecylpolyoxyethylene sulfates. [From Takiwa, F., and Ohki, K. (1967). J. Phys. Chem. 71, 1343–1349.]

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surfactants, persuasive evidence has been given to suggest that the core of the micelle contains no substantial amount of water and that if there is penetration it probably involves no more than two carbons. Surfactants containing oxyethylene moieties appear to be substantially hydrated in the region of these groups. This is implied in Fig. 6. A number of studies involving probes solubilized in the micelle or attached to the surfactant itself (e.g., carbonyl groups) indicate the presence of water around such probes. It is possible that perturbation of the system by the included probe can lead to significant water–hydrocarbon contact and/or that many such probes lie near the surface. Of course, as has been shown, the dynamic equilibrium between surfactants in bulk phase and micelle assures considerable motion of material across the micelle boundary. Additionally, it is quite likely that the micelle surface is not a simple two-dimensional interface but involves some protrusion of surfactants into the bulk where hydrocarbon wetting could occur (Fig. 7). In summary, one should note that whether there are 5–10 water molecules associated with each surfactant of a simple micelle or whether there is penetration to several carbons depth in the presence of solubilized material, a substantial fraction of material associated with the micelle is, indeed, water.

FIGURE 7 Schematic of the dynamic protrusion of surfactant monomers from spherical and rod-shaped micelles into surrounding bulk phase to enhance hydrocarbon–water content. [From Lindman, B., and Wennerstrom, ¨ H. (1980). Top. Curr. Chem. 87, 1–83.]

D. Surfactant Organization in Micelles Developing a three-dimensional model that accurately takes into account all the characteristics experimentally defined for the micelle—surface structure, water penetration, fluidity of the core, solubilization, and dynamic character—is not a trivial task. The initial picture given in Fig. 1, while providing an elementary concept with which to work, does not reflect either the dynamic character of micelles or the effect this should have on surface regularity. Further, it does not reflect any randomness in the coiling of the hydrocarbon chains. Attempts have been made to construct micelle structures along the lines suggested by Fig. 1, with molecular models in which some kinks are introduced to provide a much more rugged micelle, but one in which considerable water penetration is suggested. The representation in Fig. 7 approximates this approach. Other more geometrically simple models using square-lattice packing suggest alternate ways of arranging surfactants: first using parallel correlation with headgroups separated at right angles and then with the introduction of gauche conformations near the headgroup to release headgroup contact. The results of such efforts are given in Fig. 8. Averaging arrangements of this type can yield an approximation to the ideal microscopic oil drop shown in Fig. 8b. Statistical theory employing a three-dimensional lattice model has also been developed. The model emerging from such an approach (Fig. 9) provides constant radial

FIGURE 8 Space-filling model of 64 SDS units: (a) bilayer fragment model, (b) compact droplet by coalescing volume of tails up to a carbon, (c) parallel correlation of units with headgroups at right angles, and (d) release of headgroup interaction by gauche kinks near headgroup. Average wetting is two carbons per unit. [From Fromhertz, P. (1980). Chem. Phys. Lett. 77, 460–466.]

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FIGURE 9 Two-dimensional representation of lattice model from statistical treatment. [From Dill, K. A., and Flory, P. J. (1981). Proc. Natl. Acad. Sci. 78, 676–680.]

interlayer spacing and equal volumes for lattice sites. Each site can accommodate one alkyl chain segment. Configurational freedom in the surface sites considerably exceeds that of those near the core, implying an ordered center and a disordered surface region. It would also allow considerable water–hydrocarbon contact since the statistics suggest that a deep-lying CH2 could contact the water surface. While these proposed models provide us with images with which to work, it is not generally agreed that any one of them yields a totally accurate picture, for example, in the matter of water penetration. It is further difficult to portray the dynamic character of the micelle from such static representations even though they may be reported as time averages. E. Nonspherical Micelles For most conventional surfactants, the delicate balance of repulsive forces (between headgroups) with attractive forces (between tailgroups) results in discrete, quasispherical micelles at the cmc. Changing this balance in either direction can have a pronounced effect on the micelle shape and aggregation number. Increasing the intermolecular repulsion—using, for instance, bulky or multivalent headgroups or very short alkane chains—increases the cmc, decreases the aggregation number, and can suppress micelle formation altogether. Conversely, increasing the intermolecular attraction—e.g., by using double-chain tailgroups, or by screening out the headgroup repulsion— decreases the cmc, increases the aggregation number, and can favor micelles that are continuous along a single dimension (cylindrical rods, cf. Fig. 7) or along two dimensions (bilayer sheets, cf. Fig. 8a). The growth of rodlike and sheetlike micelles is not self-limiting, so these shapes no longer have well-defined aggregation numbers. For ionic surfactants, the micellar sphere-to-rod transition can be induced in a number of ways, all of which de-

Micelles

crease the repulsion between headgroups: (1) the addition of a tightly binding counterion, which decreases the degree of ionization and the effective headgroup charge; (2) the addition of an electrolyte, which increases the degree of electrostatic screening; or (3) a significant increase in the concentration of surfactant itself, which has the same effect. A sphere-to-rod transition manifests itself by, among other things, a sudden increase in the solution viscosity due to entanglement between the rodlike micelles. Some authors denote the surfactant concentration at which the viscosity jumps as a “second cmc.” The value and even existence of the second cmc depends on the degree of counterion binding; strongly binding counterions (e.g., in CTAB) display far lower second cmc’s than those with weaker binding (e.g., CTAC). As surfactant solutions approach large concentrations (of order 1 M), the micelles themselves begin to selforganize into higher-order structures termed lyotropic liquid crystals. These complex structures are highly dependent on surfactant geometry in ways that are not completely understood. For surfactants with highly repulsive headgroups, the first observed lyotropic phase is typically the “discontinuous cubic” phase, which is thought to consist of discrete micelles in a cubic lattice. Surfactants with less repulsive headgroups, which undergo a sphere-torod transition at a second cmc, give rise to a “hexagonal phase” consisting of hexagonally close-packed cylindrical micelles. Surfactants with weakly repulsive headgroups (or double tails), which tend to form sheetlike bilayers, give rise to “lamellar phases” consisting of stacked bilayer sheets separated by water. If surfactant concentration is increased past the first lyotropic phase, more complex bicontinuous and inverted phases are often observed, culminating finally in hydrated crystals of the solid surfactant.

IV. MICELLAR KINETICS Although the micelle may appear to be a clump of material, insoluble in the water phase, the exchange of material with the bulk phase is actually very fast. While early experimental measurements seemed to indicate two different time domains—one in the microsecond and one in the millisecond region—for material exchange rates between micelle and solution, it is now agreed that overall micelle kinetics are governed by the mechanism k+

S1 + Sn−1  Sn , k−

(2)

where k + and k − are rate constants for inclusion of a surfactant molecule into and loss of a surfactant from the aggregate, and n indicates the number of monomers in the

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TABLE II Rate Constants and Micelle Size Distribution for a Series of Sodium Alkyl Sulfates at 25◦ Ca Surfactant

FIGURE 10 Distribution of a representative micelle population as a function of aggregation number i. The dashed curve indicates the redistribution after a small perturbation of the system that leads to dissociation and decrease of the mean aggregation number of micelles. The dotted line is the final equilibrium distribution, t 1 and t 2 are the characteristic relaxation times of the process indicated by the arrows. [From Schelly, Z. A. et al. (1979). In “Solution Chemistry of Surfactants,” pp. 323–325. Plenum, New York, as discussed in Fendler, J. H. (1982). “Membrane Mimetic Chemistry,” Wiley, New York.]

aggregate. Because multiple equilibria are involved there is certainly a range of micelle sizes, but the cooperative nature of micelle formation produces size distributions for micelles that cluster about optimum values. This behavior is illustrated in Fig. 10. This mechanism allows one to neglect the interaction of submicelle aggregates: say, S j + Sk → S j+k in the kinetic description. When a micelle close to its thermodynamically optimal size is perturbed, the initial step in relaxation to the new equilibrium level reflects the process described in Eq. (2) and exhibits a relaxation time τ1 that may be expressed by τ1−1

−

−

= k /σ + ak /n, 2

(3)

where σ is the variance of micelle size distribution and a the equilibrium ratio of micellar and monomeric surfactant. This represents a change in the size of micelles but not in number. Measurements of τ as a function of a can yield the other parameters if n is known independently. Under the condition where concentrations Sn Sn+1 , that is, σ > 1, the relationship k − /k + = cmc may be used to extract both forward and reverse rate constants. In Table II, values of k + and k − are given for a series of sodium alkyl sulfates, along with values of σ, n, and the cmc. It shows a quite dramatic dependence of exitrate constants on chain length. On the other hand, k + values are slower than would be predicted by simple diffusion considerations, and it has been suggested that this is due to long-term repulsions by the surface potential field. The measurements of such processes may be made by various techniques, (e.g., p-jump and ultrasonic absorption). After the initial process (relaxation to the quasiequilibrium state) just described, a change in the number of micelles occurs, representing relaxation to the true

cmc (M)

nb σ c

k− (s−1 )

k+ (M−1 s−1 )

NaC6 SO4

0.42

17

6

1.32 × 109

3.2 × 109

NaC7 SO4

0.22

22 10

7.3 × 108

3.3 × 109

NaC8 SO4

0.13

27

—

1.0 × 108

7.7 × 108

NaC9 SO4 NaC10 SO4 d

6.10−2 3.3 × 10−2

33 41

— —

1.4 × 108 9 × 107

2.3 × 109 2.7 × 109

NaC11 SO4

1.6 × 10−2

52

—

4 × 107

2.6 × 109

NaC12 SO4

8.2 × 10−3

64 13

1.0 × 107

1.2 × 109

NaC14 SO4 NaC16 SO4 e

2.05 × 10−3 4.5 × 10−4

80 16.5 100 11

9.6 × 105 6 × 104

4.7 × 108 1.3 × 108

a Data from Aniansson, E. A. G. et al. (1976). J. Phys. Chem. 80, 905–917, as compiled by Lindman, B., and Wennerstr¨om, H. (1980). Top. Curr. Chem. 87, 1–83. b Mean aggregation number. c Standard deviation for the micelle size distribution. d 40◦ C. e 30◦ C.

equilibrium. The relaxation time τ2 is heavily dependent on the sum of dissociation rates kn− × Sn for aggregation numbers (n) between monomer and proper micelles. Characteristically, τ2 values range from 10−3 to l s−1 , increasing with temperature and decreasing with added salt. The effects of both relaxation steps on micelle size distribution are illustrated in Fig. 10.

V. SOLUBILIZATION In terms of utility, the most important feature of the micelle is its ability to take up or solubilize nonsurfactant materials. A large number of substances not readily soluble in water may be dissolved in surfactant solutions (e.g., hydrocarbon substances). A principal application of this characteristic is to household detergent, by which waterinsoluble “dirt and grease” are suspended in the micellar pseudo-phase and separated from the material to be cleaned. One technological variation of this use involves the recovery of petroleum, adherent to underground rock surfaces, by pumping detergent solutions into oil wells. Under appropriate economic conditions, this could bring into production once more, sites that have been abandoned when conventional techniques ceased to give adequate yield. It may be seen that in surfactant solutions, the solubility of various hydrophobic substances can increase rapidly only above the cmc and that association does not occur significantly with individual surfactant monomers. Figure 11 illustrates changes in decanol solubility as a function of surfactant concentration in several systems. Solubility of

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FIGURE 11 Solubility of decanol in solutions of (a) sodium octanoate (20◦ C), (b) sodium decanoate (40◦ C), (c) sodium dodecanoate (40◦ C), and (d) sodium tetradecanoate (40◦ C). [From Ekwall, P. et al. (1969). Mol. Cryst. Liq. Cryst. 8, 157–213.]

such substances in aqueous solution has been used as a measure of the cmc. Because the extent of solubilization is dependent upon the quantity of micellar surfactant, such behavior has been interpreted in terms of the phase separation model where the surfactant is considered to be a separate or pseudo-phase. Indeed, various spectroscopic observations indicate that many solubilizates move in a liquidlike environment, an observation that is consistent with the idea of a separate phase. One consequence of this approach has been to show the validity of a relationship analogous to Henry’s law: cmc(Xa) = cmc(Xa = 0) − k · Xa

(4)

for micellar systems, where Xa is the mole fraction of the material to be solubilized. As described previously, the micelle exhibits a surface region with significant polar character, while the interior is essentially hydrocarbon in nature. It is to be expected that materials associating with micelles would distribute themselves between these two regions depending on polarity or polarizability. Indeed, those materials such as aliphatic hydrocarbons that have no surface activity are to be found in the micellar core while the more polarizable aromatic hydrocarbons have been found to associate with the surface region. Many such species give evidence of lying in an alcohol-like environment. There are further energetic considerations related to the effects of surfactant packing on the micellar structure itself that influence the distribution

of a solubilizate. As an illustration, solute near the surface can produce a gain in energy by decreasing the ratio of gauche to trans configurations in hydrocarbon chains in that region. Rigid molecules such as planar aromatic hydrocarbons may perturb packing in the core, which may contribute to their unfavorable location in that domain. While various techniques, such as stopped flow, have been used to follow substrate kinetics, many kinetic measurements have involved the photophysical properties of solubilized probes. Because of the luminescent properties of their excited states, the aromatic hydrocarbons provide opportunities for monitoring movement of such probes across the micelle boundary. For example, long-lived phosphorescence of aromatic hydrocarbons has been monitored in micellar solutions containing ionic quenchers that themselves are repelled by the surfactant head groups. Since quenching must take place in the aqueous phase, phosphorescence lifetimes may be interpreted to provide rate constants for exit of the probe from the micelle. Some typical values obtained by this technique are given in Table III. Fluorescence data have also been used to obtain such information.

VI. MICELLAR CATALYSIS It has been shown that a number of chemical processes can be kinetically altered in the presence of micelles. Either

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k+ (M−1 s−1 ) 1 × 1010

k− (s−1 ) 5 × 107

Benzene

4.4 × 106

Naphthalene

2.5 × 105

Anthracene Pyrene

1.7 × 104 4.1 × 103

1-Bromonaphthalene

3.3 × 104

Perylene

4.1 × 102

a

Compilation taken from Fendler, J. H. (1982). “Membrane Mimetic Chemistry,” Wiley, New York.

rates of reaction or the distribution of products can be significantly altered when either reactants or products associate with surfactant micelles. This recognition has led to increased investigation into such behavior, and some simple examples are presented here. Several aspects of micelle–reactant interaction are suggested as responsible for the phenomenon called micellan catalysis. It is suggested that the micellar medium can affect transition states of reactions, thereby altering their rates; and some cases are cited in the literature. Seemingly, however, the most important effect in micellar catalysis relates to the control of local concentrations of components in bimolecular reactions. If a reactant exhibits some hydrophobic character, it may preferentially solubilize in the micellar pseudo-phase where its concentration can be controlled. For charged micelles, ionic reaction components can be localized at or repelled from the micelle surface. One may note, for example, that experiments with pH-sensitive dyes have shown that pH at the micelle surface can be lowered by a couple of units for negatively charged micelles such as SDS and raised by two units in positively charged systems such as CTAB. One of the most widely discussed processes involving micellar catalysis deals with hydrolysis reactions of the form O O || || −

R C O R + OH R C O− + R OH

(5)

Of course, for esters that solubilize, it is expected that reaction rates will depend on concentrations of OH− local to the micelle. One would expect that such concentrations will be enhanced in the presence of positively charged micelles and inhibited by negatively charged ones. Figure 12 gives examples of systems that follow those expectations. It may be seen that the response to concentrations of surfactant depend on the length of the hydrocarbon chain in the ester, suggesting some variation in extent to which the esters are solubilized.

FIGURE 12 Plots of kobs for the hydrolysis of p-nitrophenyl acetate (curves A), mono- p-nitrophenyl dodecanedioate (curves B), and p-nitrophenyl octanoate (curves C) versus (a) concentration of sodium dodecanoate at pH 9.59 = 0.1 and 50◦ and versus (b) concentration of n-dodecyltrimethyl ammonium bromide (LTAB) at pH 10.49 = 0.2 and 50◦ . Values of kobs for the reaction of A with sodium laurate have been divided by 2.0 to bring the curve on scale. [From Menger, F. M., and Portnoy, C. E. (1967). J. Am. Chem. Soc. 89, 4698–4703.]

The data presented by such reactions were analyzed in terms of the simple kinetic scheme

Sn  E kw

P

SnE km ,

(6)

P

where Sn is the micelle, E is the substrate, and P is the product. It may be shown that the rate constant for the reaction in the absence of micelles kw and the observed rate of reaction kobs may be related to the concentration of micelles by the expression

1 1 1   k w  kobs k w  km (k w  km)K[Sn] ,

(7)

which, it may be noted, bears resemblance to treatments of enzyme kinetics. Under the right conditions, plots of (kw + kobs )−1 versus [Sn ]−1 may be seen to yield rate constants for the micelle catalyzed reaction km and the binding constant, K s , for the equation. However, the simple conditions defined in Eq. (6) are not met under conditions described by Fig. 12b where both OH− and substrate can bind to the micelle. One should note that overall parallels drawn between micellar catalysis and enzyme behavior must be treated with care because enzymes exhibit a degree of site selectivity not seen in micellar systems. It may be noted that such a simple unimolecular treatment holds when we consider partitioning one reaction component. In the late 1980s, much effort has been expended to develop sophisticated models for catalysis that deal with bimolecular reactions and address kinetic dependencies on surface charge, pH, buffers, chain length, and specific salt effects. The literature in this area has become rather extensive.

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In systems where more than one reaction pathway is possible, localization of reactant by micelles may govern the kinetics and determine product yield. An example of this may be nicely seen in the photolysis of ketone:

O CH2

C

CH2

CH3

For convenience, one may use the notation

A

CH2

and

B

CH2

CH3

and write the reaction

ACB

h

AA  AB  BB

(8)

O The distribution of products in homogeneous solution is governed by simple statistics as shown in Fig. 13. However, with the addition of surfactant up to a concentration that localizes each reactant molecule in a micellar “cage,” the yield of AB is totally dominant. A large number of studies have been carried out with fast-reaction kinetics techniques to characterize micellar

effects on reaction rates of excited states and radiolytically generated radicals. While these might appear to be highly specialized measurements, the time resolution that may be achieved provides insight into the catalysis of many micelle-related events on a very short time scale. For example, the movement of charged radicals, such as hydrated electrons, to reaction sites at positively charged micellar surfaces has been shown to increase up to two orders of magnitude over diffusion control. Disproportionation of Br− 2 radicals has been shown to occur much more rapidly on a CTAB micelle surface than in solution. Initial steps in the radical processes governing lipid peroxidation, a mechanism of particular biological interest, have been studied in micelles to determine the effects of a membrane-mimic environment on radical behavior. Micellar influence on kinetics of excited-state quenching by adsorbed anions, photoionization of sequestered chromophores, and many other photoprocesses have been investigated.

VII. MICELLES AT INTERFACES For all the structural and behavioral complexity that micelles present, they are the simplest assembly of amphiphilic molecules with which to deal experimentally. In most cases one merely dissolves the surfactant of interest in water, and the gods of thermodynamics do the rest. While only a limited number of biological amphiphiles actually aggregate in this way (bilesalts, fatty acid salts, and lysolecithin), micelles do present accessible model systems with which to approach phenomena governing more complex, extended assemblies such as biological membranes. Although the limitations of micelle–membrane comparisons must be kept in focus, a wide range of information characterizing hydrophobic interaction and water– lipid interfacial phenomena, which are highly relevant to biological systems, has been built up from the collective study of micelles.

VIII. SURFACTANT AGGREGATES AT SOLID-LIQUID INTERFACES

FIGURE 13 Dependence of product distribution from photolysis of dissymmetrical dibenzylketones on CTACI concentration. [From Turro, N. S., and Cherry, W. P. (1978). J. Am. Chem. Soc. 100, 7431–7432.]

Surfactants are generally attracted to solid-liquid and liquid-air interfaces, and this interfacial enrichment is vital to a large number of industrial applications. (Even the term surfactant—short for surface-active agent—betrays the central importance of interfaces.) One such application is foam flotation in the mining industry, in which surfactant adsorption to ore microparticles causes them to flocculate at the surfaces of air bubbles and rise to the foam layer, where they are skimmed from the remaining matrix.

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(hydrophilic surfaces)

Adsorption density

(hydrophobic surfaces)

CMC Surfactant concentration FIGURE 14 Schematic of “two-step” absorption isotherms for surfactants on hydrophobic surfaces (dashed line, right vertical scale) and on hydrophilic surfaces (solid line, left vertical scale). The absorption models are inferred from the measured amounts of surfactant absorbed per unit surface area.

Another is particulate detergency, in which surfactant infiltration between a soil particle and the substrate eventually desorbs the soil and prevents its readsorption. A final example is tertiary oil recovery, in which oil is dislodged from the microchannels of porous rocks by competitive surfactant adsorption at the pore surfaces and by emulsification of the oil. Despite the importance of these applications, knowledge of surfactant behavior at interfaces has historically lagged far behind that in bulk solutions. Which morphologies surfactant aggregates assumed at interfaces, or even whether well-defined aggregates analogous to bulk micelles existed at interfaces, remained open questions until fairly recently. The first evidence for hydrophobic association at interfaces came from adsorption isotherms—i.e., measurements of surface adsorption density as a function of surfactant concentration in solution. In a landmark 1955 paper, A. M. Gaudin and D. N. Fuerstenau noted that the surface density of SDS on alumina increased sharply as the concentration approached the cmc, and they interpreted this as evidence for hydrophobic association into interfacial aggregates termed hemimicelles. Confirming evidence for interfacial aggregation has since come from hundreds of adsorption studies on a vari-

ety of hydrophobic and hydrophilic surfaces. At the risk of overgeneralizing, adsorption isotherms typically follow a two-step pattern (Fig. 14) for both hydrophobic and hydrophilic surfaces. In both cases, a low-density plateau at low concentrations gives way to a high-density plateau as the concentration approaches the cmc; this final adsorption density remains approximately constant up to very high surfactant concentrations. The surfactant organization at each plateau is inferred from the measured surface density and from known interaction sites. Hydrophobic surfaces, which interact with tailgroups via hydrophobic interactions, display a low-density plateau consistent with a horizontal monolayer and a high-density plateau consistent with a vertical monolayer, with tailgroups facing the surface. This is in contrast with charged hydrophilic surfaces, which interact electrostatically with oppositely charged headgroups. Here the low-density plateau approximately corresponds to a vertical monolayer (headgroups facing the surface), whereas the high-density plateau is consistent with a vertical bilayer above the cmc. However, while these flat morphologies served as the standard models of interfacial aggregation, uncertainties in surface area determination (which can approach 30%) could not definitively exclude curved interfacial

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aggregates resembling bulk micelles. (Indeed, many workers found in adsorption isotherms consistent evidence for defective or incomplete layers or even for adsorbed micelles.) While hydrophobic association was agreed to be the mechanism for interfacial adsorption near the cmc, the corresponding surfactant structure was unknown until recently. A. Imaging Interfacial Aggregates by Atomic Force Microscopy In 1995, the high-density structures above the cmc were imaged directly by atomic force microscopy (AFM) for both hydrophilic and hydrophobic surfaces. These results showed for the first time that interfacial micelles existed in well-defined shapes and sizes, that they generally possessed spherical or cylindrical curvature (in contrast to the standard models), and that this curvature was a compromise between the spontaneous curvature of bulk micelles and the constraints imposed by the flat surface. Briefly, AFM is a technique that maps the topography of a surface by plotting (on a color scale) the measured force between the surface and a small tip attached to a sensitive cantilever spring. In most applications, the tip is in direct contact with the surface, and the AFM performs as a sensitive contact profilometer. For imaging interfacial surfactant structures, however, contact forces disrupt the liquid crystalline aggregates. Therefore the repulsive colloidal stabilization forces between the surfactant layers adsorbed to the tip and sample are used as the contrast mechanism during imaging. A simplified schematic of the imaging mechanism is shown in Fig. 15 for ionic surfactants. The tip and sample are immersed in surfactant solution above the cmc (where the high-density plateau occurs in the adsorption isotherm). Surfactant adsorption on the tip and sample charges both with the same sign, resulting in a longranged, screened electrostatic repulsion between the two (Fig. 15a). By fixing the imaging force (using a feedback loop) in this noncontact regime, the AFM “flies” the imaging probe above the aggregate layer while obtaining a “surface map” of the colloidal stabilization force. This map of the tip-sample repulsion reveals the surfactant aggregate structure at the interface (Fig. 15b). Comparing AFM images with other data (e.g., adsorption isotherms) usually fixes the interfacial micelle structure uniquely. AFM imaging has been used to identify interfacial aggregate structure above the cmc for a variety of ionic, nonionic, and zwitterionic surfactants on both hydrophobic and hydrophilic surfaces. (Structures far below the cmc, corresponding to the low-density adsorption plateau, cannot be imaged readily because the tip-sample force is strongly hydrophobic and attractive in this regime.) The

B FIGURE 15 AFM imaging mechanism for interfacial surfactant aggregates. (a) Surfactant adsorption on the tip and sample creates a long-range repulsion between the tip and sample, down to separations (around 5 nm) where the opposing surfactant layers touch and fuse together. Imaging in the noncontact regime allows the AFM tip to obtain a map of the surfactant aggregate structure. (b) A sample AFM image (200 × 200 nm) of spherical micelles in a hexagonal pattern at the mica-solution interface. The surfactant is a divalent cationic surfactant with a C18 tail.

most popular substrates have been layered crystals, owing to the ease of surface preparation (cleaving by adhesive tape) and the variety of available surface properties. A summary of observed aggregate structures follows. B. Aggregates at Hydrophobic Surfaces AFM results (Fig. 16a) show that the morphology of surfactant aggregates at hydrophobic surfaces depends

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Micelles

A

B FIGURE 16 AFM images and schematics of alkyltrimethylammonium surfactant aggregates on (a) the hydrophobic graphite surface and (b) the hydrophilic mica surface. In (a), the bottom row of tails are aligned parallel to a surface symmetry axis, orienting the half-cylinder perpendicular to this axis. In (b), the smaller contact area between the aggregate and surface allows the cylindrical aggregate to meander over the surface.

heavily on the crystalline anisotropy of the substrate, with the surfactant geometry itself playing a comparatively minor role. Almost all surfactants—ionic, nonionic, and zwitterionic, with tail lengths ranging from 12 to 18 carbon atoms—aggregate in the form of half-cylindrical aggregates on cleaved crystals of graphite and MoS2 . This is evidenced by AFM images in the form of rigid, parallel stripes, separated by a little over twice the surfactant length, with stripe axes running perpendicular to the underlying lattice symmetry axes. Parallel half-cylinders are consistent with the observed stripe spacing and with the known adsorption density (roughly equivalent to a vertical monolayer) from isotherms. The cylindrical curvature is, however, initially surprising considering that these surfactants form spherical

micelles in bulk. The orientation of the aggregates with respect to the surface lattice suggests that the crystalline anisotropy of the substrate plays a central role in determining this curvature. This has been further confirmed by control experiments on amorphous hydrophobic surfaces; these show half-spherical micelles above the cmc, in agreement with the spontaneous curvature in bulk solution. The current understanding of the adsorption and aggregation process is as follows. At very low concentrations, hydrophobic attraction causes the tailgroup to adsorb horizontally on the surface, and an anisotropic interaction with the surface lattice causes the tailgroup to orient itself parallel to an underlying symmetry axis. Because the tail-surface interaction for this configuration is typically

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676 stronger than that between two tails, the molecule does not gradually desorb to a vertical orientation (as in Fig. 14) as the surfactant concentration is increased. Instead, each strip of horizontal molecules, arranged head-to-head and tail-to-tail along a symmetry axis, serves as a foundation for a half-cylindrical aggregate above the cmc (see the schematic of Fig. 16a). Apparently, the original alignment of tails parallel to a symmetry axis not only determines the direction of the final half-cylindrical aggregate, but also the fact that half-cylinders are the favored arrangement. Where the lattice alignment is absent—i.e., for isotropic substrates such as hydrophobized silica—the same surfactants form half-spherical aggregates. Both graphite and MoS2 —despite marked differences in lattice symmetry, spacing, and surface groups—give rise to similar half-cylindrical aggregates at the same perpendicular orientation relative to the underlying lattice. This suggests that strong tailgroup alignment, leading to half-cylindrical aggregates, may be a feature common to all crystalline hydrophobic adsorbents. Since hydrophobic surfaces interact with the entire length of the tailgroup, this large interaction area evidently leads to a high degree of surface control of the aggregate structure. C. Aggregates at Hydrophilic Surfaces Hydrophilic surfaces, on the other hand, interact with the much smaller surfactant headgroup. It is therefore natural to expect “full” aggregate structures, whose curvature is controlled as much by intermolecular interactions as by the surface. This is exactly what is observed by AFM. The vast majority of experiments in this category have investigated ionic surfactants on oppositely charged surfaces. On the anionic surface of silica, single-chain cationic surfactants self-assemble into spherical aggregates (resembling bulk micelles) above the cmc. These are thought to originate from the electrostatic binding between charge sites on the surface and individual surfactant molecules; the latter then serve as nucleation sites for micellar aggregation above the cmc. Similar results have been observed with anionic surfactants on the cationic surface of alumina. In both cases, the spherical interfacial aggregates are consistent with the curvature found in bulk solution. Micelle curvature is expected to be relatively unperturbed as long as the substrate charge density (i.e., adsorption site density) falls short of the charge density on the outer surface of the micelle. Most surfaces satisfy this requirement. A notable exception is the anionic cleavage plane of mica, where exchangeable surface ions give rise to a far higher adsorption density than on silica. Alkyltrimethylammonium surfactants on mica self-assemble into parallel cylindrical aggregates (see Fig. 16b)—a higher-density

Micelles

structure than close-packed spheres and a flatter curvature than is found for bulk micelles. The mica surface has been likened to a highly charged “planar counterion,” which induces a sphere-to-rod transition at the interface, in a similar way that multivalent counterions induce sphere-to-rod transitions in bulk micelles. In cases where the surfactant headgroups are highly repulsive or bulky, even the mica surface is unable to bring headgroups close enough to effect a sphere-to-cylinder transition, and interfacial micelles remain spherical. This is the case for divalent surfactants, as shown in Fig. 15b. In summary, the geometry of surfactant aggregates at charged surfaces is highly sensitive to “charge density matching” between the surface and free (unperturbed) micelles. The interfacial aggregate can have a flatter curvature than free micelles in cases where the adsorption density becomes comparable to the micelle charge density.

IX. CONCLUSION In the technological realm, an application that has generated much excitement is the synthesis of mesoscopic materials using surfactant micelles as templates, as first reported in 1992 by Beck et al. This process relies on inorganic polymerization at the interfacial region between a surfactant aggregate and a solution in which the inorganic precursors (usually silicate ions) are initially dispersed. By restricting the polymerization reaction to the micelle–solution interface, a complex nanocomposite is formed consisting of surfactant micelles embedded in an ordered array within a continuous inorganic (e.g., silica) “scaffold.” Pyrolyzing the surfactant finally results in a mesoporous material, with pore sizes of order 5 nm, which can serve as, for example, a molecular filter, catalytic support, or laser waveguide. However, aside from such applications and, of course, the industrial interest in detergent action, one should note that of the wide range of experimentalists and theoreticians that have been drawn to investigation of micelles, most have been attracted by the unique intellectual challenge such systems offer. The literature provides ample evidence that they have not been disappointed.

SEE ALSO THE FOLLOWING ARTICLES CHEMICAL KINETICS, EXPERIMENTATION • CHEMICAL THERMODYNAMICS • ELECTROPHORESIS • HYDROGEN BOND • KINETICS (CHEMISTRY) • MACROMOLECULES, STRUCTURE • PRECIPITATION REACTIONS • SILICONE

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Micelles

(SILOXANE) SURFACTANTS • SURFACTANTS, INDUSTRIAL APPLICATIONS • SURFACE CHEMISTRY

BIBLIOGRAPHY Beck, J. S., et al. (1992). J. Am. Chem. Soc. 114, 10834–10843. Fendler, J. H. (1982). “Membrane Mimetic Chemistry,” Wiley, New York. Fendler, J. H., and Fendler, E. J. (1975). “Catalysis in Micellar and Macromolecular Systems,” Academic Press, New York. Gaudin, A. M., and Fuerstenau, D. W. (1955). Trans. AIME 202, 958– 962.

677 Klafter, J., and Drake, J. M. (1989). “Molecular Dynamics in Restricted Geometries,” Wiley, New York and Chichester. Lindman, B., and Wennerstr¨om, H. (1980). Top. Curr. Chem. 87, 1–83. Manne, S., and Gaub, H. E. (1995). Science 270, 1480–1482. Manne, S., and Warr, G. G. (1999). In “Supramolecular Structure in Confined Geometries” (S. Manne and G. G. Warr, eds.), pp. 2–23. American Chemical Society, Washington DC. Menger, F. M. (1977). In “Bioorganic Chemistry III. Macro- and Multimolecular Systems” (E. E. van Tamelen, ed.), pp. 137–152. Academic Press, New York. Tanford, C. (1980). “The Hydrophobic Effect: Formation of Micelles and Biological Membranes,” Wiley, New York. Wennerstr¨om, H., and Lindman, B. (1979). Phys. Rep. 52, 1–86.

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Physical Chemistry Douglas J. Henderson

Charles T. Rettner

Brigham Young University

IBM Almaden Research Center

I. II. III. IV.

Classical Mechanics Quantum Mechanics Statistical Thermodynamics Kinetics and Dynamics

GLOSSARY Activated complex Short-lived transition state that occurs at the point of maximum energy along a reaction path when the molecules in a chemical reaction can no longer be considered as reactants or products. Adsorption Adhesion of a gas or liquid at a surface resulting in an increased concentration of the gas in the vicinity of the surface; to be distinguished from absorption, which occurs throughout the solid or liquid. Critical point Point where two phases become identical and form one phase. Degrees of freedom Variables which must be determined to specify the state of a system. Elementary reaction Reaction concerning a single chemical step, such as dissociation or recombination, as distinct from complex reactions which occur through a series of separate elementary reactions. Equation of state Relation between the thermodynamic properties of a system. Equilibrium State of an isolated system which is specified by quantities which are independent of time. Isotherm Curve joining states for which the temperature is constant.

Kinetics Study of how chemical systems change, concerning the rate at which change occurs and the factors on which this rate depends. Also used to refer to the sequence of reactions by which a complex reaction occurs. Molecular beam Stream of molecules all traveling in the same direction in vacuum, used in studies of isolated molecules and to examine the dynamics of single molecular collisions. Normal mode One of a set of coordinates of a system that can be excited while the others remain at rest. Order of a transition Transition from one thermodynamic phase to another is of order n if the first discontinuous derivative of the free energy with respect to the thermodynamic variables is of order n. Phase, thermodynamic Region of the space specified by the thermodynamic degrees of freedom of system separated from the remainder by a clearly defined surface and within which the thermodynamic properties differ from those of the remainder. Rate constant Constant that gives a measure of the rate of a chemical reaction; the proportionality constant between the rate of product formation and the product of the reagent concentrations. If the rate expression involves N molecules of the same reagent, the concentration must be raised to the power of N .

59

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160 Reversible process A process in which a system changes from one thermodynamic state to another is reversible if the thermodynamic variables in the inverse process pass through the same values but in the inverse order and in which all exchanges of heat, work, etc., with the surroundings occur with reverse sign and in inverse order. Spectroscopy Analytical technique concerned with the measurement of the interaction of energy and matter, the development of instruments for such measurements, and the interpretation of such information for analysis of the structure or constituents of a system. Techniques, such as mass spectrometry, which do not involve energy are often called spectroscopic because they also yield output scans in the form of spectra. Spectrum Intensity of a signal due to a process such as optical absorption or emission displayed as a function of some varying characteristic such as wavelength, energy, or mass. Also used in quantum mechanics and applied mathematics to specify the pattern of eigenvalues of a linear operator and in electrodynamics to specify the range of frequencies of electromagnetic radiation. State function In thermodynamics a variable is a state function if, when all the thermodynamic variables are specified, it has a unique value. As a result, the change in any state function in a reversible cyclic process must be zero. Thermodynamics Study of the changes in the properties of a system, usually as a result of changes in temperature or pressure.

PHYSICAL CHEMISTRY is the branch of chemistry in which experimental and theoretical techniques of physics are used to investigate and interpret chemical phenomena. Physical chemistry has its origins in the late nineteenth century, where it was largely concerned with the application of classical thermodynamics to chemistry. Modern physical chemistry is based more on quantum and statistical mechanics, which were developed only during the twentieth century. The branch of physical chemistry that employs twentieth century physical techniques is sometimes called chemical physics, with physical chemistry being regarded as concerned only with classical techniques. However, the distinction is artificial. Physical chemistry and chemical physics are really the same field and are considered as such here. Experimental physical chemistry has been revolutionized by relatively recent advances in electronic instrumentation, vacuum technology, and by the introduction of lasers. Equally, advances in computer power have had a great impact on theoretical studies, with an increasing emphasis on computer simulations and the detailed modeling of chemical systems.

Physical Chemistry

I. CLASSICAL MECHANICS The dynamics (i.e., motion and energetics) of molecules and atoms and, at a more fundamental level, electrons, are the origin of chemical phenomena. Prior to the twentieth century it was believed that all of the dynamics of a system, whether astronomical or molecular, were described by Newton’s equation of motion. dv , (1) dt where F is the force, v the velocity, t the time, and the proportionality factor, m, the mass of the particle or object. The force and velocity are vectors, whose direction and magnitude are both of importance. In complex problems it is often preferable to reformulate classical mechanics in terms of a scalar, such as the energy, which is characterized only by its magnitude. This gives rise to the Lagrangian and Hamiltonian equations of motion. The latter equations are of most interest here and are ∂qi ∂ = , ∂t ∂ pi (2) ∂ pi ∂ , =− ∂t ∂qi F=m

where pi and qi are generalized momenta and positions, respectively, and , the Hamiltonian, is the total energy of the system using momenta and position as variables. The space spanned by the pi and qi is called phase space. The dynamics of a system are described by a path in phase space. If the system is periodic, as is the case for electrons in an atom or molecule, then the path is a closed orbit in phase space. The advantage of the Hamiltonian formulation in physical chemistry is the fact that all variables are treated on an equal footing. However, the Hamiltonian and Newtonian formulations of classical mechanics are completely equivalent.

II. QUANTUM MECHANICS A. Duality of Matter and Energy; Uncertainty Principle During the nineteenth century it was established that matter consists of atoms and chemically bound aggregates of atoms called molecules. At first, it was thought that atoms were structureless. However, by about the turn of the century, it was shown that atoms were miniature solar systems consisting of a positively charged nucleus, whose structure is irrelevant for chemical phenomena, which contains nearly all the atomic mass, and negatively charged “planetary” electrons which orbit the nucleus.

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Because of this, nuclei are slow moving, virtually motionless, on the timescale of electronic motions. For many chemical phenomena, the nuclei can be regarded as fixed in space. Only the electronic dynamics need be considered. This simplification is called the Born–Oppenheimer approximation. At first it was thought that the electronic motions could be described by classical mechanics. However, it is impossible to describe the microscopic world by classical mechanics. It became apparent, from for example the photoelectric effect, that electromagnetic radiation was not always wavelike, but, in some circumstances, consisted of discrete, particle-like, units of magnitude called quanta, E = hν,

(3)

where ν is the frequency of the radiation and h = 6.626 ×10−34 J sec is Planck’s constant. Conversely, it became apparent, through for example the diffraction of electrons, that matter was not always particle-like but, in some circumstances consisted of waves of wavelength λ = h/ p.

(4)

Such waves can be used to probe the structure of crystal surfaces, through low-energy electron diffraction (LEED) or atomic beam diffraction. The latter is usually confined to He atoms, but even Ar atom diffraction can be discerned in favorable cases. In other words, there is a duality of matter and energy. Whether the particle-like or the wavelike character of matter/energy is dominant depends on the experiment. In fact, the experiment itself interacts with the matter/energy and defines some aspect of the system at the cost of indefiniteness of some other aspect. This uncertainty principle was made precise by Heisenberg who showed that even under the most ideal circumstances pi qi = h/4π.

(5)

If the experiment defines the particle character of the system, the uncertainty of the positions, qi , is small and the uncertainty in momenta, pi , or frequency is large. However, if the experiment defines the wave character of the system, the reverse is true. The momenta pi and positions qi are called conjugate variables. Energy and time are also conjugate variables so that Et = h/4π.

(6)

Classical mechanics, where there is no uncertainty, is a limiting case in which the magnitudes of the variables are large compared to h. As a result, classical mechanics is appropriate for large macroscopic bodies.

B. Wave Equation In the earliest formulation of quantum mechanics, classical mechanics was assumed valid with the exception that some periodic variables were quantized (i.e., had discrete values). Their values could be determined by integrating the pi over their orbits in phase space, according to  pi dqi = n i h, (7) where n i is an integer called a quantum number. The integral in Eq. (7) over a closed path is called a phase integral. However, as the implications of the duality of matter and energy and the uncertainty principle were accepted, it became apparent that one could refer only to the probability of finding the system in some configuration. Just as the wave nature of radiation meant that there was a wave equation for radiation, the wave nature of matter implied the existence of a new wave equation. This wave equation, called the Schr¨odinger equation, is formulated as an eigenvalue equation (eigen ≡ characteristic or proper) where the Hamiltonian operator “operates” on the probability function or wave function or eigenfunction, ψ, to give the energy eigenvalue, E, times ψ. Thus, the wave equation is ψ = Eψ

(8)

The wave function has the property that |ψ|2 gives the probability of the system having the eigenstate whose energy is E. The Hamiltonian operator is formed by replacing p j in the classical Hamiltonian by √ the operator −(h/i)(∂/∂q j ), where h = h/2π and i = − 1. The q j remain unchanged. Interestingly, the earlier phase integral formulation [Eq. (7)] becomes the approximate Wentzel–Kramers–Brillouin (WKB) method of solution of Schr¨odinger’s equation and remains useful in many problems in the sense that differences between quantized values of a phase integral are integral multiples of h except that there may be a zero point value of the phase integral given by a fractional value of h. C. Hydrogen-Like Atom; Electronic Transitions One of the first systems to which quantum mechanics was applied was the hydrogen-like atom consisting of a single electron orbiting a nucleus of charge Z e0 . The energy eigenvalues or levels are obtained by solving Schr¨odinger’s equation and are given by 2π 2 m 0 Z 2 e04 hc R Z 2 = − , (9) n2h2 n2 where e0 is the charge of an electron, n is an integer, c is the velocity of light, and R is called Rydberg’s constant. Strictly speaking we should not use the electronic mass m 0 E =−

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FIGURE 1 Energy-level diagram for the hydrogen atom showing allowed transitions. Insert displays the Balmer series spectrum that can be observed in the visible.

in this formula but the reduced electronic mass. However, the effect is small. This means that electronic transitions between states characterized by n 1 and n 2 emit or adsorb energy or radiation whose wavelength is given by   1 1 1 2 (10) =Z R 2− 2 . λ n1 n2 An energy-level diagram for the hydrogen atom is shown in Fig. 1, which also displays some of the allowed transitions. The energy levels of the hydrogen-like atom are degenerate because more than one state corresponds to a specific value of n (the principal quantum number). These degenerate states are characterized by the quantum numbers l and m, which characterize the spherical harmonics of the wave function. For each value of n there are n values of l(l = 0, . . . , n − 1) and for each value of l there are 2l + 1 values of m(m = − l, −l + 1, . . . , 0, . . . , l − 1, l) giving n 2 values of l and n for each value of n. D. Many-Electron Atoms; Pauli Principle; Electron Spin To understand more complex atoms containing many electrons, we must solve the many-electron Schr¨odinger equation. Even in classical mechanics, many-body problems are difficult, so it is not surprising that many-electron quantum mechanics, usually called quantum chemistry,

Physical Chemistry

is an active research field today. However, an understanding of the electronic structure of atoms can be understood in terms of the aufbau (building up) principle whereby electrons are added one at a time to the atom. However, two additional facts should be mentioned. First, the quantum numbers n, l, and m are not sufficient to specify the state of an electron. The spin of an electron must also be specified. Electrons can have one of two spins (say, up or down). This is specified by the spin quantum number, s = ± 12 (so that there are 2|s| + 1 = 2 spin states). Thus, the state of an electron is specified by n, l, m, and s. For historical reasons the values l = 0, 1, 2, 3, 4, . . . are specified by the spectroscopic notation s, p, d, e, f, . . . . Thus, an electron might be said to be in a 1s(n = 1, l = 0) or a 2 p(n = 2, l = 1), state. Similarly, states for the whole atom are termed, S, P, D, E, . . . corresponding to L = 0, 1, 2, 3, . . . , where L is the total orbital angular momentum for the atom, which is arrived at by combining the orbital angular momenta of the individual electrons. Second, electrons obey the Pauli exclusion principle. This means that only one electron can occupy a quantum state. Thus, as the aufbau principle is employed, the electrons are added one at a time to the state of lowest energy, each state being filled by one electron. From these principles a simple understanding of the periodic table is gained. Each electronic shell is specified by n and contains 2n 2 states. Thus, the first row of the periodic table corresponds to n = 1 and contains 2 elements (H and He). The electronic configurations of these elements are denoted 1s (H) and 1s 2 (He). The superscript indicates the number of electrons with the given value of l. The second row corresponds to n = 2 and contains 8 elements with configurations 1s 2 2s, 1s 2 2s 2 , 1s 2 2s 2 2 p, . . . , 1s 2 2s 2 2 p 6 . The subsequent rows contain 8 columns even though 2n 2 exceeds 8 because the energy of the electronic states is ordered (approximately) 1s/2s 2p/3s 3p/4s 3d4p/5s 4d 5p/6s 4f 5d 6p/7s . . . so that the third row is still filled with 8 elements. With potassium (Z = 19) the nineteenth electron goes into a 4s rather than a 3d level. The transition elements are regarded as occupying one position in the table since the outer shell configuration does not change as the d electrons are added and, as a result, they have similar chemical properties. As n increases, not only must the d electrons be accommodated in single positions, but the f electrons must also be accommodated in a single position, so that the table becomes more complex. However, the underlying principles are simple. E. Molecular Systems: Chemical Bond Many-electron molecular systems are even more complex than atomic systems. The theory of such systems

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is an active field of research. The potential energy terms in the wave equation for the molecule involve not only the Coulomb repulsions between the electrons and the Coulomb attractions of the electrons and nuclei but also the Coulomb repulsion between the nuclei. Given the repulsion between the nuclei, we are inclined to ask how atoms form a chemical bond in a molecule. A simple answer can be obtained by considering the one electron hydrogen molecule ion H2+ . The wave functions of each H atom separately are ψls (A) and ψls (B). An approximation to the H2+ wave function can be obtained by forming a molecular orbital from a linear combination of these two wave functions ψb = ψls (A) + ψls (B)

and they are termed doublet states. The former three have zero spin, a multiplicity of 1, and are termed singlet states, while states with a multiplicity of 3 and 4 are termed triplet and quartet states, respectively. Thus, the ground electronic state of NO is written as 2 1/2,3/2 , where the 2 refers to the doublet nature of the state, the  to  = 1, and the 12 and 32 refer to values of  which arise from the two possible ways that the spin and orbital angular momentum can add. In addition to electronic energy states, molecules posses both rotational and vibrational energy levels. Assuming a fixed distance between two atoms (rigid rotor approximation), the Schr¨odinger equation yields for the allowed rotational energy levels of a diatomic molecule h2 J (J + 1), (12) 8π 2 I where I is the moment of inertia of the molecule, J the rotational quantum number and the quantity (h 2 /8π 2 I ) is termed the rotational constant for that particular electronic state of the molecule (usually given the symbol B). Vibrational energy levels can be estimated by inserting the Hooke’s law potential energy, U (r ) = 0.5k(r − re )2 , in the Schr¨odinger equation (harmonic oscillator approximation). This yields eigenvalues, E v , for the permissible energy levels, of      E v = (h/2π ) k/µ v + 12 = hν0 v + 12 , (13) Er ≈

or ψa = ψls (A) − ψls (B).

(11)

In the first (bonding) orbital, the electron is concentrated between the nuclei, and is simultaneously attracted by both nuclei resulting in a lower electronic energy which more than offsets the repulsion of the nuclei. In other words when the electron is between the nuclei it acts as a cement holding them together. There is zero probability of finding the electron between the nuclei in the second (antibonding) orbital. As a result, a chemical bond is not formed by this orbital. F. Molecular Systems; Energy Levels When atoms combine to form molecules, the individual atomic energy levels give rise to discrete electronic energy levels or states of the molecule. The number of these molecular electronic states far exceeds those of the individual atoms because of the many different ways in which the atomic states can be combined. Electronic states of molecules are classified in terms of several molecular quantum numbers in a manner analogous to atomic electronic states. For a diatomic molecule these include the electronic orbital angular momentum, l, its component along the internuclear axis, λ, and the corresponding quantities for the molecule as a whole, L and . Just as l = 0, 1, 2, . . . gives rise to s, p, d, . . . electron states and L = 0, 1, 2 . . . gives S, P, D, . . . atomic states, so λ = 0, 1, 2 . . . yield σ, π, δ . . . electron states and  = 0, 1, 2 . . . correspond to , ,  . . . molecular states. The component of the total electronic angular momentum along the internuclear axis, , is also of importance. The ground electronic states of H2 , O2 , and N2 are  states, while those of OH and NO are  states. These latter two molecules have open valence shells with net spin of 12 , so that the multiplicity, S, of these states is 2 (S = 2S + 1),

where µ is the reduced mass of the system [µ = m 1 m 2 /(m 1 + m 2 )] and ν0 is known as the fundamental vibrational frequency. The smallest amount of vibrational energy a molecule can possess is thus ν0 /2, termed the zero-point energy. Rather than the simple Hooke’s law potential we may consider more realistic molecular potential energy curves such as a Morse potential given by U (r ) = De {1 − exp[−β(r − re )]}2 ,

(14)

where De is the dissociation energy of the molecule and β is related to De and ν0 . This leads to a similar expression for E v , but with an additional quadratic term in (v + 12 ), which is negligible for low vibrational energies. Similarly, an accurate treatment of molecular rotation leads to additional terms in higher powers of the quantity {J (J + 1)}. G. Spectroscopy Atoms and molecules can adsorb and emit radiation to change their internal energy states. The electronic transitions of the hydrogen-like atoms have already been mentioned. The quantization of the energy levels restricts the possible wavelengths of the radiation to discrete spectral lines. Only certain transitions are allowed and these are given by separate selection rules for electronic,

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164 vibrational, and rotational transitions. For example, in atomic transitions l can only take values of ±1. In addition, each allowed transition has an associated transition linestrength. In electronic spectroscopy involving transitions between electronic states, these are termed transition moments, Franck–Condon factors and H¨onl– London factors for electronic, vibrational, and rotational transitions, respectively. The spectrum of absorption lines associated with a given species can be used as a fingerprint for identification and quantification in the process of absorption spectroscopy. An absorption spectrum is usually recorded with an absorption spectrometer or, more recently, with a laser. Similarly, emission spectroscopy is concerned with the radiation that is emitted by an excited atom or molecule. Here radiation is generally spectrally resolved using a spectrograph. In the various spectral regions, these techniques may be referred to as vacuum ultraviolet (VUV), ultraviolet (UV), infrared (IR), and microwave spectroscopies. Microwave photons have energies of less than 1 meV and are associated with rotational transitions, while infrared photons have energies of up to 1 eV and are involved with vibrational transitions. More energetic radiation from the visible to VUV is usually associated with electronic transitions. In addition to identifying species, these and related spectroscopic techniques are frequently employed to obtain detailed structural information, such as bond lengths and bond angles, and to study the flow of energy in chemical reactions. Other spectroscopic techniques include: 1. Raman spectroscopy, which concerns the spectral analysis of radiation scattered by an atom or molecule. Recent developments are coherent anti-Stokes Raman spectroscopy (CARS) and surface enhanced Raman spectroscopy (SERS), both of which are sensitive laser-based techniques, and examples of laser spectroscopy. 2. Photoelectron spectroscopy, which is based on kinetic energy analysis of electrons ejected from an atom or molecule by an enegetic photon and provides information on the binding energies or ionization potentials of the ejected electrons. Recent developments include X-ray photoelectron spectroscopy (XPS) of surfaces, and the use of lasers as radiation sources. 3. Auger electron spectroscopy (AES), which involves the electron impact ionization of an atom to give an excited electronic state that decays by emission of a second electron whose energy is characteristic of the atom. This is most commonly used in surface analysis. 4. Spin resonance spectroscopy, which concerns the application of a magnetic field to split energy states associated with electron or nuclear spin orientations. Electron spin resonance (ESR) involves absorption of microwave

Physical Chemistry

radiation, while nuclear magnetic resonance (NMR) is a radio-frequency technique. 5. M¨ossbauer spectroscopy, which involves the resonant absorption of a γ -ray photon by a nucleus. The resonant condition is achieved via the Doppler effect, by sweeping the velocity of a sample relative to the source. The chemical environment of the nucleus causes characteristic frequency shifts.

III. STATISTICAL THERMODYNAMICS A. First and Second Laws of Thermodynamics; Entropy The first law of thermodynamics states the conservation of energy, δ Q = dU + δW,

(15)

where δ Q is the heat absorbed by the system, dU is the change in internal energy of the system, and δW is the work done by the system. The second law of thermodynamics states that heat cannot pass from a cold reservoir to a hot reservoir without the application of work. The change in entropy, d S, is just δ Q/T , where T is the temperature. The factor 1/T is an integrating factor that transforms δ Q into an exact differential just as 1/v 2 transforms vdu − udv into the exact differential d(u/v). Because the change in entropy, d S = δ Q/T , is an exact differential, the change in entropy in a reversible cyclic process is zero. The entropy of a thermodynamic state is a well-defined single-valued function and the entropy is said to be a state function. An equivalent statement of the second law of thermodynamics is S ≥ 0,

(16)

where the change in entropy is zero for a reversible cyclic process. The entropy increases in an irreversible process. B. Free Energy; Experimental Measurements The first and second laws of thermodynamics can be combined to give T d S = d E + δW.

(17)

If the only work done by the system results from an expansion d V or a change in the amount d Ni of the constituents, then Eq. (14) becomes T d S = dU + pd V −

m 

µi d Ni ,

(18)

i=1

where p is the pressure, V the volume, µi the chemical potential of constituent i, and Ni the amount or concentration

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of constituent i. The chemical potential of a gas is the value of the Gibbs function for one mole. In a purely mechanical system, equilibrium is achieved when the energy is a minimum. In a thermodynamic system, entropy changes as well as energy changes must be considered. At constant temperature, volume, and concentration, the Helmholtz free energy,

the so-called gas constant (R) divided by Avogadro’s number, N A = 6.022 × 1023 mol−1 , the number of molecules in a mole. The thermodynamic properties are related to the energy levels E i through the partition function Z defined by  Z = e−β A = exp(−β E i ), (23)

A = U − T S,

where the sum is over all the energy levels of the system. Thus, to obtain the thermodynamics of a system, all that is required is that Schr¨odinger’s equation for the system be solved and the partition function summed. For most systems this is a difficult task, often impossible without some approximations. There is also a relation between the entropy and the microscopic configurations of the system. The entropy is proportional to the logarithm of the number of accessible states, , of the system. Thus,

(19)

is a minimum at equilibrium, whereas at constant pressure and temperature the Gibbs free energy G = A + pV =

m 

µi Ni

(20)

i=1

is a minimum at equilibrium. For a dilute gas, where the perfect gas law ( pV = n RT ) applies, the value of µ per mole at a pressure p is µ( p) = µ◦ + RT ln( p/atm),

(21)

where µ◦ is the value of µ at 1 atmosphere, which is the pressure at which the standard state is established. The value of µ◦ is often taken as zero for the elements. In Eqs. (20) and (21) R( = 8.3144 J mol−1 K−1 ) is the so-called gas constant. A mole of any substances is the amount with a mass in grams equal to its molecular weight in atomic mass units, so that a mole of molecular hydrogen has a mass of 0.002 kg. The value of G or µ can be determined for some pressure p by measuring the volume of the gas as a function of pressure up to p and integrating. Thus,  1 p µ( p) = µ◦ + V ( p)d p, (22) n p0 where p0 is the pressure of the standard state (usually 1 atm). The free energy of a condensed phase can be related to that of a dilute gas through the vapor pressure, the pressure of the gas in equilibrium with the condensed phase. Once the free energy of the condensed phase has been established, values for other states can be obtained by measuring pressure or energy through a sequence of states leading to the desired state. C. Statistical Mechanics; Partition Function The thermodynamic properties of a system result from the dynamics of its molecules. Since even a three-body system is difficult, statistical methods must be employed to treat the large number of molecules in a thermodynamic system. The fundamental result in statistical mechanics is the fact that the probability of a system occupying the energy level E i is proportional to the Boltzmann factor, exp(−β E i ), where β = 1/kT and k = 1.3804 ×10−23 J K−1 is the Boltzmann constant. The Boltzmann constant is

i

S = k ln .

(24)

Equation (24) is called Boltzmann’s relation. At absolute zero, the system is in its ground state, and the number of accessible states is unity. Thus, the entropy of a system tends to zero as the temperature goes to zero. This is called the third law of thermodynamics. The Boltzmann relation provides a statistical interpretation of the entropy. The greater the number of accessible states, the less our knowledge of the system and the more randomness or disorder in the system. This entropy is a measure of disorder. The tendency of the entropy to increase reflects the tendency of thermodynamic systems to increase in disorder just as an initially ordered deck of cards increases in disorder during a game of cards. If the system is classical, the energy levels merge into a continuum and an important simplification results. The sum in the partition function becomes an integral. Moreover, if the kinetic energy degrees of freedom (i.e., the momenta) are independent of the potential energy or internal degrees of freedom (i.e., the generalized positions) then the momenta can be integrated immediately. For the particular case in which there is only translational motion.  λ− 3N Z= exp(−β) dr1 · · · dr N , (25) N! where λ = h/(2π mkT )1/2 , N is the number of molecules in the system, and  = (r1 , . . . , r N ) is the potential energy. The factor N ! is required because states that differ only by an interchange of molecules are not distinguishable. From Eq. (20), it follows that the average kinetic energy of the system is KE = 32 N kT

(26)

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In other words, there is a statistical relation between the temperature and the average motion of the molecules. The greater the temperature, the more rapidly the molecules move and the greater their kinetic energy. The problem of predicting the thermodynamic properties of such a classical system becomes the problem of evaluation of the configuration integral, the integral over exp(−β). This is still a difficult task. In general, it can be done only through computer simulations (Monte Carlo and molecular dynamics methods). However, there are a few simple approximations which are helpful. D. Perfect and Imperfect Gases The simplest system is the perfect gas in which the molecules do not interact, i.e.,  = 0. Thus, the configuration integral is just the volume raised to the power N . Using Stirling’s approximation, N ! = (N /e) N , Z = λ− 3N (eV /N ) N

(27)

and pV = N kT.

(28)

If the molecules interact, then the problem is more complex. The gas is called imperfect because there are deviations from the perfect gas result. These deviations can be written as a power series in the density, ρ = N /V , called a virial series. For example, if the molecules are hard spheres such that the molecules collide elastically but exert no attractive forces on each other, then βp /ρ = 1 + ρb + 58 (ρb)2 + · · · .

(29)

For hard spheres, the coefficients of ρ , called virial coefficients, are independent of the temperature. For more complex gases the virial coefficients are temperature dependent. The virial coefficients can be related to the forces between the molecules. However, both the relation itself and the evaluation of the resultant integrals rapidly become complex as the power n of ρ n increases. In general, it is difficult to go beyond n = 4. The pressure of the hard-sphere gas exceeds that of the perfect gas at the same temperature and density. To a first approximation, this can be thought to be a result of a reduction in the volume available to the molecules because of the volume occupied by the molecules themselves. The hard spheres can be said to have less free volume than the perfect gas. The hard-sphere gas cannot be liquified. Liquification requires attractive forces. Attractive forces can also cause the pressure to be less than the perfect gas result. Interestingly, attractive forces are not required for the existence of a solid phase. If the hard sphere gas is compressed,

computer simulations show that it will freeze and exist as a close-packed solid. E. Liquids; van der Waals Theory; Critical Point; Renormalization Group In contrast to a gas, a liquid need not fill space but can exist in equilibrium with its vapor with a surface separating the liquid and vapor. The pressure at which the equilibrium occurs is called the vapor pressure. Below the vapor pressure, liquid will evaporate until equilibrium is reached. For pressures greater than the vapor pressure, there is no interface between liquid and vapor. The liquid fills the container and there is no clear distinction between liquid and gas. The liquid under pressure can be heated at constant volume to a temperature greater than the critical temperature (the highest temperature at which liquid–vapor coexistence can occur), then allowed to expand and cool to the original temperature and pressure without any transition from liquid to gas being observed. A continuity of states between liquid and gas is said to exist. This is illustrated in Fig. 2. The liquid–gas phase can be referred to by the single term fluid. Thus, a theory of the liquid state is of necessity also a theory of an imperfect gas. The earliest theory of the liquid state is that of van der Waals. Although more than a century old, with slight modifications it is viable today. The idea of van der Waals was that a liquid behaved as a hard sphere gas except that the pressure must include the internal pressure due to the attractive forces of the molecules in the liquid. It is reasonable to assume that the contribution of the internal pressure to the free energy is proportional to the density. Thus,

n

FIGURE 2 Phase diagram of a typical simple liquid. The shaded region is not thermodynamically stable.

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p = p0 − ρ 2 a ,

(30)

where p0 is the pressure of the hard-sphere gas and a is a constant depending on the nature of the attractive forces. In its original formulation, the van der Waals theory was only qualitatively successful because van der Waals approximated p0 by the perfect gas expression with a reduced free volume, i.e., βp0 1 = . (31) ρ 1 − ρb This expression gives only the second hardsphere virial coefficient correctly and seriously overestimates p0 . Much more satisfactory results can be obtained from the approximation βp0 1 = (32) ρ (1 − η)4 where η = 14 ρb. The van der Waals theory predicts that the equation of state of a liquid can be expressed in a universal form if the following reduced variables, T ∗ = bkT /a, p ∗ = b2 p/a, and ρ ∗ = ρb, are used. This is called the law of corresponding states. As is illustrated in Fig. 3, the theory also predicts that below the critical temperature there is a first-order phase transition between the liquid and vapor accompanied by a discontinuous change in the density ρ. At the critical temperature the transition becomes second order since the liquid and vapor have become identical. For temperatures above the critical temperature, there is no phase transition. In the van der Waals theory, the critical point occurs when    2  ∂p ∂ p = = 0, (33) ∂ρ T ∂ρ 2 T i.e., the pressure isotherms have a point of inflection at the critical point. Modern theories show that the van der Waals theory is a first approximation to a systematic approach, called perturbation theory, in which the pressure is obtained as a power series in 1/T . In the van der Waals theory, the first two derivatives of p at constant T with respect to the density vanish at the critical point. This is not just a prediction of the van der Waals theory. Any theory in which the equation of state is analytic at the critical point will yield this result. By analytic, it is meant that the pressure can be expanded as a power series about the critical point. Experimentally, the equation of state is not analytic at the critical point. The exponents in an expansion near the critical point are generally not integers. At least one, and possibly two, more derivatives of p with respect to the density at constant T vanish near the critical point. There has been a great deal of work on the fascinating properties of the equation of state in the vicinity of the critical point. The most far

FIGURE 3 Typical pressure isotherms as a function of the volume V in the van der Waals theory. The shaded region is not thermodynamically stable. Here Tc is the critical temperature.

reaching is the renormalization group approach in which a group of successive transformations is applied to the liquid, yielding ultimately a renormalized system in which only the long-range correlations typical of the critical point remain. In this system the critical point properties can be examined. F. Mixtures Mixtures of two gases or liquids can be treated by the same techniques as liquids. The analog of a perfect gas is the ideal mixture, where molecules of the components are very similar so that the partition function can be written (for the two-component case) as Z=

N! Z1 Z2 N1 !N2 !

where N = N1 + N2 and Ni is the number of molecules of species i. From this, it can be deduced that the partial pressures of the components are proportional to their concentrations. This result is known as Raoult’s law. The factor N !/N1 !N2 ! gives rise to the entropy of mixing S = −N k[x1 ln x1 + x2 ln x2 ], where xi = Ni /N .

(34)

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For a nonideal mixture, the original approach of van der Waals is still useful. In this approach N! Z  N1 !N2 !

Z=

(35)

where Z  is obtained from the van der Waals equation of state of a liquid, preferably with the more satisfactory expression for p0 , with a and b replaced by the concentration-dependent quantities  a  = xi x j ai j (36) ij

and b =



xi x j bi j

(37)

ij

√ with ai j  aii a j j taken as a parameter. Assuming that the spherical cores of the molecules do not overlap, then  bi j =

1/3 3

1/3

bii + b j j 2

.

(38)

This approach is satisfactory for mixtures of nonelectrolytes. The situation is a little more complex for mixtures of electrolytes because of the long range of the Coulomb potential. However, each ion in the mixture tends to be surrounded by ions of opposite charge, which causes the potential to decay exponentially with a decay factor, κ, the Debye parameter, which is proportional to the square root of the product of the density and T −1 . As a result, an appropriate expansion parameter for electrolytes is κ, or T −1/2 , which is different from nonelectrolytes, where T −1 is the expansion parameter. These ideas become quantitative in the Debye–H¨uckel theory. G. Solids In contrast to the disorder of gases and liquids, there is translational order in crystals. Disordered or amorphous solids (i.e., glasses) exist which lack this order. However, they are really highly viscous liquids. This translational order is such that the entire structure, or lattice, can be generated by repeated replication of a small regular figure, termed the unit cell. The planes of any crystalline structure can be specified using Miller indices, which also serve to identify single crystal faces. Miller indices are obtained by determining the intercepts of the plane with the unit cell axes in terms of the length of the cell in that direction, taking the reciprocal, and normalizing so the indices are all integers. The ordered structure, or lattice, of a solid can be determined by X-ray or neutron diffraction studies, in which a beam of X-rays of neutrons is scattered from the sample to

produce a diffraction pattern, which can be analyzed to reveal the crystal structure of the sample. All crystal lattices can be classified into 14 Bravais lattices belonging to seven systems. For example, the simple cubic, face-centered cubic and body-centered cubic lattices are the 3 lattices of the cubic system. Cubic and hexagonal close-packed structures have the structure of tightly packed spheres where each sphere touches 12 neighbors, 6 in the same plane and 3 above and 3 below. These two close-packed structures differ in the placement of successive planes or layers. For the cubic case, a third layer is laid down to reproduce the first layer, so that the structure could be represented by ABABAB. . . . For hexagonal close packing, the third layer is again displaced, corresponding to ABCABC. . . . No theory of freezing exists. That is, there is no partition function that encompasses both the solid and fluid phases. However, separate theories of solids and fluids can be developed and their solid–fluid coexistence examined. To that extent theories of melting or freezing exist. Since freezing can occur in the hard-sphere system, no critical point is expected for freezing. This transition is expected to be first order at all temperatures, as illustrated in Fig. 1. If a solid were classical, the heat capacity would be 3N k. This is indeed the case at high temperatures and is called the law of Dulong and Petit. However, the experimental heat capacity goes to zero at low temperatures. This can be explained by regarding the solid as a collection of quantized oscillators. The only difficulty is to determine the spectrum of frequencies of the oscillators. For many purposes, the solid can be regarded as an elastic continuum. The result is the Debye theory. If something more sophisticated is needed one must solve for the normal modes of the crystal, i.e., the method of lattice dynamics. The conduction of electricity in a metal is due to the presence of free or quasi-free electrons in the metal. Classically, free electrons would contribute 3nk/2 to the heat capacity, n being the number of free electrons. However, experimental evidence indicates that the electrons do not contribute significantly to the heat capacity of a metal. The reason for this is the exclusion principle. Although the electronic gas is in its ground state, because of the exclusion principle the electrons can each occupy one energy level. The electrons occupy the levels up to a maximum energy, called the Fermi energy, εF . Only the small number of electrons with energies near εF can be thermally excited and, as a result, the electronic heat capacity is small. If the exclusion principle is taken into account, treating the conduction electrons as free describes many of the electronic properties of a metal. To treat metals in a more sophisticated manner and to account for semiconductors, the structure of the solid must be included. If this is done, the electrons are not free but are restricted to bands of energy.

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an electrolyte, for example, electrons may leave it to reduce cations in the solution, giving it a net positive charge and making the solution slightly negative. These charges arrange themselves at the interface in two layers, known as the double layer. The most important property of this double layer is the variation of potential in its vicinity. The potential governs the rate at which ions can be transported through the interface, and so controls the rate of electrochemical processes.

FIGURE 4 Image of a single atom of xenon sitting on the surface of a platinum single crystal at 4 K, obtained with a scanning tunneling microscope. Xenon has an atomic radius of ∼1.2 × 10−10 m. Picture provided by Don Eigler of the Almaden Research Center.

H. Interfaces The study of interfaces is becoming an increasingly important area of physical chemistry. Of particular interest are gas–solid and gas–liquid interfaces. Both of these can now be imaged directly with scanning tunneling microscopy (STM). Here the contours of a surface may be determined by sensing the small current that tunnels across a vacuum gap to an atomically sharp tip as it is scanned across it. Figure 4 illustrates the remarkable resolution of this instrument. It shows a single atom of xenon sitting on the surface of a platinum crystal at 4 K. For the case of the liquid–solid interface, STM can be used to study biological samples or the electrodes of an electrochemical cell. Other important interfaces are those between a liquid and its vapor and between two imiscible liquids. Consider, for example, the physical adsorption of a gas by a solid. If the solid is regarded as a giant sphere, the adsorption of the gas can be regarded as the interaction of a gas with a single infinitely large molecule dissolved in that gas. If the simplest form of the van der Waals theory of mixtures is applied to that system, then the adsorption isotherm is just  = ρβ

 a12 , β(∂ p /∂ρ)T

(39)

 is a constant. At low densities β(∂ p /∂ρ)T = 1 where a12 and the adsorption is proportional to the density (Henry’s law). However, at higher densities β∂ p /∂ρ is a function of the density, and deviations from Henry’s law are observed. Especially interesting is the region near the critical point of the gas where (∂ p /∂ρ)T → 0 and singularities in the adsorption are observed. Interfaces between dissimilar materials may also become electrically polarized, with a separation of charge occurring at the interface. When a metal is placed into

IV. KINETICS AND DYNAMICS The previous sections have dealt only with the equilibrium properties of a system of molecules. Such properties tell us nothing of the time required for equilibrium to be reached or about the dynamical properties of these systems. The rate at which change occurs is the province of kinetics. The detailed manner in which chemical forces act to bring about atomic and molecular motion is the province of chemical dynamics. This section deals with the motions of atoms and molecules and the processes associated with chemical change. A. Kinetic Theory of Gases The kinetic theory of gases assumes that molecules have negligible size compared to their separation, are in continuous random motion, and interact only via elastic scattering. These postulates permit the calculation of molecular speed and velocity distributions. The probability that a molecule has a speed between v and v + dv is found to be d F(v) = 4π (m/2π kT )3/2 v 2 exp(−mv 2 /2kT )dv,

(40)

where T is the gas temperature and m the molecular mass. This is the Maxwell distribution of molecular speeds. Figure 5 displays this distribution for nitrogen gas at 25 and 500◦ C. Notice that the velocities are in the range of hundreds to thousands of meters per second, which are typical of those for small molecules at ambient temperatures. Recent experiments using light pressure from lasers to slow down atoms have resulted in atoms moving with velocities comparable to walking speed (1 m/s) and below. Such slow species are ideal for spectroscopic studies, since their adsorption spectra are not blurred by the Doppler effect due to their motion. The Maxwell distribution of molecular speeds permits the evaluation of such important quantities as the pressure p exerted by a dilute gas and the collision frequency Z in the gas under given conditions. The pressure is then given by 2 p = 13 ρmvrms ,

(41)

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metals, obtaining the result that the ratio of the thermal and electrical conductivities is a universal constant times the temperature (the Wiedemann–Franz law). B. Transport Properties

FIGURE 5 Maxwell distributions of speeds for molecular nitrogen at 25◦ C (298 K) and 500◦ C (773 K). Arrows indicate v¯ and vrms for each case. The most probable velocity has been arbitrarily scaled to unity in each case.

while the collision frequency for a one-component gas is given by √ Z = 2σ v¯ ρ, (42) where ρ is the number of molecules per unit volume and σ is the collision cross section. The quantity vrms is the average square speed and is related to the average kinetic energy and thus to the temperature, so that  3kT vrms = v 2  = , (43) m where m is the molecular mass. This quantity differs from the average speed, 8kT v¯ = v = . πm Equation (43) is equivalent to Eq. (26). Knowing the collision frequency and the molecular speed, it is possible to estimate the mean free path between collisions, λ = v/Z , so that √ λ = 1/ 2ρσ, (44) which shows the expected behavior that λ must decrease as the diameter of the molecule increases or as the density of the gas increases. Helium and nitrogen have estimated self-collision cross sections of 0.13 and 0.31 nm2 and at a pressure of 1 Torr ( = 133.3 N m−2 = 1.32 × 10−3 atm) there are about 3 × 1022 molecules m−3 , giving mean free paths of ∼1.8 × 10−4 and 7.6 × 10−5 m for helium and nitrogen, respectively. At 25◦ C, these species have respective velocities of v¯ = 1254 and 474 m/sec, giving collision frequencies of 6.9 and 6.2 × 106 sec−1 per molecule. The kinetic theory of gases can also be applied to the free-electron gas to describe the transport properties of

The kinetic motion of molecules may cause them to change their spatial distribution through successive random movements. This is the process of diffusion, which is a transport property. Other transport properties include viscosity, electrical conductivity, and thermal conductivity. While diffusion is concerned with the transport of matter, these are associated with the transport of momentum, electrical charge, and heat energy, respectively. Transport is driven in each case by a gradient in the respective property. Thus, the diffusion rate of species A is given by Fick’s law, Jz (A) = −D[dρ(A)/dz]

(45)

where Jz (A) is the net flux of A molecules crossing unit area in the z direction and D is the diffusion coefficient; simply kinetic theory leads to D ≈ 13 v¯ λ

(46)

Derivation of other transport properties follow from similar relationships. The viscosity coefficient or viscosity of a gas is given by η ≈ 13 v¯ ρmλ,

(47)

while the thermal conductivity coefficient κ is given by κ ≈ 13 v¯ ρλCv = ηCv /m,

(48)

where Cv is the molar heat capacity of the gas at constant volume. Notice that since λ is inversely proportional to ρ, both η and κ are independent of the gas density. This will be true so long as λ is small compared to the dimensions of the apparatus. In solution, D is given by the Stokes–Einstein relation which relates D to the viscosity coefficient of the solution, η, and the effective hydrodynamic radius a, where D = kT /6π ηa

(49)

and by the Einstein–Smoluchowski relation: D = d 2 /2τ,

(50)

where τ is the characteristic time between jumps of distance d. More elaborate theories of transport phenomena make use of the Boltzmann transport equation or computer simulations.

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C. Chemical Kinetics; Activated Complex Theory The quantitative study of chemical reaction rates and the factors on which these rates depend constitutes the field of chemical kinetics. Chemical reaction rates can be expressed in terms of the rate of production of any reaction product or as the rate of decrease in the concentration of any reactant. The individual steps in a chemical reaction sequence are termed elementary reactions. A number of consecutive elementary reactions may be responsible for a given chemical change. If the observed rate is found to be proportional to the concentration of a given reactant raised to some power, α then α is said to be the order of the reaction with respect to that reagent. The sum of the orders over all reagents gives the overall order of the reaction. A complex reaction, involving a number of elementary steps, may have a noninteger order. Thus the order should not be confused with the molecularity of an elementary reaction, which is the number of reagent molecules involved in a single reaction step. The sequence of elementary reactions by which a reaction proceeds is termed the reaction mechanism, a term also used to describe the detailed process of bond breaking and formation in a single reactive collision. By way of illustration, consider the formation of nitrogen dioxide from nitric oxide and oxygen. This reaction is found to be third order, corresponding to 2NO + O2 = 2NO2

(51)

with a third-order rate law corresponding to d[NO2 ] (52) = keff [NO]2 [O2 ]. dt Here the constant keff is the reaction rate constant and square brackets indicate concentrations. If concentrations are given in moles per liter, the rate constants will have units of (mol/L)1−n sec−1 , where n is the order of the reaction. A likely mechanism for this process can be written in terms of the elementary steps: k1

NO + NO → N2 O2 k−1

N2 O2 → NO + NO k2

N2 O2 + O2 → 2NO2 ,

(53a) (53b) (53c)

which leads to the observed rate law if the first two steps are assumed to come to equilibrium prior to the third reaction, or if the steady-state approximation, which assumes that the rate of change of all concentrations is zero, is invoked. Reaction (53a) has a molecularity of 2 and is a bimolecular reaction, while reaction (53b) is an example of a unimolecular reaction, involving a single species. An important class of reaction mechanisms are those in which a reaction product from one step is a reagent

in a prior step. The species concerned is often a highly reactive molecule with a vacancy in its outermost shell of electrons, termed a free radical. Such processes are termed chain reactions. Chain reactions are very important in polymerization reactions, where a radical may add to another reactant to form another (larger) radical. In cases where more than one reagent species is formed as a product of a later step, the chain is said to be branched, and such branching chain reactions often lead to explosions. In other cases, explosions may occur as a result of a fast exothermic reaction which yields a net excess of energy in the form of heat and in a time too short for the energy to be dissipated. The increase in temperature then causes an increase in rate, and the cycle ends in a thermal explosion. In some mechanisms a species may be consumed in one step of a reaction only to be regenerated in a subsequent step. In cases where the presence of this species increases the overall reaction rate, it is termed a catalyst, which is defined as a species that increases the rate of a reaction without being consumed or changing the reaction products. A catalyst must increase the rate of both forward and backward reactions in any system at equilibrium and can be thought of as lowering E 0 (see below). An expression for bimolecular rate constants can be obtained by observing that along a reaction coordinate the energy surface consists of two wells, representing the reactants and products, separated by a saddle point, representing the maximum energy required to pass along the minimum energy path between reactants and products. If the height of this maximum, relative to the reactant well, is E 0 then only collisions where the energy exceeds E 0 can lead to reaction. Integrating the Boltzmann distribution of energies over all energies exceeding E 0 , shows that probability of a collision with energy in excess of E 0 is proportional to exp(−βE 0 ). This is consistent with the rate law of Arrhenius: k = A exp(−E 0 /kB T ),

(54)

where A is known as the pre-exponential factor, and the Boltzmann constant is written as kB here to avoid confusion; A can readily be estimated from collision theory, using the expression for the collision frequency for one reagent with another, Z 12 Z 12 = ρ1 ρ2 σ12 [8kB T /µπ ]1/2

(55)

which leads to A = Pσ12 N A [8kB T /µπ ]1/2 ,

(56)

where NA is Avogadro’s number, which converts ρ to molar units, and P is the so-called steric factor, which accounts for the fact that not all collisions lead to reaction. Alternatively, we can replace the product Pσ12 with σreac , where σreact is termed the reactive cross section.

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172 A more general treatment of detailed reaction rates is available in the activated complex theory of Eyring, which assumes that there is an intermediate state between the reactants and the products, called the activated complex or transition state which can be regarded as at least somewhat stable and which is in thermodynamic equilibrium with the reactants, thus permitting thermodynamics to be applied. Instead of an energy, we must use the free energy G (because the pressure is constant) in the exponential. This treatment yields   kT G ‡0 k=κ exp − h RT     kT S0‡ H0‡ =κ exp − exp − , (57) h R RT where R is the gas constant, κ is the transmission coefficient and G ‡0 S0‡ , and H0‡ refer to differences between the activated complex and the reactants. Extensions of this statistical thermodynamical approach to estimating reaction rates include the RRK and RRK M theories of unimolecular decay rates, and the information theoretic formulation of reaction dynamics. These theories are remarkably successful, although generally more successful at interpreting experimental data and correlating results than at deriving results a priori. D. Reaction Dynamics; Inelastic Collisions Kinetic measurements and knowledge of reaction products and yields can provide only rather limited insight into the molecular dynamics of chemical reactions. To understand the detailed manner in which atoms and molecules move together and come apart in the process of a chemical reaction, it is necessary to study the isolated elementary reactions in as much detail as possible. Such isolation is most often provided by a dilute gas environment. Ultimately, the hope is to understand reaction dynamics in terms of electronic structure and to be able to calculate this for a chosen system. The electronic structure or potential energy surface is the meeting ground between theory and experiment. Currently most studies seek to probe those factors, or states, which influence the rate of chemical reactions, such as vibration and translational energy, and to examine the manner in which energy and angular momentum are disposed among the product states for various processes. This is the area known as state-to-state chemistry. Molecular photodissociation is an ideal process for such studies and has been examined in considerable detail. This unimolecular event is sometimes considered as a “half collision,” where the absorption of a photon excites the system to a repulsive state that flies apart. A number of radiation sources have been employed for such photoly-

Physical Chemistry

sis experiments, including discharge lamps, flash lamps, and synchrotrons. However, most recent studies have concerned laser photolysis. The photofragments are detected, for example, by emission or laser spectroscopy, which provides information on the velocity and quantum-state distribution of the fragments, with respect to rotational, vibrational, and electronic states. It is even possible with the aid of femtosecond lasers to follow the photofragmentation process in real time. Such measurements can provide information on the shape of the excited state potential energy surface. In the last decade, researchers have taken these ideas one step further to use the coherent nature of laser light to control the outcome of a photochemical reaction such as photodissociation. In this work, one or more pulse of laser energy is used to drive a reaction to a desired outcome, opening up exciting possibilities for new methods of chemical synthesis. Bimolecular reactions are often studies by firing collimated streams of reagents at each other in the form of crossed molecular beams. The scattered reagents and products can be detected by a rotatable mass spectrometer in order to measure angular distributions. Such experiments have shown that many reactions occur in essentially a single encounter in a direct mechanism, while others proceed through a long-lived complex mechanism. In other experiments, spontaneous light emission from the unrelaxed, or nascent, products, termed chemiluminescence, has been analyzed to yield quantum-state distributions. Lasers are often used to probe internal states of products, for example, by inducing emission as in laser-induced fluorescence (LIF) detection, and to prepare molecules in specific states and with chosen orientations. Vibrational energy is often found to be more efficacious in promoting reaction than is translational or rotational energy, since it is more strongly coupled to the reaction coordinate, or path in phase space along which reaction takes place. Product distributions are frequently observed to be far from equilibrium. For example, in direct reactions, high vibrational levels are often found to be more populated than low ones. This so-called population inversion forms the basis of the chemical laser. Reaction rate constants cannot be used to describe such detailed processes.  Instead the differential  reaction cross section, σreact n 1 , n 2 , n 3 , . . . n 1 , n 2 , n 3 , is employed, where n i are various quantum numbers and the primed quantities refer to reaction products. Such cross sections represent the effective collision area for reagents with given n 1 , n 2 , . . . , to give specific products. Rate constants represent the effective average of the product of the cross section with the approach velocity taken over the calculated distribution of reagent quantum states. Cross sections can be predicted from semiclassical trajectory calculations, in which equations of motion are

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solved by numerical integration, or they can be obtained from quantum calculations via the time-dependent Schr¨odinger equation. Both approaches require a previously calculated potential energy surface. However, accurate potentials are currently available only for the H + H2 reaction and its isotopic analogues, for which precise quantum calculations can be made. For other systems approximate surfaces can be obtained either semiempirically (e.g., using the LEPS or BEBO functions), or from approximate a priori calculations. Statistical theories are also employed. These are based on the assumption that different reaction channels are populated in proportion to the volume of phase space with which they are associated, which is consistent with conservation of energy and linear and angular momentum. Dynamical factors may cause deviations from such statistical behavior, providing information on the reaction mechanism. This is recognized in the information theoretic approach where the fully statistical outcome, or prior distribution, is compared with observations in so-called surprisal plots which indicate the degree to which the data deviate from statistical behavior. This approach has proven very valuable in the correlation and extension of a wide body of data. Since reaction rates can depend not only on reagent energy, but also on the form in which it is available, a full understanding of chemical behavior requires knowledge of the manner in which energy in various forms is redistributed by collisions. This information must be obtained by studies of energy transfer associated with inelastic collisions. Experimental studies vary from kinetic measurements of decay rates, to full state-to-state studies. It is found that rotational energy is readily transferred from one molecule to another, occurring on almost every collision. Transfer from rotation to translation can be 102 times slower, while transfer to vibration may be 104 times slower. Transfer between translation and vibration occurs only about once in a million collisions at room temperature. In general, the rate of energy transfer decreases rapidly as the amount of energy transferred increases, following an approximate exponential gap rule. E. Reactions in Solution In principle, reactions in solution occur in a similar manner to those in the gas phase and in some favorable cases the observed rate constant is the same in both phases. For example, the unimolecular decomposition of N2 O5 yields similar A and E 0 values in the gas phase and in a large range of solvents. However, there are many important differences. Reactions of ionic species and of large molecules such as proteins and polymers are rare in gas-phase studies but are common in solution. Reactions in solution are

often catalyzed, for example, by protons in acid catalysis and by enzymes in many biological systems. Moreover, interactions with solvent molecules may grossly alter the potential energy surface on which reaction occurs, compared to the isolated gas-phase system. Such interactions are strongest for polar reagents and solvents. Reactions in solution are often diffusion controlled, where the limiting step is the rate at which reagents can find each other. In the absence of strong interactions such as those between ions, the rate constant may be estimated from Fick’s law [Eq. (45)] together with the Stokes– Einstein relation [Eq. (49)] giving: k = 8RT /3η,

(58)

where R is the gas constant. Since reactants are also slow to drift apart, the time-averaged collision frequency per molecule may be close to the gas-phase value, so that reactions with rate constants much smaller than given by Eq. (58) may be relatively insensitive to diffusion effects (in practice this means E 0 ≥ 40 kJ/mol). Reaction products may also be slow to move apart, thus in liquid-phase photodissociation, where the adsorption of light causes a molecule to fall apart, the surrounding solvent cage may hold the products together long enough for recombination to occur. If two reactions are in equilibrium with an equilibrium constant K , and the back reaction is held constant at the diffusion limit, kd , then the forward rate constant will be equal to K kd . More generally, it is often found that for a given reaction involving a series of similar reagents, k ∝ K α, log k = b + α log K .

(59) (60)

This is the Brønstead equation, and is an example of a linear free-energy relationship, since log K ∝ G 0 and log k ∝ G ‡0 , then Eq. (60) could be written as G ‡0 = b + αG 0

(61)

Related to the Brønstead equation is the Hammett equation, which expresses the rate constant k of one of a series of related reactions in terms of a specific reference reaction with rate k0 , giving log(k/k0 ) = ρσ,

(62)

where ρ is a characteristic of the type of reaction and σ is a characteristic of the specific system. Expressions such as the Brønstead and Hammett equations are particularly useful since the complex nature of the environment makes absolute rate theories such as the activated complex theory difficult or impossible to apply in solution. The rate constant for a bimolecular reaction in solution can be expressed in terms of the activity coefficients of

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the reagents, γ (A) and γ (B), and of the transition state, γ (AB)‡ , by k = k0 [γ (A)γ (B)/γ (AB)‡ ],

(63)

where an activity coefficient is defined as the ratio of the effective concentration to the actual concentration, as required to obey Raoult’s law. The Debye–H¨uckel limiting law gives γ in terms of the ionic charges z of the species: √ log γ (i) = −sz i2 I , (64) where I is the ionic strength: I =

=∞ 1 i xi z i2 , 2 i =0

in which xi indicates molar concentration, giving: √ log k = log k0 + 2sz A z B I .

(65)

(66)

This indicates that the rate constant for reactions in solution depends on the ionic strength of the solution, behavior referred to as the kinetic salt effect. Thus, addition of inert ions will increase the rate of reaction between ions of like charge and vice versa. Recent advances in theory √ have shown that log k ceases to be a linear function of I at large ionic strength. The reaction rate continues to increase with increasing ionic strength, but less rapidly than that predicted by the above relation. A careful examination of the experimental data confirms this prediction. F. Reactions at Surfaces; Heterogeneous Catalysis; Corrosion Atoms and molecules frequently adsorb on surfaces, where they may decompose and/or react with other adsorbed species. Modern technology is increasingly dependent on surface chemistry which underlies many industrially important processes as well as destructive processes such as corrosion. It is useful to distinguish two types of adsorption: physical adsorption, or physisorption, and chemical adsorption, or chemisorption. Physisorption is similar in nature to condensation and involves little chemical interaction with the surface, being associated with van der Waals forces. Chemisorption involves a true chemical interaction with the surface, with the formation of a chemical bond. Thus, the enthalpy of physisorption is usually of the order of 20 kJ mol−1 , while for chemisorption values are in the region of 200 kJ mol−1 . A chemisorbed molecule may either remain intact in molecular chemisorption, or fall apart in dissociative chemisorption. In an important recent development, it is now possible to identify individual molecular bonds of adsorbed molecules using STM

(see above). In this method, the STM current shows features associated with the vibration frequencies of chemical bonds. Figure 6(A) shows an STM image of the molecule HC2 D on a copper surface. Figure 6(B) shows an image of the CD bond of this molecule. The fraction of a surface covered by a gas, , is given in terms of monolayers (ML), where 1.0 ML represents complete coverage by a single layer. The pressure dependence of  at a given temperature is termed an adsorption isotherm. If the rate of adsorption is ka p(1 − ), and the rate of desorption is kd , we obtain the Langmuir isotherm: Kp = (67) 1 + Kp where K = ka /kd . Thus, when K p is small,  is simply proportional to the pressure. An adsorption dependence on 1 −  arises in the ideal case in which each molecule adsorbs at and occupies a single surface site. If two adjacent sites are required for adsorption, a (1 − )2 dependence might hold. The probability of adsorption for a single collision is termed the sticking probability, which usually implies chemisorption and may range from unity, for say oxygen on a clean metal surface, to close to zero for an inert system. At low temperatures, even inert gases may trap into a physisorption state, in proportion to their trapping probability. With a sticking or trapping probability of unity, a monolayer will form in about 1 sec at 10−6 Torr, which means that studies of clean surfaces must be carried out under conditions of ultra high vacuum (UHV), or below ∼10−9 Torr. The ability of surfaces to promote chemical reactions stems largely from their ability to cause dissociation. Consider the decomposition of ammonia to nitrogen and hydrogen: 2NH3 → N2 + 3H2

(68)

In the gas phase, this process has an activation energy of ∼330 kJ mol−1 , whereas in the presence of on a tungsten surface this falls to ∼160 kJ mol−1 . By providing an alternative low-energy path for reaction, the surface causes a large increase in the reaction rate. This is an example of heterogeneous catalysis. (See Section IV.C for a definition of catalysis.) For such a reaction to occur at a surface, the ammonia must first diffuse to the surface, it must dissociatively chemisorb, the hydrogen and nitrogen atoms must then recombine, and they must desorb and diffuse away from the surface. The recombination and desorption may actually occur as one step, as the reverse of dissociative adsorption. Either the adsorption or desorption steps are rate-limiting in gas–surface reactions, although fast liquid–surface reactions may be diffusion

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FIGURE 6 Scanning tunneling microscope image of a singly deuterated acetylene molecule (HC2 D) molecule on a Cu(100) surface. (A) An image of the C–D bond obtained by setting the second differential of the tunneling current to 269 meV. [See Stipe, B. C., Rezaei, M. A., and Ho, W. (1999). Phys. Rev. Lett. 82, 1724; courtesy B. C. Stipe, with permission.]

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176 limited. This is consistent with the fact that dissociative chemisorption and recombinative desorption are often activated processes. Chemical reactions at the gas–surface interface can be followed by monitoring gas-phase products with, for example, a mass spectrometer, or by directly analyzing the surface with a spectroscopic technique such as Auger electron spectroscopy (AES), photoelectron spectroscopy (PES), or electron energy loss spectroscopy (EELS), all of which involve energy analysis of electrons, or by secondary ionization mass spectrometry (SIMS), which examines the masses of ions ejected by ion bombardment. Another widely used surface probe is low-energy electron diffraction (LEED), which can provide structural information via electron diffraction patterns. At the gas–liquid interface, optical reflection ellipsometry and optical spectroscopies are employed, such as Fourier transform infrared (FTIR) and laser Raman spectroscopies. Elastic and inelastic collisions of atoms and molecules at surfaces are also of importance. The scattering of hydrogen and helium from surfaces leads to diffraction patterns in the same manner as with LEED, but since the atoms penetrate the surface far less deeply than even low-energy electrons, the structures obtained reflect the very surface of the sample. The inelastic surface scattering of molecules can be examined in detail using laser and mass spectrometric detection for the scattered molecules. Such measurements can be used to model the form of the gas–surface interaction potential, knowledge of which is a prerequisite for any detailed picture of gas–surface reaction dynamics. Not all surface chemistry is catalytic. In many cases the surface itself may be consumed, in processes such as etching and corrosion. Etching is employed to fabricate devices where it provides for the controlled removal of material. In the semiconductor industry, for example, discharges containing fluorine are used to etch silicon by volatilization as silicon fluorides. Corrosion is generally an unwanted process whereby items are destroyed through dissolution and/or oxidation. Metals may corrode through many different (usually electrochemical) processes. For example, in the presence of oxygen, a metal may displace protons as water or reduce oxygen to OH− , in acid and alkaline environments, respectively. In principle, this process requires the additional presence of a second metal, with a lower electrochemical potential. However, all samples have regions of high and low strain, which will have slightly different potentials. A given metal can be protected by contact with another (sacrificial) metal with a more negative potential, which will be preferentially corroded. This is applied in the galvanizing of iron by zinc.

Physical Chemistry

SEE ALSO THE FOLLOWING ARTICLES ADSORPTION • ATOMIC SPECTROMETRY • CHEMICAL THERMODYNAMICS • CRYSTALLOGRAPHY • KINETICS (CHEMISTRY) • LIQUIDS, STRUCTURE AND DYNAMICS • MECHANICS, CLASSICAL • PERIODIC TABLE (CHEMISTRY) • QUANTUM CHEMISTRY • QUANTUM MECHANICS • STATISTICAL MECHANICS • SURFACE CHEMISTRY

BIBLIOGRAPHY Adamson, A. W. (1980). “Physical Chemistry of Surfaces,” Wiley, New York. Adamson, A. W. (1979). “A Textbook of Physical Chemistry,” 2nd ed., Academic Press, New York. Atkins, P. W. (1982). “Physical Chemistry,” 2nd ed., Freeman, San Francisco. Bernstein, R. B. (1982). “Chemical Dynamics via Molecular Beam and Laser Techniques,” Oxford Univ. Press, New York. Berry, R. S., Rice, S. A., and Ross, J. (1980). “Physical Chemistry,” Wiley, New York. Eyring, H. (1944). “Quantum Chemistry,” Wiley, New York. Eyring, H., Henderson, D., and Jost, W. (1967). “Physical Chemistry—An Advanced Treatise,” 15 vols., Academic Press, New York. Eyring, H., Henderson, D., Stover, B. J., and Eyring, E. M. (1982). “Statistical Mechanics and Dynamics,” 2nd ed., Wiley, New York. Herzberg, G. (1950). “Molecular Spectra and Molecular Structure: I. Spectra of Diatomic Molecules,” 2nd ed., Van Nostrand-Reinhold, New York. Kauzmann, W. (1957). “Quantum Chemistry,” Academic Press, New York. Laidler, K. J. (1965). “Chemical Kinetics,” 2nd ed., McGraw-Hill, New York. Levine, R. D., and Bernstein, R. B. (1974). “Molecular Reaction Dynamics,” Oxford Univ. Press, New York. Levine, R. D., and Bernstein, R. B. (1987). “Molecular Reaction Dynamics and Chemical Reactivity,” Oxford Univ. Press, New York. Linnett, J. W. (1960). “Wave Mechanics and Valency,” Methuen, London. McQuarrie, D. A. (1976). “Statistical Mechanics,” Harper and Row, New York. Moore, W. J. (1983). “Basic Physical Chemistry,” Prentice Hall, New York. Partington, J. R. (1954). “An Advanced Treatise on Physical Chemistry,” 5 vols., Wiley, New York. Rowlinson, J. S. (1982). “Liquids and Liquid Mixtures,” 3rd ed., Butterworths, London. Smith, I. W. M. (1980). “Kinetics and Dynamics of Elementary Gas Reactions,” Butterworths, London. Smith, R. A. (1961). “Wave Mechanics of Crystalline Solids,” Wiley, New York. Steinfeld, J. I. (1974). “Molecules and Radiation: An Introduction to Modern Molecular Spectroscopy,” MIT Press, Cambridge, Massachusetts.

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Potential Energy Surfaces Donald G. Truhlar University of Minnesota

I. Introduction II. Quantum Mechanical Basis for Adiabatic Potential Energy Surfaces III. Topology of Adiabatic Potential Energy Surfaces IV. Breakdown of the Adiabatic Approximation V. Shapes of Potential Energy Surfaces

GLOSSARY Adiabatic representation Representation in which the electronic wave functions are calculated for fixed (i.e., nonmoving) nuclei. Avoided intersection Case in which two potential energy surfaces come together but do not intersect. Conical intersection Case in which two potential energy surfaces intersect such that their separation decreases to zero linearly in the relevant nuclear coordinates. Diabatic representation Representation in which the electronic wave function is not adiabatic. Dunham expansion Taylor series expansion of a potential energy curve in the vicinity of its minimum. Electron affinity Binding energy of an electron to a neutral atom or molecule. Equilibrium configuration Geometry of a molecule’s nuclear framework corresponding to the minimum adiabatic energy. Force field The gradient of the potential energy surface.

Glancing intersection Case in which two potential energy surfaces intersect such that their separation decreases to zero quadratically in the relevant nuclear coordinates. Ionization energy Energy required to remove an electron from an atom or molecule.

A POTENTIAL ENERGY SURFACE is an effective potential function for molecular vibrational motion or atomic and molecular collisions as a function of internuclear coordinates. The concept of a potential energy surface is basic to the quantum mechanical and semiclassical description of molecular energy states and dynamical processes. It arises from the great mass disparity between nuclei and electrons (a factor of 1838 or more) and may be understood by considering electronic motions to be much faster than nuclear motions. (When we say nuclear motions and nuclear degrees of freedom in this article, we refer to motions of the nuclei considered as wholes, i.e., to atomic motions.) This difference in timescales leads to the so-called electronic adiabatic approximation and to

9

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10 electronic adiabatic potential energy surfaces. Under some circumstances, however, it is convenient to use more general definitions; this results in potential energy surfaces variously known as diabatic or nonadiabatic, the latter adjective being a useful double negative (it is convenient to use this term because, except for a few small terms, adiabatic surfaces may be defined uniquely but in general diabatic surfaces are not; nonadiabatic has the useful connotation then of “anything except the adiabatic”). The concept of potential energy surfaces may be generalized to systems in liquids, in which case one obtains potentials of mean force.

I. INTRODUCTION The separation of electronic and nuclear motions according to time scales and the consequent introduction of an effective potential energy surface for nuclear motion was first considered by Born and Oppenheimer. Although their method is seldom used in modern work, the modern equivalents are still commonly called Born–Oppenheimer approximations and Born–Oppenheimer potential energy surfaces. The modern form of the derivation, which is summarized below, dates to a later paper by Born and to work by Born and Huang. Occasionally the phrases Born–Oppenheimer and Born–Huang are used to specify whether certain small terms are included in the potential energy surfaces, although, as mentioned above, it is more common to refer to any adiabatic surfaces as Born– Oppenheimer surfaces. A potential energy surface is an effective potential energy function for the relevant nuclear degrees of freedom. The latter are usually defined as all nuclear degrees of freedom minus three overall translations and two or three rotations of the nuclear subsystem. If an atom has N nuclei, it is common to consider the potential energy as a function of 3N − 5 coordinates for N = 2 (since two nuclei always lie on a line and therefore, when considered as point masses, have only two rotational degrees of freedom) and 3N − 6 coordinates otherwise. Thus for N = 2 we actually have a potential energy curve (i.e., a function of one variable), whereas for N ≥ 3 we have a potential energy hypersurface (i.e., a function of three or more variables). A potential energy surface would strictly speaking denote the potential energy as a function of two coordinates in a two-dimensional cut through the (3N − 6)-dimensional internal-coordinate space. In this article, however, we follow the very common language by which any potential energy hypersurface or potential energy function is referred to as a surface.

Potential Energy Surfaces

II. QUANTUM MECHANICAL BASIS FOR ADIABATIC POTENTIAL ENERGY SURFACES The Schr¨odinger equation for a system of N nuclei and n electrons is (H − E)(r, R) = 0,

(1)

where H is the Hamiltonian or energy operator of the molecule: h2 2 h2 2 H =− ∇ − ∇ + V (r, R) + Hrel (r, R). (2) 2M R 2m r In these equations, h is Planck’s constant divided by 2π, and R denotes a 3N -dimensional vector of scaled nuclear coordinates:   M1 1/2 R1 = X 1, (3a) M   M1 1/2 R2 = Y1 , (3b) M .. . R3N =



MN M

1/2 ZN,

(3c)

where M j , X j , Y j , and Z j are the mass and Cartesian coordinates of the nucleus j, M is any of the nuclear masses or a convenient reduced nuclear mass, m is the electronic mass, r is a 3n-dimensional vector of electronic Cartesian coordinates {xk , yk , z k }nk=1 : r1 = x 1 ,

(4a)

r2 = y1 ,

(4b)

.. . r3n = z n ,

(4c)

V (r, R) is the sum of all coulomb forces between the particles, Hrel (r, R) is the energy operator for relativistic effects including mass-velocity and spin–orbit coupling, E is the total energy, and ψ(r, R) is the wave function. Notice that the first two terms in Eq. (1) represent the nonrelativistic kinetic energy of the nuclei and the electrons. Usually the change in ψ(r, R) is about the same order of magnitude when we move a nucleus a small amount as when we move an electron the same small amount. When this is the case, ∇R2 ψ(r, R)/M is smaller than ∇r2 ψ(r, R)/m by a factor of order m/M, which is less than about 10−3 ; thus the first term in Eq. (2) may be neglected to a first approximation. The physical interpretation of this is that because of their larger masses, the nuclei move so slowly compared

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with the electrons that the dependence of the wave function on electronic coordinates is essentially the same as if the nuclei were completely stationary (i.e., as if their mass were infinite so that their kinetic energy was zero). Then Eq. (1) becomes   h2 2 − ∇r + V (r, R) + Hrel (r, R) − Uα (R) 2m × ψα (r; R) = 0,

(5)

where Uα (R) and ψα (r; R) denote the eigenvalue and eigenfunction, respectively, with index α. Notice that both of these depend parametrically on the nuclear positions R and also that the ψα (r, R) form a complete set of functions of r for any particular value of R. In general the spectrum {Uα (R)} contains a continuous part as well as a discrete part, but the discrete part is the important part for the concept of potential energy surfaces. We shall therefore use only a discrete index α. This is adequate for almost all practical work since the continuous part is usually neglected in calculations. For a more rigorous derivation, however, all sums over discrete α in the equations below must be replaced by a sum and an integral. To obtain the complete system wave function we choose a trial function  ψ(r, R) = ψα (r; R)χα (R). (6) α

Since the {ψα (r; R)} are complete, this trial function yields the exact ψ(r, R) if we retain the complete set and solve for {χα (R)} by the variation method. The best {χα (R)} by this method satisfy the equation   dr ψβ∗ (r; R)(H − E) ψα (r; R)χα (R) = 0, α

β = 1, 2, . . . , ∞.

(7)

If we carry out the indicated operations, using Eq. (5) and the orthogonality of the {ψα (r; R)} in r at fixed R, Eqs. (7) become   h2 − ∇R2 χα (R) + 2 Fαβ (R) · ∇R χβ (R) 2M β  + G αβ (R)χαβ (R) + [Uα (R) − E]χα (R) = 0, β

α = 1, 2, . . . , ∞, (8) where

 Fαβ (R) =

and

 Gαβ (R) =

dr ψα∗ (r; R)∇R ψβ (r; R)

(9)

dr ψα∗ (r; R)∇R2 ψβ (r; R).

(10)

By the same argument given above for the variation of ψ(r, R) with respect to r and R, we expect that the terms containing Fαβ (R) and G αβ (R) are usually much smaller than the term containing Uα (R). When this is so, we may neglect the small terms, and the set of coupled Eqs. (8) simplifies to a separate uncoupled equation for each χα (R), namely,   h2 2 − (11) ∇ + Uα (R) − E χα (R) = 0. 2M R This has the form (Hnuc − E)χα (R) = 0,

(12)

where Hnuc is an effective Hamiltonian for nuclear motion given by h2 2 ∇ + Uα (R). (13) 2M R Since a Hamiltonian is usually the sum of a kinetic energy operator and a potential energy operator, we may interpret Uα (R) as an effective potential for nuclear motion. In fact, Uα (R) is the potential energy surface that we sought to derive. Recalling the origin of Uα (R), we see that it represents the total energy of the electrons, both kinetic and potential, plus all the rest of the potential energy, when the electrons are in state α. Alternatively, it represents the entire (coulombic plus relativistic) potential energy of all particles plus the electrons’ kinetic energy. When the equations for the χα (R) decouple, as in Eq. (12), the electronic state is preserved during the nuclear motion. The resulting quantized energy requirement of the electrons plus the rest of the potential energy (given in the absence of relativistic effects by the nuclear–nuclear coulombic interaction energy) together constitute an effective potential for nuclear motion. When nuclear motion can be approximated by classical mechanics (which is often reasonable, especially for atoms heavier than hydrogen), Eqs. (12) and (13) are replaced by Hnuc = −

¨ = −∇R Uα (R), MR

(14)

¨ is the nuclear acceleration, and the right-hand where R side is the force on the nuclei. Since Uα (R) generates the force function, it is sometimes called the force field. Inclusion of the spin–orbit and other relativistic terms in Eq. (5), as we have done, is, strictly speaking, the most correct approach. This yields, as we have seen, a set of nuclear wave functions χα (R) whose uncoupled motion is governed by the potentials Uα (R) and which are coupled only by the nuclear-derivative terms Fαβ (R) and G αβ (R). In practice, though, Hrel (R) is difficult to treat on an equal footing with the coulombic terms in the Hamiltonian. Therefore one sometimes works with

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12 nonrelativistic adiabatic potential surfaces, although the adjective nonrelativistic is seldom stated explicitly. In this approximation one temporarily neglects Hrel (r, R) to solve Eq. (5). This yields a new set of nuclear wave functions χα (R) and Uα (r, R) that are easier to work with, but the effect of Hrel (r, R) must be included later. The matrix elements  Hrel,αβ (R) = dr ψα∗ (r; R)Hrel (r, R)ψβ (r; R) (15) provide both a perturbative correction to the nonrelativistic potential energy surfaces, for α = β, and an additional coupling mechanism that must be considered along with Fαβ (R) and Uαβ (R) for α = β. In the rest of this article we restrict our attention to the nonrelativistic approximation, and we assume that Hrel (r, R) has been neglected in solving Eq. (5).

III. TOPOLOGY OF ADIABATIC POTENTIAL ENERGY SURFACES As shown in Eq. (5), an adiabatic potential energy surface is an eigenvalue of a Hermitean operator, that is, one that has only real eigenvalues. In most cases these eigenvalues are nondegenerate. At some geometries R, however, two or more eigenvalues may be equal, which is called a degeneracy. Such degeneracies may be mandated by symmetry or may be accidental. Points where two or more eigenvalues are equal are particularly interesting, and we may categorize some features of the potential energy surfaces in the neighborhoods of these points on the basis of symmetry. The nuclear configuration R, which appears as a parameter in the eigenvalue Eq. (5), may be classified by a symmetry point group, for example, D∞h for a homonuclear diatomic molecule or another symmetric linear molecule, C∞v for a heteronuclear diatomic molecule or other nonsymmetric linear molecule, Td for a tetrahedral molecule, C2v for a symmetric nonlinear triatomic molecule, and Cs for a planar molecule. Since the operations of such a group commute with −(h 2 /2M)∇r2 + V (r, R), the eigenfunctions ψα (r, R) of this operator can be taken to transform as irreducible representations of the group. We may thus classify both the eigenfunctions ψα (r, R) and eigenvalues Uα (R) by these irreducible representations, e.g.,

+ g , u , or g for D∞h or A1 or B2 for C 2v . First consider the case of two nuclei. As already mentioned, the potential energy surfaces in this case are really curves; they depend on only one scalar variable, the internuclear distance, which we may call R. One can show, on general grounds, that for a system with an even number of electrons, two Uα ( R) may be accidentally equal

Potential Energy Surfaces

FIGURE 1 An avoided crossing for a diatomic molecule. Ua is the potential energy for electronic state a , R the internuclear distance, and R ∗ the distance corresponding to an avoided crossing.

at isolated values of R if they correspond to different symmetry but not if they correspond to the same symmetry. When two Uα ( R) are equal, that is called a curve crossing. Sometimes two Uα ( R) approach very closely as if they are about to cross but then avoid crossing. This is called an avoided crossing. An example is shown in Fig. 1. When the system has an odd number of electrons and exists in a magnetic-free region, the possibilities are the same except that all Uα ( R) occur in degenerate pairs. Imposition of a magnetic field removes the degeneracy. Now consider the case of N ≥ 3 for which the potential energy surfaces depend on three or more variables. Here we also find potential surface intersections of states belonging to different symmetries and avoided intersections of states belonging to the same symmetry, but there is also a third possibility, namely, intersections even of potential energy surfaces belonging to the same symmetry. Such intersections in general occur in subspaces of dimension 3N − 8 or lower. If we consider a subset of two degrees of freedom in which the intersection occurs at a single point, the shape of the surfaces in the vicinity of the intersection is as shown in Fig. 2, that is, the two surfaces form a double cone. Such intersections are called conical intersections. Another shape of intersection may occur at linear geometries of polyatomic molecules. In this case, the two surfaces may have zero slope at the point of intersection. Such intersections are called glancing rather than conical.

FIGURE 2 Portions of two potential energy surfaces exhibiting a conical intersection. Ua is the potential energy for electronic state a .

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In general, for systems of three or more nuclei, some potential energy surfaces that intersect when one neglects spin–orbit coupling avoid intersecting when one includes it.

IV. BREAKDOWN OF THE ADIABATIC APPROXIMATION We are now in a position to understand the limitations of the adiabatic potential energy surface concept. First, however, we should understand the physical origin of avoided crossings and avoided intersections. We begin by considering the diatomic molecule NaCl, and we let R denote the distance between the nuclei. At R = ∞, the energy of two neutral atoms is lower than the energy of a Na+ ion and a Cl− ion by the difference E of the ionization potential of Na and the electron affinity of Cl− . As R is decreased, however, the energy of the ionic state decreases rapidly because of the long-range coulomb attraction, which may be represented by −e2 / R where e is the electronic charge, while the energy of the neutral state stays approximately constant until much shorter distances where the covalent interaction becomes appreciable. Thus at some distance R ∗ given approximately by e2 E ∼ (16) = R ∗ the hypothetical purely ionic state and the hypothetical purely covalent state would have the same energy. Actually though, at this R the corresponding eigenfunctions of Eq. (5) have mixed character, partly covalent and partly ionic, with about

50% partial ionic character. Since both states have 1 + g character, their eigenvalues are different. We call the energies of the hypothetical states with pure valence characteristics U1d ( R) and U2d ( R), where d denotes diabatic (or nonadiabatic). Although U1d ( R) and U2d ( R) cross, the adiabatic curves U1 ( R) and U2 ( r ) avoid crossing, having the shapes shown in Fig. 1. For this case α = 1 corresponds to a covalent state to the right of R ∗ but to an ionic state to the left of R ∗ and vice versa for α = 2. Now recall the argument given above for neglecting F12 (R) and G 12 (R). At R = R ∗ , the wave function is changing character very rapidly as a function of R, and the action of ∇R on ψ(r, R) is unusually large; this means that F12 ( R) and G 12 ( R) need not be negligible. In such a case, the dynamics governed by the nuclearmotion wave functions χ1 (R) and χ2 (R) do not decouple into independent motions governed by effective potentials (i.e., the adiabatic potential energy surface concept breaks down). Although we have given the argument for a particular diatomic molecule, the effect is general, and the

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13 adiabatic approximation is expected to break down in the vicinity of any avoided crossing or avoided intersection, and also at actual intersections for which the symmetry of the two states is the same at the intersection. There are other circumstances under which the adiabatic approximation may break down. We have considered the case where ψ(r, R) varies rapidly with R because of the factor ψα (r; R). A second case where the term containing Fαβ (R) in Eq. (8) may be significant occurs when χα (R) varies unusually rapidly, for instance, in highenergy collisions of Na+ with Cl− . When the nuclear speed is large, χα (R) must vary significantly on the scale of a very small de Broglie wavelength. We may summarize the two cases in a simple but approximate way as follows: When the nuclear kinetic energy is much smaller than the spacings between the adiabatic electronic energy surfaces Uα (R), these surfaces serve as potential energy surfaces for nuclear motion. When the nuclear kinetic energy is comparable to or larger than the spacings between the Uα (R), the adiabatic approximation may, and often does, break down. The adiabatic potential energy surfaces need not be abandoned completely when the adiabatic approximation breaks down, especially if the region of breakdown is fairly localized, as it often is when the breakdown is due to an avoided or conical intersection. If the nonadiabatic behavior is localized to a small region, we often employ the model of surface hopping. In this model the nuclear motion is assumed to be governed by an adiabatic potential energy surface until a nonadiabatic region is reached. In such a region there is a nonzero quantum mechanical probability that the system “hops” to another surface. Based on this probability one portion of the quantum mechanical probability density exits the nonadiabatic region in one of the adiabatic electronic states, and the other portion exits in the other one or more coupled adiabatic electronic states. After this the nuclear motions again proceed independently as governed by single potential energy surfaces until another nonadiabatic region is reached. Although this model neglects certain coherency effects that may be important for quantitative work, it is often useful for qualitative discussions and semiquantitative calculations. Another concept often invoked for qualitative discussions and for calculations when the adiabatic approximation breaks down is that of diabatic potential energy surfaces. There are several nonequivalent ways of defining such surfaces, each of which may be useful under some circumstances. The simplest way is that already illustrated above in conjunction with the NaCl example: namely, a diabatic state is the effective potential energy function for nuclear motion when the electronic state is artificially constrained to a state of prespecified pure valency.

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A second way to define diabatic electronic states and potential energy surfaces is more mathematical. Notice that the valency-character prescription leads to states coupled by the operator h2 2 (17) ∇ + V (r, R) + Hrel (r, R) 2m r and also by −(h 2 /2M)∇R2 . From the point of view of the nuclear motion, the former is sometimes called potential coupling (since it involves only multiplicative operators in the nuclear coordinates), and the latter is called derivative coupling. Since electronically adiabatic states have derivative coupling but no electronic coupling, a natural question is whether useful diabatic states can be defined to have potential coupling but not derivative coupling. Unfortunately, this leads to states that are completely independent of R and are not useful. It is possible, however, to make one component (in one or another coordinate system) of the vector coupling operator Fαβ (R) vanish for all α, β. Furthermore, if Fαβ (R) is approximated as the gradient of a scalar (which can be a good approximation when nonadiabatic effects are dominated by a narrowly avoided intersection), then it is possible to make all components of Fαβ (R) vanish for all α, β. Both of these prescriptions are sometimes employed to obtain diabatic states. Consider, for example, the case where Fαβ (R) is the gradient of a scalar for all R; then it has zero curl. We define diabatic electronic states by  φαd (r; R) = φβ (r; R)Tβα (R). (18) Hel = −

β

The states {φαd (r; R)χαd (R)} will be uncoupled by nuclear derivative operators if we choose Tβα (R) for all β and α such that  ∇R Tαβ (R) = Fαγ (R)Tγβ (R), (19) γ

 −

 h2 2 ∇r + V (r, R) − Uαn (R) ψαn (r, R) = 0, (22) 2m

where the superscript n denotes nonrelativistic. The true adiabatic states are coupled only by −(h 2 /2M)∇R2 , but these nonrelativistic adiabatic states are coupled by both this operator and Hrel (r, R). Because of the latter coupling, the nonrelativistic adiabatic electronic states ψαn (r; R) and their associated potential energy curves Uαn (R), which are the most widely employed states and potential energy surfaces in quantum chemistry, are actually diabatic. They are nevertheless usually called adiabatic although nonrelativistic is adiabatic is technically more appropriate.

V. SHAPES OF POTENTIAL ENERGY SURFACES A. Diatomics A schematic illustration of some typically shaped adiabatic potential energy curves for a diatomic molecule is shown in Fig. 3. All five curves shown become large and positive at small internuclear distance R. This represents a repulsive force between the nuclei and is due to internuclear repulsion and the unfavorability of overlapping the atomic charge clouds of the two different centers. All five curves tend to constants at large R. This is because the atomic interaction energy eventually decreases to zero as the distance between the atoms is increased. The constant spacings between the curves at large R are equal to the atomic excitation energies. Curves 1 and 2 are effective potentials for the interaction of ground-state atoms, and curves 3–5 represent effective potentials for the case where at least one of the atoms is excited. The figure shows an avoided crossing between curves 2 and 3 and

and this set of coupled partial differential equations does have a solution if Fαβ (R) has zero curl. Furthermore, if the state expansion α, β, . . . is truncated to a finite number of computationally important states, then the diabatic electronic basis is not independent of R. In this way, one can define a diabatic basis by a transformation from an adiabatic one, and it spans the same space. The diabatic potential surfaces are given by  d Uαα (R) = |Tγ α (R)|2 Uγ (R), (20) γ

and the potential couplings are given by  Uαβ (R) = Tγ∗α (R)Uγ (R)Tγβ (R).

(21)

γ

As mentioned in Section II, the usual treatments of potential energy surfaces neglect Hrel (r, R) in Eq. (5). Thus one solves

FIGURE 3 Typical potential energy curves for a diatomic molecule ordinarily thought of as bound. Ua and R are as in Fig. 1; De is the equilibrium bond energy of the ground state. Curves 1–5 are discussed in the text.

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true crossings of curves 3 and 4 by curve 5. Curves with deep enough minima, such as 1 and 3, may possess bound states of nuclear motion. Classically, a pair of nuclei whose motion is governed by one of these curves could show simple, almost-harmonic motion in the vicinity of the minimum of the curve. Quantally, there could be bound vibrational states localized in these regions of R. Curve 1 has the typical shape for the ground electronic state of a stable molecule such as H2 , N2 , or HCl. Such a curve is often represented in the vicinity of the minimum by a quadratic potential: 1 Uα ( R) ∼ (23) = k( R − Re )2 , 2 where k is the force constant (actually the quadratic force constant) and Re the equilibrium internuclear distance. An analytic representation valid over a wider range of R is given by 1 1 Uα ( R) ∼ = k11 ( R − Re )2 + k111 ( R − Re )3 2 2 1 + k1111 ( R − Re )4 + · · · , (24) 2 where k has been renamed k11 , and k111 and k1111 are anharmonic (cubic and quartic) force constants. Sometimes the constant coefficient of ( R − Re ) j is written 1/j! instead of 12 so care must be exercised when using anharmonic force constants. Equation (24) is called a Dunham expansion. An approximate representation of a potential curve like curve 1 in Fig. 3 that gives its approximate shape over the full range of R is Uα ( R) = De {1 − exp[−α( R − Re )]}2 .

(25)

This is called a Morse curve, De the equilibrium dissociation energy, and α the Morse range parameter. More complicated analytic forms with more parameters are also used. Information about the various parameters (Re , k11 , k111 , . . . , De , and α) comes primarily from spectroscopy, scattering or kinetics experiments, and quantum machanical electronic structure calculations. These are also the sources for information about potential energy surfaces of systems with three or more atoms. Figure 4 shows, to about the same scale as Fig. 3, some typically shaped potential curves for a diatomic system usually thought of as unbound (e.g., He2 , HeNe, or ArH). Notice that the lowest potential energy curve has only a very small minimum at large R. When this minimum is important for the problem at hand, such a potential energy curve is often represented by a Lennard–Jones 12–6 potential:     σ 12 σ 6 Uα ( R) = 4ε , (26) − R R

FIGURE 4 Typical potential energy curves for a pair of atoms ordinarily thought of as unbound. Ua and R are as in Fig. 1. See text for discussion of the curves.

where ε is the well depth and σ the collision diameter. Notice that the minimum of Uα (R) occurs at R = Rm where Rm = 21/6 σ

(27)

Uα ( R = Rm ) = −ε.

(28)

and that

When the minimum of a predominantly repulsive potential curve is considered negligible, it may be represented by a so-called anti-Morse curve: Uα ( R) = D AM {exp[−2β( R − R AM )] + 2 exp[−β( R − R AM )]} + C AM ,

(29)

where D AM , β, R AM , and C AM are constants, or even by the simpler   β Uα ( R) = exp(−α R). (30) R Equation (29) or (30) could be applied to curve 4 of Fig. 3, to curve 1, 3, or 4, of Fig. 4, or even to curve 2 of Fig. 3, for which it might be useful in the region to the right of the avoided crossing and to the left of the shallow, large- R minimum. Notice that the zero of energy is arbitrary for potential energy surfaces as long as it is chosen consistently throughout a given calculation. In Fig. 3 we placed the zero of energy at the bottom of the lowest potential curve. In Fig. 4 we placed it at the energy of two separated groundstate atoms. B. Larger Molecules Potential energy surfaces for systems with three or more atoms are harder to illustrate because they depend on three or more internal coordinates. Analytic representations are also more complicated than for diatomics.

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Potential energy surfaces may be roughly classified into those with deep enough minima to support one or more strongly bound vibrational states and those without such minima. We shall call the former attractive surfaces because of the role played by the attractive forces between the atoms in creating the minimum. The latter will be called repulsive surfaces. Because spacings between potential energy curves are usually much larger than thermal energies, most molecular motions under ordinary conditions are usually governed by the lowest energy potential surface. Thus molecules that usually exist as bound, stable entities (e.g., H2 O, CO2 , or CH4 ) have attractive surfaces in their ground (i.e., lowest energy) electronic states, whereas molecules usually observed only as transient species during collisions (e.g., FH2 which exists during F + H2 or H + HF collisions or NeH2 O, which exists during Ne + H2 O collisions) have repulsive surfaces. Systems like Ne2 or NeH2 O may exist as stable but very weakly bound (and thus easily and usually dissociated) species because of shallow potential wells in predominantly repulsive potential energy surfaces. Such weakly bound species are called van der Waals molecules. Systems with attractive surfaces in their ground electronic states may have repulsive surfaces in excited (i.e., higherenergy) electronic states, and vice versa. Examples of van der Waals wells and repulsive excited states for the easily illustrated special case of diatomic molecules may be seen in Figs. 3 and 4. The most well understood region of attractive potential energy surfaces is usually the region near the minimum. One usually describes the potential energy surface in such a vicinity by a Taylor’s series about the minimum: Uα (R) = Ue +

1  k i j qi q j 2 i j

+

1  ki jk qi q j qk 2 i j k

+

1  ki jkl qi q j qk ql + · · · , 2 i j k l (31)

where Ue , ki j , kik j , . . . are constants, and the {q j } are suitable internal coordinates defined to vanish at the location of the minimum. As written, Eq. (31) contains no terms linear in the {q j }, but if these are not related to Cartesian coordinates by a linear transformation, it may be necessary to include linear terms. Equation (31) is called an anharmonic force field. If the coordinates are linear combinations of Cartesians and terms beyond the quadratic are neglected, it becomes a harmonic force field. In the harmonic approximation it is always possible to define the {q j } in such a way that the cross terms vanish (i.e., ki j = 0

FIGURE 5 Perspective view of potential energy surface for collinear H + HCl → H2 + Cl. The vertical axis is potential energy, and axes in the horizontal plane are nearest-neighbor distances.

if i = j). If this is done and cross terms vanish in the kinetic energy operator as well, Eq. (31) is called a normal-mode expansion. In the vicinity of the minimum of the surface, a twodimensional cut through an attractive potential energy surface has the shape of a distorted paraboloid of revolution. Figure 5 shows a perspective view of a cut through a potential energy surface for a chemical reaction; in particular it is based on an approximate surface for the reaction H + HCl → H2 + Cl. To represent the potential as a function of two internal coordinates, the three atoms are restricted for this figure to lie on a straight line. If the hydrogens are labeled Ha and Hb , the left–right axis is the Hb -to-Cl distance with large values at the left, and the front–back axis is the Ha -to-Hb distance with large values in the foreground; the third interpair distance is the sum of these two. The vertical axis is potential energy. The figure clearly shows the existence of a minimum-energy reaction path from reactants in the foreground to products at the back left. The highest energy point along the minimumenergy path is a saddle point. This point is sometimes called the transition state, and it primarily determines the threshold energy for reaction to occur. The shape of the reaction path is important for determining the reaction probability as a function of the vibrational and relative translational energy of the reactants. Figure 6 shows the same information as in Fig. 5 but in the form of a contour map (i.e., a set of isopotential contours). The horizontal axis is the distance from Cl to

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the center of mass of H2 , and the vertical axis is [(MCl + 2MH /4MCl ]1/2 times the Ha -to-Hb distance. Because of the mass scaling factor, the nuclear-motion Hamiltonian, Eq. (13), in this coordinate system becomes   h2 ∂ 2 ∂2 Hnuc = − + Uα (x , y), + (32) 2µ ∂ x 2 ∂ y2 where µ=

2MCl MH . (MCl + 2MH )

(33)

Since the coefficient of both derivative operators is the same, this Hamiltonian is the same as that for a single particle in two dimensions under the influence of a potential function Uα (x , y). This analogy is very helpful in mentally visualizing the motion of a polyatomic system whose dynamics are governed by a multidimensional potential surface. In Fig. 6 the H + HCl reaction is initiated at the lower right, and products are formed when the system, having passed through or near the saddle point (denoted + in the figure), reaches the top. For solutes in the liquid phase (e.g., an organic molecule in aqueous solution), one can obtain an effective potential

FIGURE 6 Contour map of a potential energy surface for collinear H + HCl → H2 + Cl. x is the distance from H to the center of mass of HCl and y the mass-scaled distance from Cl to its nearest H. Both axes are given in units of a 0, where 1a 0 = 1 bohr = 0.5292 × 10−10 m.

function of the solute coordinates by adding the free energy of solution to the gas-phase potential surface. The resulting potential function may be used in Eq. (14), and it is called a potential of mean force.

SEE ALSO THE FOLLOWING ARTICLES ATOMIC AND MOLECULAR COLLISIONS • ORGANIC CHEMICAL SYSTEMS, THEORY • QUANTUM MECHANICS • SURFACE CHEMISTRY

BIBLIOGRAPHY Gao, J., and Thompson, M. A. (1998). “Combined Quantum Mechanical and Molecular Mechanical Models,” American Chemical Society, Washington, DC. Herzberg, G. (1966). “Molecular Spectra and Molecular Structure. III. Electronic Spectra and Electronic Structure of Polyatomic Molecules,” Van Nostrand Reinhold, New York. Kondratiev, V. N., and Nikitin, E. E. (1981). “Gas-Phase Reactions: Kinetics and Mechanisms,” Springer-Verlag, Berlin. L¨owdin, P.-O., and Pullman, B., eds. (1983). “New Horizons of Quantum Chemistry,” D. Reidel, Dordrecht, Holland. Maitland, A., Rigby, M., Smith, E. B., and Wakeham, W. A. (1981). “Intermolecular Forces,” Clarendon, Oxford. Michl, J., and Bonaˇci´c-Kouteck´y. (1990). “Electronic Aspects of Organic Photo Chemistry,” John Wiley & Sons, New York. Murrell, J. N., Carter, S., Farantos, S. C., Huxley, P., and Varandas, A. J. C. (1984). “Molecular Potential Energy Functions,” Wiley, Chichester. Salem, L. (1982). “Electrons in Chemical Reactions: First Principles,” Wiley-Interscience, New York. Simons, J. (1983). “Energetic Principles of Chemical Reactions,” Jones and Bartlett, Boston. Smith, I. W. M. (1980). “Kinetics and Dynamics of Elementary Gas Reactions,” Butterworths, London. Truhlar, D. G., ed. (1981). “Potential Energy Surfaces and Dynamics Calculations: For Chemical Reactions and Molecular Energy Transfer,” Plenum, New York. Truhlar, D. G., and Morokuma, K., eds. (1999). “Transition State Modeling for Catalysis,” American Chemical Society, Washington, DC.

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Solid-State Chemistry Bahne C. Cornilsen Michigan Technological University

I. The Structure–Property Relationship II. Solid-State Structure and Structural Characterization III. Preparation Techniques and Property Variation

GLOSSARY Condensed matter sciences Include solid-state physics, ceramic science and engineering, metallurgical engineering, polymers, and materials science, as well as solid-state chemistry. Sintering General process by which powders react and densify to form polycrystalline compacts or ceramics. Such compacts densify by grain growth and porosity reduction. Sintering can be reactive, i.e., involve the reaction of two or more solid components to form a product or products. Structure Used herein to refer to the molecular-level, crystallographic, three-dimensional arrangement of atoms, as controlled by the chemical bonding. Both long-range and short-range order/disorder, including crystal imperfections and defects, must be considered.

SOLID-STATE CHEMISTRY concerns the preparation, structure, and properties of solid materials, often focusing on the relationship between structure and properties. Unique solid-state properties are taken advantage of for numerous practical, technological applications. Many properties are structure sensitive, i.e., they are controlled by the chemical bonding and molecular-level structure.

The syntheses and fabrication procedures themselves often play a key structure-controlling role in the preparation of materials with optimum properties. Detailed characterization of the structure and bonding is of major concern to the solid-state chemist. Control of structure and bonding during preparation and processing allows control of the critical, technologically important properties. Such control is necessary to optimize material performance.

I. THE STRUCTURE–PROPERTY RELATIONSHIP Solid-state chemists characterize materials with respect to their chemical and physical properties, structure, and bonding, as well as define how these properties are controlled by the chemical bonding and microscopic structure. They Vplay an increasingly important role as part of a team of condensed matter specialists whose common goal is to design materials with optimum properties for critical applications. Improved economic performance in a globally competitive economy is strongly dependent upon materials development. Materials problems limit the development of a variety of technologies, including high-temperature superconductors, high-energy density batteries, structural and insulating materials able to

295

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296 withstand extremely high temperatures, improved heterogeneous catalysts, improved electronic materials, optics compatible with very high laser intensities, stabile nuclear waste containment materials, and longer lasting electrodes for fuel cells and magnetohydrodynamic generators. To solve such problems and to optimize performance, it is necessary to control the molecular-level structure and bonding and to understand the structure–property relationship. The syntheses and fabrication processes themselves are therefore critical for preparation of solid-state materials which display the desired properties. Properties of interest include the chemical reactivity as well as the electronic, magnetic, optical, and mechanical properties.

II. SOLID-STATE STRUCTURE AND STRUCTURAL CHARACTERIZATION Characterization of the detailed chemical structure and bonding of a solid is a prerequisite to the understanding and control of the chemical and physical properties. In general the properties of a solid are controlled by the macroscopic and the microscopic structures. The macroscopic characteristics (e.g., dislocations) predominantly influence the mechanical properties, such as strength for metals and ceramics. The microscopic structure (interatomic or molecular-level structure) is controlled by the chemical bonding. Solids are classified according to their chemical bonding as metals, semiconductors, or insulators. Complete structural characterization of a material involves not only the elemental composition for major components and a study of the crystal structure, but also the impurity content (impurities in solid solution and/or additional phases) and stoichiometry. Noncrystalline materials can display unique behavior, and noncrystalline second phases can alter properties. Both the long-range order and crystal imperfection or defects must be defined. For example, the structural details which influence properties of oxides include the impurity and dopant content, nonstoichiometry, and the oxidation states of cations and anions. These variables also influence the point-defect structure, which in turn influences chemical reactivity, and electrical, magnetic, catalytic, and optical properties. Point defects are imperfections in the actual crystalline architecture as compared to the ideal lattice in which each atom site is filled with the appropriate element. They can influence properties at extremely low levels (ppb or ppm). Typical point defects include crystal sites with missing atoms (vacancies), atoms positioned in sites that are not filled by the crystal structure in question (interstitials), crystal sites containing impurity atoms (dopants), and cations with different oxidation states. Because mass,

Solid-State Chemistry

charge, and the number of lattice sites must be conserved, unusual oxidation states can be introduced (in dopants or predominant cations) and nonstoichiometric compositions stabilized. For example, Ni(III) can be formed in NiO, which nominally contains Ni(II) ions, and nickel vacancies (VNi ) are formed according to Eq. 1. 3+  Ni2+ Ni(1−3x) NiNi(2x) VNi(x)

(1)

Point-defect ordering (e.g., vacancy-dopant pairs) leads to interesting complications. Preparation conditions themselves (e.g., oxygen partial pressure and temperature in oxides) thermodynamically define and control this defect content and structure. It is important to realize that point defects are thermodynamically allowed and defined; they are not anomalous in the least. Therefore, undoped, highpurity compounds may exhibit sizable nonstoichiometry due to intrinsic point defects. Doping (intentional addition of an impurity) allows one to precisely control the point-defect content and nonstoichiometry and, thereby, the properties. Transport properties are influenced by the point defects. Electrical conduction (hole or electron transport) and solid state diffusion of atoms generally vary with the quantity and type of point defects. The determination of how nonstoichiometry is accommodated (i.e., by what type and amount of defect) is an active research area. Nonstoichiometry can also be accommodated by subtle changes in structure known as extended defects or crystallographic shear. Crystallinity, impurity levels, point-defect structure, and nonstoichiometry are each controlled by or influenced by the preparation method; therefore, it is discussed further in Section III. Surface properties can differ from the bulk structurally, both as clean surfaces or because of products formed on reactive surfaces (physisorbed or chemisorbed). The former can experience relaxation, that is, surface reconstruction due to the distortion in bonding for surface atoms which are lacking bonds. Impurity segregation at a surface can further alter properties, as can second phases formed on a surface. The activity of heterogeneous catalysts and corrosion is controlled by such surface properties and by the bulk and surface point-defect structures. Phases formed on semiconductor surfaces can change the electrical properties in an uncontrolled, deleterious fashion. Oxide passivation layers on compound semiconductors (e.g., mercury cadmium telluride IR detectors or gallium arsenide solar cells) can be grown to impart protection to the surfaces and to stabilize electrical properties by preventing uncontrolled reactions. Interfaces between two bulk phases, between the bulk and a surface, or at grain boundaries can further complicate the chemistry. Grain boundaries in polycrystalline materials can contain second phases (crystalline or noncrystalline) and have significant width. This is termed an

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interphase. On the other extreme, highly ordered (coherent) interfaces can occur between two microcrystallites (or grains). A second phase in a boundary can impart unique electrical properties, significantly influencing the characteristics of a capacitor, for example. Furthermore, dopants may segregate to a grain boundary phase, affecting boundary and bulk phase properties. Positive temperature coefficient (PTC) resistors and boundary layer capacitor operation are based on such effects. Characterization of the preceding structural variables is complex and challenges an extensive array of modern analytical instrumentation. Diffraction techniques (X-ray, neutron, and electron) are basic to the study of crystal structures. Improved data analysis techniques make these methods even more powerful for the study of powders. Nuclear magnetic resonance (NMR) spectroscopy has become a powerful tool for the study of solids with the advent of magic angle spinning techniques. Neutron inelastic scattering, Raman scattering, and IR vibrational spectroscopic analyses have been traditionally used to study lattice dynamics and solid-state phase transformations. They can also provide information about dopants and point-defect structures through studies of local modes as well as the extensive crystal structure information. High-resolution electron microscopy lattice imaging has proven to be a powerful tool for the study of crystal structures and extended defects. Electron spin resonance remains an effective tool for the study of paramagnetic solids, including impurities or low-level dopant structures. A variety of X-ray and electron spectroscopic techniques have been developed. These are particularly useful for providing information about elemental composition, surface structures, and cation oxidation states.

III. PREPARATION TECHNIQUES AND PROPERTY VARIATION The crystal structure of a solid can influence the properties of a material, for example, the structure must be noncentric for a material to demonstrate antiferromagnetic, ferromagnetic, ferroelectric, or piezoelectric behavior. Rapid cooling of a sample from high temperature and/or high pressure can quench in a structure that is not stable at room temperature or atmospheric pressure. High-pressure oxide polymorphs, which are more dense, have been studied to model the earth’s interior. Furthermore, unique crystal structure characteristics of a material can allow structure– property variation, for example, insertion compound formation in layered materials. Solids can be prepared as single crystals, glasses, thin films, powders, or sintered powder compacts (ceramics). Powders may be noncrystalline or polycrystalline, exhibit-

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297 ing varying degrees of order. Powders may be used as catalysts or as precursors for crystal growing, glass formation, or ceramic formation. Numerous technologically useful components are produced by sintering powders, taking advantage of the unique electrical, magnetic, optical, or mechanical properties. Particle size is also an important powder variable. Production of powders with a controlled particle size distribution allows production of ceramics with uniform microstructures, improving mechanical properties. Strength of zirconia and zirconia–alumina refractories can be improved another way. Phase transitions which involve large atomic displacements lead to microcracking during heating and cooling, which implement fracture. Smallparticle-sized zirconia stabilizes a high-temperature phase at lower temperatures. This behavior, called “phase transformation toughening,” has been explained on the basis of higher surface free energies for these systems. Since a phase transition from this high-temperature phase is eliminated, the structural properties are improved. Research has shown that novel chemical preparation methods allow the production of unique materials, demonstrating properties unattainable through more traditional methods. Traditional solid-state synthesis techniques require high temperatures to increase the kinetics and allow reaction in reasonably short times. It is common to react solid powders after mixing by grinding or ball-milling. To ensure complete reaction of two powdered reactants it may be necessary to carry out repeated grinding and heating cycles. The grinding is necessary to reduce diffusion distances and increase product homogeneity. Repeated grinding and high-temperature treatments introduce undesirable impurities. Low-temperature solid preparation methods (meaning from ∼900◦ C to cryogenic temperatures) can produce powders having fewer impurities, high surface areas, and other unique characteristics which have useful applications, such as reactive surfaces for sintering or for catalysis. Higher treatment temperatures can actually reduce such activity by changing the bulk and/or surface structures. Low-temperature syntheses can sometimes allow unique surface phases to be stable. Tetragonal barium titanate, prepared at ∼700◦ C, has hexagonal barium titanate on the surface, which is stabilized by a higher surface free energy. Normally this hexagonal phase is not formed below 1460◦ C. This hexagonal surface was also found to reversibly adsorb CO2 as a surface carbonate. A variety of solution (water or organic solvent) techniques have been devised to control composition (e.g., the Ca:Mn ratio in an oxide such as CaMnO3 ). Control of this ratio is important in terms of the compositionsensitive properties. This is especially true for transition

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298 metal–containing systems because several oxidation states are stable and available for most of them. Unusual oxidation states introduced by variation in cation ratio and oxygen nonstoichiometry can influence catalytic and electronic properties. The “solid-state precursor” method involves precipitation of a compound or solid solution (e.g., oxalate or carbonate) with the desired stoichiometry which is then thermally decomposed to form the desired product oxide. Atomic scale mixing of the components precludes long calcining times at high temperatures. Powders formed are extremely homogeneous and have high surface areas. The composition of the products, however, does depend on the solution solubilities, and the precision of these precursor methods is not always high. The so-called “liquid mix technique” does allow better precision as well as homogeneity because no precipitation occurs; rather, the solution is heated to a viscous, glasslike form which is then decomposed to the product oxide. No solution– solid partitioning occurs, giving a composition equal to that of the starting mixture. The latter can be weighed to high accuracy (hundreds of ppm or better for major components), providing precise control of product cation ratios. The sol–gel technique is a low temperature, solutionpreparation method which has been applied extensively to produce glasses, fibers, coatings (protective or dielectric), abrasive particles, and controlled-pore-size catalyst– substrates. This method is used to prepare glasses, for example, that cannot be obtained upon quenching from the melt. Either colloidal or polymeric gels are formed by gelation of a precursor solution, involving hydrolysis and condensation of colloidal sols of metal salt or hydroxide solutions or of metal alkoxides. Drying, solvent removal, and firing conditions are then chosen to provide the desired microstructures and properties. Another important solution technique which should be mentioned is the “homogeneous precipitation technique.” It favors formation of a more ordered, crystalline product when two solutions are mixed to form an insoluble compound. The principle is slow precipitation, avoiding instantaneous formation of a disordered product. The benefits of all of the low-temperature solution techniques include homogeneity (atomic scale mixing) and minimal introduction of impurities. Other methods of preparation include chemical vapor deposition (CVD) and electrochemical methods. The latter are used to form thin films and protective coatings as well as battery electrodes. Since this is generally a lowtemperature method, the structure can differ from that of the same material prepared at a higher temperature. It can be disordered or amorphous. Chemical vapor deposition involves vapor phase transport of volatile organometallics

Solid-State Chemistry

or other metal-containing species to the reaction site, and is used in the production and development of semiconductor devices. In some instances materials with potentially useful properties have not been exploited until prepared as pure crystals and films. Examples of this include doped polythiazyl, (SN)x , and polyacetylene, (CH)x , which have metallic properties, including electrical conductivity. The use of polymer precursors for ceramics (e.g., silicon carbide) is another interesting solid-state preparation technique. An exciting example, demonstrative of every aspect of solid-state chemistry, is the development of hightemperature superconducting oxides, which has followed the 1986 discovery of superconducting YBa2 Cu3 O7−x . This oxide will conduct at temperatures much higher than previous superconducting metal alloys, thereby reducing cooling expense. The synthesis, purification, characterization, extension to other metal-oxide systems, and the eventual application of these oxides in devices is certain to become a classic example of solid-state chemical science and technology. Control of the structure during preparation and processing allows one to control the properties and to optimize material performance for particular applications. Thorough structural characterization is a prerequisite. Based on the knowledge of how synthesis and processing influence structure and of how structure controls properties, the structure can be tailored and materials can be designed for optimum performance.

SEE ALSO THE FOLLOWING ARTICLES ANALYTICAL CHEMISTRY • BONDING AND STRUCTURE IN SOLIDS • CRYSTALLOGRAPHY • LASERS, SOLID-STATE • MICROSCOPY (CHEMISTRY) • PHASE TRANSFORMATIONS, CRYSTALLOGRAPHIC ASPECT • PRECIPITATION REACTIONS • SOLID-STATE IMAGING DEVICES • SUPERCONDUCTORS, HIGH TEMPERATURE • SURFACE CHEMISTRY

BIBLIOGRAPHY Brinker, C. J., and Scherer, G. W. (1990). “Sol–Gel Science,” Academic Press, Boston. Corbett, J. K., ed. (1985). Symposium on metal–metal bonding in solidstate clusters and extended arrays. J. Solid State Chem. 57(1), 1. Etourneau, J. (1999). Novel synthesis methods for new materials in solidstate chemistry. Bull. Mater. Sci. 22(3), 165–174. Fischer, J. E. (1997). Fulleride solid-state chemistry: Gospel, heresies and mysteries. J. Phys. Chem. Solids 58(11), 1939–1947, Grasselli, R. K., and Brazdil, J. F., eds. (1985). “Solid State Chemistry in Catalysis,” Series 279, American Chemical Society, Washington, D.C.

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Solid-State Chemistry Holt, S. L., Milstein, J. B., and Robbins, M., eds. (1980). “Solid State Chemistry: A Contemporary Overview,” Advances in Chemistry Series No. 186, American Chemical Society, Washington, D.C. Honig, J. M., and Rao, C. N. R., eds. (1982). “Preparation and Characterization of Materials,” Academic Press, New York. Nelson, D. L., and George, T. F. (1988). “Chemistry of High-Temperature Superconductors II,” American Chemical Society, Washington, D.C. Pimentel, G. C., and Coonrod, J. A. (1987). “Opportunities in Chemistry, Today and Tomorrow,” National Academy Press, Washington, D.C.

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299 Snyder, R. L., Condrate, R. A., and Johnson, P. F., eds. (1985). “Advances in Materials Characterization II,” Vol. 19, Materials Science Research, Plenum, New York. Sorensen, O. T., ed. (1981). “Nonstoichiometric Oxides,” Academic Press, New York. State of the art symposium: Solid state. (1980). J. Chem. Educ. 57, 531– 590. West, A. R. (1984). “Solid State Chemistry and its Applications,” Wiley, New York. Zelinski, B. J., and Uhlmann, D. R. (1984). Gel technology in ceramics. J. Phys. Chem. Solids 45, 1069.

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Solid-State Electrochemistry Gunter ¨ Holzapfel ¨ Dortmund University

I. Introduction II. Disorder in Solids III. Transport Processes—Diffusion, Mobility, and Partial Conductivity IV. Solid Electrolytes, Solid Ionic Conductors, and Solid–Solution Electrodes V. Galvanic Cells with Solid Electrolytes VI. Technical Applications of Solid Electrolytes

GLOSSARY Disorder Deviations from the regular crystal structure in a crystalline compound. Galvanic cell Arrangement for the conversion of chemical energy to electrical energy using electrodes in electrolytes. Interstitial ion Ion in excess compared to the ideal lattice of a solid. Ionic couductivity Electrical conductivity caused by ions. Solid electrolyte Solid compound in which the electrical current is carried out by ions, associated with mass transfer. Solid solution electrode Solid compounds exhibiting mixed (ionic and electronic) conductivity, where it is possible to dissolve or remove additional ions. Structural disorder Special kind of disorder in a crystal in which practically all ions of one kind are mobile and statistically distributed among their available lattice sites.

Superionic conductor Solid electrolyte with very high ionic conductivity, comparable to liquid electrolytes. Vacancy Missing ion in comparison to the ideal lattice of a solid.

THE FIELD of solid-state electrochemistry deals with research on physical, chemical or electrochemical problems using solid electrolytes. Solid electrolytes, also called solid ionic conductors, are solid, generally crystalline compounds in which the electrical current is carried by ions. Therefore, the passage of current is associated with mass transfer. Solid electrolytes enable us to build galvanic cells similar to those in the electrochemistry of liquids. Such galvanic cells with solid electrolytes play a most important role for scientific investigations in the field of thermodynamics and kinetics as well as for technical applications like batteries, sensors, fuel cells, and chemotronic components.

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I. INTRODUCTION The field of solid-state electrochemistry—the electrochemistry of solids—has developed rapidly since about the middle of the twentieth century. This was caused by the discovery of new solid electrolytes with high ionic conductivity and some fundamental publications which pointed out the importance of solid ionic conductors for the thermodynamic investigations. Between the electrochemistry of solids and the electrochemistry in liquids there exist large analogies in theoretical point of view as well as in experimental methods. This is shown in Table I. The theoretical treatment of the disorder in solids shows extensive analogies to the theory of electrolytic dissociation. In the case of liquid electrolytes, the dissociated ions enable the conduction of the electric current. In solid electrolytes the conductivity is caused by the thermodynamic disorder of the crystals. This means that there are vacancies or interstitial ions in the crystal lattice which enable electrical conductivity. A further analogy exists regarding galvanic cells. From a thermodyamic point of view the treatment of galvanic cells with solid electrolytes corresponds to that in the liqud phase. Furthermore, the applied measuring methods largely correspond to those known in the electrochemistry of liquids. In spite of these analogies the electrochemistry of solids is more complex than the electrochemistry in aqueous solutions. So it must be noted that apart from ionic conduction, solids often show an electronic conductivity, caused by electrons or electron defects, which may be predominant in many cases over the ionic conduction. In good solid electrolytes the conduction of the electrical current is caused exclusively by the ions—in most cases practically by only one kind of ion present in a crystal. To explain the ionic conductivity of the solid the components that influence it must be examined. For the ionic conductivity σi of an ionic species i to be valid: σi = z i Fu i ci .

(1)

Here F is the Faraday constant, z i is the valence, ci is the concentration, and u i is the electrical mobility of the ions. From this equation it can be seen that two important quan-

tities influence the partial conductivity σi . These quantities are the concentration ci of the particle i and the electrical mobility u i . The concentration ci corresponds to the disorder of the solid and the mobility u i to transport processes.

II. DISORDER IN SOLIDS Under real conditions all ordered compounds exhibit deviations from a regular crystal structure. These are of great importance particularly for the understanding of the thermodynamic and kinetic properties of crystalline compounds. Such deviations may be: 1. interstitial ions, which are ions in excess as compared to the ideal lattice; 2. charged vacancies, which are missing ions compared to the ideal lattice; 3. foreign ions, which may be on either interstitial or regular lattice sites; and 4. electron disorder, which means the presence of quasi-free electrons and electron defects. These so-called point defects can be described by structure elements according to Kr¨oger or by building units according to Schottky. The symbols used for these descriptions are summarized in Tables II and III. Structure elements are defined relative to the empty space, and building units are defined relative to the ideal lattice. One building unit according to Schottky corresponds to a combination of structure elements (generally two) according to Kr¨oger. This is shown in the third column of Table III. Possible defects are illustrated schematically in Fig. 1 for an AB crystal using Kr¨oger symbols. The two-dimensional section through the AB lattice shows A and B particles mainly in their normal positions. These particles have not been assigned any electrical charge because it is only meaningful to express charge relative to the unperturbed lattice, and A and B ions in their normal sites are electrically neutral relative to the unperturbed lattice. The figure also shows two A ions at interstitial sites; these ions bear an TABLE II The Kroger ¨ Symbols for Neutral Structure Elements in an AB Latticea

TABLE I Analogies between the Electrochemistry of Liquids and Solids Liquids

Solids

Electrolytic dissociation Galvanic cells with liquid electrolytes Electrochemical kinetics, particularly electrode kinetics

Disorder in solid compounds Galvanic cells with solid electrolytes Reactions in and on solids

Particle A

Particle B

Vacancy V

Foreign particle C

A site B site

AA AB

BA BB

VA VB

CA CB

Interstitial

Ai

Bi

Vi

Ci

Sites

a The examples chosen are A and B particles, foreign particles C, and vacancies at different sites.

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Solid-State Electrochemistry TABLE III Notation of the Lattice Molecule and Typical Defects in an AB Lattice in Terms of Building Units Using the Old and New Schottky Symbols Compared to the Corresponding Structure Elements According to Kroger ¨ and Vink Schottky old

Schottky new

Kr¨oger/Vink

AB lattice molecule Neutral A particle on interstitial site Neutral A vacancy Neutral B particle on A-site

AB AO

AB A

Ai − Vi

A B•(A)

|A| B|A|

V A − AA BA − AA

Neutral C particle on A-site Quasi-free electron Electron hole

C•(A)  ⊕

C|A| e |e· |

CA − AA e h

excess of positive charge denoted with. Two unoccupied sites can be seen; these vacancies are negatively charged relative to the unperturbed lattice and are denoted with  . Structure elements with the corresponding symbols proposed by Kr¨oger and Vink are easy to memorize; this notation is now also widely used in the literature. However, it is very important to know that the numbers of the various structure elements in a crystalline compound are not independent of one another. This is due to the fixed ratio of the number of A and B sites in an AB lattice. It is therefore generally impossible to change the concentration of only one type of structure element in a crystal. Thus, to create a vacancy on an A site we must either remove an A particle or simultaneously add a B particle to a new B site, so that the ratio of A to B sites remains unchanged. The generation of an interstitial particle leads to the destruction of a vacancy in the interstitial lattice. Thus, if we increase the size of a crystal or change the number of defects contained in it, one must either add or remove combinations of structure elements, this means that in general building units have to be used. Similarly, the use of structure elements to describe reactions of defects generally involves the use of building units.

FIGURE 1 Possible defects in an AB crystal using Kroger ¨ symbols.

The concentrations of charged atomic defects—point defects—follow the law of mass action. The considerations of thermodynamic equilibria can be applied to disorder equilibria in solid crystalline compounds, the socalled ordered mixtures. Point defects can be regarded as quasi-chemical species with which chemical reactions can be formulated. This has led to the so-called imperfection chemistry. As an example, the disorder equilibrium between vacancies and interstitial particles—the so-called Frenkel equilibrium—will be regarded. In this case a particle A moves from an A lattice site to an interstitial site whereby, for example, with respect to the unperturbed lattice a single positively charged interstitial . particle Ai is formed, a vacancy Vi in the interstitial lattice is destroyed, and a negatively charged vacancy VA on an A site is left. This exchange process can be written in the form of a chemical reaction, a so-called disorder reaction . A + |A| = 0 (2) using the new Schottky notation, we have . AA + Vi = Ai + VA .

(3)

Here the reaction is formulated in terms of structure elements, which for a thermodynamical treatment must be combined with building units as follows: . (Ai − Vi ) + (VA − AA ) = 0. (4) The concentration of the interstitial particles or vacancies as building units is identical with the concentration as structure elements. Symbolizing the concentrations by square brackets the law of mass action corresponding to Eq. (4) can thus be formulated as follows: . [Ai ][VA ] = const. (5) For all crystals there exists at equilibrium a constant product of the concentrations of interstitial particles and the corresponding vacancies, the so-called Frenkel equilibrium which depends on temperature and pressure. For higher concentrations of defects the equation must be written in terms of activities instead of concentrations. Frenkel disorder occurs in the silver halides, for example, in AgCl. Another kind of disorder equilibrium exists between A vacancies and B vacancies in a binary AB crystal—the so-called Schottky equilibrium. In this case the exchange of particles between A and B vacancies and the crystal is considered. This means that either A or B particles are transferred to the surface of the crystal, which is thus enlarged, while A or B vacancies are generated, or vice versa. For example, the A vacancies may be singly negative and the B vacancies singly positive. When the particles are brought to the surface, two of the particles there will be

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converted to volume particles, that is, the exchange effectively leads to the generation of a new lattice molecule and two vacancies. This can be written in terms of the new Schottky notation as . O = |A| + |B| + AB, (6) or using structure elements: . AA + BB = VA + VB + AB.

(7)

Here AB denotes the lattice molecule. This equation can be written in terms of building units: . O = (VA − AA ) + (VB − BB ) + AB. (8) Since the concentration of AB molecules can be regarded as constant at low defect concentration it can be included into the constant of the law of mass action, and we obtain: . [VA ][VB ] = const, (9) which means the product of A and B vacancies in an AB crystal is constant in this case. At higher concentrations of the defects activities must be used again. Schottky disorder is to be found in alkali metal halides such as NaCl and KCl. Aside from the described kinds of disorder there exist various other types. As an example in which electrons and the environment are playing a role in disorder, let us consider ZnO. At a temperature of 900◦ C the most important disorder centers in this compound are single positively charged zinc ions on interstitial sites and free electrons. The following equation representing the incorporation of oxygen, for example, from the air, into ZnO may therefore be formulated: . 1 O (g) + Zni + e = Vi + ZnO, (10) 2 2 or using building units: 1 O (g) 2 2

. + (Zni − Vi ) + e = ZnO.

(10a)

Because of electrical neutrality it follows that in ZnO . [Zni ] ∼ (11) = [e], that is, the concentration of zinc ions on interstitial sites is virtually equal to that of free electrons. The law of mass action corresponding to Eq. (10) is given by . 1/2 pO2 [e][Zni ] = const, (12) which may be simplified using Eq. (11) to yield 1/2

pO2 [e]2 = const

(13)

or −1/4

[e] ∼ pO2 .

(14)

According to Eq. (14) the concentration of free electrons is proportional to the inverse fourth root of the oxygen partial pressure; this is shown in Fig. 2. The conductivity

FIGURE 2 Dependence of the concentration of the electrons of ZnO at 900◦ C on the oxygen partial pressure.

of ZnO, which is mainly due to the partial conductivity of the free electrons, decreases with increasing oxygen partial pressures according to Eq. (14). The concentration of imperfection centers increases with rising temperature. The limiting case is crystals in which the concentrations of vacancies and interstitial particles become comparable; there then occurs a statistical distribution of particles in normal lattice positions and in interstitial positions. In this case it is no longer reasonable to distinguish between regular and interstitial lattice positions. The total number of positions a single type of particle can occupy may be several times higher than the number of such particles in the crystal. There exist crystals which have much more equivalent lattice sites available for one type of ion than ions are present in the lattice. At sufficiently high temperatures all ions are mobile and may be statistically distributed among these lattice sites. In this case we say a partial lattice of the crystal is in a quasi-molten state, the crystal has now a structural disorder. If such a disorder is present, the natural limit of the concentration of the mobile species is reached because all ions of one kind are now mobile. The best electrolytes known have such a disorder. Examples for such types are among others RbAg4 I5 at room temperature and AgI above 149◦ C. At this temperature AgI makes a transition from the β to the α phase; the partial lattice of Ag becomes quasi-molten, and there exist regions throughout which the silver ions virtually perform random motions as shown in Fig. 3. The idea that the silver ions in α-AgI are already in a quasi-molten state is supported by thermodynamic values and the diffusion coefficients, which are of the order of 10−5 cm2 /s similar to a liquid. The entropy for the transition from β-AgI to α-AgI amounts to 14.5 J/K mol, the entropy of fusion is 11.3 J/K mol. It is assumed that silver iodide melts in two stages. In the β–α transition the silver partial lattice passes into

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j = −D dc/d x,

(15)

where D is called Fick’s diffusion coefficient. If there is a concentration coefficient in any direction in space then we must use the general form of Fick’s first law:   dc dc dc , , . (16) j = −D grad c = −D d x dy dz

FIGURE 3 Lattice of silver iodide with regions in which the silver ions can move.

the quasi-molten state (structural disorder), whereas the iodide partial lattice does not become liquid below the actual melting point. Comparable crystals that do not show the special behavior of structural disorder before they melt have an entropy of fusion about as large as the sum of the transition entropy and entropy of fusion in AgI, that is, about twice as large as the residual entropy of fusion in AgI. For completeness let us mention that besides the point defects discussed in this chapter there exist still other defects in crystals. These include 1. one-dimensional defects such as edge of screw dislocations, 2. two-dimensional defects such as surfaces or grain boundaries, and 3. three-dimensional defects such as cavities. These defects are not discussed here because they are not properties of the thermodynamic equilibrium.

III. TRANSPORT PROCESSES—DIFFUSION, MOBILITY, AND PARTIAL CONDUCTIVITY In this section the transport of ions in an electrical field and their diffusion in a concentration or activity gradient will be treated. The expressions derived are valid for the fluxes of each type of ion or electron separately. From these expressions equations for an interconnected transport of different types of particles can be derived. In the following a phenomenological treatment and an outlook on the statistical treatment will be given. A. Transport by Diffusion Fick stated an empirical relationship for the diffusion flux j in a concentration gradient dc/d x in the x direction. This is known as Fick’s first law:

In what follows only gradients in one direction (the x direction) will be regarded. The time dependence of the concentration is given by Fick’s second law: ∂c ∂ 2c (17) = D 2. ∂t ∂x These laws are valid in the case of ideal behavior that is, when the chemical potential µ of the particles holds: c µ = µ0 + RT ln 0 , (18) c where µ0 is the chemical potential in the standard state, c is the concentration of the particles, c0 is their concentration in the standard state, R is the general gas constant, and T is the absolute temperature. Using this equation Fick’s first law [Eqs. (15)] can be written as Dc dµ . (19) RT d x In the case of nonideal behavior, that is, for higher concentrations of the mobile particles, the more general expression jx = −

µ = µ0 + RT ln a

(20)

must be used for the chemical potential of the diffusing species, where a is the activity. A component diffusion coefficient D K is then defined in such a manner that an expression analogous to Eq. (19) is obtained: D K c dµ . (21) RT d x A relationship between D and D K can be reached in the following way. Since dc is equal to c d ln c, Eq. (15) can be expressed as j =−

d ln c . (22) dx From Eqs. (21) and (20) an expression for the particle flux which contains the component diffusion coefficient D K is obtained: d ln a j = −D K c . (23) dx A comparison of Eqs. (22) and (23) gives the relationship between D K and Fick’s diffusion coefficient D: d ln a D = DK . (24) d ln c j = −Dc

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As can be seen from Eq. (24), D = D K if the activity a is proportional to the concentration c. The factor d ln a/d ln c is called the thermodynamic factor. Another diffusion coefficient—the tracer—or self-diffusion coefficient DTr is defined, which can be measured at electrochemical equilibrium by using a radioactive isotope. Under certain conditions, for example, for vacancy diffusion, the tracer diffusion coefficient DTr is related to the component diffusion coefficient by: DTr = f D K ,

B. Transport in an Electrical Field If an electrical field only is present in the crystal and no gradient in the concentration or activity of the particles, the electrical current density i can be related to the electrical field strength E by Ohm’s law i = σE

(26)

i = −σ (dϕ/d x),

(27)

or where ϕ denotes the electrical potential and σ the electrical conductivity. The electrical current density i is connected with the current density j of the particles (28)

Here F denotes the Faraday constant, and z is the charge of the particles. From Eqs. (27) and (28) it follows that j =−

σ dϕ . zF dx

(29)

σi dϕ D K ,i ci dµi − . RT d x zi F d x

Eq. (30) may be reduced to give   ci D K ,i dµi z i F dϕ ji = − + . RT dx dx With the definition of the electrochemical potential

(34)

or ji =

σi 2 2 zi F

dηi dx

(34a)

or using the electrical mobility according to Eq. (1): ji =

u i ci dηi zi F d x

(35)

From Eqs. (34) and (35) it can be seen that the discussion of mobility of the particles can be reduced to the discussion of the diffusion coefficient D K ,i . D. Chemical Diffusion Under certain conditions the fluxes of ions and electrons are related to each other by the conditions of electrical neutrality. This holds especially in the case when local differences in stoichiometry equilibrate. Here metal ions and electrons or nonmetal ions and electrons diffuse simultaneously. These transient phenomena are described ˜ In the by the so-called chemical diffusion coefficient D. ˜ following the result for the relationship between D and the component diffusion coefficient will be given for a compound whose partial conductivity σX− of the nonmetal ions is negligible in comparison with that of metal ions σMe+ , and at the same time their electron partial conductivity is much larger than that of metal ions: (36)

d ln aMe D˜ = D K ,Me , d ln cMe

(37)

where aMe and cMe are the activity of the metal and the concentration, respectively. Examples are the compounds FeO and Ag2 S.

(30) E. Atomistic Interpretation of the Transport of Ions in Solids

Using Eq. (1) and the Nernst–Einstein equation D K ,i = (u i /z i F)RT.

ci D K ,i dηi RT d x

In this case we obtain:

If besides a gradient of the electrical potential, ϕ, a gradient of the chemical potential µ or activity a of the particles is also present in the compound, we must write the more general equation ji = −

ji = −

σi  σMe+  σX− .

C. Transport in an Electrical Field and in a Concentration- or Activity-Gradient

(33)

the expression in parentheses in Eq. (32) can be summarized to dηi /d x and we get:

(25)

where f is the so-called correlation factor. The correlation factor f is of the order of one for most cases.

j = i/z F.

ηi = µi + z i Fϕ

(31)

(32)

To understand the high ionic conductivity of some solid compounds an atomistic interpretation of the transport of ions in solids should be regarded. From the atomistic point of view the movement of ions in solids can be regarded as successive jumps between lattice sites or interstitial sites. For random motion of a

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particle in an isotropic crystal the diffusion coefficient can be expressed by D K ,i = 16 νa 2

(38)

if there is no correlation between the jumps; ν denotes the jump frequency and a denotes the distance for each jump. Using Eqs. (38), (31), and (1) we obtain for the partial conductivity  σi = ci νa 2 z i2 F 2 6RT (39) and for the electrical mobility:

 u i = z i F νa 2 6RT .

(40)

From Eqs. (39) and (40) we can see that the important quantity for the conductivity and the mobility is the product of the jump frequency of an ion and the square of the jump distance. The jump distance is of the order of the lattice parameter. It is possible to estimate an upper limit of the jump frequency. The maximum jump frequency νmax , that is, the highest frequency of change of a lattice site, results if the particles move with thermal speed v between the lattice sites without performing oscillations on such a site. Thus νmax = v/a .

(41)

For a given temperature this natural limit cannot be exceeded. The maximum conductivity denoted as σi (max), is then given according to Eqs. (40) and (41) as ci vaz i2 F 2 . (42) 6RT Herewith we can calculate a maximum possible partial conductivity for a substance, for example, silver iodide. Assuming that the silver ions migrate with thermal velocity v from one lattice site to another without oscillating at each lattice site, we get a jump frequency ν = v/a = 3.4 × 1012 s−1 at a temperature of 300◦ C, a diffusion coefficient of D K ,i = 5.6 × 10−5 cm2 /s, and the maximum conductivity is σ(max) = 2.8 −1 cm−1 for a jump distance ˚ The measured conductivity is σ = 1.97 −1 cm−1 , of 1 A. which is not much less than the calculated value. Many efforts have been made to improve this very simple but already good model. Besides jump and lattice gas models continuous models have been made. These models rely on the fact that the diffusion of an ion is not represented by instantaneous jumps from an equilibrium site to another one but by a continuous motion in between. From these considerations it can be seen that there is a natural upper limit for the value of the ionic conductivity of solid compounds. This upper limit is between 1 and 10 −1 cm −1 corresponding to a component diffusion coefficient of about 2 to 20 × 10−5 cm2 s. These values correspond to those in liquid electrolytes. Good σi (max) =

solid electrolytes reach these values or come near them. They are sometimes called super ionic conductors.

IV. SOLID ELECTROLYTES, SOLID IONIC CONDUCTORS, AND SOLID–SOLUTION ELECTRODES Having discussed in Sections II and III concentration and mobility, which influence the conductivity of solid electrolytes, a compilation of solid ionic conductors will be given in this section. This compilation does not presume to be complete because new solid electrolytes are discovered and developed continuously. In Fig. 4, the conductivities of some of the most important ones are shown as a function of temperaure and reciprocal temperature. The conductivity of liquid sulfuric acid is included for comparison. In the following, several important solid electrolytes will be treated according to the type of mobile ions that cause the ionic conductivity. A. Silver Ion Conductors One of the first solid electrolytes exhibiting a very high ionic conductivity, found in 1914, is α-Agl. This conducting α-phase is stable above 149◦ C and its high conductivity is caused by structural disorder. A similar disorder exists in RbAg4 I5 . This solid electrolyte exhibits the highest silver ion conductivity at room temperature at present. Therefore, it is of great technical interest. A

FIGURE 4 Conductivity of some very common solid electrolytes; H2 SO4 included for comparison.

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Solid-State Electrochemistry TABLE IV Conductivity of Several Silver Ion Conductors Compound

Temperature (◦ C)

σ ×102 /Ω cm) (×

150–500 −15–270

130–260 14–100

Agl RbAg4 I5 Ag3 SI Ag3 SBr Ag2 Hgl4 C5 H5 NHAg5 I6

TABLE VI Conductivity of Several Sodium Ion Conductors Compound

250–440

84–100

β-NaAl11 O17 Na5 GdSi4 O12 Na3 Zr2 Si2 PO12

20–300

0.1–8.2

NaSbO3 · 16 NaF

50–90 −20–480

Temperature (◦ C)

σ ×102 /Ω cm) (×

20–640 40–210

1.5–55 0.3–10

40–220

0.3–7.5

40–220

0.001–0.7

0.001–0.4 0.1–440

number of other silver ion conductors have been developed. Some of them and their conductivities for the given temperature ranges are listed in Table IV.

to obtain three-dimensional conductivity. This electrolyte has become of great interest for building sodium sulfur batteries, as will be discussed in Section VI.B. The conductivities of β-Al2 O3 and some other sodium ion conductors are shown in Table VI.

B. Oxygen Ion Conductors

D. Copper Ion Conductors

Doped zirconia dioxide and thorium dioxide are important solid electrolytes that owe their conductivity to transport of oxygen ions. They can be used between 600 and 1600◦ C. They are also an interesting example of how high ionic conductivity can arise by processes other than structural disorder. The disorder centers responsible for the ionic conductivity of zirconium dioxide are charged oxygen ion vacancies. These are produced by dissolution of CaO, MgO, or Y2 O3 in the zirconium dioxide. The calcium is incorporated at zirconium positions, since, however, only one oxygen ion is introduced with each calcium ion, one oxygen position remains unoccupied for each calcium atom introduced. The amount of doping is of the order of 10 mol%; in this way it can be seen why doping produces a very large number of vacancies in the zirconium dioxide. In Table V the conductivities of several oxygen conductors in given temperature ranges are listed.

The first solid electrolytes with high copper ion conductivity at room temperature were discovered in 1973. An example is 7CuBrC6 H12 N4 CH3 Br, whose conductivity at room temperature is 0.017 −1 cm−1 . Several other copper ion conductors have since been described. One of these conductors represented by the formula Rb4 Cu16 I7 Cl13 has a conductivity of 0.34 −1 cm−1 at 25◦ C. This is the solid electrolyte with the highest conductivity at room temperature known at present. In Table VII the conductivity of some copper ion conductors are listed.

C. Sodium Ion Conductors The most important Na+ ion conductor is Na2 O 11Al2 O3 , the so-called β-Al2 O3 . The mobile sodium ions are incorporated into planes of the lattice and can therefore move only in two dimensions; polycrystalline material is used TABLE V Conductivity of Several Oxygen Ion Conductors Compound

Temperature σ ×102 /Ω cm) (× (◦ C)

E. Proton Conductors Solid-state proton conductors with high ionic conductivity are eagerly sought because they could have important practical applications, for example, in fuel cells, water electrolyzers, and sensors. Substances with an appreciable proton conductivity known today are hydrogen uranyl phosphate tetrahydrate (HUP) HUO2 PO4 ·4H2 O— usable above 1◦ C—with an ionic conductivity of σ = 4 × 10−3 −1 cm−1 at 20◦ C and hydrogen uranyl arsenate tetrahydrate HUO2 AsO4 ·4H2 O—usable above 29◦ C— with an ionic conductivity of σ = 6 × 10−3 −1 40◦ C. Another kind of solid proton conductor is the protonic β-alumina. It can be produced by exchanging the TABLE VII Conductivity of Several Copper Ion Conductors Compound

ZrO2 (10 mol% Sc2 O3 ) ZrO2 (10 mol% Y2 O3 ) ZrO2 (13 mol% CaO) ThO2 (7.5 mol% Y2 O3 )

600–1400 600–1400

2–100 0.3–50

640–1400

0.2–30

1000–1500

Bi2 O3 (20 mol% Er2 O3 )

270–730

1.3–12 0.001–45

Temperature σ ×102 /Ω cm) (◦ C) (×

Rb4 Cu16 I7 Cl13 7CuBr·C6 H12 N4 CH3 Br

10–110

28–62

10–130

1.5–14

7CuCl·C6 H12 N4 HCl 17Cul·3C6 H12 N4 CH3 I

20–110

0.4–5

20–140

0.1–2.2

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whole sodium content of the solid sodium ion conductor Na–β-A12 O3 (see Section IV.C) for H+ , H+ (H2 O)n , or NH+ 4 ions. The ionic conductivity of these compounds is in the region of 10−4 –10−5 −1 cm−1 at room temperature. Various possible mechanisms of the conduction process in the different proton conductors are being discussed. The longtime stability of these compounds, which is especially important for possible technical applications, has not been clearly established. There exist still other solid proton conductors but as their conductivity in general is not as high as in the compounds mentioned earlier they will not be treated here. In addition to the solid electrolytes already described there exist solids in which other ions (e.g., lithium, fluoride or chloride ions) are mobile. These solid ionic conductors will not be treated here. F. Solid–Solution Electrodes Another class of important solids is mixed conducting solids exhibiting both fast ion transport and electronic conductivity. If, in addition to these properties a sufficiently large range of stoichiometry is present, these mixed conducting solids can be used as electrodes, for example, for batteries, the so-called solid–solution electrodes. One of the most promising solid–solution electrodes is based on titanium disulfide, in which it is possible to dissolve relatively large amounts of lithium metal in the TiS2 phase. There is a continuous range of nonstoichiometry from TiS2 to LiTiS2 . The structure of TiS2 and other similar chalcogenides of transition metals can be described as a sequence of layers held together by van der Waals forces only. The lithium is dissolved between the layers whereby the distance between the layers is slightly increased. Other changes in the crystal structure do not occur. Compounds of this kind are sometimes called insertion or intercalation compounds. Many other solid–solution electrodes have been investigated. Examples of other layer compounds besides TiS2 that are able to dissolve certain alkali metals and in some cases Cu+ or Ag+ ions include TiSe2 , MoS2 , WS2 , TaS2 , ZrS2 , NbS2 , VSe2 , MoSe2 , WSe2 , and CrS2 . In this regard we should also mention a related series of certain transition metal oxides that do not exhibit layered structures but are able to insert alkali metals or copper or silver, for example, TiO2 , MnO2 , MoO3 , WO3−y , V2 O5 , and Ta2 O5 .

V. GALVANIC CELLS WITH SOLID ELECTROLYTES The existence of solid ionic conductors has made possible the development of the electrochemistry of solids. In the

electrochemistry of solids galvanic cells with solid electrolytes play a very important role. In analogy to galvanic cells with liquid electrolytes, those with solid electrolytes consist of at least two electrodes separated by an electrolyte, which in this case is a solid ionic conductor. The important properties of such cells will be discussed. For this purpose the following galvanic cell will be considered as an example: pO , Pt / ZrO2 (+Y2 O3 ) / Pt, pO 2 2

←−−− ←−−−−−−− ←−−−. 4e−

2O2−

4e−

I It consists of doped zirconium dioxide as solid electrolyte with practically pure ionic conduction for oxygen ions. On the two sides there are porous, electronically conducting electrodes (e.g., consisting of porous platinum), surrounded by gaseous oxygen at different partial pressures. The zirconium dioxide must separate the electrode spaces from one another in gas-tight manner. For example, a tube of zirconium dioxide can be used that carries one electrode on the inner side and the second on the outer side, the outside and the inside being surrounded by gaseous oxygen at different partial pressures. If in this galvanic cell a positive electrical current flows from the left to the right electrode, 1 mol of O2 is transported from the right to the left electrode space by the passage of 4 faradays (4f). The following cell reaction occurs: O2 (right electrode) → O2 (left electrode).

(43)

Considering this cell reaction, we can obtain two properties of cell I; the first property is a thermodynamic one and the second a kinetic one. The thermodynamic property is the following. The electrical work that in the reversible case of the galvanic cell can be performed with the passage of 4 faradays amounts to 4EF, where E is the emf of the galvanic cell, defined as the electrical potential of the right electrode minus that of the left electrode. This work is equal to the negative Gibbs energy G of cell reaction (43): G(cel1 reaction) = −4EF.

(44)

Here G can be related to the chemical potentials µO2 of oxygen, and we then have, for Eq. (44), the following expression: µO2 (left electrode) − µO2 (right electrode) = −4EF. (45) The chemical potential µO2 is connected with the oxygen partial pressure pO2 by the following equation:    µO2 = µ0O2 + RT ln pO2 pO0 2 . (46) Here µ0O2 denotes the standard chemical potential of oxygen, corresponding to a partial pressure pO0 2 of 1 atm; R is

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the gas constant and T is the absolute temperature. Substitution of Eq. (46) into Eq. (45) gives: E=

pO RT pO (right electrode) RT ln 2 = ln  2 . (47) 4F pO2 (left electrode) 4F p O2

This shows a relation existing between E, the emf of the galvanic cell, and the ratio of oxygen partial pressures at the two electrodes. If one oxygen partial pressure is known the other one can be determined. According to Eqs. (45)–(47) the emf of a galvanic cell contains different thermodynamic information. The emf makes it possible to determine the Gibbs energy of the cell reaction and the chemical potentials of the electrode components or the partial pressures of gases. It shall be mentioned here that reaction enthalpies and reaction entropies can also be deduced from the temperature dependence of the emf. More details on thermodynamic investigations will be given in Section V.A. The kinetic properties of galvanic cell I with doped zirconia as solid electrolyte arise from the fact that the flux of current through a cell such as cell I is a measure of the reaction rate by which oxygen is passed from one side of the cell to the other. Only oxygen ions can flow through the electrolyte when the electrical circuit is closed. For the rate of transport of the O2− particles in moles per unit time J through the electrolyte we can write: J = I /z F,

(48)

where I is the electrical current; in the case of oxygen the valence z is −2. The transport of mass in the form of ions through the electrolyte can often be attributed to a chemical reaction or a transport process at an electrode. In this way reaction rates can be measured electrically. It is often possible to analyze reaction mechanisms in detail by a combination of rate measurements by means of the electrical current with measurements of thermodynamic quantities—in particular, chemical potentials—by means of the emf of the galvanic cell. More details on kinetic investigations using galvanic cells will be given in Section V.B. A. Thermodynamic Investigations As a typical example for thermodynamic investigations using solid electrolytes the determination of the Gibbs energy G 0NiO at temperatures of 800–1000◦ C will be considered. The following cell with doped ZrO2 as solid electrolyte for oxygen ions can be used: Pt, Ni, NiO / ZrO2 (+Y2 O3 ) / Pt, pO2 = 1atm

2f: ←−−−−− ←−−−−−−− ←−−−−−−−. 2e−

O2−

II

2e−

One way to obtain a relation between the Gibbs energy and the emf of the cell is to regard the so-called virtual cell reaction. We assume that a certain amount of charge is passed through the cell as a current. In this example, the cell reaction is the formation of 1 mole NiO from solid nickel and oxygen by passing a flow of electricity of 2 f through the cell: Ni + 12 O2 = NiO · · · G NiO . This reaction does not take place under open-circuit conditions. It would take place under current flow, but then, in general, polarization effects will occur. The maximum possible electrical energy that we could obtain from the cell for the virtual reaction is the measured emf under open-circuit conditions multiplied by 2 f in this example. This electrical energy is related to the Gibbs energy G of the cell reaction; in this case to the Gibbs energy of formation of NiO from solid nickel and oxygen by: G 0NiO = −2EF.

(49)

Here G 0NiO is written because NiO as well as Ni and O2 are in their standard states. Similar investigations have been carried out on many other systems, for example, Cu2 O, FeO, PbO, In2 O3 , WO2 , ZnO, SiO2 , MoO2 , NiCl2 O4 , FeCr2 O4 , NiAl2 O4 , MgF2 , ThF4 , UF3 , and AlF3 . Furthermore, enthalpies and entropies of reaction as well as partial molar enthalpies and entropies and partial molar volumina can be measured using similar cells. B. Kinetic Investigations According to the electrochemistry of liquids kinetic investigations using solid electrolytes can be carried out in different ways. 1. Measurements at Zero Current In the case of zero-current measurements the electrical potential difference between the two end phases of the cell is measured under open circuit conditions. Information about thermodynamic quantities of reaction systems, for example, about chemical potentials, activities, or partial pressures, is obtained from such measurements. This was already described. 2. Measurements under Steady-State Conditions In this case there is no time dependence of currents and potentials. Therefore, it is experimentally unimportant whether the potentials or the currents are controlled. Here the steady-state current that represents a reaction rate may be measured as a function of the potential or vice versa.

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3. Measurements under Controlled Potential In this case a constant potential difference is applied to the cell and the resulting current is measured as a function of time. These investigations are called potentiostatic. 4. Measurements under Controlled Current In this case a constant current is passed through the cell and the resulting potential difference is measured as a function of time. These investigations are called galvanostatic. If galvanostatic or potentiostatic measurements or investigations under steady-state conditions are carried out it is often preferable to use separate cells of the same type under zero current to measure potential differences, so that the reference electrode is not polarized by a current. In the following, a few examples of kinetic investigations using solid electrolytes will be discussed. C. Electrochemical Measurements of Oxygen Diffusion in Metals The principle of the electrochemical measurement of oxygen diffusion in a metal consists in bringing the metal from a well-defined state into another well-defined state and following the diffusion-controlled relaxation process electrochemically. For example, the metal sample is placed on one side of the solid electrolyte ZrO2 and functions as one electrode of a galvanic cell. On the other side of the electrolyte there is a practically unpolarizable electrode such as porous platinum in contact with air, or an Fe/FeO electrode, which has a fixed oxygen partial pressure of about 10−19 atm at 800◦ C. The following cell may be used: Fe, FeO / ZrO2 (+Y2 O3 ) / Me + O(dissolved)

←−−− ←−−−−−−− ←−−−−−−−−− 2e−

O

O2−

←−−−−−−−−−. 2e−

III In cell III there is an Fe/FeO electrode on one side and a metal containing dissolved oxygen on the other side. The emf of the cell before beginning the experiment is a measure of the initial activity or concentration of the dissolved oxygen. At a certain time an emf is applied to the cell to make the oxygen activity at the metal/electrolyte interface very small. Then oxygen diffuses out of the metal and is carried as an electrical current through the electrolyte to the other side of the cell. In this way the diffusion current is transformed into an electrical current and can be measured. From the time dependence of the current the diffusion coefficient can be calculated using suitable diffusion equa-

TABLE VIII Determination of the Diffusion Coefficient of Oxygen in Various Solid and Liquid Metals

Metal

Solid or liquid

Temperature range (◦ C)

Diffusion coefficient (cm2 /s)

Ag Ag

s l

760–900 970–1200

1.5 × 10−5 –2.9 × 10−5 8.2 × 10−5 –1.7 × 10−4

Cu

l

990–1220

1.4 × 10−4 –2.2 × 10−4

Cu

s

800–1030

9.3 × 10−6 –3.5 × 10−5

Sn Ni

l s

730–930 1393

4.5 × 10−5 –7.4 × 10−5 1.3 × 10−6

Pb

l

800–1100

1.0 × 10−5 –1.7 × 10−5

Fe

l

1620

1.5 × 10−4

Sb Bi

l l

750–950 750–950

1.4 × 10−6 –2.9 × 10−6 8.6 × 10−6 –1.4 × 10−5

tions. Several systems have been investigated in this way. Some results are shown in Table VIII. D. Measurements of Chemical Diffusion Coefficients The process of attaining a uniform composition, which occurs in compounds where an existing gradient of stoichiometry is allowed to equalize, can be described by ˜ discussed in Secthe chemical diffusion coefficient D tion III.D. For such equilibration processes it is necessary, on the grounds of electrical neutrality, that both ions and electrons or electron defects must migrate simultaneously whereby the fluxes of ions and electrons are related to one another. The electrochemical method for the determination of chemical diffusion coefficients D˜ will be shown here as an example for the mixed conductor w¨ustite FeO. The basic element for the investigation of w¨ustite Fe1−δ O (δ = deviation from ideal stoichiometry) is the solid-state galvanic cell pO 2 , Pt/ZrO2 (+Y2 O3 )/Fe1−δ O)/Pt IV with doped ZrO2 as solid electrolyte, an electrode consisting of porous platinum in contact with air at one side and the w¨ustite being investigated as electrode at the other side. The experimental setup is shown in Fig. 5. The principle of the measurement is that, starting from a suitable initial state, the potential difference E of cell IV or the current I , respectively, are varied systematically. The other variable I or E is measured as a function of time. From the obtained results the chemical diffusion coefficient D˜ of w¨ustite can be calculated. In a potentiostatic experiment a definite value of the chemical potential of oxygen, corresponding to a certain deviation δ from the ideal stoichiometry,

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The chemical diffusion coefficients D˜ of Ag2+δ S measured at 200 and 300◦ C as a function of the deviation from the ideal stoichiometry are shown in Fig. 6. Another method for determining chemical diffusion coefficients is to use the frequency dependence of the cell impedance, which is obtained by ac measurements. This will not be treated here. E. Electrochemical Investigations of Formation of Surface Layers on Metals FIGURE 5 Experimental setup for the measurement of the chemical-diffusion coefficient of wustite: ¨ W, working electrode; R, reference electrode; C, counter electrode.

is set up in the w¨ustite before starting the measurement. Then a sudden change of the potential of the cell stipulated potentiostatically causes a current flow that is measured as a function of time. This current is primarily a measure of the addition or removal of oxygen at the phase boundary ZrO2 /w¨ustite. As a consequence of this, iron diffuses within the w¨ustite to or from the phase boundary. Thus, the current is equivalent to the diffusion current of iron, During this process the stoichiometry in the compound changes with time until a new δ value is attained. From the solution of the diffusion equations for this problem the chemical diffusion coefficient D˜ can be calculated. The result obtained for w¨ustite at 1000◦ C and a deviation δ = 0.106 from ideal stoichiometry is D˜ = 3.2 × 10−6 cm2 /s. Similar investigations with an improved experimental setup allowed the determination of the chemical diffusion coefficient D˜ of Ag2+δ S over the total range of stoichiometric composition of this compound.

The formation of nickel sulfide on nickel will be discussed as an example. The experimental arrangement is shown in Fig. 7. Silver iodide was used as the solid electrolyte, being a pure Ag+ ionic conductor under the experimental conditions. The negative pole of a power source was connected to the left-hand side of the arrangement and the positive pole to the right-hand side. The electrolytic cell itself consisted of tablets pressed together in a glass tube furnace flushed with nitrogen. An electrical current passing through the cell is a measure of the rate at which silver is removed from the silver sulfide, since silver ions migrate through the AgI and electrons through the external circuit. In this case, however, the rate of loss of the silver corresponds to the rate of the formation of nickel sulfide on nickel, since in this reaction nickel displaces the silver from the Ag2 S. The experiments were carrid out using the galvanostatic and potentiostatic methods. The important quantities are the current, which is a measure of the reaction rate (here of the formation of nickel sulfide), and the cell emf, which is not only a measure of the chemical potential of the silver in silver sulfide, but,

FIGURE 6 Chemical-diffusion coefficients of Ag2+δ S as a function of the deviation δ from stoichiometry at 200 and 300◦ C.

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FIGURE 7 Galvanic cell for the electrochemical investigation of the formation of NiS on nickel.

voltage U . The voltage is proportional to the ratio of mass to electrical charge of the silver ions and to the length of the rod. To verify Eq. (52) we should know the acceleration a at every time. In the experiments, however, the acceleration itself was not measured, rather it was the velocity before striking the ground that was measured. The accleration a and the velocity v are related by a = d v/dt ,

in view of the Gibbs–Duhem equation, also is a measure of the chemical potential of the sulfur in this compound. Thus, because of the high mobility of the silver in Ag2 S, it is also a measure of the chemical potential of the sulfur at the Ag2 S/NiS phase boundary. In this way the parabolic constant k for the rate of formation of NiS on nickel at 400◦ C could be measured as a function of the chemical potential of the sulfur on the outside of NiS. The results provided information about the disorder in NiS. F. Investigations on the Forces of Inertia of the Mobile Ions in Solid Ionic Conductors As an example, a rod of RbAg4 I5 with silver electrodes at both ends is exposed to an acceleration in the longitudinal direction. This acceleration may be produced, for example, by the braking of the rod dropped from different heights to a plastic material on the ground. Because of the high mobility of the silver ions it may be assumed that in the moment of striking the silver ions will be shifted only a little against the iodide lattice. Thus, an electrical field having the field strength E is built up with the result that the silver ions will be decelerated in the same way as the rigid iodide lattice. The sum of all forces acting on the silver ions must be zero. In this case the essential forces are the electrical force and the force of inertia. Quantitatively the electrical charge e multiplied by the electrical fields strength E is equal to the mass m of the silver ions multiplied by the acceleration a: eE = ma.

(53)

where t is the time. By integration of Eq. (53) over the whole time of striking we obtain the negative velocity −v:  after strike a dt = −v (54) before strike

because the rod RbAg4 I5 has velocity v = 0 after the decleration. Using this we obtain by integration of Eq. (52)  after strike ml v. (55) U dt = e before strike This equation was used for the interpretation of the experiments. Figure 8 shows a voltage–time curve measured at 25◦ C with a rod of RbAg4 I5 , which has a length of 7.4 cm. Thevelocity before striking was 1.44 m/s. The voltage pulse U dt is given by the area below the curve and has a value of 1.22 × 10−7 V s. According to Eq. (55) the area below the curve depends only on the velocity before striking for a given length of the rod. Figure 9 shows the measured voltage pulses from dropping experiments at 25◦ C using a RbAg4 I5 rod of 7.4-cm length as a function of the velocity before striking. The full line holds for the limiting case of inelastic striking, calculated with the help of Eq. (55) where m is the mass of

(50)

Since it can be assumed that the electrical field strength is constant along the rod of length l, an integration of E over l delivers the negative volgate −U , which can be measured between the ends of the rod:  l E dl = El = −U . (51) 0

Inserting Eq. (51) in Eq. (50) we obtain: ml a. (52) e Equation (52) shows the relationship between the acceleration acting on the rod of RbAg4 I5 and the electrical −U =

FIGURE 8 Example of a measured voltage-time curve of a dropping experiment (T = 25◦ C; length of the rod = 7.4 cm; velocity before striking = 1.44 m/s).

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FIGURE 10 Zirconium dioxide sensor for the measurement of partial pressures of oxygen. The housing and packing are electrically conducting.

FIGURE 9 Voltage pulse as a function of the v before striking at 25◦ C. (Length of the rod = 7.4 cm; ——, calculated; ×, measured).

a silver atom. It is seen that the results of the measurements and the calculated voltage pulses are in agreement. Furthermore, it should be noticed that neither in Eq. (55) nor in Eq. (52) is the termperature included as a variable. That means the values of the voltage pulses should be independent of the temperature. This could be confirmed by measurements at 25 and 120◦ C and may be of interest for practical applications, as will be seen. In addition to the investigations described, other kinetic experiments have been carried out with the help of solid-electrolyte galvanic cells. The investigations include phase-boundary reactions at the solid–gas phase boundary (including measurements of evaportion and condensation rates) and phase-boundary reactions at the solid–solid phase boundary. These investigations will not be discussed here.

VI. TECHNICAL APPLICATIONS OF SOLID ELECTROLYTES Solid electrolytes are also of great technological importance. Some examples of applications will be described in the following sections. A. Sensors Galvanic cells with solid electrolytes can be used for direct measurement of partial pressures in gases and concentrations in liquids and melts. An important example is cell I, which contains doped zirconium dioxide as solid electrolyte. By using cells of this type a wide range of oxygen partial pressures in gases (down to 10−16 atm) can be determined. The zirconium dioxide probe for such work is used at temperatures between about 500 and 1000◦ C.

The wide range of oxygen partial pressures makes such cells an excellent analytical instrument for measurements of gases. It should be mentioned that the measured value is obtained almost instantaneously. In such measurements a reference atmosphere, for example, air or a metal–metal oxide mixture, is located on one side of the zirconium dioxide, so that the oxygen partial pressure is already known at one electrode; the oxygen partial pressure at the other electrode is then measured by the emf in accordance with Eq. (47). In the measurement the gas is passed to the sensor. Interesting industrial applications lie in the analysis of exhaust gases from furnaces or combustion engines. It is advisable to arrange the zirconium dioxide probe in the exhaust gas stream near the reaction space. Figure 10 shows such a sensor that can be used for the control and regulation of the combustion process in automobile engines. The possibility of very exact control of gasoline– air mixtures is of special interest in connection with the control of air pollution. A similar probe can be used to measure the concentration or thermodynamic activity of oxygen in liquid metals. For example, if the probe is dipped in the melt during steel production, the oxygen activity can be measured directly, which is of appreciable advantage. A further possibility to measure partial pressures of gases is given by the use of galvanic cells, in which another kind of ions than the corresponding species in the gas is mobile in the solid electrolyte. As an example, the following galvanic cell for measuring chlorine partial pressures will be regarded   Ag/RbAg4 I5 /AgCl, Cl2 pCl2 , V where RbAg4 I5 is used as a silver ion conducting solid electrolyte. Cell V is in principle a concentration (activity) cell for silver ions analogous to cell I delivering the emf

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E=

RT aAg (AgCl) ln , F aAg (Ag)

(56)

where aAg (AgCl) denotes the activity of silver in the righthand AgCl-electrode, and aAg (Ag) denotes the activity of silver in the left-hand silver electrode, respectively, which has a value of one in this case because of the pure Ag metal. The activity of silver in AgCl, which is in equilibrium with the gas-atmosphere, having a certain chlorine partial pressure pCl2 , is given by   −1/2   aAg (AgCl) = pCl2 /po exp G oAgCl RT , (57) where p o = 1 atm and G oAgCl is the Standard-Gibbs energy of formation of AgCl. Inserting Eq. (57) into Eq. (56), we obtain a relation between the chlorine partial pressure to be determined and the measured emf E   RT  E= (58) ln pCl2 /po − G oAgCl F . 2F Also, the partial pressures of other gases, for example, NO2 , O2 , or sulfur, can be measured by such sensors. Another method of determining the partial pressure of a gas is given by measuring the current flowing through a suitable galvanic cell with a solid electrolyte. This principle will be discussed in the following exemplified at an oxygen sensor. In principle, cell I discussed in Section V can be used, where pO 2 is the oxygen partial pressure to be measured. In this case, a current is passed through the cell so that oxygen is transported from the right-hand side of the cell to the left-hand side. For this an outer electrical voltage has to be applied to the cell. To get a defined correlation between pO 2 and the electrical current flowing through the cell, a so-called “diffusion-barrier” has to be arranged in front of the electrode at the right-hand side. This may, for example, consist of a porous ceramic material. The external electrical voltage is chosen in such a way that each oxygen molecule reaching the surface of the right-hand electrode immediately reacts electrochemically to an oxygen ion, which is then transported through the solid electrolyte. Under these conditions, the flowing current is proportional to the partial pressure pO 2 and so can be used to measure pO 2 . Another type of sensor—an accelerometer—can be constructed by applying the principles used in the case of the investigations on the forces of inertia of the mobile ions in solid ion conductors, described in Section V.F. As shown in Eq. (52) the voltage U measured at both ends of a rod of a solid electrolyte is a direct measure of the acceleration acting on this rod at every time; this means that such an arrangement, consisting of a rod of RbAg4 I5 as solid electrolyte with silver wires at both ends as electrodes for measuring the voltage U , can be used as an accelerometer. This is of interest for practical applications. The principle of such an accelerometer is shown in Fig. 11.

FIGURE 11 Schematic diagram of a RbAg4 I5 accelerometer.

B. Solid Electrolyte Batteries: The Sodium–Sulfur Cell The principle of a sodium–sulfur cell is shown in Fig. 12. The solid electrolyte is a Na+ ion conductor, consisting of β-Al2 O3 . It is generally used as a tube closed at one end and filled with liquid sodium as the anode. An iron sponge, which absorbs the liquid sodium, serves to improve the wetting of the electrolyte and to improve safety. A metal wire leads out of the anode to carry the current. The cathode consists of liquid sodium polysulfide and sulfur inserted in porous graphite. The working temperature of the sodium–sulfur cell is around 300◦ C. In the cell reaction sodium ions pass through the electrolyte and electrons through the external circuit, so that sodium is dissolved in sodium polysulfide. In this way electrical energy can be liberated. The energy density of the sodium–sulfur cell is many times greater than that

FIGURE 12 Sodium–sulfur cell.

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of the customary lead batteries, and the materials needed for the electrolytes and electrodes are available in large quantities. The cell can be recharged by changing the direction of the current. The sodium–sulfur cell is of great interest for large-scale energy storage and for electrotraction for electric vehicles. Prototypes have already been built. In addition to the sodium–sulfur cell other cell systems have been developed using other solid electrolytes. C. Fuel Cells and Electrolyzers Figure 13 shows schematically a high-temperature fuel cell incorporating zirconium dioxide as the solid electrolyte. The zirconium dioxide in a tube or disk form separates two electrode compartments, one containing air or pure oxygen and the other a fuel gas (e.g., hydrogen). The zirconium oxide carries two porous electrodes: nickel can be used on the fuel side, and on the oxygen side lanthanum–nickel oxide or some other electronconducting oxide. In a high-temperature fuel cell the oxygen takes up electrons at one electrode, then passes through the electrolyte as ions and combines at the other electrode with H2 to give H2 O, whereby the electrons are given up and flow again to the other electrode in the external current circuit. In this way energy is made available to the user. The advantages of the high-temperature fuel cell are that little or no polarization occurs at the electrodes and high current densities can be achieved. By reversing the direction of the current flow in a hightemperature fuel cell, that is, by supplying the cell with electrical energy, steam can be decomposed and the cell can thus be used as an electrolyzer. The hydrogen produced can be stored or conducted by pipelines to remote sites where it can serve for the production of energy in a

high-temperature fuel cell if required. This principle is being discussed in connection with large-scale energy storage and transport of energy. D. Chemotronic Components Galvanic cells containing solid electrolytes, which find use in electrical circuits, are often called chemotronic components. Next we describe as an example coulometers and time switches. However, there exist more chemotronic building units containing solid electrolytes such as analog memories and capacitors. They will not be described here. The galvanic cell Ag/RbAg4 I5 /Au VI can be used as a coulometer or time switch. Here the electrolyte RbAg4 I5 is a good Ag+ ion conductor even at room temperature. By passing a current through this cell with the negative pole at the silver side, silver is deposited on the gold electrode. The time switch is then in the loaded state. The silver can be transported back to the original silver electrode by a current flowing in the reverse direction. In this stripping process the cell potential is determined mainly by ohmic losses in the solid electrolyte that lie in the millivolt range. When all the silver has been stripped from the gold electrode the cell shows a sudden rise in potential, which can be used as a signal. Such electrochemical switches are suitable for times in the region of seconds to months. Cell VI can also be used as a coulometer; the amount of a current flowing through the cell during charging is then determined by the discharge process.

SEE ALSO THE FOLLOWING ARTICLES CRYSTALLOGRAPHY • ELECTROCHEMISTRY • ELECTROLYTE SOLUTIONS, TRANSPORT PROPERTIES • LASERS, SOLID-STATE • SOLID-STATE CHEMISTRY • SOLID-STATE IMAGING DEVICES

BIBLIOGRAPHY

FIGURE 13 High-temperature fuel cell.

Bard, A. J., and Faulkner, L. R. (2001). “Electrochemical Methods: Fundamentals and Applications,” Wiley, New York. Bruce, P. G. (1995). “Solid-State Electrochemistry,” Cambridge University Press, Cambridge. Gellings, P. J., and Boumeester, H. J. M. (eds.) (1997). “The CRC Handbook of Solid-State Electrochemistry,” CRC Press, Boca Raton, Florida. Stimming, U., Singhal, S. C., Tagawa, H., and Lehnert, W. (eds.) (1997). “Proceedings of the Fifth International symposium on Solid Oxide Fuel Cells (SOFC-V),” Electrochemical Society, Pennington, New Jersey.

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Superacids George A. Olah G. K. Surya Prakash University of Southern California

I. Acid Strength and Acidity Scale II. Superacid Systems III. Application of Superacids

GLOSSARY Br¨onsted superacids Proton donor acids stronger than 100% sulfuric acid. Carbenium ions Compounds containing a trivalent, tricoordinate carbon bearing a positive charge. Also called “classical cations.” Carbocations Compounds containing carbon bearing a positive charge which encompass both carbenium and carbonium ions. Carbonium ions Compounds containing high coordinate carbon bearing a positive charge with multicenter bonding. Also called “nonclassical” cations. Conjugate Br¨onsted–Lewis superacids Superacidic proton donor acids comprised of a combination of Br¨onsted and Lewis acids. Hammet’s acidity constant, H 0 A logarithmic thermodynamic scale used to relate acidity of proton donor acids. Immobilized superacids Superacids (both Br¨onsted and Lewis types) bound to inert supports such as graphite, fluorinated graphite, etc. Lewis superacids Electron acceptor acids stronger than aluminum trichloride. Solid superacids Solid materials possessing superacid

sites. May be of the Br¨onsted or the Lewis superacid type. Superacids Acid systems that encompass both Br¨onsted and Lewis superacids as well as their conjugate combinations. Superelectrophiles Electrophiles that are further activated by Br¨onsted or Lewis superacid complexation.

CHEMISTS long considered mineral acids such as sulfuric and nitric acids to be the strongest protic acids to exist. More recently this view has changed considerably with the discovery of extremely strong acid systems that are hundreds of millions, even billions, of times stronger than 100% sulfuric acid. Such acid systems are termed “superacids.” The term “superacids” was first suggested by Conant and Hull in 1927 to describe acids such as perchloric acid in glacial acetic acid that were capable of protonating certain weak bases such as aldehydes and ketones. Superacids encompass both Br¨onsted (proton donor) and Lewis (electron acceptor) acids as well as their conjugate pairs. The concept of acidity and acid strength can be defined only in relation to a reference base. According to an arbitrary but widely accepted suggestion

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176 by Gillespie, all Br¨onsted (protic) acids stronger than 100% sulfuric acid are classified as superacids. Various methods are available to measure protic superacid strengths (vide infra). Lewis acids also cover a wide range of acidities extending beyond the strength of the most frequently used systems such as AlCl3 and BF3 . Olah et al. (1985) suggested the use of anhydrous aluminum trichloride, the most widely used Friedel–Crafts catalyst, as the arbitrary unit to define Lewis superacids. Lewis acids stronger than anhydrous aluminum trichloride are considered Lewis superacids. There remain, however, many difficulties in measuring the strength of Lewis acid (vide infra). The high acidity and the extremely low nucleophilicity of the counterions of superacidic systems are especially useful for the preparation of stable, electron-deficient cations, including carbocations. Many of these cations, which were formerly suggested only as fleeting metastable intermediates and were detectable only in the gas phase in mass spectrometric studies, can be conveniently studied in superacid solutions. New chemical transformations and syntheses that are not possible using conventional acids can also be achieved with superacids. These include transformations and syntheses of many industrially important hydrocarbons. The unique ability of superacids to bring about hydrocarbon transformations, even to activate methane (the principal component of natural gas) for electrophilic reactions, has opened up a fascinating new field in chemistry.

I. ACID STRENGTH AND ACIDITY SCALE The chemical species that plays the key role in Br¨onsted acids is the hydrogen ion, that is, the proton: H+ . Since the proton is the hydrogen nucleus with no electron in its 1s orbital, it is not prone to electronic repulsion. The proton consequently exercises a powerful polarizing effect. Due to its extreme electron affinity, proton cannot be found as a free “naked” species in the condensed state. It is always associated with one or more molecules of acid or the solvent (or any other nucleophile present). The strength of protic acid thus depends on the degree of association of the proton in the condensed state. Free protons can exist only in the gas phase and represent the ultimate acidity. Due to the very small size of a proton (105 times smaller then any other cation) and the fact that only 1s orbital is used in bonding by hydrogen, proton transfer is a very facile reaction, reaching diffusion-controlled rates, and does not necessitate important reorganization of the electronic valence shells. Understanding the nature of the proton is important when generalizing quantitative relationships in acidity measurements.

Superacids

A number of methods are available for estimating acidity of protic acids in solution. The best known is the direct measurement of the hydrogen ion activity used in defining pH [Eq. (1)]. pH = log aH+ .

(1)

This can be achieved by measuring the potential of a hydrogen electrode in equilibrium with a dilute acid solution. In highly concentrated acid solutions, however, the pH concept is no longer applicable, and the acidity must be related very closely to the degree of transformation of a base with its conjugate acid, keeping in mind that this will depend on the base itself and on medium effects. The advantage of this method was shown in the 1930s by Hammett and Deyrup, who investigated the proton donor ability of the H2 O–H2 SO4 system over the whole concentration range by measuring the extent to which a series of nitroanilines were protonated. This was the first application of the very useful Hammett acidity function [Eq. (2)]. BH+ . (2) B The pKBH+ is the dissociation constant of the conjugate acid (BH+ ) and BH+ /B is the ionization ratio, which is generally measured by spectroscopic means [ultraviolet, nuclear magnetic resonance (NMR), and dynamic NMR]. Hammett’s “H0 ” scale is a logarithmic scale on which 100% sulfuric acid has an H0 value of −12.0. Various other techniques are also available for acidity measurements of protic acids. These include electrochemical methods, kinetic rate measurements, and heats of protonation of weak bases. Even with all these techniques it is still difficult to measure the acidity of extremely acidic superacids, because of the unavailability of suitable weak reference bases. In contrast to protic (Br¨onsted) acids, a common quantitative method to determine the strength of Lewis acids does not exist. Whereas the Br¨onsted acid–base interaction always involves a common denominator—the proton (H+ ) transfer, which allows direct comparison—no such common relationship exists in the Lewis acid–base interaction. The result is that the definition of “strength” has no real meaning with Lewis acids. The “strength” or “coordinating power” of different Lewis acids can vary widely against different Lewis bases. Despite the apparent difficulties, a number of qualitative relationships have been developed to characterize Lewis acids. Schwarzenbach and Chatt classified Lewis acids into two types: class a and class b. Class a Lewis acids form their most stable complexes with the donors in the first row of the periodic table—N, O, and F. Class b acids, on the other hand, complex best with donors in the second or subsequent row—Cl, Br, I, P, S, etc. Guttmann has H0 = pKBH+ − log

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introduced a series of donor numbers (DN) and acceptor numbers (AN) for various solvents in an attempt to quantify complexing tendencies of Lewis acids. Based on a similar premise, Drago came up with parameter E, which measures the covalent bonding potential of each series of Lewis acids as well as bases. Pearson has proposed a qualitative scheme in which a Lewis acid and base are characterized by two parameters, one of which is referred to as strength and the other as softness. Thus, the equilibrium constant for a simple Lewis acid–base reaction would be a function of four parameters, two for each partner. Subsequently, Pearson introduced the hard and soft acids and bases (HSAB) principle to rationalize behavior and reactivity in a qualitative way. Hard acids correspond roughly in their behavior to Schwarzenbach and Chatt’s class a acids. They are characterized by small acceptor atoms that have outer electrons that are not easily excited and that bear a considerable positive charge. Soft acids, which correspond to class b acids, have acceptor atoms of a lower positive charge and a large size, with easily excited outer electrons. Hard and soft bases are defined accordingly. Pearson’s HSAB principle states that hard acids prefer to bind to hard bases and soft acids prefer to bind to soft bases. The principle has proved useful in rationalizing and classifying a large number of chemical reactions involving acid–base interactions in a qualitative manner, but it gives no basis for quantitative treatment. Many attempts have been made in the literature to rate qualitatively the activity of Lewis acid catalysts in Friedel–Crafts-type reactions. However, such ratings depend largely on the nature of the reaction for which the Lewis acid catalyst is employed. Thus, the classification of Lewis superacids as those stronger than anhydrous aluminum trichloride is only arbitrary. Just as in the case of Gillespie’s classification of Br¨onsted superacids, it is important to recognize that acids stronger than conventional Lewis acid halides exit, with increasingly unique properties. Another area of difficulty is measuring the acid strength of solid superacids. Since solid superacid catalysts are used extensively in the chemical industry, particularly in the petroleum field, a reliable method for measuring the acidity of solids would be extremely useful. The main difficulty to start with is that the activity coefficients for solid species are unknown and thus no thermodynamic acidity function can be properly defined. On the other hand, because the solid by definition is heterogeneous, acidic and basic sites can coexist with variable strength. The surface area available for colorimetric determinations may have acidic properties widely different from those of the bulk material; this is especially true for well-structured solids such as zeolites.

The complete description of the acidic properties of a solid requires the determination of the acid strengths as well as the number of acid sites. The methods that have been used to answer these questions are basically the same as those used for the liquid acids. Three methods are generally quoted: (1) rate measurement to relate the catalytic activity to the acidity, (2) the spectrophotometric method to estimate the acidity from the color change of adequate indicators, and (3) titration by a strong enough base for the measurement of the amount of acid. The above experimental techniques vary somewhat, but all the results obtained should be interpreted with caution because of the complexity of the solid acid catalysts. The presence of various sites of different activity on the same solid acid, the change in activity with temperature, and the difficulty of knowing the precise structure of the catalyst are some of the major handicaps in the determination of the strength of solid superacids.

II. SUPERACID SYSTEMS Following Conant’s early work, the field of superacids, which had been dormant till the late 1950s, started to undergo rapid development in the early 1960s, involving the discovery of new systems and an understanding of their nature as well as their chemistry. As mentioned, superacids encompass both Br¨onsted and Lewis types and their conjugate combinations. A. Bronsted ¨ Superacids Using Gillespie’s arbitrary definition, Br¨onsted superacids are those with an acidity exceeding that of 100% sulfuric acid (H0 , −12). These include perchloric acid (HClO4 ), fluorosulfuric acid (FSO3 H), trifluoromethanesulfonic acid (CF3 SO3 H), and higher perfluoroalkanesulfonic acid (Cn Fn +2 SO3 H). Physical properties of some of the most commonly used superacids are listed in Table I. Studies by Gillespie have shown that truly anhydrous hydrogen fluoride (HF), which is extremely difficult to obtain in the pure form, has a Hammett acidity constant (H0 ) of −15.1 rather than the −11.0 found for the usual anhydrous acid. However, traces of water impurity drop the acidity to the generally observed value. Thus for practical purposes, hydrogen fluoride, which always contains some water impurity, is not discussed here, as its acidity of H0 = −11.0 is lower than that of H2 SO4 . Teflic acid (TeF5 OH) has been suggested to have an acidity comparable to that of fluorosulfuric acid. However, no concrete acidity measurements are available to support such a claim. A number of carbocationic salts bearing carborane anions [CB11 H6 Cl− 6 , etc.] have been studied. However, their parent Bronsted acids,

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Superacids TABLE I Physical Properties of Bronsted ¨ Superacids Property Melting point (◦ C) Boiling point (◦ C) Density (25◦ C), g/cm3 Viscosity (25◦ C), cP Dielectric constant Specific conductance (20◦ C), −1 · cm−1 −H0 (neat) a b

HClO4

ClSO3 H

HSO3 F

CF3 SO3 H

−112

−81

−89

−34

110 (Explosive)

151–152 (Decomposing)

162.7

162

1.767a — — —

1.753 3.0b 60 ± 10 0.2–0.3 × 10−3

1.726 1.56 120 1.1 × 10−4

1.698 2.87 2.0 × 10−4

≈13.0

13.8

15.1

14.1

At 20◦ C. At 15◦ C.

which can be considered as potential superacids, are still unknown. 1. Perchloric Acid (HClO4 ) Commercially, perchloric acid is manufactured by either reaction of alkali perchlorates with hydrochloric acid or direct electrolytic oxidation of 0.5 N hydrochloric acid. Another commercially attractive method is the direct electrolysis of chlorine gas (Cl2 ) dissolved in cold, dilute perchloric acid. Perchloric acid is commercially available in a concentration of 70% (by weight) in water, although 90% perchloric acid also had limited availability (due to its explosive hazard, it is no longer provided at this strength); for 70–72% HClO4 , an azeotrope of 28.4% H2 O, 71.6% HClO4 , boiling at 203◦ C is safe for usual applications. It is a strong oxidizing agent, however, and must be handled with care. Anhydrous acid (100% HClO4 ) is prepared by vacuum distillation of the concentrated acid solution with a dehydrating agent such as Mg(ClO4 )2 . It is stable only at low temperatures for a few days, decomposing to give HClO4 · H2 O (84.6% acid) and ClO2 . Perchloric acid is extremely hygroscopic and a very powerful oxidizer. Contact of organic materials with anhydrous or concentrated perchloric acid can lead to violent explosions. For this reason, the application of perchloric acid has serious limitations. The acid strength, although not reported, can be estimated to be around H0 = −13 for the anhydrous acid. Although various cation salts can be prepared with perchlorate gegen ions, the ionic salts tend to be unstable (explosive) due to their equilibria with covalent perchlorates. The main use of perchloric acid is in the preparation of − its salts, such as NH+ 4 ClO4 , a powerful oxidant in rocket fuels and pyrotechniques. 2. Chlorosulfuric Acid (ClSO3 H) Chlorosulfuric acid, the monochloride of sulfuric acid, is a strong acid containing a relatively weak sulfur–chlorine

bond. It is prepared by the direct combination of sulfur trioxide and dry hydrogen chloride gas. The reaction is very exothermic and reversible, making it difficult to obtain chlorosulfuric acid free of SO3 and HCl. On distillation, even in a good vacuum, some dissociation is inevitable. The acid is a powerful sulfating and sulfonating agent as well as a strong dehydrating agent and a specialized chlorinating agent. Because of these properties, chlorosulfuric acid is rarely used for its protonating superacid properties. Gillespie and co-workers have measured systematically the acid strength of the H2 SO4 –ClSO3 H system using aromatic nitro compounds as indicators. They found an H0 value of −13.8 for 100% ClSO3 H. 3. Fluorosulfuric Acid (HSO3 F) Fluorosulfuric acid, HSO3 F, is a mobile colorless liquid that fumes in moist air and has a sharp odor. It may be regarded as a mixed anhydride of sulfuric and hydrofluoric acid. It has been known since 1892 and is prepared commercially from SO3 and HF in a stream of HSO3 F. It is readily purified by distillation, although the last traces of SO3 are difficult to remove. When water is excluded, it may be handled and stored in glass containers, but for safety reasons the container should always be cooled before opening because gas pressure may have developed from hydrolysis. HSO3 F + H2 O   H2 SO4 + HF Fluorosulfuric acid generally also contains hydrogen fluoride as an impurity, but according to Gillespie the hydrogen fluoride can be removed by repeated distillation under anhydrous conditions. The equilibrium HSO3 F   SO3 + HF always produces traces of SO3 and HF in stored HSO3 F samples. When kept in glass for a long time, SiF4 and H2 SiF6 are also formed (secondary reactions due to HF). Fluorosulfuric acid is employed as a catalyst and chemical reagent in various chemical processes including

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alkylation, acylation, polymerization, sulfonation, isomerization, and production of organic fluorosulfates. It is insoluble in carbon disulfide, carbon tetrachloride, chloroform, and tetrachloroethane, but it dissolves most organic compounds that are potential proton acceptors. The acid can be dehydrated to give S2 O5 F2 . Electrolysis of fluorosulfuric acid gives S2 O6 F2 or SO2 F2 + F2 O, depending on the conditions employed. HSO3 F has a wide liquid range (mp = −89◦ C, bp = +162.7◦ C), making it advantageous as a superacid solvent for the protonation of a large variety of weak bases.

TABLE II Characteristics of Perfluoroalkanesulfonic Acids bp (◦ C) (760 mm Hg)

Compound CF3 SO3 H C2 F5 SO3 H C4 F9 SO3 H C5 F11 SO3 H C6 F13 SO3 H C8 F17 SO3 H

Cl2

Cl2

CF3 SSCF3 −→ CF3 SCl −→ CF3 SO2 Cl H2 O

aq. KOH

−−−→ CF3 SO3 H CF3 SO3 H is a stable, hygroscopic liquid that fumes in moist air and readily forms the stable monohydrate (hydronium triflate), which is a solid at room temperature (mp, 34◦ C; bp, 96◦ C/1 mm Hg). The acidity of the neat acid as measured by UV spectroscopy with a Hammett indicator indeed shows an H0 value of −14.1. It is miscible with water in all proportions and soluble in many polar organic compounds, such as dimethylformamide, dimethylsulfoxide, and acetonitrile. It is generally a very good solvent for organic compounds that are capable of acting as proton acceptors in the medium. The exceptional leaving-group properties of the triflate anion, CF3 SO− 3, make triflate esters excellent alkylating agents. The acid and its conjugate base do not provide a source of fluoride ion even in the presence of strong nucleophiles. Furthermore, as it lacks the sulfonating properties of oleums an HSO3 F, it has gained a wide range of application as a catalyst in Friedel–Crafts alkylation, polymerization, and organometallic chemistry. 5. Higher Homologous Perfluoroalkanesulfonic Acids Higher homologous perfluoroalkanesulfonic acids (see Table II) are hygroscopic oily liquids or waxy solids. They are prepared by the distillation of their salts from H2 SO4 , giving stable hydrates that are difficult to dehydrate. The acids show the same polar solvent solubilities as trifluoromethanesulfonic acid but are quite insoluble in benzene,

H0 (22◦ C)

161

1.70

−14.1

170 198

1.75 1.82

−14.0 −13.2

212 222

−12.3

249 241

4. Trifluoromethanesulfonic Acid (CF3 SO3 H) Trifluoromethanesulfonic acid (CF3 SO3 H, triflic acid), the first member in the perfluoroalkanesulfonic acid series, has been studied extensively. Besides its preparation by electrochemical fluorination of methanesulfonyl halides, triflic acid may also be prepared from trifluoromethanesulfenyl chloride.

Density (25◦ C)

257

heptane, carbon tetrachloride, and perfluorinated liquids. Many of the perfluoroalkanesulfonic acids have been prepared by the electrochemical fluorination reaction of the corresponding alkanesulfonic acids (or conversion of the corresponding perfluoroalkane iodides to their sulfonyl halides). α,ω-Perfluoroalkanedisulfonic acids have been prepared by aqueous alkaline permanganate oxidation of the compounds, Rf SO2 (CF2 CF2 )n –SO2 F. C8 F17 SO3 H and higher perfluoroalkanesulfonic acids are surface-active agents and form the basis for a number of commercial fluorochemical surfactants. B. Lewis Superacids Lewis superacids are arbitrarily defined as those stronger than anhydrous aluminum trichloride, the most commonly used Friedel–Crafts catalyst. Some of the physical properties of the commonly used Lewis superacids are given in Table III. 1. Antimony Pentafluoride (SbF5 ) Antimony pentafluoride is a colorless, highly viscous liquid at room temperature. Its viscosity is 460 cP at 20◦ C, which is close to that of glycerol. The pure liquid can be handled and distilled in glass if moisture is excluded. TABLE III Physical Properties of Some Lewis Superacids Property mp (◦ C) bp (◦ C) Specific gravity at 15◦ C (g/cc) a

At the bp.

SbF5

AsF5

TaF5

NbF5

B(OSO2 CF3 )3

7.0

−79.8

97

72–73

43–45

142.7 3.145

−52.8 2.33a

229 3.9

236 2.7

68–83 (0.5 Torr) —

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180 The polymeric structure of the liquid SbF5 has been established by 19 F NMR spectroscopy and is shown to have the following frameworks: a cis-fluorine bridged structure is found in which each antimony atom is surrounded by six fluorine atoms in an octahedral arrangement.

Superacids

ciation, but it is a monomeric covalent compound with a high degree of coordinating ability. It is prepared by reacting fluorine with arsenic metal or arsenic trifluoride. As a strong Lewis acid fluoride, it is used in the preparation of ionic complexes and, in conjunction with Br¨onsted acids, forms conjugate superacids. It also forms, with graphite, stable intercalation compounds that show an electrical conductivity comparable to that of silver. Great care should be exercised in handling any arsenic compound because of its potential high toxicity. 3. Tantalum and Niobium Pentafluoride

Antimony pentafluoride is a powerful oxidizing and a moderate fluorinating agent. It readily forms stable intercalation compounds with graphite, and it spontaneously inflames phosphorus and sodium. It reacts with water to form SbF5 · 2H2 O, an unusually stable solid hydrate (probably a hydronium salt, H3 O+ SbF5 OH) that reacts violently with excess water to form a clear solution. Slow hydrolysis can be achieved in the presence of dilute NaOH and forms Sb(OH)− 6 . Sulfur dioxide and nitrogen dioxide form 1:1 adducts, SbF5 :SO2 and SbF5 :NO2 , as do practically all nonbonded electron-pair donor compounds. The exceptional ability of SbF5 to complex and subsequently ionize nonbonded electron-pair donors (such as halides, alcohols, ethers, sulfides, and amines) to carbocations, first recognized by Olah in the early 1960s, has made in one of the most widely used Lewis halides in the study of cationic intermediates and catalytic reactions. Vapor density measurements suggest a molecular association corresponding to (SbF5 )3 at 150◦ C and (SbF5 )2 at 250◦ C. On cooling, SbF5 gives a nonionic solid composed of trigonal bipyramidal molecules. Antimony pentafluoride is prepared by the direct fluorination of antimony metal or antimony trifluoride (SbF3 ). It can also be prepared by the reaction of SbCl5 with anhydrous HF, but the exchange of the fifth chloride is difficult, and the product is generally SbF4 Cl. As shown by conductometric, cryoscopic, and related acidity measurements, it appears that antimony pentafluoride is by far one of the strongest Lewis acids known. Antimony pentafluoride is also a strong oxidizing agent, allowing, for example, preparation of arene dications. At the same time, its easy reducibility to antimony trifluoride represents a limitation in many applications, although it can be easily refluorinated. 2. Arsenic Pentafluoride (AsF5 ) Arsenic pentafluoride (AsF5 ) is a colorless gas at room temperature, condensing to a yellow liquid at −53◦ C. Vapor density measurements indicate some degree of asso-

The close similarity of the atomic and ionic radii of niobium and tantalum are reflected by the similar properties of tantalum and niobium pentafluorides. They are thermally stable white solids that may be prepared either by the direct fluorination of the corresponding metals or by reacting the metal pentachlorides with HF. Surprisingly, even reacting metals with HF gives the corresponding pentafluorides.They both are strong Lewis acids, complexing a wide variety of donors such ethers, sulfides, amines, and halides. They both coordinate with fluoride ions to form anions of the type (MF6 )− . TaF5 is a somewhat stronger acid than NbF5 , as shown by acidity measurements in HF. The solubility of TaF5 and NbF5 in HF and HSO3 F is much more limited than that of SbF5 or other Lewis acid fluorides, restricting their use to some extent. At the same time, their high redox potentials and more limited volatility make them catalysts of choice in certain hydrocarbon conversions, particularly in combination with solid catalysts. 4. Boron tris(Trifluoromethanesulfonate) [B(OSO2 CF3 )3 ] Boron tris (trifluoromethanesulfonate) was first prepared by Engelbrecht and Tschager in trifluoromethanesulfonic acid solution (vide infra) as a conjugate acid system. Olah and co-workers have isolated B(OSO2 CF3 )3 in pure form by treating boron trihalides (chlorides, bromides) with 3 equiv of triflic acid in Freon 113 or SO2 ClF solution. BX3 + 3CF3 SO3 H → B(OSO2 CF3 )3 + 3HX Boron tris(trifluoromethanesulfonate) is a colorless low-melting compound [mp, 43–45◦ C; bp, 68–73◦ C (0.5 Torr)] which decomposes on heating above 100◦ C at atmospheric pressure. It is extremely hygroscopic and is readily soluble in methylene chloride, 1,1,2trifluorotrichloroethane (Freon 113), SO2 , and SO2 ClF. Boron tris(trifluoromethanesulfonate) is a strong nonoxidizing Lewis acid and an efficient Friedel–Crafts catalyst. Apart from the discussed Lewis acids, other highly acidic systems such as Au(OSO2 F)3 , Ta(OSO2 F)5 , Pt(OSO2 F)4 , and Nb(OSO2 F)5 have been reported as

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their conjugate acids in HSO3 F solution. All the above conjugate superacids were found to be highly conducting and strongly ionizing over the entire conecentration range. C. Conjugate Bronsted–Lewis ¨ Superacids 1. Oleums: Polysulfuric Acids SO3 -containing sulfuric acid (oleum) has long been considered the strongest mineral acid and one of the earliest superacid systems to be recognized. The concentration of SO3 in sulfuric acid can be determined by weight or by electrical conductivity measurement. The most accurate H0 values for oleums so far have been published by Gillespie and co-workers (Table IV). The increase in acidity on the addition of SO3 to sulfuric acid is substantial, and an H0 value of −14.5 is reached with 50 mol% SO3 . The main component up to this SO3 concentration is pyrosulfuric (or disulfuric) acid H2 S2 O7 . On heating or in the presence of water, it decomposes and behaves like a mixture of sulfuric acid and sulfur trioxide. In sulfuric acid, it ionizes as a stronger acid: + − H2 S2 O7 + H2 SO4   H3 SO4 + HS2 O7

(K = 1.4 × 10−2 ) At higher SO3 concentrations, a series of higher polysulfuric acids such as H2 S3 O10 and H2 S4 O13 is formed and a corresponding increase in acidity occurs. However, as can be seen from Table IV, the acidity increase is very small after reaching 50 mol% of SO3 , and no data are available beyond 75%. Despite its high acidity, oleum has found little application as a superacid catalyst, mainly because of its strong oxidizing power. Also, its high melting point and viscosity have considerably hampered its use for spectroscopic study of ionic intermediates and in synthesis, except as an oxidizing or sulfonating agent. 2. Tetra(hydrogensulfato)Boric Acid–Sulfuric Acid HB(HSO4 )4 prepared by treating boric acid [B(OH)3 ] with sulfuric acid ionizes in sulfuric acid as shown by acidity measurements.

+ − HB(HSO4 )4 + H2 SO4   H3 SO4 + B(HSO4 )4

The increase in acidity is, however, limited to H0 = −13.6 as a result of insoluble complexes that precipitate when the concentration of the boric acid approaches 30 mol%. 3. Fluorosulfuric Acid–Antimony Pentafluoride (Magic Acid) Of all superacids, “Magic Acid,” a mixture of fluorosulfuric acid and antimony pentafluoride, is probably the most widely used medium for the spectroscopic observation of stable carbocations. The fluorosulfuric acid–antimony pentafluoride system was developed in the early 1960s by Olah for the study of stable carbocations and was studied by Gillespie for electron-deficient inorganic cations. The name Magic Acid originated in Olah’s laboratory at Case Western Reserve University in the winter of 1966. The HSO3 F:SbF5 mixture was used extensively by his group to generate stable carbocations. J. Lukas, a German postdoctoral fellow, put a small piece of Christmas candle left over from a lab party into the acid system and found that it dissolved readily. He then ran a 1 H NMR spectrum of the solution. To everybody’s amazement, he obtained a sharp spectrum of the t-butyl cation. The long-chain paraffin, of which the candle was made, had obviously undergone extensive cleavage and isomerization to the more stable tertiary ion. It impressed Lukas and others in the laboratory so much that they started to nickname the acid system Magic Acid. The name stuck, and soon others started to use it too. It is now a registered trade name and has found its way into the chemical literature. The acidity of the Magic Acid system as a function of the SbF5 content has been measured successively by Gillespie, Sommer, Gold, and their co-workers. The increase in acidity is very sharp at a low SbF5 concentration (≈10%) and continues up to the estimated value of H0 = −26.5 for a 90% SbF5 content. The initial ionization of HSO3 F:SbF5 is as follows. 2HSO3 F + SbF5   H2 SO3 F+ + SbF5 (SO3 F)− At higher concentrations of SbF5 , complex polyantimony fluorosulfate ions are formed. SbF5 + SbF5 (SO3 F)−   Sb2 F10 (SO3 F)−

TABLE IV H0 Values for the H2 SO4 –SO3 System Mol% SO3

H0

Mol% SO3

H0

Mol% SO3

H0

1.00 2.00 5.00 10.00 15.00 20.00

−12.24 −12.42 −12.73 −13.03 −13.23 −13.41

25.00 30.00 35.00 40.00 45.00 50.00

−13.58 −13.76 −13.94 −14.11 −14.28 −14.44

55.00 60.00 65.00 70.00 75.00

−14.50 −14.74 −14.84 −14.92 −14.90

Due to these equilibria, the composition of the HSO3 F: SbF5 system is very complex and depends on the SbF5 content. Aubke and co-workers have investigated the structures of complex anions in the Magic Acid system by modern 19 F NMR studies. The major reason for the wide application of the Magic Acid system compared with others (besides its very high acidity) is probably the large temperature range in which

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182 it can be used. In the liquid state, NMR spectra have been recorded from temperatures as low as −160◦ C (acid diluted with SO2 F2 and SO2 ClF) and up to +80◦ C (neat acid in a sealed NMR glass tube). Glass is attacked by the acid very slowly when moisture is excluded. The Magic Acid system can also be an oxidizing agent that results in reduction to antimony trifluoride and sulfur dioxide. On occasion this represents a limitation. 4. Fluorosulfuric Acid–Sulfur Trioxide Freezing-point and conductivity measurements have shown that SO3 behaves as a nonelectrolyte in HSO3 F. Acidity measurements show a small increase in acidity that is attributed to the formation of polysulfuric acids HS2 O6 F and HS3 O9 F up to HS7 O21 F. The acidity of these solutions reaches a maximum of −15.5 on the H0 scale for 4 mol% SO3 and does not increase any further. 5. Fluorosulfuric Acid–Arsenic Pentafluoride AsF5 ionizes in FSO3 H, and the AsF5 FSO− 3 anion has an octahedral structure. The H0 acidity function increases up to 5 mol% AsF5 , with a value of −16.6. 6. Hydrogen Fluoride–Antimony Pentafluoride (Fluoroantimonic Acid) The HF:SbF5 (fluoroantimonic acid) system is considered the strongest liquid superacid and also the one that has the widest acidity range. Due to the excellent solvent properties of hydrogen fluoride, HF:SbF5 is used advantageously for a variety of catalytic and synthetic applications. Anhydrous hydrogen fluoride is an excellent solvent for organic compounds with a wide liquid range. The acidity of HF, initially estimated as H0 ≈ −11, has now been revised to an H0 of −15.1 for highly purified anhydrous HF. A dramatic increase in acidity (H0 ≈ −20.5) is observed when 1 mol% SbF5 is added to anhydrous HF. The initial sharp increase in acidity is apparently due to the removal of residual moisture impurity. For more concentrated solutions, only kinetic data are available, mainly from the work of Brouwer and co-workers, who estimated the relative acidity ratio of 1:1 HF:SbF5 and 5:1 HSO3 F:SbF5 to be 5 × 108 :1. This means an H0 value in excess of −30 on the Hammett scale for the 1:1 composition. The acidity may increase still further for higher SbF5 concentrations. It has been shown by infrared measurements that an 80% SbF5 solution has the maximum concentration of H2 F+ . In any case, even for the composition range of 1–50% SbF5 , this is the largest range of acidity known. The same infrared study has also shown that the predominant cationic species (i.e., solvated proton) in 0–40 mol% SbF5

Superacids + is the H3 F+ 2 ions. The H2 F ion is observed only in highly concentrated solutions (40–100 mol% SbF5 ), contrary to the widespread belief that it is the only proton-solvated species in HF:SbF5 solutions. Ionization in dilute HF solutions of SbF5 (1–20% SbF5 ) is thus − + SbF5 + 3HF   SbF6 + H3 F2

The structure of the hexafluoroantimonate and of its − higher homologous anions Sb2 F− 11 and Sb3 F16 , which are formed when the SbF5 content is increased, have been determined by 19 F NMR studies.

7. HSO3 F:HF:SbF5 When Magic Acid is prepared from fluorosulfuric acid not carefully purified (which always contains HF), on addition of SbF5 the ternary superacid system HSO3 F:HF:SbF5 is formed. Because HF is a weaker Br¨onsted acid, it ionizes fluorosulfuric acid, which, on addition of SbF5 , results in a high-acidity superacid system at low SbF5 concentrations. 19 F NMR studies on the system have indicated the − presence of SbF− 6 and Sb2 F11 anions, although these can result from the disproportionality of SbF5 (FSO3 )− and Sb2 F10 (FSO3 )− anions. 8. HSO3 F:SbF5 :SO3 When sulfur trioxide is added to a solution of SbF5 in HSO3 F, there is a marked increase in conductivity that continues until approximately 3 mol of SO3 has been added per mol of SbF5 . This increase in conductivity has been attributed to an increase in H2 SO3 F+ concentration arising from the formation of a much stronger acid than Magic Acid. Acidity measurements have confirmed the increase in acidity with SO3 :SbF5 in the HSO3 F system. This has been attributed to the presence of a series of acids of the type H[SbF4 (SO3 F)2 ], H[SbF3 (SO3 F)3 ], H[SbF2 (SO3 F)4 ] of increasing acidity. Of all the fluorosulfuric acid-based superacid systems, sulfur trioxide-containing acid mixtures are, however, difficult to handle and cause extensive oxidative side reactions on contact with organic compounds.

9. HSO3 F–Nb(SO3 F)5 and HSO3 H–Ta(SO3 F)5 The in situ oxidation of niobium and tantalum metals in HSO3 F by bis(fluorosulfuryl)peroxide, S2 O6 F2 , gives the solvated Lewis acids M(SO3 F)5 , M = Nb or Ta. These acid systems have been shown to be highly acidic by conductivity studies.

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10. HSO3 F–Au(SO3 F)3 and HSO3 F–Pt(SO3 F)4 These superacids based on gold and platinum have been developed. They show a high acidity and good thermal stability. However, the high cost of the metals involved precludes their widespread use.

tion studies indicate that the HF:TaF5 system is a weaker superacid than HF:SbF5 . 13. Hydrogen Fluoride–Boron Trifluoride (Tetrafluoroboric Acid) Boron trifluoride ionizes anhydrous HF as follows:

11. Perfluoroalkanesulfonic Acid-Based Systems a. CF3 SO3 H:SbF5 . CF3 SO3 H:SbF5 (n = 1) was introduced by Olah as an effective superacid catalyst for isomerizations and alkylations. The composition and acidity of systems where n = 1, 2, 4 have been studied by Commeyras and co-workers. The change in composition of the triflic acid–antimony pentafluoride system depending on the SbF5 content has been studied. For the 1:1 composition, the main counteranion is [CF3 SO3 SbF5 ]− , and for the 1:2 composition [CF3 SO3 (Sb2 F11 )]− is predominant. With increasing SbF5 concentration, the anionic species grow larger and anions containing up to 5 SbF5 units have been found. In no circumstances could free SbF5 be detected. b. CF3 SO3 H:B(SO3 CF3 )3 . The acidity of triflic acid can also be substantially increased by the addition of boron triflate B(OSO2 CF3 )3 as indicated by Engelbrecht and Tschager. The increase in acidity is explained by the ionization equilibrium: B(OSO2 CF3 )3 + 2HSO3 CF3 +   2HSO3 CF3 + − B(SO3 CF3 )4

The measurements were limited due to the lack of a suitable indicator base, and even 1,3,5-trinitrobenzene the weakest base used, was fully protonated (H0 ≈ −18.5) in a 22 mol% solution of boron triflate. The acid system has found many synthetic applications, due mainly to the efforts of Olah and co-workers. 12. Hydrogen Fluoride–Tantalum Pentafluoride HF:TaF5 is a catalyst for various hydrocarbon conversions of practical importance. In contrast to antimony pentafluoride, tantalum pentafluoride is stable in a reducing environment. The HF:TaF5 superacid system has attracted attention mainly through the studies concerning alkane alkylation and aromatic protonation. Generally, heterogeneous mixtures such as 10:1 and 30:1 HF:TaF5 have been used because of the low solubility of TaF5 in HF (0.9% at 19◦ C and 0.6% at 0◦ C). For this reason, acidity measurements have been limited to very dilute solutions, and an H0 value of −18.85 has been found for the 0.6% solution. Both electrochemical studies and aromatic protona-

− BF3 + 2HF   BF4 + H2 F+

The stoichiometric compound exists only in an excess of HF or in the presence of suitable proton acceptors. The HF:BF3 (fluoroboric acid)-catalyzed reactions cover many of the Friedel–Crafts type reactions. One of the main advantages of this system is the high stability of HF and BF3 and their nonoxidizing nature. Both are gases at room temperature and are easily recovered from the reaction mixtures. Acidity measurements of the HF:BF3 system have been limited to electrochemical determinations, and a 7 mol% BF3 solution was found to have an acidity of H0 = −16.6. This indicates that BF3 is a much weaker Lewis acid compared with either SbF5 or TaF5 . Nevertheless, the HF:BF3 system is strong enough to protonate many weak bases and is an efficient and widely used catalyst. 14. Conjugate Friedel–Crafts Acids (HBr:AIBr3 , HCl:AICl3 , Etc.) The most widely used Friedel–Crafts catalyst systems are HCl:AlCl3 and HBr:AlBr3 . These systems are indeed superacids by Gillespie’s definition. However, experiments directed toward preparation from aluminium halides and hydrogen halides of the composition HAlX4 were unsuccessful in providing evidence that such conjugate acids are formed in the absence of proton acceptor bases. D. Solid Superacids The acidic sites of solid acids may be of either the Br¨onsted (proton donor, often OH group) or the Lewis type (electron acceptor). Both types have been identified by IR studies of solid surfaces absorbed with pyridine. Various solids displaying acidic properties, whose acidities can be enhanced to the superacidity range, are listed in Table V. 1. Immobilized Superacids (Bound to Inert Supports) Ways have been found to immobilize and/or to bind superacidic catalysts to an otherwise inert solid support. These include graphite intercalated superacids. Graphite possessing a layered structure can form intercalation compounds with Lewis acids such as AsF5 and SbF5 . These

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TABLE V Solid Acids

III. APPLICATION OF SUPERACIDS

1. Natural clay minerals: kaolinite, bentonite, attapulgite, montmorillonite, clarit, Fuller’s earth, zeolites, synthetic clays or zeolites 2. Metal oxides and sulfides: ZnO, CdO, Al2 O3 , CeO2 , ThO2 , TiO2 , ZrO2 , SnO2 , PbO, As2 O3 , Bi2 O3 , Sb2 O5 , V2 O5 , Cr2 O3 , MoO3 , WO3 , CdS, ZnS 3. Metal salts: MgSO4 , CaSO4 , SrSO4 , BaSO4 , CuSO4 , ZnSO4 , CdSO4 , Al2 (SO4 )3 , FeSO4 , Fe2 (SO4 )3 , CoSO4 , NiSO4 , Cr2 (SO4 )3 , KHSO4 , (NH4 )2 SO4 , Zn(NO3 )2 , Ca(NO3 )2 , K2 SO4 , Bi(NO3 )3 , Fe(NO3 )3 , CaCO3 , BPO4 , AlPO4 , CrPO4 , FePO4 , Cu3 (PO4 )2 , Zn3 (PO4 )2 , Mg3 (PO4 )2 , Ti3 (PO4 )4 , Zr3 (PO4 )4 , Ni3 (PO4 )2 , AgCl, CuCl, CaCl2 , AlCl3 ,TiCl3 , SnCl2 , CaF2 , BaF2 , AgClO4 , Mg(ClO4 )2 4. Mixed oxides: SiO2 :Al2 O3 , SiO2 :TiO2 , SiO2 :SnO2 , SiO2 :ZrO2 , SiO2 :BeO, SiO2 :MgO, SiO2 :CaO, SiO2 :SrO, SiO2 :ZnO, SiO2 :Ga2 O3 , SiO2 :Y2 O3 , SiO2 :La2 O3 , SiO2 :MoO3 , SiO2 :WO3 , SiO2 :V2 O5 , SiO2 :ThO2 , Al2 O3 :MgO, Al2 O3 :ZnO, Al2 O3 :CdO, Al2 O3 :B2 O3 , Al2 O3 :ThO2 , Al2 O3 :TiO2 , Al2 O3 :ZrO2 , Al2 O3 :V2 O5 , Al2 O3 :MoO3 , Al2 O3 :WO3 , Al2 O3 :Cr2 O3 , Al2 O3 :Mn2 O3 , Al2 O3 :Fe2 O3 , Al2 O3 :Co3 O4 , Al2 O3 :NiO, TiO2 :CuO, TiO2 :MgO, TiO2 :ZnO, TiO2 :CdO, TiO2 :ZrO2 , TiO2 :SnO2 , TiO2 :Bi2 O3 , TiO2 :Sb2 O5 , TiO2 :V2 O5 , TiO2 :Cr2 O3 , TiO2 :MoO3 , TiO2 :WO3 , TiO2 :Mn2 O3 , TiO2 :Fe2 O3 , TiO2 :Co3 O4 , TiO2 :NiO, ZrO2 :CdO, ZnO:MgO, ZnO:Fe2 O3 , MoO3 :CoO:Al2 O3 , MoO3 :NiO:Al2 O3 , TiO2 :SiO2 :MgO, MoO3 :Al2 O3 :MgO 5. Cation-exchange resins, polymeric perfluorinated resinsulfonic acids 6. Heteropolyacids (Keggin type) 7. Bis(perfluorosulfonyl)imides, bis- and tris(trifluoromethylsulfonyl) methanes

intercalates are not very stable, however, as the Lewis acid tends to leach out. Similar intercalates have been obtained with other Lewis acids such as AlCl3 , AlBr3 , NbF5 , and TaF5 and conjugate acid systems such as HF:SbF5 . Flourine-complexed acids such as SbF5 -fluorinated graphite and SbF− 5 -fluorinated alumina have been used for hydrocarbon isomerizations. 



RH2



ArH2

RHX

A. Preparation of Stable Trivalent Carbocations Superacids such as Magic Acid and fluoroantimonic acid have made it possible to prepare stable, long-lived carbocations, which are too reactive to exist as stable species in more basic solvents. Stable superacidic solutions of a large variety of carbocations, including trivalent cations (also called carbenium ions) such as t-butyl cation 1 (trimethylcarbenium ion) and isopropyl cation 2 (dimethylcarbenium ion), have been obtained. Some of the carbocations, as well as related acyl cations and acidic carboxonium ions and other heteroatom stabilized carbocations, that have been prepared in superacidic solutions or even isolated from them as stable salts are shown in Fig. 1.

CH3 

H3C

OH

RCH

OH



CH3

RX

B. Aromatic and Homoaromatic Cations and Carbodications According to H¨uckel’s (4n + 2) electron rule, if a carbocation has an aromatic character, it is stabilized by resonance. 



ROH2

R2OH



ROH

R2O

RCHCH2

HSO3F-SbF5 or HF-SbF5



R2CX2

RSH2

R2S

X

RCONR2 RCOOR′

OH



RSH

RCHO

CH3 2

Spectroscopic techniques such as 1 H and 13 C NMR, infrared, ultraviolet, and X-ray photoelectron spectroscopy have been employed to characterize carbocations. In many cases cation salts can be isolated with the superacid gegen ion, and some of them are structurally characterized by X-ray crystallography.

R2CO



H3C

1

(RO)2CO

(RO)2C

C

RCHCH3 RH

R2C



C

ArH 

H

RCOOH 

RCOOH2 X∆

R

  RCONR2 RCOOR′ R2CX R H H



R2SH

R

X





RCOH2O

R FIGURE 1 Some ways of generating carbocations generated in superacids.

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FIGURE 2 Aromatically stabilized cations and dications and some bridgehead dications.

Some aromatically stabilized H¨uckeloid systems generated in superacid media along with some carbodications are shown in Fig. 2. C. Static-Bridged or Equilibrating Carbocations Some carbocations tend to undergo fast degenerated rearrangements through intramolecular hydrogen or alkyl shifts to the related identical (degenerate) structures. The question arises whether these processes involve equilibrations between limiting “classical” ion intermediates (trivalent carbenium ions), whose structures can be adequately described by using only Lewis-type two-electron, twocenter bonds separated by low-energy transition states, or whether intermediate “nonclassical” hydrogen- or alkylbridged carbonium ions (higher coordinate carbonium ions) are involved, which also require the presence of twoelectron bonds between three or more centers for their description. It is difficult to answer this question by NMR spectroscopy because of its slow time scale; however, NMR has been used to delineate structures where degenerate rearrangements lead to averaged shifts and coupling constants. Solid-state 13 C NMR (using cross-polarization magicangle spinning techniques), isotopic substitution, and faster methods such as infrared, Raman, and, especially, X-ray photoelectron spectroscopy (ESCA) are particularly useful in investigating these systems. Some typical examples are depicted in Fig. 3. D. Hydrocarbon Transformations The astonishing acidity of Magic Acid and related superacids allows protonation of exceedingly weak bases. Not only all conceivable π -electron donors (such as olefins, acetylenes, and aromatics) and n-donors (such

FIGURE 3 Degenerate classical (carbenbium) and nonclassical (carbonium) carbocations.

as ethers, amines, and sulfides) but also weak σ -electron donors such as saturated hydrocarbons including the parent alkane and methane are protonated. The ability of superacids to protonated saturated hydrocarbons (alkanes) rests on the ability of the two-electron, two-center covalent bond to share its bonded electron pair with empty orbitals ( p or s) of a strongly electron-deficient reagent such as a protic acid:

R–H  H

R

H H

Superacids are suitable reagents for chemical transformation, particularly of hydrocarbons. E. Isomerization The isomerization of hydrocarbons is of practical importance. Isomeric dialkylbenzenes, such as xylenes, are starting materials for plastics and other products. Generally, the need is for only one of the possible isomers, and thus there is a potential for intraconversion (isomerization). Straightchain alkanes with five to eight carbon atoms have considerably lower octane numbers than their branched isomers, and hence there is a need for higher-octane branched isomers. Isomerizations are generally carried out under thermodynamically controlled conditions and lead to equilibria. The ionic equilibria in superacid systems generally favor increasing amounts of the higher-octane branched isomers at lower temperatures. Lewis-acid-catalyzed isomerization of alkanes can be effected with various systems. Superacid-catalyzed reactions can be carried out at much lower temperatures, even at or below room temperature, and thus provide more of the branched isomers. This is of particular

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importance in preparing lead-free gasoline. Increasing the octane number by this means is preferable to the addition of higher-octane aromatics or olefins, which may pose environmental or health-hazard problems. Of the many important superacid-catalyzed isomerizations, the isomerization tricyclo [5.2.1.02,6 ] decane to adamantane is unique, a reaction discovered by Schleyer.

Isomerization of alkylaromatics can also be effectively carried out with superacids. F. Alkylation Alkylation of aromatics is carried out industrially on a large scale; an example is the reaction of ethylene with benzene to produce ethylbenzene, which is then dehydrogenated to styrene, the monomer used in producing polystyrene. Traditionally, these alkylations have been carried out in solution with a Friedel–Crafts acid catalyst such as AlCl3 . However, these processes are quite energy consuming and form a complex mixture of products requiring large amounts of catalyst, most of which is tied up as complexes and can be difficult or impossible to recover. The use of solid superacidic catalyst permits clean, efficient heterogeneous alkylations with no concomitant complex formation.

 CH2

CH2

AlCl3–HCl

CH2CH3

Aliphatic alkylation is widely used to produce highoctane gasolines and other hydrocarbon products. Conventional paraffin (alkane)–olefin (alkene) alkylation is an acid-catalyzed reaction; it involves the addition of a tertiary alkyl cation, generated from an isoalkane (via hydride abstraction) to an olefin. An example of such a reaction is the isobutane–ethylene alkylation, yielding 2,3-dimethylbutane. The great interest in strong-acid chemistry is further exemplified by the discovery that lower alkanes such as methane and ethane can be polycondensed in Magic Acid at 50◦ C, yielding mainly C4 to C10 hydrocarbons of the gasoline range. The proposed mechanism (Fig. 4) necessitates the intermediacy of protonated alkanes (pentacoordinate carbonium ions), at least as high-lying intermediates or transition states. Hydrogen must be oxidatively removed (by either the excess superacid or added oxidants) to make the condensation of methane thermodynamically feasible.

FIGURE 4 Mechanism of oxidative methane oligocondensation.

Because of the high reactivity of primary and secondary ions under these conditions, the alkylation reaction is complicated by hydride transfer and related competing reactions. However, in this mechanism it is implicit that an energetic primary cation will react directly with methane or ethane. This opens the door to new chemistry through activation of these traditionally passive molecules. A convenient way to prepare an energetic primary cation is to react ethylene with superacid. This has been used with HF–TaF5 catalyst to achieve ethylation of methane in a flow system at 50◦ C. With a methane–ethylene mixture (85:14), propane is the major product. G. Polymerization The key initiation step in cationic polymerization of alkenes is the formation of a carbocationic intermediate, which can then interact with excess monomer to start propagation. The mechanism of the initiation of cationic polymerization and polycondensation has been extensively studied. Trivalent carbenium ions play the key role, not only in acid-catalyzed polymerization of alkenes, but also in polycondensation of arenes (π -bonded monomers), as well as in cationic polymerization of ethers, sulfides, and nitrogen compounds (nonbonded electron-pair donor monomers). Pentacoordinated carbonium ions, on the other hand, play the key role in the electrophilic reactions of σ -bonds (single bonds), including the oligocondensation of alkanes and the cocondensation of alkanes and alkenes. Alkylation and oligocondensation reactions of alkanes giving higher molecular weight alkanes have been achieved under superacid conditions. H. Superacids in Organic Syntheses and Superelectrophilic Activation Since the discovery of stable carbocations, they were known to be readily quenched by various nucleophiles. These reactions, which were first used to confirm the structure of the ions, proved to be very useful in organic synthesis. The selectivity of the reactions is based on the fact that generally only thermodynamically more stable ions are formed under the reaction conditions, resulting in

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a high selectivity. The new functional group created in a superacid medium will itself undergo protonation and, thus, be protected against any further electrophilic attack. In this way, a number of new selective reactions were achieved in a high yield, as shown by the examples below. Furthermore, electrophiles that contain nonbonded electron pairs, π -bonds, or even σ -bonds can be further activated by protonation or Lewis acid complexation leading to superelectrophiles. Such activated species react with many deactivated aromatic as well as aliphatic compounds. 1. Phenol–Dienone Rearrangement This isomerization is of substantial importance in natural product syntheses, usually catalyzed by a strong base. The reaction occurs with good yields in polycyclic systems under superacidic conditions, as shown by Gesson and Jacquesay. CH3 O

CH3 O HF-SbF5

H

H

H O

HO

2. Reduction Hydride ion transfer to carbocations is a well-known reaction in hydrocarbon chemistry. This reaction has been used successfully in superacid to reduce α,β-unsaturated ketones with methylcyclopentane as the hydride donor. Superacid-catalyzed reduction of aromatics, as shown by Wristers, requires both a hydride donor and hydrogen.

H HF-SbF5

O

O

H

H HF-SbF5

COOCH3

CO/CH3OH

O

O

H

Olah et al. have developed direct carbonylation of isoalkanes that lead to ketones in high conversion and high selectivity under HF:BF3 catalysis. The chemistry is unlike the Koch reaction and involves activated formyl cation inserting directly into the C–H σ -bond of isoalkanes, followed by strong acid-catalyzed rearrangement.

O H3C H3C H3C

H

HF:BF3 CO

H3C CH3 CH3

4. Oxidation Novel oxidations of hydrocarbons in superacids with ozone or hydrogen peroxide have been investigated. Proto+ nated ozone (O+ 3 H) or hydrogen peroxide (H3 O2 ) attacks the single σ -bond, resulting in oxygen insertion. These can be followed by protolytic transformation, such as the conversion of isobutane into acetone and methyl alcohol. (CH3)3C

(CH3)3CH H2O2 Magic acid

(CH3)2C

O  CH3OH



H

O3 Magic acid

H2O

(CH3)2C

[(CH3)3C



H (CH3)3C

H2O2

O3H



O]

H2O

OOH

1,2CH3 shift



O CH3

By similar procedures aromatics are also hydroxylated in high yields at low temperatures. H2O2 HSO3F/SO2ClF

HF-TaF5 isopentane H2

CH 5. Superelectrophilic Activation

3. Carbonylation The reaction between carbocations and carbon monoxide affording oxocarbenium ions (acyl cations) is a key step in the well-known Koch–Haaf reaction for preparing carboxylic acids from alkenes. This reaction has been extensively studied under superacidic conditions. An example is indicate below.

+ + Electrophiles such as NO+ 2 , CH3 CO , and H3 O can be further activated in strong protic acids to their respective dications: NO2 H2+ , CH3 COH2+ , and H4 O2+ . Such superelectrophiles are responsible for the high electrophilic reactivity in superacids. For example, acetyl cation is a poor acetylating agent for chlorobenzene in trifluoroacetic acid. However, in superacidic trifluoromethanesulfonic acid medium, acetylation takes place with ease.

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Cl

Cl  CH3COSbF6

COCH3

Olah and co-workers have shown hydrogen–deuterium exchange of molecular H2 and D2 , respectively, with 1:1 HF:SbF5 and HSO3 F:SbF5 at room temperature. The facile formation of HD does indicate that protonation or deuteration occurs involving 3 at least as a transition state in the kinetic exchange process. The H3 ion, 3, is the simplest two-electron, three-center bonded entity.

in CF3COOH at 60°C, 24 hr, 1% in CF3SO3H at 60°C, 30 min, 81%

H H

Similar activations have been proposed for Lewis acid complexations. I. Miscellaneous Reactions Many acid-catalyzed reactions can be advantageously carried out using solid superacids instead of conventional acid systems. The reactions can be carried out in either the gaseous or the liquid phase. Using the example Nafion-H (a perfluoroalkane resin sulfonic acid, developed by DuPont) solid acid, several simple procedures were reported to carry out alkylation, transbromination, nitration, acetalization, hydration, and so on. J. Superacids in Inorganic Chemistry 1. Halogen Cations It has often been postulated that the monoatomic ions I+ , Br+ , and Cl+ are the reactive intermediates in halogenation reactions of aromatics and alkenes. The search for the existence of such species has led to the discovery of I+ 2 and other related halogen cations, which are stable in superacids. The I+ 2 cation may be generated by the oxidation of I2 with S2 O6 F2 in HSO3 F solution, − 2I2 + S2 O6 F2 → 2I+ 2 + 2SO3 F

and a stable blue solution of this cation can also be obtained by oxidizing iodine with 65% oleum. In a less acidic medium, the I+ 2 cation disproportionates to more stable oxidation states. The electrophilic Br+ 2 cation is obtainable only in the very strong superacid Magic Acid or fluoroantimonic acid, and it disproportionates in HSO3 F. The Cl+ 2 cation, which is much more electrophilic, has not yet been observed in solution. Monoatomic halogen cations seem to be too unstable for direct observation. 2. The Trihydrogen Cation, H+ 3 The H3 ion, 3, was first discovered by Thompson in 1912 in hydrogen discharge studies. Actually, it was the first observed gaseous ion–molecule reaction product.

H 3

3. Cations of Other Nonmetallic Elements Elemental sulfur, selenium, and tellurium give colored solutions when dissolved in a number of strongly acidic me2+ 2+ 2+ 2+ dia. It has been shown that S2+ 16 , S8 , S4 , Se8 , Te4 , 2+ and Te6 are present in such solutions. These cations are formed by the oxidation of elements by H2 S2 O7 or S2 O6 F2 ; for example, − 4S + 6H2 S2 O7 → S2+ 4 + 2HS3 O10 + 5H2 SO4 + SO2

Like the halogen cations, the sulfur, selenium, and tellurium cations are highly electrophilic and undergo disproportionation in media with any appreciable basic properties, although, as would be anticipated, the ease of disproportionation increases in series tellurium < selenium < sulfur. 4. Noble Gas Cations Noble gas cationic salts of xenon and krypton have also been isolated from superacid medium. The examples + + + include XeF+ , Xe2 F+ 3 , HCNXeF , XeOF3 , KrF , and + Kr2 F3 .

SEE ALSO THE FOLLOWING ARTICLES NOBLE-GAS CHEMISTRY • ORGANOMETALLIC CHEMISTRY • PHYSICAL ORGANIC CHEMISTRY

BIBLIOGRAPHY Gillespie, R. J., and Peel, T. E. (1971). Adv. Phys. Org. Chem. 9, 1. Jost, R., and Sommer, J. (1988). Rev. Chem. Int. 9, 171. Olah, G. A. (1993). Angew. Chem. Int. Ed. Engl. 32, 767. Olah, G. A., Prakash, G. K. S., and Sommer, J. (1985). “Superacids,” Wiley, New York. Tanabe, K. (1970). “Solid Acids and Bases; Their Catalytic Properties,” Academic Press, New York. Vogel, P. (1985). “Carbocation Chemistry,” Elsevier, Amsterdam.

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Surface Chemistry Simon R. Bare G. A. Somorjai University of California, Berkeley, and Lawrence Berkeley Laboratory

I. Surface Structure of Clean Surfaces II. Surface Structure of Adsorbates on Solid Surfaces III. Thermodynamics of Surfaces IV. Electrical Properties of Surfaces V. Surface Dynamics

GLOSSARY Adsorbate Adsorbed atom or molecule. Adsorption Process by which molecules are taken up on the surface by chemical or physical action. Chemisorption Binding of molecules to surfaces by strong chemical forces. Desorption Process by which molecules are removed from the surface. Heat of adsorption Binding energy of the adsorbed species. Physisorption Binding of molecules to surfaces by weak chemical forces. Sticking probability Ratio of the rate of adsorption to the rate of collision of the gaseous molecule with the surface. Surface free energy Energy necessary to create a unit area of surface. Surface reconstruction Equilibration of surface atoms to new positions that changes the bond angles and rotational symmetry of the surface atoms.

.

Surface relaxation Equilibration of surface atoms to new positions that changes the interlayer distance between the first and second layers of atoms. Surface state Electronic state localized at the surface. Surface unit cell Two-dimensional repeating unit that fully describes the surface structure. Work function Minimum energy required to remove an electron from the surface into the vacuum outside the solid.

SURFACES constitute the boundaries of condensed matter, solids, and liquids. Surface chemistry explores the structure and composition of surfaces and the bonding and reactions of atoms and molecules on them. There are many macroscopic physical phenomena that occur on surfaces or are controlled by the electronic and physical properties of surfaces. These include heterogeneous catalysis, corrosion, crystal growth, evaporation, lubrication, adhesion, and integrated circuitry. Surface chemistry examines the science of these phenomena as well.

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I. SURFACE STRUCTURE OF CLEAN SURFACES A. Introduction To the naked eye the surface of a single crystal of a metal looks perfectly planar with no imperfections. If this crystal is now examined with an optical microscope features of the surfaces down to the wavelength of visible light ˚ can be resolved. The surface will look gran(∼5000 A) ular with distinct regions of crystallinity separated from each other by boundaries or dislocations. These dislocations indicate areas on the surface where there is a mismatch of the crystalline lattice, and they can take several forms, for example, edge dislocations or screw dislocations. The presence of these dislocations or defects can dominate certain physical properties of the material. Dislocation densities of the order of 106 –108 cm−2 are commonly found on metal single crystals, whereas the number is lower (104 –106 cm−2 ) on semiconductor surfaces due to the different nature of the bonding. These defect densities must be compared with the total concentration of surface atoms (about 1015 cm−2 ). On further magnification, for example, using a scanning electron microscope, ˚ and our features can be resolved down to about 1000 A, view of the surface changes further. The surface will look pitted, with distinct planar areas (terraces) bounded by walls many atomic layers in height. Thus, on the microscopic and submicroscopic scale the surface morphology appears to be heterogeneous, with many different surface sites that differ by the number of neighboring atoms surrounding them. What about the nature of the surface on an atomic scale? In order to be able to discuss and understand the structure of surfaces it is necessary to understand the techniques that are capable of viewing the surface on an atomic scale. We briefly describe such techniques, illustrating their capabilities with pertinent examples. The techniques more commonly used are field-ion microscopy (FIM), lowenergy electron diffraction (LEED), helium atom diffraction, and high-energy ion scattering. In addition, the relatively new technique of scanning tunneling microscopy (STM) is proving to be a very promising tool. Only brief descriptions are given here and the reader is referred to some of the excellent books on the subject given in the bibliography.

Surface Chemistry

vented by M¨uller in 1936. The basic microscope can be very simple. In an ultrahigh vacuum cell, a potential of about 10,000 V is applied between a hemispherical tip of refractory metal of radius ∼10−4 cm and a fluorescent screen. The tip is charged positively, and a gas (usually helium) is allowed to impinge on the surface. Under the influence of the very strong electric field helium atoms that are incident on the tip are ionized. The positive ions thus created are repelled radially from the surface and accelerated onto the fluorescent screen, where a greatly magnified image of the crystal tip is displayed. The ionization probability depends strongly on the local field variations induced by the atomic structure of the surface—protruding atoms generate stronger ionization than atoms embedded in close-packed planes, thereby producing individual bright spots on the screen. The small radius of the tip is needed to produce the large fields necessary for ionization, but it also permits the immense magnification of the microscope. The tip surface is directly imaged with magnification of about 107 . Figure 1 depicts the image of a tungsten field-ion tip. Well-defined atomic planes of the crystal tip can be readily identified, indicating that there is order of the atomic scale, i.e., most of the surface atoms in any crystal face are situated in ordered rows separated by well-defined interatomic distances. The technique is limited to the refractory metals (W, Ta, Ir, and Re) which can withstand the strong electric field at the tip without desorption or evaporation from the surface. However, its great advantage is that individual atoms can be imaged on the screen, which also allows studies of surface diffusion.

B. Techniques Sensitive to Surface Structure 1. Field-Ion Microscopy Field-ion microscopy is one of the oldest techniques used for surface structure determination, having been in-

FIGURE 1 Field-ion micrograph of a tungsten tip. Various crystal planes are labeled. (Courtesy of Lawrence Berkeley Laboratory.)

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2. Low-Energy Electron Diffraction Another method which has demonstrated that crystal surfaces are ordered on an atomic scale is low-energy electron diffraction (LEED). This method is the most frequently used technique, such that virtually all modern surface science laboratories now rely on it for surface structural information. In order to obtain diffraction from surfaces, the incident wave must satisfy the condition λ ≤ d, where λ is the wavelength of the incident beam and d the interatomic distance. In addition, the incident beam should not penetrate much below the surface plane but should backdiffract predominantly from the surface so that the scattered beam reflects the properties of the surface atoms and not those of the bulk. The deBroglie wavelength λ of electrons is given √ ˚ = 150/E, where E is in electron volts. Thus, by λ (in A) in the energy range 10–500 eV, the wavelength varies from ˚ which is smaller or equal to most interatomic 3.9 to 0.64 A, distances, and the escape depth of the backscattered elec˚ thereby providing trons in this energy range is 5–10 A, surface sensitivity. These elastically scattered low-energy electrons yield surface structural information. The technique of LEED is depicted schematically in Fig. 2, while a schematic of the LEED apparatus is shown in Fig. 3. A collimated primary beam of electrons with a diameter of 0.1–1 mm at energies of 15–350 eV is impinged on a surface and the elastically backscattered electrons, after traveling through a field-free region, are spatially analyzed. This is achieved most commonly (see Fig. 3) by passing the scattered electrons through four hemispherical grids. The first grid is at the crystal potential while the second and third are at a retarding voltage to eliminate inelastic electrons, and the fourth is at ground. After passing through these grids the diffracted beams are accelerated onto a hemispherical phosphor screen.

If the crystal surface is well-ordered, a diffraction pattern consisting of bright, well-defined spots will be displayed on the screen. The sharpness and overall intensity of the spots are related to the degree of order of the surface. When the surface is less ordered, the diffraction beams broaden and become less intense, while some diffuse intensity appears between the beams. A typical set of diffraction patterns from a well-ordered surface is shown in Fig. 4. The presence of the sharp diffraction spots clearly indicates that the surface is ordered on an atomic scale. Similar LEED patterns have been obtained from solid single-crystal surfaces of many types including metals, semiconductors, alloys, oxides, and intermetallics. Due to the importance of LEED in surface chemistry we briefly discuss other aspects of the technique which make it one of the most powerful surface sensitive tools. It is convenient to subdivide the technique into two-dimensional LEED and three-dimensional LEED. In two-dimensional LEED we observe only the symmetry of the diffraction pattern on the fluorescent screen. The bright spots which correspond to the two-dimensional reciprocal lattice belonging to the repetitive crystalline surface structure yield immediate information about the size and orientation of the surface unit cell, i.e., the geometry of the surface layer. This is important information since reconstructioninduced and adsorbate-induced new periodicities are immediately visible. The diffuse background intensity also contains information about the nature of any disorder present on the surface. In three-dimensional LEED the information gained from the two-dimensional pattern is supplemented by the intensities of the diffraction spots which are measured as a function of incident electron energy. By comparing these intensity-versus-voltage curves [I (V ) curves] with those simulated numerically with the help of a suitable theory, the precise location of atoms or molecules in the surface with respect to their neighbors is determined. Thus, the bond length and bond angles in the surface layer are calculated. It should be mentioned, however, that the analysis of the LEED beam intensities requires a theory of the diffraction process which is a nontrivial point due to multiple scattering of LEED electrons by the surface, and this is not simple to represent in a theory. 3. Atomic-Beam Diffraction Another technique that utilizes the principle of diffraction is atomic- or molecular-beam diffraction. The deBroglie wavelength λ associated with helium atoms is given by the following:

FIGURE 2 Scheme of the low-energy electron diffraction experiment. (Courtesy of Lawrence Berkeley Laboratory.)

˚ = λ(A)

h 0.14 = , 1/2 (2ME) E(eV)1/2

(1)

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FIGURE 3 Scheme of the low-energy electron diffraction apparatus employing the postacceleration technique.

FIGURE 4 LEED pattern from a Pt(111) crystal surface at (a) 51 eV, (b) 63.5 eV, (c) 160 eV, and (d) 181 eV incident electron energy. (Courtesy of Lawrence Berkeley Laboratory.)

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where h is Planck’s constant and M and E are the mass and energy, respectively, of the helium atom. Atoms with ˚ and can reada thermal energy of ∼20 meV have λ = 1 A ily diffract from surfaces. The information obtained from atomic-beam diffraction is similar to that from LEED, but there are differences between the two techniques. In LEED the relatively high-energy (20–200 eV) electrons used penetrate the crystal, multiple scattering events are important, and the LEED electrons are scattered primarily from the ion cores of the crystal lattice. In atom diffraction, there is virtually no penetration of the low-energy (10–200 meV) atomic beam, making it much more surface sensitive than the electron beam. The atomic beam is primarily scattered from the valence electrons of the surface atoms. In fact, their scattering is usually simulated by a “hard wall” around the atoms in the top layer of the surface so that diffraction is from a “corrugated hard wall” with the periodicity of the surface mesh. As in LEED the location of the diffracted beams indicates the surface periodicity. Their intensities are related to the structure of the scattering potential within a unit mesh, in this case to the relative amplitude and positions of corrugations around the surface atoms. The essential elements of the apparatus necessary to perform atomic-beam diffraction are an atomic beam of gas and a detector. The atomic beam is usually generated from a nozzle source incorporating several skimmers. The energy (wavelength) of the beam is varied by either heating or cooling the nozzle. The detector usually employed is a mass spectrometer, mounted on a rotatable device to enable it to be movable over a large range of scattering angles. The atomic beam is chopped with a variable-frequency chopper before it impinges on the surface. In this way, an alternating intensity of the beam is generated at the mass spectrometer detector, which is readily separated from the noise due to helium atoms in the background. To illustrate the type of data that is obtained, Fig. 5 shows the He diffraction traces from a Au(110)-(1 × 2) surface at two different wavelengths. Helium diffraction is especially sensitive to surface order on an atomic scale. On scattering from a well-ordered single crystal surface nearly 15% of the scattered helium atoms appear in the specular helium beam whereas this fraction can drop to 1% when the surface is disordered. Measurements of the fraction of specularly scattered helium can therefore provide information on the degree of atomic disorder in the solid surface. 4. Scanning Tunneling Microscopy The relatively new technique of scanning tunneling microscopy also clearly demonstrates order on an atomic

FIGURE 5 Helium diffraction traces for Au(110)-(1 × 2) at a surface temperature of 100 K with incident angle i = 48◦ . The wave˚ length λHe is (a) 1.09 A˚ and (b) 0.57 A.

scale on single-crystal surfaces. It images surface topogra˚ and phies in real space with a lateral resolution of ∼2 A ˚ vertical resolution of ∼0.05 A. The technique utilizes the tunnel effect. Due to the wave nature of electrons, they are not strictly confined to the interior bounded by the surface atoms. Therefore, the electron density does not drop to zero at the surface but decays exponentially on the outside with a decay length of a few angstroms. If two metals are approached to within a few angstroms, the overlap of their surrounding electron clouds becomes substantial, and a measurable current can be induced by applying a small voltage between them. This tunnel current is a measure of the wave-function overlap, and depends very strongly on the distance between the two metals. This is the physical basis of the scanning tunneling microscope. Experimentally one of the electrodes is sharpened to a pointed tip which is scanned over the surface to be investigated (the other electrode) at constant tunnel current. The tip thus traces contours of constant wave-function overlap, and in the case of constant decay length, the trace is an almost true image of the surface atomic positions, i.e., the surface topography. An example is shown in Fig. 6.

5. Ion Scattering Ion scattering from surfaces is usually subdivided into two scattering regions: low-energy ion scattering (LEIS), energies typically ∼1 keV, and high-energy scattering (HEIS), energies 0.1–1 MeV. High-energy ion scattering is a probe that tests the local position of surface atoms relative to their bulklike sites. In HEIS the velocity of the ion is such that it is moving fast compared to the thermal motions of the

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FIGURE 6 Scanning tunneling microscope picture of a clean (1 × 5) reconstructed Au(100) surface with monatomic ˚ with approximately 1.5 A˚ from scan to scan. The inset shows the LEED steps. Divisions on the crystal axes are 5 A, pattern of the predominant (1 × 5) corrugation. (Courtesy of Lawrence Berkeley Laboratory.)

atoms in the solid, thus the beam senses a frozen lattice. If the target is amorphous each atom would sense a uniform distribution of impact parameters of the ions and diffuse scattering results. However, the scattering spectrum from a single crystal aligned with a major symmetry axis parallel to the beam is drastically modified from that of the amorphous target. The impact parameter distribution is also uniform at the first monolayer, but the first atom shadows the second from the beam, and small angle scattering events determine the impact parameter distribution at the second atom. This results in a unique (nonuniform) flux distribution at the second atom. Figure 7 illustrates the effect of small-angle scattering in a two-atom model. Ions incident at the smallest impact parameter undergo largeangle scattering, those at large impact parameter suffer small deflections which determine the flux distribution of ions near the second atom. The closest approach of the ion to the second atom, R, can be approximated assuming Coulomb scattering as follows:   1/2 R = 2 Z 1 Z 2 e2 d E ,

proximation, can be written analytically which leads to an estimate of the two-atom surface peak intensity I : 

R2 I =1+ 1+ 2 2ρ





 −R exp , 2ρ 2

(3)

where ρ is the two-dimensional root mean square thermal vibrational amplitude. The first term represents the unit contribution from the first atom in the string, the second term represents the variable contribution from the second

(2)

where Z 1 and Z 2 are the masses of the incident and target atoms, respectively; d is the atomic spacing; and E is the incident ion energy. This gives rise to a shadow cone beneath the surface atom as illustrated in Fig. 7. The flux distribution at the second atom, within the Coulomb ap-

FIGURE 7 Schematic showing the interactions at the surface of an aligned single crystal and the formation of the shadow cone. The energy spectra for the aligned and nonaligned case are also shown.

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atom. While the two-atom Coulomb approximation is not adequate enough to compare to experiment, it illustrates that the surface peak intensity is a function of one parameter, ρ/R. The intensity of the surface peak is thus sensitive to the atomic arrangement on the surface, i.e., the positions of the surface atoms with respect to their bulklike positions. The effect on the surface peak of different surface structures is depicted in Fig. 8. The nature of reconstructed, relaxed, and adsorbate-covered surfaces are discussed in the sections below. High-energy ion scattering is also a sensitive tool for answering other important questions in surface chemistry, namely what type of atoms are present on the surface and how many are present. All of the techniques discussed so far indicate that the solid surface is ordered on an atomic scale. Most of the surface atoms occupy equilibrium atomic positions that are located in well-defined rows separated by equal interatomic positions. This atomic order is predominant despite the fact that there are large numbers of atomic positions on the surface where atoms have different numbers of neighbors. A pictorial representation of the topology of a monatomic crystal on an atomic scale is shown in

FIGURE 9 Model of a heterogeneous solid surface, depicting different surface sites. These sites are distinguishable by their number of nearest neighbors.

Fig. 9. The surface may have atoms in any of the positions shown in the figure. There are atoms in the surface at kink positions and in ledge positions, and there are adatoms adsorbed on the surface at various sites. Atomic movement from one position to another proceeds by surface diffusion. To the first approximation, the binding energy of the surface atoms is proportional to the number of nearest and next-nearest neighbors. Therefore, for example, atoms at a ledge are bound more strongly than are adatoms. In equilibrium there is a certain concentration of all these surface species, with those species predominating whose binding energies are greatest. Thus, the adatom concentration on clean well-equilibrated surfaces should be very small indeed. However, while these surfaces are ordered on an atomic scale their structure is not always one of simple termination of the bulk unit cell, relaxation or reconstruction being common. C. Surface Relaxation

FIGURE 8 Schematic of the dependence of the intensity of the surface peak (SP) on different crystal surface structures. (a) The ideal crystal SP from “bulklike” surface, (b) enhanced SP observed in normal incidence for a reconstructed surface, (c) enhanced SP observed in nonnormal incidence for a relaxed surface, and (d) reduced SP observed in normal incidence for a registered overlayer.

Generally, the surface unit cells of clean metal surfaces have been found to be those expected from the projection of the bulk X-ray unit cell onto the surface, referred to as a (1 × 1) structure (in Miller index notation), and the uppermost layer z spacing (spacing in the direction normal to the surface plane) is equal to the bulk value within about 5%. Such surfaces include the (111) crystal faces of face centered cubic aluminum, platinum, nickel, and rhodium, and the (0001) crystal faces of hexagonal close packed cadmium and beryllium. This information has almost exclusively been determined by a detailed intensity analysis of the diffraction beams in LEED as a function of incident electron energy, and the interatomic positions ˚ The in the surface layer are calculated to within 0.1 A. Al (110) surface shows a 5–15% contraction, the Mo(100) surface a 11–12% contraction, and the W(100) surface a 6% contraction of the top-layer z spacing with respect to the bulk, while retaining the (1 × 1) surface unit cell. Generally, crystal planes whose atoms are less densely packed [for example, bcc (100) and fcc (110) planes] will be more likely to show relaxation than the more densely

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380 packed planes. In forming a surface of the less densely packed planes it is necessary to remove a larger number of nearest-neighbor atoms. Thus, to minimize the surface free energy a relocation of the surface atoms from their bulk positions is quite likely. There are several explanations as to why surface relaxation is prevalent on the more open surfaces. First, it can be imagined that the electron cloud attempts to smooth its surface, thereby producing electrostatic forces that draw the surface atoms toward the substrate, the effect being stronger the less closely packed the surface. Second, with fewer neighbors the two-body repulsion energy is smaller, allowing greater atomic overlap and therefore more favorable bonding at shorter bond lengths. Third, for surface atoms the bonding electrons are partly redistributed from the broken bonds to the remaining unbroken bonds, thereby the charge content of the latter is increased, reducing the bond length. D. Surface Reconstruction Surface atoms in any crystal are in an anisotropic environment which is very different from that around bulk atoms. The crystal symmetry that is experienced by each surface atom is markedly lower than when the atom is in the bulk. This symmetry change and lack of neighbors in the direction perpendicular to the surface allows displacement of surface atoms in ways which are not allowed in the bulk. Surface relaxation is one consequence of this, the other major consequence being surface reconstruction. Here the two-dimensional surface unit cell is different from that given by the termination of the bulk structure on the plane of interest. Surface reconstruction can give rise to a multitude of different structures depending on the electronic structure of a given substance. The phenomenon is more frequent on semiconductor surfaces than on metal surfaces. While the geometry is readily observed in the LEED pattern, the actual elucidation of the real-space reconstructed surface structure is often extremely difficult, and in some cases even after years of study and many proposals of the structure, the true structure is still not known. Such a system is the Si(111) surface. Upon cleaving in UHV at room temperature, the surface exhibits a (2 × 1) surface structure. On heating to about 700 K the surface structure changes to one with (7 × 7) periodicity. This (7 × 7) structure is then the stable structure of the (111) face. While this surface has been studied by a multitude of techniques, including LEED, STM, and He atom diffraction there is still no generally accepted structure of either the (2 × 1) or (7 × 7) reconstructions. One of the best known examples of reconstruction of metallic surfaces is that for the (100) faces of three 5d transition metals that are neighbors on the periodic table:

Surface Chemistry

gold, platinum, and iridium. The ideal unreconstructed surfaces have a square net of atoms. Surface reconstruction produces a superlattice that is basically five times larger in one direction than for the ideal surface. For Ir(100) the superlattice is denoted (5 × 1), for Pt(100) by the ma1 trix notation ( −15 14 ), and for Au(100) by the superlattice (5 × 20). From LEED I (V ) analyses, evidence indicates that the nature of the reconstruction is similar on all three metals and consists of a close-packed hexagonal top layer that is positioned in slightly different ways on the square net substrate. These reconstructions are consistent with the knowledge that the (111) face of fcc metals is energetically the most favorable. It is worth noting here that the adsorption of gases such as oxygen, carbon monoxide, or hydrogen, or the presence of impurities can inhibit these surface reconstructions. On the other hand, the presence of such adsorbates can also induce different surface reconstructions. The nature and cause of these surface phase transformations are not well established at present. The case of structural change from metastable to stable on adsorption or removal of adsorbates indicates the likelihood of electronic transitions that accompany reconstruction. At the surface there are fewer nearest neighbors as compared to atoms in the bulk. The electronic structure that is the most stable in this reduced-symmetry environment may be substantially different from that of the bulk metal. Since the surface atoms are surrounded by atoms only on one side and there is vacuum on the other, they may change their coordination number by slight relocation with simultaneous changes in the electronic structure. It is indeed surprising that more surfaces do not show reconstruction. E. Stepped and Kinked Surfaces The close-packed faces of solids (low-Miller-index faces) have the lowest surface free energy, and therefore they are the most stable with respect to rearrangement on disordering up to or near the melting point. However, stepped and/or kinked surfaces (high-Miller-index faces), although of higher surface free energy, are very important. They are known to play important roles during evaporation, condensation, and melting. Steps and kinks are sites where atoms break away as an initial process leading to desorption, or where atoms migrate during condensation to be incorporated into the crystal lattice. Theories of crystal growth, evaporation, and the kinetics of melting have identified the significance of these lower coordination-number sites in controlling the rate processes associated with phase changes. In addition, studies of chemisorption and catalysis using single-crystal surfaces have revealed different binding energies and enhanced chemical activity at steps and kinks on high-Miller-index

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FIGURE 10 LEED patterns (left) and surface structures (right) of (a) flat, (b) stepped, and (c) kinked platinum surfaces. (Courtesy of Lawrence Berkeley Laboratory.)

transition metal surfaces as compared to low-Millerindex surfaces. Adsorption of diatomic or polyatomic molecules frequently leads to dissociation with greater probability of steps and kinks than on flat atomic terraces. The presence of steps on a single crystal surface is readily discernible by LEED. The LEED patterns differ from those expected from crystals with low-index faces in that the diffraction beams are split into doublets. This splitting (Fig. 10) is a function of ordered steps on the surface. The distance between the split beams is inversely related to the distance between the steps, i.e., the terrace width. From the variation of the intensity maximum of the doublet spots with electron energy the step height can be determined. Many stepped surfaces exhibit high thermal stability. In particular the one-atom-height step periodic terrace configuration appears to be the stable surface structure of many high-Miller-index surfaces. While most of the stepped surfaces are stable when clean in their one-atomheight step ordered terrace configuration, in the presence of a monolayer of carbon or oxygen many stepped surfaces undergo structural rearrangement. The step height and terrace width may double, or faceting may occur. Faceting occurs when the step orientation becomes as prominent as that of the terrace and new diffraction features become rec-

ognizable in LEED. Upon removing the impurities from the surface, the original one-atom-height step ordered terrace surface structure is usually regenerated.

II. SURFACE STRUCTURE OF ADSORBATES ON SOLID SURFACES A. Introduction While the knowledge of the structure of clean solid surfaces is important in its own right for determining various properties of those surfaces, many phenomena are associated with the presence of adsorbates on the surfaces. In fact, in the natural environment of our planet, surfaces are never truly free of adsorbates. On approaching a surface each atom or molecule encounters a net attractive potential. This results in a finite probability that it will be trapped on the surface. This trapping, adsorption, is always an exothermic process. At the low pressure of 1 × 10−6 torr, approximately 1 × 1015 gas molecules collide with each square centimeter of surface per second. Since the surface concentration of atoms is about 1015 cm−2 , at this pressure the surface may be covered with a monolayer of gas within seconds; this is the major reason why surface studies are performed under ultrahigh vacuum conditions (P < 1 × 10−8 torr). The very low

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382 pressure is needed to maintain clean surface conditions for a time long enough to perform experimental measurements. At atmospheric pressure the surface will be covered within a fraction of a second. The constant presence of the adsorbate layer influences the chemical, mechanical, and surface electronic properties. Adhesion, lubrication, and resistance to mechanical or chemical attack or photoconductivity are just a few of the many macroscopic surface processes that are controlled by various properties of monolayers. Two macroscopic experimentally determinable parameters characterize the adsorbed monolayer: the coverage and the heat of adsorption. The coverage  is defined as the ratio of the number of adsorbed atoms or molecules to the total number of adsorption sites (usually taken as the number of atoms in the surface plane). The heat of adsorption Hads , is implicitly linked to the strength of the adsorbate–substrate bond. Knowledge of both parameters often reveals the nature of bonding in the adsorbed layer. Atoms or molecules that impinge on the solid surface from the gas phase will have a residence time τ on the surface. If the impinging molecules achieve thermal equilibrium with the surface atoms τ = τ0 exp Hads /RT , where τ0 is related to the average vibrational frequency associated with the adsorbate. The heat of adsorption is always positive and is defined as the binding energy of the adsorbed species. A larger Hads and lower temperature increase the residence time. For a given incident flux, larger Hads and lower temperature yield higher coverages. It is conventional to divide adsorption into two categories: physisorption and chemisorption. Physisorption (or physical adsorption) systems are characterized by weak interactions ( Hads < 15 kcal mol−1 , accompanied by short residence times) and require adsorption studies to be performed at low temperature and relatively high pressure (high flux). Adsorbates that are characterized by stronger chemical interactions ( Hads ≥ 15 kcal mol−1 ), where near-monolayer adsorption commences even at room temperature and at low pressures (≤10−6 torr), are called chemisorption systems. While the two names imply two distinct types of adsorption, there is a gradual change from the physisorption to the chemisorption regime. One of the most fascinating facts about the structure of these physisorbed and chemisorbed overlayers in the submonolayer to few monolayer regime is the preponderance of the formation of long-range ordered structures. Well over 1000 two-dimensional unit cells have been documented in the literature. While only the shape, size, and orientation of the cells are known for most of them, the adsorption site and bond lengths have been determined for about 500 of them.

Surface Chemistry

B. The Ordering of Adsorbed Monolayers The ordering process in the adlayer is due to an interplay of the bonding with the substrate and the bonding between the adatoms or admolecules. Once a molecule adsorbs it may diffuse on the surface or remain bound at a specific site during most of its residence time. Thermal equilibration among the adsorbate and between the adsorbate and substrate atoms (i.e., adsorption) is assured if Hads and ∗ , the activation energy for bulk diffusion, are high E D(bulk) enough as compared to kT (≥10kT ). However, ordering primarily depends on the depth of the potential energy barrier that keeps an atom or molecule from hopping to a neighboring site along the surface. The activation en∗ ergy for surface diffusion, E D(surface) , is an experimental parameter that is of the magnitude of this potential energy barrier. E D∗ can be experimentally determined on well-characterized surfaces by field-ion microscopy, for example, and for Ar and W adatoms and O atoms on tungsten surfaces has the value 2, 15, and 10 kcal mol−1 , ∗ respectively. For small values of E D(surface) ordering is restricted to low temperatures, since as the temperature is increased the adsorbate becomes very mobile. As the value ∗ of E D(surface) increases, ordering cannot commence at low temperature since the adsorbate atoms need to have a considerable mean free path along the surface to find their equilibrium position once they land on the surface at a different location. Naturally, if the temperature is too high, the adsorbate will desorb or vaporize. The binding forces of adsorbates on substrates have components perpendicular and parallel to the surface. The perpendicular component is primarily responsible for the binding energy ( Hads ), while the parallel component often determines the binding site on the surface. The binding site may also be affected by adsorbate–adsorbate interactions, which produce ordering within an overlayer. These interactions may be subdivided into direct adsorbate– adsorbate interactions (not involving the substrate at all) and substrate mediated interactions. The latter are complicated many-atom interactions, for example dipole–dipole interactions. The adsorbate–adsorbate interactions may be repulsive; they always are repulsive at sufficiently small adsorbate– adsorbate separations. At larger separations they may be attractive, giving rise to the possibility of island formation. They may also be oscillatory, moving back and forth between attractive and repulsive as a function of adsorbate– adsorbate separation, with a period of several angstroms giving rise, for example, to non-close-packed islands. Except for the strong repulsion at close separations, the adsorbate–adsorbate interactions are usually weak compared to the adsorbate–substrate interactions, even when we consider only the components of the forces parallel

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to the surface. In the case of chemisorption, where the adsorbate–substrate interaction dominates, the adsorbates usually choose an adsorption site that is independent of the coverage and of the overlayer arrangement (the positions the other adsorbates choose). This adsorption site is usually that location which provides the largest number of nearest substrate neighbors, which is independent of the position of other adsorbates. Adsorbates with these properties normally do not accept close packing; the substrate controls the overlayer geometry and imposes a unique adsorption site. Close packing of an adsorbate layer is also observed. In this case the overlayer chooses its own lattice (normally a hexagonal close-packed arrangement) with its own lattice constant independent of the substrate lattice and results in the formation of incommensurate lattices. In this case no unique adsorption site exists: each adsorbate is differently situated with respect to the substrate. This situation is especially common in the physisorption of rare gases. Their relatively weak adsorbate–substrate interactions allow the adsorbate–adsorbate interactions to play the dominant role in determining the overlayer geometry. The chemisorption case is exemplified by oxygen and sulfur on metals; the physisorption case by krypton and xenon on metals and graphite. Intermediate cases exist. Although undissociated CO on metals is not physisorbed but chemisorbed, it sometimes produces incommensurate close-packed hexagonal overlayers. 1. The Effect of Temperature on Ordering Temperature has a major effect on the ordering of adsorbed monolayers: all of the important ordering parameters (the rates of desorption and surface and bulk diffusion) are exponential functions of the temperature. The influence of temperature on the ordering of C3 –C8 saturated hydrocarbons on the Pt(111) crystal face is shown in Fig. 11. At the highest temperatures, adsorption may not take place, since under the exposure conditions the rate of desorption is greater than the rate of condensation of the vapor molecules. As the temperature is decreased, the surface coverage increases and ordering becomes possible. First, one-dimensional lines of molecules form; then at lower temperatures ordered two-dimensional surface structures form. Not surprisingly, the temperatures at which these ordering transitions occur depend on the molecular weights of the hydrocarbons, which also control their vapor pressure, heats of adsorption, and activation energies for surface diffusion. As the temperature is further decreased, multilayer adsorption may occur and epitaxial growth of crystalline thin films of hydrocarbon commences. Figure 11 clearly demonstrates the controlling effect of temperature on the ordering and the nature of ordering of

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FIGURE 11 Monolayer and multilayer phases of the n-paraffins C3 to C8 on Pt(111) and the temperatures at which they are observed at 10−7 torr.

the adsorbed monolayer. Although changing the pressure at a given temperature may be used to vary the coverage by small amounts and thereby change the surface structures in some cases, the variation of temperature has a much more drastic effect on ordering. Temperature also markedly influences chemical bonding to surfaces. There are adsorption states that can only be populated if the molecule overcomes a small potential energy barrier. The various bond-breaking processes are similarly activated. The adsorption of most reactive molecules on chemically active solid surfaces takes place without bond breaking at sufficiently low temperatures. As the temperature is increased, bond breaking occurs sequentially until the molecule is atomized. Thus, the chemical nature of the molecular fragments will be different at various temperatures. There is almost always a temperature range for the ordering of intact molecules in chemically reactive adsorbate–substrate systems. It appears that for these systems ordering is restricted to low temperatures below 150 K, and consideration of surface mobility becomes, perhaps, secondary. 2. The Effect of Surface Irregularities on Ordering A solid surface exhibits a large degree of roughness on a macroscopic scale. Therefore, it is to be expected that if nucleation is an important part of the ordering process, surface roughness is likely to play an important role in preparing ordered surface structures. The transformation

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temperature or pressure at which one adsorbate surface structure converts into another can also be affected by the presence of uncontrolled surface irregularities (surface defects). Other causes that could influence ordering are the presence of small amounts of surface impurities that block nucleation sites or interfere with the kinetics of ordering, or impurities below the surface that are pulled to the surface during adsorption and ordering. The effect of surface irregularities on ordering can be investigated in a more controlled way using stepped crystal surfaces. In general, the smaller the ordered terrace between the steps, the stronger the effect of steps on ordering. For instance, the ordering of small molecular adsorbates on a high-Miller-index Rh(S)-[6(111) × (100)] is largely unaffected by the presence of steps whereas on the Rh(S)[3(111) × (111)] the ordering is influenced by the higher step density. Steps can also affect the nucleation of ordered domains. It is frequently observed on W and Pt stepped surfaces that when two or three equivalent ordered domains may form in the absence of steps, only one of the ordered domains grows in the presence of steps. 3. The Effect of Coadsorbates on Ordering It has been found that although certain molecules may not order when present on their own on a surface they can be induced to order by coadsorption of either carbon monoxide or nitric oxide. For example Table I summarizes the ordered structures that have been observed by the coadsorption of alkylidynes, acetylene, aromatics, and alkalis with CO on Rh(111). At low temperature both Na and ethylidyne form (2 × 2) overlayers on Rh(111) but with increasing temperature begin to disorder. If CO is coadsorbed then the adsorbates can be reordered into a c(4 × 2) unit cell. It is thought that the nature of this type of ordering process is due to adsorbate–adsorbate interactions: A molecule that might not otherwise order due to weak adsorbate–adsorbate interactions is ordered by coadsorbTABLE I Ordered Structures Induced by CO on Rh(111) Type of molecule

LEED pattern

System

Alkylidynes

c(4 × 2) √ √ (2 3 × 2 3)R30◦ c(4 × 2)

3CCH2 CH3 + CO C2 H2 + CO

Acetylene Aromatics

Alkalis

(3 × 3) (3 × 3) √ c(2 3 × 4)Rect √ c(2 3 × 4)Rect √ ( 3 × 7)Rect c(4 × 2)

CCH3 + CO

C6 H5 F + 2CO C6 H6 + 2CO C6 H6 F + CO C6 H6 + CO Na + 7CO Na + CO

ing a molecule such as CO, which has interactions strong enough to induce ordering in the overlayer. Similar phenomena have been observed on Pt(111), and it is thought that this coadsorbate-induced ordering may prove to be a very general phenomenon. C. Ordered Adsorbate Structures As mentioned in the introduction to this section, well over 1000 ordered adsorbate structures have been observed with LEED. A full listing and discussion of these structures is outside the scope of this article. Instead, one example of each of the three following categories of adsorption are presented to give an illustrative indication of the types of structures found: (i) an ordered monolayer of atoms, (ii) an ordered organic monolayer, and (iii) an ordered molecular monolayer. First, a few generalities of the ordered adsorbate structures are discussed, based on the large number of LEED observations: the so-called “rules of ordering”: 1. The rule of close-packing. Adsorbed atoms or molecules tend to form surface structures characterized by the smallest unit cell permitted by the molecular dimensions and adsorbate–adsorbate and adsorbate–substrate interactions. They prefer close-packing arrangements. Large reciprocal unit meshes are uncommon and the most frequently observed meshes are the same size as the substrate mesh, i.e., (1 × 1) or are approximately √ √ twice as large, e.g., (2 × 2), c(2 × 2), (2 × 1), ( 3 × 3). 2. The rule of rotational symmetry. Adsorbed atoms or molecules form ordered structures that have the same rotational symmetry as the substrate surface. If the surface unit mesh has a lower symmetry than the substrate, then domains of the various possible mesh orientations are to be expected on different areas of the surface with a resulting increase in symmetry. 3. The rule of similar unit cell vectors. Adsorbed atoms as molecules in monolayer thickness tend to form ordered surface structures characterized by unit cell vectors closely related to the substrate unit cell vectors. The surface structure bears a closer resemblance to the substrate structure than to the structure of the bulk condensate. These are not hard-and-fast rules but rather are generalizations of a great many systems. 1. Ordered Atomic Monolayers Some important conclusions can be drawn from the known structures of atomic adsorbates on single-crystal surfaces. First, the adsorbed atoms tend to occupy sites where they are surrounded by the largest number of substrate atoms

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(largest coordination number). This site is usually the one that the bulk atoms would occupy in order to continue the bulk lattice into the overlayer. The tendency toward occupying the site with the largest coordination number during adsorption on metals holds independently of the crystallographic face of a given metal, the metal for a given crystallographic face, and the adsorbate for a given substrate. Second, the adsorbed atom–substrate atom bond lengths are similar to the bond lengths in organometallic compounds that contain the atom pairs under consideration. The most common adsorption geometries are displayed in Fig. 12. The threefold hollow sites on the fcc(111) and hcp(0001) and bcc(110) are shown both in top and side views. Similarly, the fourfold hollow sites on the fcc(100) and bcc(100) crystal faces are shown. Finally, the center, long-bridge, and short-bridge sites on the fcc(100) crystal face and the location of atoms in an underlayer in the hcp(0001) crystal face are also displayed. In addition to the situations discussed, there exist some unique atomic adsorbate geometries. For example, small atoms such as nitrogen and hydrogen often prefer to sit below the surface, as in the case of titanium single-crystal surfaces. Also in the presence of strong chemical inter-

FIGURE 13 Structure of the p (2 × 2) and c (2 × 2) sulphur overlayers on Ni(100).

actions there may be a rearrangement of the substrate layer (an adsorbate-induced reconstruction). One example is oxygen on the Fe(100) crystal face. As an example of ordered atomic adsorption Fig. 13 portrays the two structures of sulfur on Ni(100). At a coverage of one-quarter of a monolayer of S, a p(2 × 2) overlayer is formed, and at one-half of a monolayer a c(2 × 2) structure is observed. In both cases the S sits in the fourfold hollow site (highest coordination). A LEED intensity analysis has been performed for both structures, and within experimental error the bond lengths are the same for both structures.

FIGURE 12 Top and side views (in top and bottom sketches of each panel) of adsorption geometries on various metal surfaces. Adsorbates are drawn shaded. Dotted lines represent clean surface atomic positions; arrows show atomic displacements due to adsorption.

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2. Ordered Molecular Monolayers Molecules adsorbed on surfaces may retain their basic molecular identity, bonding as a whole to the substrate. They may dissociate into their constituent atoms, which bond individually to the substrate. Alternatively, molecules may break up into smaller fragments which become largely independent or recombine into other configurations. There are also cases of intermediate character where relatively strong bonding distorts the molecule. One example of ordered molecular monolayers is CO on Pd(111). The (111) surface of fcc metals is the closepacked plane and shows similar ordering for adsorbed √ √CO for a variety of transition metals. That is the ( 3 × 3)R30◦ structure, formed at a coverage of one-third of a monolayer on the (111) faces of Pd, Ni, Pt, Ir, Cu, and Rh. This similarity in ordering is probably due to their surfaces being rather smooth with respect to variations in the CO adsorption energy. Smaller diffusion barriers between different adsorption sites are to be expected, and for large coverages repulsive interactions will be mainly responsible for the arrangement of the adlayer. Figure 14 √ √ shows a schematic representation of this ( 3 × 3)-R30◦ CO structure on Pd(111) and the corresponding observed

LEED pattern. A LEED structural analysis has not been performed for this structure, but supporting evidence using infrared spectroscopy indicates that the CO molecules sit in the threefold hollow sites. As the coverage  is increased to one-half of a monolayer the LEED pattern transforms to a c(4 × 2). In this structure the CO molecules all sit in twofold bridge sites. If the adsorption takes place at low temperature (90 K), incrcasing the CO coverage beyond  = 0.5 leads to the appearance of a series of LEED patterns arising from hexagonal superstructures, which by a continuous compression and rotation of the c(4 × 2) unit cell lead to a (2 × 2) coincidence pattern at a coverage of  = 0.75. These transformations are also shown schematically in Fig. 14.

3. Ordered Organic Monolayers The adsorption characteristics of organic molecules on solid surfaces are important in several areas of surface science. The nature of the chemical bonds between the substrate and the adsorbate and the ordering and orientation of the adsorbed organic molecules play important roles in adhesion, lubrication, and hydrocarbon catalysis.

FIGURE 14 Schematic representation of the CO on Pd(111) system. Structure models and observed LEED structures for the various CO coverages  are shown.

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There are many examples in the literature of ordered structures observed by LEED, but only a few of these structures have been calculated from the diffraction beam intensities. However, the ordering characteristics and size and orientation of the unit cells have been determined from the geometry of the LEED patterns. By studying the systematic variation of their shape and bonding characteristics correlations can be made between these properties and their interactions with the metal surfaces. Examples of ordered organic monolayers are normal paraffins on platinum and silver (111) surfaces. If straightchain saturated hydrocarbon molecules from propane (C3 H8 ) to octane (C8 H18 ) are deposited from the vapor phase onto Pt or Ag (111) between 100 and 200 K ordered monolayers are produced. As the temperature is decreased a thick crystalline film can condense. The paraffins adsorb with their chain axis parallel to the platinum substrate, and their surface unit cell increases smoothly with increasing chain length as shown in Fig. 15. Multilayers condensed on top of the ordered monolayers maintain the same orientation and packing found in the monolayers. The monolayer structure determines the growth orientation and the surface structure of the growing organic crystal. This phenomenon is called pseudomorphism, and as a result, the surface structures of the growing organic crystals do not correspond to planes in their reported bulk crystal structures.

FIGURE 15 Observed surface unit cells for n-paraffins on Pt(111).

D. Vibrational Spectroscopy Vibrational frequencies have been used for many years by chemists to identify bonding arrangements in molecules. Each bond has its own frequency, so the vibrational spectrum yields information on the molecular structure. This same information can now be obtained when molecules are adsorbed on single-crystal surfaces and, when combined with another surface-structure-sensitive technique (e.g., LEED), gives a very powerful combination of surface-structure determination. Vibrational spectroscopy also provides significant information on the identity of the surface species; its geometric orientation; the adsorption site; the adsorption symmetry; the nature of the bonding involved; and, in some cases, bond lengths, bond angles, and bond energies. For example, if CO is adsorbed and we observe the C O stretching mode the adsorption is molecular, whereas if the individual modes of metal-C and metal-O are observed then dissociation has taken place. In addition, each of these vibrational modes (C, O, and CO) has a different frequency for each bonding site. The intensities also relate to the concentration of each species on the surface. Electrons scattering off surfaces can lose energy in various ways. One of these ways involves excitation of the vibrational modes of atoms and molecules on the surface. The technique to detect vibrational excitation from surfaces by incident electrons is called high-resolution electron energy loss spectroscopy (EELS). This is the most common type of vibrational spectroscopy used for studying surface–absorbate complexes on single-crystal surfaces. Experimentally, a highly monoenergetic beam of electrons is directed toward the surface, and the energy spectrum and angular distribution of electrons backscattered from the surface is measured. In a typical experiment the kinetic energy of the incident electron beam is in the range of 1–10 eV. Under these conditions the electrons penetrate only the outermost few layers of the crystal, and the backscattered electrons contain only surface information. The incident electrons, monochromatized typically between 3 and 10 meV (∼25–80 cm−1 , 1 meV = 8.065 cm−1 ) and with energy E i , can lose energy hω upon exciting a quantized vibrational mode. These backscattered electrons of energy E i − hω produce the vibrational spectrum. There are several designs of electron monochromator and electron energy analyzers for performing EELS, and one of the most common designs, that of a single-pass 127◦ cylindrical electrostatic deflector, is shown in Fig. 16. A typical EELS spectrum, that of CO on Rh(111) is shown in Fig. 17. The sensitivity of EELS in detecting submonolayer quantities of adsorbates on the sample depends on the particular parameters of the spectrometer, the sample, and the

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FIGURE 16 Schematic diagram of an EELS spectrometer of the single-pass 127◦ cylindrical electrostatic deflector type.

Surface Chemistry

adsorbate. However, typical sensitivity is quite high due in part to the high inelastic electron cross section. A detection limit of ∼10−4 monolayers can be achieved for a strong dipole scatterer such as CO. In addition, unlike many other surface spectroscopies, EELS is also capable of detecting adsorbed hydrogen, although at a lower sensitivity (typically 10−1 –10−2 monolayers). It is a nondestructive technique and can be used to explore the vibrational modes of weakly adsorbed species and those susceptible to beam damage, such as hydrocarbon overlayers. The spectral range accessible with high-resolution EELS is quite large. Typical experiments examine between 200 and 4000 cm−1 , but much larger regions can be analyzed. Vibrational modes as far out as 16,000 cm−1 have been examined. Besides fundamentals, energy losses due to overtones, combination bands, and multiple losses are distinguishable. A distinct advantage of EELS is that electrons can excite the vibrational modes of the surface by three different mechanisms: dipole scattering, impact scattering, and resonance scattering. By analyzing the angular dependence of the inelastically scattered electrons a complete symmetry assignment of the surface–adsorbate complex can be made. The restrictions on the adsorption system are minimal: ordered or disordered overlayers can be examined, as can either well-structured single crystal samples or optically rough surfaces. Hence, chemisorption on evaporated films can be studied, as can the nature of metal overlayer– semiconductor interactions. In addition, coadsorbed atoms and molecules can be studied without difficulty. The major disadvantage of EELS, especially compared to optical techniques, is the relatively poor instrumental resolution, which usually varies between 3 and 10 meV (25–80 cm−1 ). The spectral resolution hinders assignment of vibrations due to individual modes, although peak assignments can be made to within 10 cm−1 . The high sensitivity of EELS coupled with the advantages discussed above has encouraged rapid development and use of this technique, despite resolution limitations, such that it has now been used to study hundreds of adsorptions systems. As an example of the type of surface chemistry that can be followed using EELS, Fig. 18 shows a series of EELS spectra of the adsorption and thermal decomposition of ethylene on Rh(111).

III. THERMODYNAMICS OF SURFACES FIGURE 17 Electron energy loss spectrum of CO adsorbed on Rh(111). The loss peak at 468 cm−1 is due to the Rh–CO symmetric stretch, and that at 2036 cm−1 is due to the C–O symmetric stretch. The spectrum was recorded at a resolution of 30 cm−1 .

A. Introduction The environment of atoms in a surface is substantially different to that of atoms in the bulk of the solid. Surface

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Therefore, E S is the excess of total energy that the solid has over E 0 , which is the energy that the system would have if the surface were in the same thermodynamic state as the interior. The other surface thermodynamic functions are defined similarly, for example, the specific surface free energy G S is given by the following: G S = H S − T SS, (5) S S where H and S are the specific surface enthalpy and entropy, respectively. B. Surface Tension in a One-Component System Creating a surface involves breaking chemical bonds and removing neighboring atoms, and this requires work. Under conditions of constant temperature and pressure at equilibrium, the surface work δW S is given by the following: S δWT,P = d(G S A), (6) where A is the increase in the surface area. If G S is independent of the surface area, surface work is as follows: S δWT,P = G S d A.

FIGURE 18 EELS spectra of the adsorption and decomposition of ethylene (C2 H4 ) on Rh(111) at (a) 77 K, (b) 220 K, and (c) 450 K.

atoms are surrounded by fewer nearest neighbors than bulk atoms, and these neighbors are not distributed evenly around the surface atoms. An atom in the interior experiences no net forces, but these forces become unbalanced at the surface. Consequently the thermodynamic parameters used to describe surfaces are defined separately from those that characterize the bulk phase. The specific surface energy E S , the energy per surface area, is related to the total energy E by the following equation: E = NE 0 + AE S ,

(4)

where A is the surface area of a solid composed of N atoms, and E 0 is the energy of the bulk phase per atom.

(7)

In a one-component system the specific surface free energy, G S , is frequently called the surface tension or surface pressure and is denoted by γ . Here γ may be viewed as a pressure along the surface opposing the creation of new surface. It has dimensions of force per unit length (dynes per centimeter, ergs per square centimeter, or newtons per meter). The surface tension γ for an unstrained phase is also equal to the increase of the total free energy of the system per unit increase of the surface area as follows:   ∂G S γ =G = . (8) ∂ A T,P The free energy of formation of a surface is always positive, since work is required in creating a new surface, which increases the total free energy of the system. In order to minimize their free energy solids or liquids assume shapes in equilibrium with the minimum exposed surface area as possible. For example, liquids tend to form a spherical shape and crystal faces which exhibit the closest packing of atoms tend to be the surfaces of lowest free energy of formation and thus the most stable. Surface tension is one of the most important thermodynamic parameters characterizing the condensed phase. Table II lists selected experimentally determined values of surface tensions of liquids and solids that were measured in equilibrium with their vapor. Comparing the surface tension values of metals and oxides in Table II it can be seen that oxides have in general

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Surface Chemistry TABLE II Selected Values of Surface Tension of Solids and Liquids Material He (1) N2 (1) Ethanol (1) Water Benzene n-Octane Carbon tetrachloride Bromine W (s) Nb (s) Au (s) Ag (s) Ag (l) Fe (s) Fe (l) Pt (s) Cu (s) Ni (s) Hg (l) NaCl (s) KCl (s) CaF2 (s) MgO (s) SiO2 (s) Al2 O3 (s) Polytetrafluoroethylene Polyethylene Polystyrene Poly(vinyl chloride)

γ (ergs cm−2 )

T (◦ C)

0.308

−270.5

9.71 22.75 72.75 28.88 21.80 26.95 41.5 2900 2100 1410 1140 879 2150 1880 2340 1670 1850 487 227 110

−195 20 20 20 20 20 20 1727 2250 1027 907 1100 1400 1535 1311 1047 1250 16.5 25 25

450 1200

−195 25

307

1300

690 18.5 31 33 39

2323 20 20 20 20

a low surface tension. Therefore, a reduction in the total free energy of the system can be achieved by oxidation of the surface and a uniform oxide layer covering the surface is expected under conditions near thermodynamic equilibrium. Similarly, deposition and growth of a metal film on a metallic substrate of higher surface tension should yield a uniform layer that is evenly spread to completely cover the substrate surface. Likewise, a very poor spreading of the film is expected on deposition of a metal of high surface tension on a low-surface-tension substrate. This latter condition results in “island growth” and the deposited highsurface-tension metal will grow as whiskers to expose as much of the low-surface-tension substrate during the growth as possible. These, of course, are surface thermodynamic predictions and may be overidden by the presence of impurities at the surface or difficulties of nucleation. Since atomic bonds must be broken to create surfaces, it is expected that the specific surface free energy will be

related to the heat of vaporization, which is related to the energy input necessary to break all the bonds of atoms in the condensed phase. In fact, it has been found experimentally that the molar surface free energy of a liquid metal can be estimated by the following: γlm = 0.15 Hvap ,

(9)

where Hvap is the heat of vaporization of the liquid, and the molar surface free energy of a solid metal is given by the following: γsm = 0.16 Hsub ,

(10)

where Hsub is the heat of sublimation of the solid. For other materials, oxides, or organic molecules, such a simple relationship does not work due to the complexity of bonding and the rearrangement or relaxation of surface atoms at the freshly created surfaces. C. Surface Tension of Multicomponent Systems In many important surface phenomena, such as heterogeneous catalysis or passivation of the surface by suitable protective coatings, the chemical composition of the topmost layer controls the surface properties and not the composition in the bulk. Thus, investigations of the physical– chemical parameters that control the surface composition are of great importance. One of the major driving forces for the surface segregation of impurities from the bulk and for the change of composition of alloys and other multicomponent systems is the need to minimize the surface free energy of the condensed phase system. The change of the total free energy of a multicomponent system can be expressed with the inclusion of the surface term as follows:  dG = S dT + V d P + γ d A + µi dn i , (11) i

where µi is the chemical potential of the ith component and dn i is the change in the number of moles of the ith component. At constant temperature and pressure, Eq. (11) can be rewritten as follows:  dG T,P = γ d A − µi dn i , (12) i

where the minus sign indicates the decrease of the bulk concentration of the ith component. Comparing this equation with Eq. (8), the surface tension γ is no longer equal to the specific surface free energy per unit area for a multicomponent system. Using simple arguments in which number of moles of the condensed phase are transferred to the freshly created surface, the Gibbs equation can be derived as follows:  dγ = −S S dT = i dµi , (13)

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where i is the excess number of moles of compound i at the surface. Just like the free energy relations for bulk phases, the Gibbs equation predicts changes in surface tension as a function of experimental variables such as temperature and surface concentration of various components. As a result of the Gibbs equation, the surface composition in equilibrium with the bulk for a multicomponent system can be very different from the bulk composition. As an example we discuss the surface composition of an ideal binary solution. For such a solution at a constant temperature the Gibbs equation can be expressed as follows:



 (l + m) X 2b (γ1 − γ2 )a X 2S exp = exp RT RT X 1S X 1b ×



X 1b

2

 2 l  S 2  S 2 , − X 2b + X2 − X1 RT (17)

where  is the regular solution parameter and is directly related to the heat of mixing Hm by the following: =

Hm  . X 1b l − X 1b

where γ1 is the surface tension of the pure component and a the surface area occupied by one mole of component 1. Perfect behavior is assumed, i.e., the surface areas occupied by the molecules in the two different components are the same (a1 = a2 = a); X 1S and X 1b are the atom fractions of component 1 in the surface and in the bulk, respectively. It is also assumed that the surface consists of only the topmost atomic layer. For a two-component system, Eq. (15) can be rewritten in the following form:  X 1S X 1b (γ2 − γ1 )a = b exp , (16) RT X 2S X2

Here l is the fraction of nearest neighbors to an atom in the plane and m is the fraction of nearest neighbors below the plane containing the atom. In this approximation the surface composition becomes a fairly strong function of the heat of mixing, its sign, and its magnitude in addition to the surface tension difference and temperature. Auger electron spectroscopy (AES) and ion scattering spectroscopy (ISS) are two experimental techniques which are most frequently used for quantitative determination of the surface composition. Figure 19 shows the surface atom fraction of gold, determined by AES and ISS, plotted as a function of the bulk atom fraction for the Ag–Au system. The solid line gives the calculated surface composition using the regular solution model and the dashed line indicates the curve that would be obtained in the absence of surface enrichment. The regular solution model appears to overestimate somewhat the surface segregation in this case, although the surface is clearly enriched in silver. Table III lists several binary alloy systems that have been investigated experimentally by AES or ISS and

where X 1S , X 2S , X 1b , X 2b have their meaning defined above; γ1 and γ2 are the surface tensions of the pure components; and the other symbols have their usual meaning. From Eq. (16), it can be seen that the component that has the smaller surface tension will accumulate on the surface. Equation (16) also predicts that the surface composition of ideal solutions should be an exponential function of temperature. While the bulk composition of a multicomponent system is little affected by temperature, the surface concentration of the constituents may change markedly. The surface segregation of one of the constituents becomes more pronounced the larger the difference in surface tensions between the components that make up the solution. Surface segregation is expected to be prevalent for metal solutions, since metals have the highest surface tensions. In reality, however, metallic alloys are not ideal solutions since they have some finite heat of mixing. In such a case the surface composition can be approximated in the regular solution monolayer approximation

FIGURE 19 Surface phase diagram of Au–Ag alloy.

dγT = −1 dµ1 − 2 dµ2

(14)

and it has been shown that the surface tension of component 1 in an idea dilute solution is given by the following:    γ = γ1 + (RT /a) ln X 1S X 1b , (15)

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Predicted regular solution

Ag–Pd Ag–Au Au–Pd Ni–Pd Fe–Cr Au–Cu

Ag Ag Au Pd Cr Cu

Cu–Ni Au–Ni Au–Pt Pb–In Au–In Al–Cu Pt–Sn Fe–Sn Au–Sn

Cu Au Au Pb In Al Sn Sn Sn

Experimental Ag Ag Au Pd Cr Au, none, or Cu depending on composition Cu Au Au Pb In Al Sn Sn Sn

the segregating components that were experimentally observed and also predicted by the regular solution model. The agreement is certainly satisfactory. It appears that for binary metal alloy systems that exhibit regular solution behavior there are reliable methods to predict surface composition. So far we have discussed the surface composition of multicomponent systems that are in equilibrium with their vapor or in which clean surface–bulk equilibrium is obtained in ultrahigh vacuum. In most circumstances, however, the surface is covered with a monolayer of adsorbates that frequently form strong chemical bonds with the surface atoms. This solid–gas interaction can markedly change the surface composition in some cases. For example, carbon monoxide, when adsorbed on the surface of a Ag–Pd alloy, forms much stronger bonds with Pd. While the clean surface is enriched with Ag, in the presence of CO, Pd is attracted to the surface to form strong carbonyl bonds. When the adsorbed CO is removed, the composition returns to its original Ag-enriched state. Nonvolatile adsorbates, such as carbon or sulfur, may have a similar influence on the surface composition as long as their bonding to the various constituents of the multicomponent system is different. Adsorbates should therefore be viewed as an additional component of the multicomponent system. A strongly interacting adsorbate converts a binary system to a ternary

system. As a result, the surface composition may markedly change with changing ambient conditions. The mechanical properties of solids, embrittlement, and crack propagation, among others, depend markedly on the surface composition. These studies indicate that the surface composition and the mechanical properties of structural steels may change drastically when the ambient conditions are changed from reducing to oxidizing environments. D. Equilibrium Shape of a Crystal or a Liquid Droplet In equilibrium the crystal will take up a shape that corresponds to a minimum value of the total surface free energy.  In order to have the equilibrium shape, the integral γ d A over all surfaces of the crystal must be a minimum. Crystal faces that have high atomic density have the lowest surface free energy and are therefore most stable. The plot of the surface free energy as a function of crystal orientation is called the γ plot. Solids and liquids will always tend to minimize their surface area in order to decrease the excess surface free energy. For liquids, therefore, the equilibrium surface becomes curved where the radius of curvature will depend on the pressure difference on the two sides of the interface and on the surface tension as follows: (Pin − Pext ) = 2γ /r,

(18)

where Pin and Pext are the internal and external pressures, respectively, and r is the radius of curvature. In equilibrium a pressure difference can be maintained across a curved surface. The pressure inside the liquid drop or gas bubble is higher than the external pressure, because of the surface tension. The smaller the droplet or larger the surface tension, the larger is the pressure difference that can be maintained. For a flat surface r = ∞, and the pressure difference normal to the interface vanishes. Let us now consider how the vapor pressure of a droplet depends on its radius of curvature r . We obtain the following: ln (P/P0 ) = 2γ Vin /RT r,

(19)

where Vin is the internal volume. This is the well-known Kelvin equation for describing the dependence of the vapor pressure of any spherical particle on its size. Small particles have higher vapor pressures than larger ones. Similarly, very small particles of solids have greater solubility than large particles. If we have a distribution of particles of different sizes, we will find that the larger particles will grow at the expense of the smaller ones. Nature’s way to avoid the sintering of small particles that would occur according to the Kelvin equation is to produce a system

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with particles of equal size. This is the world of colloids where particles are of equal size and therefore stabilized and are usually charged either all negative or all positive or to repel each other by long-range electrostatic forces. Milk and our blood are only two examples of systems that contain colloids. E. Adhesion and the Contact Angle Let us turn our attention to the interfacial tension, that is, the surface tension that exists at the interface of two condensed phases. Let us place a liquid droplet on a solid surface. The droplet either retains its shape and forms a curved surface or it is spread evenly over the solid. These two conditions indicate the lack of wetting or wetting of the solid by the liquid phase, respectively. The contact angle between the solid and the liquid, to a large extent, permits us to determine the interfacial tension between the solid and the liquid. The contact angle is defined by Fig. 20. If the contact angle is large ( approaching 90◦ ), the liquid does not readily wet the solid surface. If  approaches zero, complete wetting of the solid surface takes place. For  larger than 90◦ the liquid tends to form spherical droplets on the solid surface that may easily run off, i.e., the liquid does not wet the solid surface at all. Remembering that the surface tension always exerts a pressure tangentially along a surface, the surface free energy balance between the surface forces acting in opposite directions at the point where the three phases solid, liquid, and gas meet is given by the following: cos  = (γsg − γsl )/γlg .

(20)

Here γlg is the interfacial tension at the liquid–gas interface and γsg and γsl are the interfacial tensions between the solid–gas and the solid–liquid interfaces, respectively. Knowing γlg and the contact angle in equilibrium at the solid–liquid–gas interface, we can determine the difference γsg − γsl but not their absolute values. Since the wet-

FIGURE 20 Definition of the contact angle between a liquid and solid and the balance of surface forces at the contact point among the three phases (solid, vapor, and liquid).

ting ability of the liquid at the solid interface is so important in practical problems of adhesion or lubrication, there is a great deal of work being carried out to determine the interfacial tensions for different combinations of interfaces. The usefulness of a lubricant is determined by the extent to which it wets the solid surface and maintains complete coverage of the surface under various conditions of use. The strength of an adhesive is determined by the extent to which it lowers the surface free energy by adsorption on the surface. The work of adhesion is defined as follows: WAs = γ1,0 + γs,0 − γsl ,

(21)

where γ1,0 and γs,0 are the surface tensions in vacuum of the liquid and solid, respectively. In general, solids and liquids that have large surface tension form strong adhesive bonds, i.e., have large work of adhesion. The work of adhesion is in the range of 40–150 ergs/cm2 for solid–liquid pairs of various types. Organic polymers often make excellent adhesives because of the large surface area covered by each organic molecule. The adhesive energy per mole is much larger than that for adhesion between two metal surfaces or between a liquid and a solid metal because of the many chemical bonds that may be formed between the substrate and the adsorbed organic molecule. F. Nucleation Another important phenomenon that owes its existence to positive surface free energy is nucleation. In the absence of a condensed phase, it is very difficult to nucleate one from vapor atoms because the small particles that would form have a very high surface area and dispersion and, as a result, a very large surface free energy. The total energy of a small spherical particle has two major components: its positive surface free energy, which is proportional to πr 2 γ , where r is the radius of the particle, and its negative free energy of formation of the particle with volume V . The volumetric energy term is proportional to −r 3 ln (P/Peq ), where P is the pressure over the system and Peq is the equilibrium vapor pressure:     4πr 3 P G(total) = − RT ln + 4πr 2 γ , (22) 3Vm Peq where Vm is the molar volume of the forming particle. Initially, the atomic aggregate is very small and the surface free energy term is the larger of the two terms. In this circumstance a condensate particle cannot form from the vapor even at relatively high saturation (P > Peq ). Similarly, a liquid may be cooled below its freezing point without solidification occurring. Above a critical size of the spherical particles the volumetric term becomes larger and dominates since it

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FIGURE 22 One-dimensional potential energy of an adatom in a physisorbed state on a planar surface as a function of its distance z from the surface.

FIGURE 21 Free energy of homogeneous nucleation as a function of particle size.

decreases as ∼r 3 , while the surface free energy term increases only as ∼r 2 . Therefore, a particle that is larger than this critical size grows spontaneously at P > Peq . This is shown in Fig. 21. Because of the difficulty of obtaining this critical size, which involves as many a 30–100 atoms or molecules, homogeneous nucleation is very difficult indeed. To avoid this problem we add to the system particles of larger than critical size that “seed” the condensation or solidification. The use of small particles to precipitate water vapor in clouds to start rain and the use of small crystallites as seeds in crystal growth are two examples of the application of heterogeneous nucleation. G. Physical and Chemical Adsorption The concepts of physical adsorption (physisorption) and chemical adsorption (chemisorption) were introduced above. The nature of the two classifications is linked to the heat of adsorption, Hads , which is defined as the binding energy of the adsorbed species. Physical adsorption is caused by secondary attractive forces (van der Waals) such as dipole–dipole interaction and induced dipoles and is similar in character to the condensation of vapor molecules onto a liquid of the same composition. The interaction can be described by the onedimensional potential energy diagram shown in Fig. 22.

An incoming molecule with kinetic energy E k must lose at least this amount of energy in order to stay on the surface. It loses energy by exciting lattice phonons in the substrate, for example, and the molecule comes to equilibrium in a state of oscillation in the potential well of depth equal to the binding energy or adsorption energy E a = Hads . In order to leave the surface (desorb) the molecule must acquire enough energy to surmount the potential-energy barrier E a . The desorption energy is equal to the adsorption energy. The binding energies of physisorbed molecules are typically ≤15 kcal mol−1 . Chemisorption involves chemical bonding; it is similar to a chemical reaction and involves transfer of electronic charge between adsorbent and adsorbate. The most extreme form of chemisorption occurs when integral numbers of electrons are transferred, forming a pure ionic bond. More usually there is an admixture of the wave functions of the valence electrons of the molecule with the valence electrons of the substrate into a new wave function. The electrons responsible for the bonding can then be thought of as moving in orbitals between substrate and adatoms and a covalent bond has been formed. Two examples of the potential energy diagrams for chemisorption are shown in Fig. 23. Some of the impinging molecules are accommodated by the surface and become weakly bound in a physisorbed state (also called a precursor state) with binding energy E p . During their stay time in this state, electronic or vibrational processes can occur which allow them to surmount the energy barrier, and electron exchange occurs between the adsorbate and substrate. The molecule, or adatom in the case of dissociative chemisorption, now finds itself in a much deeper well; it is chemisorbed. Figure 23a shows the case in which the energy barrier for chemisorption is less than E p , so there is no overall activation energy to chemisorption. Figure 23b illustrates

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the case of adsorption isotherms for physical adsorption, to determine the surface area of the adsorbing solid. Consider a uniform surface with a number n 0 of equivalent adsorption sites. The ratio of the number of adsorbed atoms or molecules n to n 0 is defined as the coverage,  = n/n 0 . Atoms or molecules impinge on the surface from the gas phase, where they establish a surface concentration [n a ]s (molecules per square centimeter). Assuming that only one type of species of concentration [n a ]g (molecules per cubic centimeter) exists in the gas phase the adsorption process can be written as follows: FIGURE 23 One-dimensional potential energy curves for dissociative adsorption through a precursor or physisorbed state: (a) adsorption into the stable state with no activation energy and (b) adsorption into the chemisorption well with activation energy E ∗ .

the case in which there is an overall activation energy E ∗ to chemisorption. In the former case the activation energy for desorption E d is equal to the heat of adsorption, while in the latter case the heat of adsorption is given by the difference between the heat of desorption and the activation energy. The occurrence of an activation energy to chemisorption is by far the exception rather than the rule. From these considerations it is expected that to a first approximation physisorption will be nonspecific, any gas will adsorb on any solid under suitable circumstances. However, chemisorption will show a high degree of specificity. Not only will there be variations from metal surface to metal surface, as would be expected from the differences in chemistries between the metals, but also different surface planes of the same metal may show considerable differences in reactivity toward a particular gas. H. Adsorption Isotherms An adsorption isotherm is the relationship at constant temperature between the partial pressure of the adsorbate and the amount adsorbed at equilibrium. Similarly an adsorption isobar expresses the functional relationship between the amount adsorbed and the temperature at constant pressure, and an adsorption isostere relates the equilibrium pressure of the gaseous adsorbate to the temperature of the system for a constant amount of adsorbed phase. Usually it is easiest from an experimental viewpoint to determine isotherms. The coordinates of pressure at the different temperatures for a fixed amount adsorbed can then be interpolated to construct a set of isosteres, and similarly to obtain an isobaric series. Adsorption isotherms can be used to determine thermodynamic parameters that characterize the adsorbed layer (heats of adsorption and the entropy and heat capacity changes associated with the adsorption process), and in

k Agas   Asurface k

and the net rate of adsorption as F (molecules cm−2 sec−1 ) = k[n a ]g − k  [n a ]s ,

(23)



where k and k are the rate constants for adsorption and desorption, respectively. Starting with a nearly clean surface far from equilibrium, the rate of desorption may be taken as zero and Eq. (23) becomes the following: F (molecules cm−2 sec−1 ) = k[n a ]g ,

(24)

where k, derived from the kinetic theory of gases, equals α(RT /2π M)1/2 cm sec−1 , α is the adsorption coefficient, and M the molecular weight of the impinging molecules. The surface concentration [n a ]s under these conditions is the product of the incident flux F and the surface residence time τ : [n a ]s = Fτ.

(25)

Here τ is the surface residence time, given by: τ = τ0 exp( Hads /RT ).

(26)

Replacing [n a ]g by the pressure using the ideal gas law, Eq. (25) can be rewritten as follows:   αPN A Hads [n a ]s = . (27) τ0 exp (2π MRT)1/2 RT The simplest adsorption isotherm is obtained from Eq. (27), which can be rewritten as  = k  P where k  =

  αNA 1 Hads . τ exp 0 n 0 (2π MRT )1/2 RT

(28)

(29)

The coverage is proportional to the first power of the pressure at a given temperature provided that there are an unlimited number of adsorption sites available and Hads does not change with coverage. The isotherm of Eq. (28) is unlikely to be suitable to describe the overall adsorption process, but the Langmuir isotherm is a simple modification which represents a more

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real situation. The Langmuir isotherm assumes that adsorption is terminated on completion of one molecular adsorbed gas layer (monolayer) by asserting that any gas molecule that strikes an adsorbed atom must reflect from the surface. All the other assumptions used to derive Eq. (28) are maintained (i.e., homogeneous surface and noninteracting adsorbed species). If [n 0 ] is the surface concentration of a completely covered surface, the number of surface sites available for adsorption, after adsorbing [n a ]s molecules is [n 0 ] − [n a ]s . Of the total flux incident on the surface, a fraction ([n a ]s /[n 0 ])F will strike molecules already adsorbed and, therefore, be reflected. Thus, a fraction (1 − [n a ]s /[n 0 ])F of the total incident flux will be available for adsorption. Equation (25) should then be modified as follows:   [n a ]s [n a ]s = 1 − Fτ, (30) [n 0 ] which can be rearranged to give [n a ]s =

[n 0 ]Fτ [n 0 ]k P = [n 0 ] + Fτ [n 0 ] + k P

(31)

from which =

k P , 1 + k P

(32)

where k  = k/[n 0 ]. Equation (32) is the Langmuir isotherm. The adsorption of CO on Pd(111) obeys the Langmuir isotherm, and typical isotherms from this system are shown in Fig. 24. It can readily be shown that in the case of dissociative adsorption the Langmuir isotherm becomes

 2  1 P=  k 1− or =

(k  P)1/2 . 1 + (k  P)1/2

(33)

A clear weakness of the Langmuir model is the assumption that the heat of adsorption is independent of coverage. Several other isotherms have been developed which are all modifications of the Langmuir model. For example, the Temkin isotherm can be derived if a linearly declining heat of adsorption is assumed, i.e., H = H0 (1−β), where H0 is the initial enthalpy of adsorption. The isotherm is =

RT ln A P, β H0

(34)

where A is a constant related to the enthalpy of adsorption. The possibility of multilayer adsorption is envisaged in the Brunauer–Emmett–Teller (BET) isotherm. The assumption is made that the first layer is adsorbed with a heat of adsorption H1 and the second and subsequent layers are all characterized by heats of adsorption equal to the latent heat of evaporation, HL . By considering the dynamic equilibrium between each layer and the gas phase the BET isotherm is obtained, p 1 c−1 p . = + V ( p0 − p) Vm c Vm c p0

(35)

In this equation V is the volume of gas adsorbed, p the pressure of gas, p0 the saturated vapor pressure of the liquid at the temperature of the experiment, and Vm the

FIGURE 24 Adsorption isotherms for CO on Pt(111) single-crystal surfaces.

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volume equivalent to an adsorbed monolayer. The BET constant c is given by the following: c = exp(H1 − HL )/RT.

(36)

The BET equation owes its importance to its wide use in measuring surface areas, especially of films and powders. The method followed is to record the uptake of an inert gas (Kr) or nitrogen at liquid nitrogen temperature (−195.8◦ C). A plot of p/V ( p0 − p) versus p/ p0 , usually for p/ p0 up to about 0.3, yields Vm , the monolayer uptake. This value is expressed as an area by assuming that ˚ and 25.6 A ˚ for the area per molecule for nitrogen is 16.2 A krypton. In general, the BET isotherm is most useful for describing physisorption for which H1 and HL are of the same order of magnitude while the preceding isotherms are more useful for chemisorption. It is worth noting that the BET isotherm reduces to the Langmuir isotherm when H1  HL . I. Heat of Adsorption An important physical–chemical property that characterizes the interaction of solid surfaces with gases is the bond energy of the adsorbed species. The determination of bond energy is usually made indirectly by measuring the heat of adsorption (or heat of desorption) of the gas. The heat of adsorption can be determined readily in equilibrium by measuring several adsorption isotherms. The Clausius– Clapeyron equation   ∂(ln P) Hads = (37) ∂T RT 2  can be integrated to give     P1 − Hads 1 1 ln . = − P2  R T1 T2 Measuring the adsorption isotherm at two different temperatures, provided that proper equilibrium is established between the adsorbed and gas phase, yields the heat of adsorption. The heat of adsorption can also be obtained by direct calorimetry. The method most commonly used consists of measuring the temperature rise caused by the addition of a known amount of gas to a film of the metal prepared by evaporation in vacuo. This measurement will yield the differential heat of adsorption qd at the particular value of . The differential heat of adsorption is related to the isosteric heat of adsorption by the following: q = qd + RT ;

(38)

the difference is only RT, which is within experimental error.

The last, and most common, method of determining the heat of adsorption is a kinetic method called temperature programmed desorption (TPD). The method is as follows. The sample is cleaned in ultrahigh vacuum and a gas is allowed to adsorb on the surface at known pressures while the surface is kept at a fixed temperature. The sample is then heated at a controlled rate, and the pressure changes during the desorption of the molecules are recorded as a function of time and temperature. The pressure–temperature profile is usually referred to as the desorption spectrum. The desorption rate F(t) is commonly expressed as follows:   E des , (39) F(t) = v f (σ ) exp − RT where v is the preexponential factor and f (σ ) an adsorbate concentration-dependent function. The various procedures for determining these parameters are well described in the literature. Assuming that v and E des are independent of the adsorbate concentration σ and t, E des can be obtained for zeroth-, first-, and second-order desorption, respectively, as follows:   E0 ν0 E (40) = exp − R σa RTp   E1 v1 E1 (41) = exp − RTp2 α RTp   E2 E2 v2 σ exp − , = RTp2 α RTp

(42)

where Tp is the temperature at which a desorption peak is at a maximum and σ is the initial adsorbate concentration. The subscripts 0, 1, and 2 denote the zeroth-, first-, or second-order desorption processes; α is a constant of proportionality for the temperature rise with time, usually T = T0 + αt; that is, the temperature of the sample is raised linearly with time. As seen from the equations, Tp is independent of σ for the first-order desorption process. Alternatively, Tp is increased or is decreased with σ0 for the zeroth- and second-order process, respectively. Equations (40)–(42) allow us to determine the activation energy and the preexponential factor and also to distinguish between zeroth-, first-, and second-order desorption processes from the measurements of the dependence of the peak temperatures on initial adsorbate concentrations and heating rate α. A typical TPD spectrum is shown in Fig. 25. The bond energy Hbond is readily extracted from the heat of adsorption. In the case of the chemisorption of a diatomic molecule X2 onto a site on a uniform solid surface M the molecule may adsorb without dissociation

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FIGURE 26 Isoteric heat of adsorption for CO on Pd(111) crystal face as a function of coverage.

FIGURE 25 Typical thermal desorption spectra of CO from a Pt(553) stepped crystal face as a function of coverage. The two peaks are indicative of CO bonding at step and terrace sites. The higher temperature peak corresponds to CO bound at step sites.

to form MX2 . In this case, the heat of adsorption, Hads is defined as the energy needed to break the M X2 bond Hads

MX2(ads) −−→ M + X2(gas) If the molecule adsorbs dissociatively, the heat of adsorption is defined as follows: Hads

2MX(ads) −−→ 2M + X2(gas) The energy of the surface chemical bond is then given by Hbond (MX2 ) = Hads for associative adsorption or   Hads + DX2 Hbond (M X ) = 2 for dissociative adsorption, where DX2 is the dissociation energy of the X2 gas molecule. The heat of adsorption is not a constant, quantity for a particular adsorbate–substrate system; there are several factors which affect the value of Hads . First, the heat of adsorption can change markedly with the coverage  of the adsorbed pahse. An example of this is shown in Fig. 26 for CO on a Pd(111) surface. Decreasing values of Hads

with increasing adsorbate coverage are commonly observed due to repulsive adsorbate–adsorbate interactions. Second, the surface is heterogeneous by nature. There are many sites where the adsorbed species have different binding energies. Perhaps the most striking effect is that for adsorption on stepped and kinked platinum and nickel single crystal surfaces where molecules dissociate in the presence of these surface irregularities while they remain intact on the smooth low-Miller-index surfaces. If a polycrystalline surface is utilized for chemisorption studies instead of a structurally well-characterized single-crystal surface the measured Hads will be an average of adsorption at the various binding sites. In fact, even on the same crystal surface molecules may occupy several different adsorption sites with different coordination numbers and rotational symmetries, and each site may exhibit a different binding energy and therefore a different heat of chemisorption. For example, on the (111) face of fcc metals the adsorbates may occupy a three-fold site, a twofold bridge site, or an on-top site. Figure 27 shows the measured heats of adsorption of CO or single-crystal surfaces for many different transition metals while Fig. 28 shows the heats of adsorption of CO on polycrystalline transition metal surfaces. The heats of chemisorption on single-crystal planes indicate the presence of binding sites on a given surface which differ by ∼20 kcal mol−1 . It is not possible to identify one value of the heat of chemisorption of an adsorbate on a given transition metal unless the binding state is specified or it is certain that only one binding state exists. A polycrystalline surface however exhibits all the adsorption sites of the faces from which it is composed. Since these sites are present simultaneously heats of chemisorption for these surfaces represent an average of the binding energies of the different surface sites. As a result the measured heats of adsorption of Fig. 28 do not show the large structural variations that can be seen in Fig. 27. The adsorbate may also change bonding as a function of temperature as well as the adsorbate concentration. For example, oxygen may be molecularly adsorbed at low temperatures while it dissociates at higher temperatures.

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FIGURE 27 Heats of adsorption of CO on single-crystal surfaces of transition metals.

IV. ELECTRICAL PROPERTIES OF SURFACES A. Introduction Many of the physical and chemical properties of solid surfaces are directly influenced by the concentration of mobile charge carriers (electrons and diffusing ions). The concentration of these free charge carrier varies widely

FIGURE 28 Heats of adsorption of CO on polycrystal-line transition-metal surfaces.

for materials of different types. Metals, which are good conductors of electricity with resistivities in the 10−4  m range, have large free electron concentrations; almost every atom contributes one electron to the lattice as a whole. For insulators, with a resistivity of 109  m, and semiconductors with intermediate values, often less than 1 of every 106 atoms may contribute a free electron. The temperature dependence of the carrier concentration and the conductivity may be different for different materials depending on the mechanism of excitation by which the mobile charge carriers are created. Under incident radiation or bombardment by an electron beam surfaces emit photons, electrons, or both. The emission properties of solid surfaces differ widely, just as their mechanisms or relaxation after excitation by highenergy radiation differ. Many surface-sensitive experimental techniques providing information related to the electronic properties of surfaces are based on these processes, for example, Auger electron spectroscopy (AES), X-ray photoelectron spectroscopy (XPS), and ultraviolet photoelectron spectroscopy (UPS). These are discussed below. The underlying reason for the differences of the conductivity mechanisms and emission properties on the surfaces of the different materials lies in the differences in their electronic band structure. The band structure model of solids has been successful in explaining many solidstate properties, and we may apply it with confidence in studies of solid surfaces. There are many excellent textbooks on the subject of solid-state physics giving detailed descriptions of the band theory of solids, and a description is not presented here. In the following section a basic understanding of electron bands is assumed. B. The Energy Level Diagram For many purposes, in analyzing the electrical properties of metals or semiconductors, we are not concerned with the detailed shape of the electronic bands. We may conveniently represent schematically the electronic bands by straight lines where the potential energy of the electron near the top of the valence band and at the bottom of the conduction band is plotted against distance x through the crystal starting from the surface (x = 0). The energy gap represents the minimum potential energy difference between the two bands. In this type of diagram the electron energy increases upward and the energy of the positive hole increases downward, as indicated in Fig. 29. For a homogeneous crystal the bands may be horizontal, as shown in this figure. At the surface the bands may vary in energy with respect to their value in the bulk of the solid since the free carrier concentrations at the surface may be different from those in the bulk of the crystal.

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FIGURE 29 Energy-level diagram as a function of distance x from the surface (x = 0).

C. Surface Dipole and Surface Space Charge The anisotropic environment of surface atoms not only gives rise to such processes as surface relaxation and surface reconstruction but also to a redistribution of charge density. For a metal this redistribution can be explained as follows. In the bulk of a metal each electron lowers its energy by “pushing” the other electrons aside to form an “exchange correlation hole.” This attractive interaction Vexch is lost when the electron leaves the solid, so there is a sharp potential barrier Vs at the surface. In a quantum mechanical description, the electrons are not totally trapped at the surface and there is a finite probability for them to spread out into the vacuum. This is depicted in Fig. 30. This charge redistribution induces a surface dipole Vdip that modifies the barrier potential. The work function φ (which will be discussed in detail below) is the minimum energy necessary to remove an electron at the Fermi energy E F from the metal into the vacuum. The magnitude of this induced surface dipole is different at various sites on the heterogeneous metal surface. For example, a step site on a tungsten surface has dipole of 0.37 Debye (D) per edge atom as measured by work function studies. At a tungsten adatom on the surface there is a dipole moment of 1 D. At semiconductor and insulator surfaces the separation of negative and positive charges leads to the formation of a space-charge region. This space-charge region near the surface is formed by the accumulation or depletion

FIGURE 30 Charge density oscillation and redistribution at a metal–vacuum interface.

Surface Chemistry

of charge carriers in the surface with respect to the bulk carrier concentration. Such a space charge may also be induced by the application of an external electric field or by the presence of a charged layer on the surface such as adsorbed ions or electronic surface states which act as a source or sink of electrons. The height of the surface potential barrier Vs and its distance of penetration into the bulk, d, depend on the concentration of mobile charge carriers in the surface region. It can be shown that   2εε0 Vs 1/2 d≈ , (43) en e (bulk) where ε is the dielectric constant in the solid, ε0 the permittivity of free space, and n e (bulk) the bulk carrier concentration. The higher the free carrier concentration in the material, the smaller is the penetration depth of the applied field into the medium. Using a typical value of ε = 16, for electron concentrations of 1017 cm−3 or larger, the space charge is restricted to distances on the order of one atomic layer or less. This is due to the large free carrier density screening the solid from the penetration of the electrostatic field caused by the charge imbalance. In most metals almost every atom contributes one free valence electron and since the typical atomic density is of the order of 1022 cm−3 the free carrier concentration in metals is in the range of 1020 –1022 cm−3 . Thus, Vs and d are so small that they can usually be neglected. For semiconductors, or insulators on the other hand, typical free carrier concentrations at room temperature are in the range of 1010 –1016 cm−3 . Therefore, at the surfaces of these materials, there is a space-charge barrier of appreciable height (several electron volts) and penetration depth that could extend over ˚ into the bulk. This thousands of atomic layers (≈104 A) is the reason for the sensitivity of semiconductor devices to ambient changes that affect the space-charge barrier height. There is an induced electric field at the surface under most experimental conditions due to the adsorption of gases or because of the presence of electronic surface states. The electronic and many other physicochemical properties of semiconductor and insulator surfaces depend very strongly on the properties of the space charge. For example, the conduction of free carriers across the solid or along its surface could become space-charge-limited. The rate of charge transfer from the solid to the adsorbed gas, which results in chemisorption or chemical reaction, can become limited by the transfer rate of electrons over the space-charge barrier. When the energy level diagram was introduced, it was assumed that the electron energy levels remained unchanged right to the surface (x = 0). However, the presence of the space charge (and also surface states) leads to a bending of the bands. If the surface region becomes

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FIGURE 31 Energy-level diagram (a) in the absence of any space charge and (b) with a surface space charge due to depletion of electrons in the surface region.

FIGURE 32 Potential energy diagram illustrating the work function. EF is the Fermi energy, φ is the work function, and W is the potential well bonding the conduction band electrons into the solid.

depleted of electrons it would require more energy to transfer an electron to the conduction band from, for example, the reference state E F , due to the space charge potential barrier. This is depicted schematically in Fig. 31. Conversely, it is now easier to transfer a hole to the surface since the difference between E F and E V becomes smaller. It is very likely that there is accumulation or depletion of charges at semiconductor or insulator surfaces under all ambient conditions. For surfaces under atmospheric conditions, adsorbed gases or liquid layers at the interface provide trapping of charges or become the source of free carriers. For clean surfaces in ultrahigh vacuum, there are electronic surface states that act as traps or sources of electrons and produce a space-charge layer of appreciable height. Thus, the mobile carriers from the surface layer are swept into the interior or are trapped at the surface as the space-charge layer consists dominantly of static charges, the one most frequently encountered in experimental situations. We have so far considered the space-charge layer properties only in the insulating solid, assuming that the surface layer that acts as a donor or the electron trap is of monolayer thickness. However, considering the properties of solid–liquid interfaces or semiconductor–insulator contacts, it should be recognized that the space-charge layer may extend to effective Debye lengths on both sides of the interface. This is a most important consideration when we investigate the surface properties of colloid systems or of semiconductor–electrolyte interfaces.

insulators it can be regarded as the difference in energy between an electron at rest in the vacuum just outside the solid (i.e., at the level of zero kinetic energy) and the most loosely bound electrons in the solid. Thus, the work function is evidently an important parameter in situations where electrons are removed from the solid. A schematic energy level diagram assuming the freeelectron model of a metal showing the work function is depicted in Fig. 32. From the figure it can be seen that the value of φ depends on W , the depth of the potential well bonding the conduction electrons into the solid. Here W is a bulk property determined by the attraction for its electrons of the lattice of positive ions as a whole; it has an energy of the order of a few electron volts. The origin of the work function itself can be considered as being due to the image potential of the escaping electron. Electrostatic theory shows that a charge −e outside a conductor is attracted by an image charge +e placed at the position of the optical image of −e in the conducting plane. If −e is a distance x from the plane the image force is e2 /16π ε0 x 2 . This force is experienced by the electron escaping into the vacuum and is negligible beyond 10−6 – 10−5 cm away from the surface. The image potential is a specific surface contribution to W , and a second surface contribution is the existence of a surface double layer or dipole layer. Surface atoms are in an unbalanced environment, they have other atoms on one side of them but not on the other; thus, the electron distribution around them will be unsymmetrical with respect to the positive ion cores. This leads to the formation of a double layer. Two important effects emanate from this; the work function is sensitive to both the crystallographic plane exposed and to the presence of adsorbates. The orientation of the exposed crystal face affects φ because the strength of the electric double layer depends on the density of positive ion cores which in turn will vary

D. Work Function and Contact Potential The work function of a solid is a fundamental physical property of the solid which is related to its electronic structure. It is defined as the potential that an electron at the Fermi level must overcome to reach the level of zero kinetic energy in the vacuum. In semiconductors and

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Tungsten

Molybdenum

(110) (112) (111) (001) (116)

4.68 4.69 4.39 4.56 4.39

5.00 4.55 4.10 4.40 —

from one face to another. The work function of various crystal planes of tungsten and molybdenum are listed in Table IV. It can be seen that there is more than 0.3 eV difference in work function values. This variation of work function from one crystal face to another can clearly be demonstrated using a field emission microscope (FEM). This microscope is identical in construction to the FIM described earlier. However, instead of having helium or another imaging gas in the vacuum, no gas is admitted. The potential on the sample tip is reversed so that electrons are accelerated out of it by a very high local electric field (∼ 4 × 107 V cm−1 ). The current emitted from the tip surface where the work function is φ is approximately proportional to exp(−Aφ 3/2 ) and is a very fast function of φ. The brightness observed on the fluorescent screen is a function of the value of φ at that place on the tip, and the FEM image will consist of darker and brighter areas, the brightness depending on the work function of each crystal face exposed. An image is shown in Fig. 33 which is the FEM image of

FIGURE 33 Field emission pattern of a tungsten tip. The (011) plane is in the center. (Courtesy of Lawrence Berkeley Laboratory.)

a tungsten tip. The changes in φ produced by adsorbed atoms or molecules can be followed in the FEM. The work function of a solid is also sensitive to the presence of adsorbates. In fact, in virtually all cases of adsorption the work function of the substrate either increases or decreases; the change being due to a modification of the surface dipole layer. The formation of a chemisorption bond is associated with a partial electron transfer between substrate and adsorbate and the work function will change. Two extreme cases are (i) the adsorbate may only be polarized by the attractive interaction with the surface giving rise to the build up of a dipole layer, as in the physisorption of rare gases on metal surfaces; and (ii) the adsorbate may be ionized by the substrate, as in the case of alkali metal adsorption on transition metal surfaces. If the adsorbate is polarized with the negative pole toward the vacuum the consequent electric fields will cause an increase in work function. Conversely, if the positive pole is toward the vacuum then the work function of the substrate will decrease. The work function is a rather complicated (and not fully understood) function of the surface composition and geometry. Nevertheless, general systematic observations of φ are quite helpful. For example, the sign of φ for atomic adsorption is mostly that implied by the magnitude of the ionization potential, electron affinity, or dipole moment of the adsorbates as one would expect. The most common usage of work function changes in surface chemistry is in the monitoring of the various stages of adsorption as a function of coverage. Often the work function change will go through a maximum or minimum at particular coverages corresponding to the completion of an ordered atomic arrangement. Experimentally, the most accurate way of measuring changes in work function is by the Kelvin method, which uses a vibrating probe as a variable capacitor. A contact potential difference is set up between two conductors connected externally and the sample and a reference electrode form a parallel plate condenser. The distance between the two is periodically varied, thus generating an alternating current in the connecting wire. If a voltage source is placed in the connecting circuit just balancing out the contact potential difference, no current will flow. Once this situation has been achieved for the clean surface, a change in work function due to adsorption is simply the additional voltage which needs to be applied to compensate the change and keep the current zero. Accuracies of φ to within ±1 meV are obtainable. Intimately linked to the concept of work function is the process of thermionic emission. Thermionic emission is, as the name suggests, the phenomenon whereby electrons are ejected from a metal when it is heated in vacuum. The electrons that require the least amount of thermal energy to overcome their binding energy in the solid and

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escape are those in the high-energy tail of their equilibrium distribution in the metal. Thermionic emission is the most frequently used method to produce electron beams, for instance, in oscilloscope tubes and electron microscopes. Refractory metals (e.g., W) have traditionally been used as filaments in electron guns, mainly due to the fact that they can be heated to high temperatures and thus produce a relatively intense thermionic current. Since the work function of W is relatively high W filaments are often coated with a metal for lower work function, for example, Th (φ = 2.7 eV) to enable them to be operated at lower temperature for the same current thereby extending their lifetime. E. Surface States In a bulk solid the infinite array of ion cores in crystallographic sites leads to a potential that varies in a three-dimensionally periodic manner. The solutions to Schr¨odinger’s equation for such a potential lead to allowed energy bands, which are occupied by the electrons in the solid, and to particular values of the wave vector k of the electron where no traveling-wave solutions exist. The absence of eigenstates for these values of k leads to the band gaps in the electronic structure of the solid. The solid, however, is not infinite but is bounded by surfaces. In turn, surface atoms have fewer nearest neighbors and are in an asymmetric environment. The introduction of such a discontinuity at the surface perturbs the periodic potential and gives rise to solutions of the wave equation that would not have existed for the infinite crystal. These are derived by using appropriate boundary conditions to terminate the crystal and are called surface-state wave functions. These special solutions are waves which can travel parallel to the surface but not into the solid. They are localized at the surface and can have energies within the band gap of the bulk band structure. These states can trap electrons or release them into the conduction band. The concentration of electronic surface states in clean surfaces can be equal to the concentration of surface atoms (∼1015 cm−2 ). Impurities or adsorbed gases can reduce the surface state density. One important consequence of the presence of electronic surface states is that the electron bands are modified at the surface even in the absence of a space charge or electron acceptor or donor species (such as adsorbed gases). The shape of the conduction band at the surface of an intrinsic semiconductor in the presence of electrondonor and electron-acceptor surface states is shown in the energy level diagrams in Figs. 34a and 34b. Surface states can be associated not only with the termination of a three-dimensional potential at a perfect clean bulk exposed plane but also with changes in the potential

FIGURE 34 Energy-level diagrams for an intrinsic semiconductor in the presence of (a) electron–donor or (b) electron–acceptor surface states.

due to relaxation, reconstruction, structural imperfections, or adsorbed impurities. If the charge associated with any of these surface states is different from the bulk charge distribution then band bending will occur. Surface states can be observed, for example, using ultraviolet photoelectron spectroscopy, which is discussed below. F. Electron Emission from Surfaces The most important methods of analyzing the surface electronic and chemical composition involve energy analysis of electrons emitted from a surface during its bombardment with electrons, ultraviolet photons, or X-ray photons. For example, part of the experimental verification of the band theory of metals comes from the measured intensity and energy distribution of electrons emitted under excitation by photons. It should be remembered that we have already mentioned two ways in which electrons can be emitted from surfaces; (i) by applying a very high electric field (∼107 V cm−1 ) which pulls electrons from the surface, as used in FEM, and (ii) by heating the solid as in thermionic emissions. Before discussing the two major electron emission techniques from surfaces, photoelectron spectroscopy and Auger electron spectroscopy (AES), it is pertinent to briefly discuss the surface sensitivity of the interaction of electrons with solids. Figure 35 shows the mean free path of electrons in metallic solids as a function of the electron energy. This curve is often called the “universal curve,” and shows a broad minimum in the energy range between 10 and 500 eV with the corresponding mean free ˚ Electron emission from solids path on the order of 4–20 A. with energy in this range must originate from the top few atomic layers. Therefore, all experimental techniques involving the incidence and/or convergence from surfaces of electrons having energy between 10 and 500 eV are surface sensitive. For incident electrons of higher energy (1–5 kV) the surface sensitivity can be enhanced by having the electron beam impinging on the surface at grazing

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FIGURE 35 Universal curve for the electron mean free path as a function of electron kinetic energy. Dots indicate individual measurements.

incidence. Photons have a much larger penetration depth into the solid due to the much smaller scattering cross section. However, electrons created by excitation below a few atomic layers from the surface cannot escape due to inelastic scattering within the solid. If a monoenergetic beam of electrons of energy E p strikes a metal surface then a typical plot of the number of scattered electrons N (E) as a function of their kinetic energy E is shown in Fig. 36. The curve is dominated by a strong peak at low energies due to secondary electrons created as a result of inelastic collisions between the incident electrons and electrons bound to the solid. Other features in the spectrum include (i) the elastic peak at E p that is utilized in LEED, (ii) inelastic peaks at loss energies of 10–500 meV which provide information about the vibrational structure of the surface–adsorbate complex utilized in EELS, (iii) inelastic peaks at greater loss energies (plasmon losses) which provide information about the electronic structure of surface atoms, and (iv) small peaks on the large secondary electron peak due to Auger electrons which provide information on the chemical composition of the surface.

5000 eV) strikes the atoms of a material, electrons that have binding energies less than the incident beam energy may be ejected from the inner atomic level. By this process a singly ionized, excited atom is created. The electron

Auger Electron Spectroscopy Auger electron spectroscopy is the most common technique for determining the composition of solid and liquid surfaces. Its sensitivity is about 1% of a monolayer, and it is a relatively simple technique to perform experimentally. Auger electron emission occurs in the following manner. When an energetic beam of electrons or X-rays (1000–

FIGURE 36 Experimental number of scattered electrons N(E) of energy E versus electron energy E curve. (Courtesy of Lawrence Berkeley Laboratory.)

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FIGURE 37 Scheme of the Auger electron emission process.

vacancy formed is filled by deexcitation of electrons from other electron energy states. The energy released in the resulting electronic transition can, by electrostatic interaction be transferred to still another electron in the same atom or in a different atom. If this electron has a binding energy that is less than the energy transferred to it from the deexcitation of the previous process that involves the filling of the deep-lying electron vacancy, it will be ejected into vacuum, leaving behind a doubly ionized atom. The electron that is ejected as a result of the deexcitation process is called an Auger electron, and its energy is primarily a function of the energy-level separations in the atom. These processes are schematically displayed in Fig. 37. To a first approximation the energy of the Auger electron depicted in Fig. 37 is given by E Auger = E K − E LI − E LIII

(44)

and is independent of the energy of the incident beam. This is an important difference between AES and photoelectron spectroscopy and means that it is not necessary to monochromatize the electron beam which adds to the experimental convenience. There are two major experimental designs for AES. One is using the retarding grid analyzer which uses the same electron optics as LEED, thus LEED and AES can be performed using the same apparatus. The second is the cylindrical mirror analyzer (CMA) which has an inherently better signal-to-noise ratio. Scanning Auger microprobes are now in widespread use in the microelectronics industry for spatial chemical analysis of surfaces. With the exception of hydrogen and helium, all other elements are detectable by Auger electron spectroscopy. The Auger spectrum is usually presented as the second derivative of intensity, d 2 I /d V 2 , as a function of electron energy (eV). This way the Auger peaks are readily separated from the background, due to other electron loss processes that occur simultaneously. A typical Auger spectrum of molybdenum is shown in Fig. 38. By suitable analysis of the experimental data, as well as by the use of suitable reference surfaces, the Auger electron spectroscopy study can provide quantitative chemi-

FIGURE 38 Typical Auger spectra from (a) a clean Mo(100) single-crystal and (b) a Mo(100) surface contaminated with sulfur.

cal analysis in addition to elemental compositional analysis of the surface. It is possible to separate the surface composition from the composition of layers below the surface by appropriate analysis of the Auger spectral intensities. In this way the surface composition as well as the composition in the near-surface region can be obtained. When chemical analysis is desired in the near-surface region, AES may be combined with ion sputtering to obtain a depth-profile analysis of the composition. Using high-energy ions, the surface is sputtered away layer by layer while, simultaneously, AES analysis detects the ˚ composition in depth. Sputtering rates of 100 A/min are usually possible and the depth resolution of the compo˚ which is mainly determined by the sition is about 10 A, statistical nature of the sputtering process. A different aspect of AES concerns shifts in the observed peak energies that are due to chemical shifts of atomic core levels (in a way analogous to X-ray photoelectron spectroscopy). For example, studies of different oxidation states of oxygen at metal surfaces have shown chemical shifts that grow with the formation of higher oxidation states. G. Photoelectron Spectroscopy Photoelectron spectroscopy is a technique whereby electrons directly ejected from the surface region of a solid by incident photons are energy analyzed and the spectrum is then related to the electron energy levels of the system. The field is usually arbitrarily divided into two classes: ultraviolet photoelectron spectroscopy (UPS) and X-ray photoelectron spectroscopy (XPS). The names derive from the energies of the photons used in the

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particular spectroscopy. Ultraviolet photoelectron spectroscopy studies the properties of valence electrons that are in the outermost shell of the atom and utilizes photons in the vacuum ultraviolet region of the electromagnetic spectrum [He I (21.22 eV), He II (40.8 eV), and Ne I (16.85 eV) resonance lamps are the most commonly used photon sources]. X-ray photoelectron spectroscopy investigates the properties in the inside shells of atoms and uses photons in the X-ray region [Mg Kα (1253.6 eV) and Al Kα (1486.6 eV) being the most common]. With the advent of synchrotron radiation, a polarized, tunable light source covering the entire useful energy range, the division is now somewhat redundant. In both types of spectroscopy, if the incident photon has enough energy hν it is able to ionize an electronic shell and an electron which was bound to the solid with energy E B is ejected into vacuum with kinetic energy E k . By conservation of energy: E k = hν − E B .

(45)

If the incident radiation is monochromatic and of known energy, and if E k can be measured using a high-resolution energy analyzer (such as either a concentric hemispherical or cylindrical mirror analyzer), then the binding energy E B can be deduced. Equation (45) gives a highly simplified relationship between the kinetic energy, E k , of the emitted photoelectrons and their binding energy; E k is modified by the work function of the energy analyzer and by several atomic parameters that are associated with the electron emission process. The ejection of one electron leaves behind an excited molecular ion. The electrons in the outermost and in other orbitals experience a change in the effective nuclear charge due to an alteration of screening by other electrons. This gives rise to satellite peaks near the main photoelectron peaks. Several other effects, including spin-orbit splitting, Jahn–Teller effect, and resonant absorption of the incident photon by the atom, influence the detected photoelectron spectra. One of the most important applications of XPS is the determination of the oxidation state of elements at the surface. The electronic binding energies for inner-shell electrons shift as a result of changes in the chemical environment. An example of these shifts can be seen in nitrogen, indicating the photoelectron energy for various chemical environments (Fig. 39). These energy shifts are closely related to charge transfer in the outer electronic level. The charge redistribution of valence electrons induces changes in the binding energy of the core electrons, so that information on the valence state of the element is readily obtainable. A loss of negative charge (oxidation)

FIGURE 39 Is electronic binding energy shifts in nitrogen, indicating the different photoelectron energies observed in various chemical environments.

is in general accompanied by an increase in the binding energy E B of the core electrons. Relative surface coverages can also be obtained with XPS by monitoring the intensities of the core level peaks. Absolute coverages can be obtained from the core level intensities, but it is usual to calibrate against another technique. As mentioned above UPS probes the valence electrons of the solid. It is these electrons which form a chemisorption bond and a knowledge of electronic density of states at a surface is of vital importance in attempts to understand the formation of chemical bonds between solid surfaces and adsorbed atoms or molecules; UPS can provide even more information about the system if the emitted electrons are both energy and spatially analyzed. This is known as angle-resolved UPS (ARUPS). Using ARUPS the band structures of clean and adsorbate-covered surfaces have been determined, mapping out the dispersion of electronic states; ARUPS also reveals directional effects due to the spatial distribution of electronic orbitals of atoms and molecules at the surface. By changing the angle of incidence and the angle of detection, the electronic orbitals from which the photoelectrons are ejected can be identified. In addition, ARUPS provides detailed information about the surface chemical bond including the direction of the bonding orbitals and the orientation of the molecular orbitals of adsorbed species on the surface.

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V. SURFACE DYNAMICS A. Atomic Vibrations Until now it has been convenient to discuss both the properties and methods in terms of rigid lattices of atoms or molecules. In reality, the atoms are in motion and this motion should be included in a complete treatment of any properties it may affect. In X-ray diffraction experiments it is well known that the intensity of the scattered rays decreases as the temperature is increased. Simultaneously, the intensity of the diffuse background of the diffraction pattern increases. The simplest explanation for this observation is that the atoms are not rigid, but are vibrating about their equilibrium positions, and as a result, the exact Bragg condition is not met. Scattered waves from the rigid lattice that were adding up in phase now have a phase difference fluctuating with time due to the atomic motion. The effect of this motion on the intensity of the elastically diffracted beams is described in most good solid-state physics texts. Briefly, if I0 is the intensity elastically diffracted by a rigid lattice then the intensity I due to scattering by the vibrating lattice in the direction determined by Bragg scattering due to a reciprocal-lattice vector g¯ is given by the following: I = I0 exp(−αu 2 |g|),

(46)

assuming that the atoms are in simple harmonic motion. u 2  is the mean-square amplitude of vibration in the direction g¯ and α is a constant related to the number of dimensions in which the atoms are allowed to vibrate. In one dimension α = 1; in three dimensions α = 13 . The exponential factor in Eq. (46) is called the Debye–Waller factor and is often denoted as exp(−2M). The same kind of effect is observed in LEED only because LEED intensities arise from the just few atomic layer of a crystal the value of u 2  is that for the surface atoms. Because of the absence of nearest neighbors on the vacuum side we expect that u 2  at the surface will be greater than in the bulk. By using the Debye model of the solid it is possible to relate the observed intensity of the elastically scattered electrons in LEED to measurable quantities. We obtain the following:    12h 2  cos φ 2 T    I00 (T ) = I00 (0) exp −  , (47)  mk λ 2D  where I00 (T ) is the temperature-dependent intensity of the (0, 0) beam resulting from a beam of electrons of wavelength λ incident on the surface at an angle  relative to the surface normal. I00 (0) is the specularly reflected intensity from a rigid lattice, h is Planck’s constant, m is the atomic

mass, k is Boltzmann’s constant, T is the temperature, and D is the Debye temperature. (The Debye temperature is associated with the energy ωmax of the highest frequency phonon mode possible in the Debye model of vibrations in the solid, hωmax = kD .) Equation (47) implies that a plot of the logarithm of the intensity at a given energy (wavelength) as a function of temperature is a straight line, the slope of which yields D , a measure of the surface vibrational amplitude perpendicular to the surface. In reality, the electron-beam penetration varies as a function of energy, so that Eq. (47) provides, at any given energy, an effective Debye temperature, which is some average of the surface and bulk layers. In empirical fashion, however, we may arrive at a surface Debye temperature from the low-energy limit of this effective Debye temperature. Adsorbates should have a marked influence on surfaceatom vibrations, since they change the bonding environment with respect to that on the clean surface. The adsorption of oxygen on tungsten increases the surface Debye temperature with respect to the bulk value due to the stronger W O bond as compared to the W W bond. Studies of surface-atom vibrations in the presence of adsorbates provide information on the nature of the surface bond. B. Surface Diffusion As discussed above, at any finite temperature the atoms at the surface of a crystal are vibrating at some frequency ν0 . Thus, ν0 times every second each atom strikes the potential-energy barrier separating it from its nearest neighbors (Fig. 40). The thermal energy causing the atoms to oscillate with increasing amplitude as the temperature is increased is not sufficient to dislodge most of them from their equilibrium positions. The thermal energy (3RT ≈ 1.8 kcal mol−1 at 300 K) tied up in lattice vibrations is only a small fraction of the total energy necessary

FIGURE 40 One-dimensional potential energy diagram parallel to the surface plane.

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to break an atom from its neighbors and to move along the surface. This bond breaking energy is of the order 15– 50 kcal mol−1 for many metal surfaces. As the temperature of the surface is increased, more and more surface atoms may acquire enough activation energy to break bonds with their neighbors and move along the surface. Such surface diffusion plays an important role in many surface phenomena involving atomic transport, e.g., crystal growth, vaporization, and adsorption. The migration of atoms or molecules along the surface is one of the most important steps in surface reactions and has proved to be the ratelimiting step for many reactions that have been studied at low pressures. A surface contains many defects on an atomic scale. Atoms in different surface sites have different binding energies. Surface diffusion can be considered as a multistep process whereby atoms break away from their lattice position (e.g., a kink site at a ledge) and migrate along the surface until they find a new equilibrium site. The frequency f with which an atom will escape from a site will depend upon the height, E D∗ , of the potential energy barrier it has to climb in order to escape as follows:   E D∗ f = zν0 exp − , (48) kB T where z is the number of equivalent neighboring sites. For a (111) face of an fcc metal, z = 6, the vibration frequency is of the order of 1012 sec−1 . Assuming that E D is 20 kcal mol−1 , at 300 K the atom makes one jump in every 50 sec, while at 1000 K one in 10−8 sec. Thus, the rate of surface diffusion varies rapidly with temperature. This is the case for a single jump to a neighboring equilibrium surface site. What is of great importance is the long-distance motion of a surface atom. The result is derived from considering a mathematical treatment of an atom executing a random walk for a time t over a mean-square distance X 2 . For a sixfold symmetrical surface, we obtain the following: X 2  = f td 2 /3.

(49)

The value of f d 2 is a property of the material that characterizes its atomic transport. Its value provides information about the mechanism of atomic transport, and it is customary to define the diffusion coefficient D as follows: D = f d 2 /2b,

(50)

where b is the number of coordinate directions in which diffusion jumps may occur with equal probability. Equation (50) can therefore be rewritten as    D = D0 exp − E D∗ kB T , (51) where D0 = (v0 d 2 /6), (v0 d 2 /4), for sixfold or fourfold symmetry, respectively; D is usually given in units of

square centimeters per second. If D is determined experimentally as a function of temperature, then a plot of ln D versus 1/T will yield us the activation energy of the diffusion process, provided that the diffusion occurs by a single mechanism. The rms distance X 2 1/2 can be expressed in terms of the diffusion coefficient by substitution of Eq. (50) into Eq. (49) to give for b = 6 as follows: X 2 1/2 = (4Dt)1/2 .

(52)

From measurements of the mean travel distance of diffusing atoms the diffusion coefficient can be evaluated. Conversely, knowledge of the diffusion coefficient allows us to estimate the rms distance or the time necessary to carry out the diffusion. For example, the diffusion coefficients of silver ions on the surface of silver bromide can be estimated to be 10−19 and 10−13 cm2 /sec at 300 and 100 K, respectively. Assuming that a rms distance of 10−4 cm is required for silver particle aggregation (printout) to commence, of what duration are the light-exposure times required? Using Eq. (52) we have t = 5 sec and t = 5 × 104 sec at 300 and 100 K, respectively. The exponential temperature dependence of D is, of course, the reason that silver bromide photography cannot be carried out at low temperatures (much below 300 K) but is easily utilized at about room temperature. We can also see that at slightly elevated temperature (∼450 K) the thermal diffusion of silver particles should be rapid enough (D ≈ 3 × 10−7 cm2 sec−1 ) so that their aggregation will take place rapidly even in the dark (t ≈ 10−2 sec) in the absence of any photoreaction. Surface diffusion has so far been discussed in terms of a single surface atom. However, on a real surface many atoms diffuse simultaneously and in most diffusion experiments the measured diffusion distance after a given diffusion time is an average of the diffusion lengths of a large, statistical number of surface atoms. A thermodynamic treatment in terms of macroscopic parameters can be followed to yield the following: D = D0 exp(−Q/RT ),

(53)

where Q is the total activation energy for the overall diffusion process and only one diffusion mechanism is involved. Experimentally, the diffusion coefficient D is obtained by using a relationship between the diffusion rate and coverage gradient, namely Fick’s second law of diffusion in one dimension: ∂c/∂t = D(∂ 2 c/∂ x 2 ),

(54)

where c is the concentration of adatoms, t the time, and x the distance along the surface. In most surface diffusion studies the surface concentration of diffusing atoms, c, is

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measured as a function of distance x along the surface, and Eq. (54) is solved by the use of boundary conditions that approximate the experimental geometry. These experiments are by no means trivial, and many novel experimental techniques have been applied to study surface diffusion on single crystals. A technique which has been used to measure surface diffusion rates is scanning Auger electron spectroscopy, which can follow adsorbate diffusion. A particular Auger transition of the adsorbate under investigation is used as a monitor of relative concentration versus distance scanned across the surface. Profiles are recorded after heating periods to observe the change in concentration profile as a function of time and temperature. While this technique monitors mass transport, and values of D and Q are averaged values, field ion microscopy can be used to follow the diffusion of individual atoms across a surface. To study diffusion, the metal is vapor deposited onto the tip. The tip is then heated to remove evaporated adatoms until only one or two remain on the surface plane of interest. The diffusion is then examined by photographically recording the position of the adatom at low temperatures, removing the applied field, and heating to the desired temperature for a given time. The tip is then cooled, the field reapplied, and the field ion image examined to see if the atom has moved to a neighboring site. This process is then repeated many times to obtain useful values of diffusion rates, and by examining the diffusion over a temperature range, the activation barrier to surface diffusion can be determined. Figure 41 shows a series of field ion images of a Rh atom on the W(112)

plane at 327 K. The field ion images are taken at 1-min intervals and the Rh atom can clearly be seen to have diffused across the surface. Unfortunately because of the high field strengths employed, adsorbates such as O or N tend to be stripped from the surface as ions, so their microscopic diffusion cannot be studied by this method. An interesting result from FIM studies of metal adatoms on metals is the recognition of clusters as important contributions to material transport. It has been found that rhenium dimers diffuse more rapidly than single Rh atoms on the W(112) plane, and Rh trimers diffuse at roughly the same rate as dimers. This is not a general trend, however, as iridium dimers move much more slowly than singles. While the single adatom diffusion technique gives us detailed microscopic information, the mass transport techniques are of use as they help to give understanding of the technologically important processes such as sintering and creep. C. Surface Reactions Heterogeneous catalysis, corrosion, photosynthesis, and adhesion are examples of chemical processes that are partially or fully controlled by reactions at surfaces. For the case of gas–solid reactions the surface reactions can be divided into two major categories: (i) stoichiometric surface reactions where the solid surface participates directly in the reaction by compound formation and (ii) catalytic surface reactions where the reaction occurs at the solid surface but the surface does not undergo any net chemical change. In both cases gaseous molecules impinge on the surface, adsorb, react, and form various intermediates of varying stability, and then the products desorb into the gas phase if they are volatile. All surface reactions involve a sequence of elementary steps that begins with the collision of the incident atoms or molecules with the surface. As the gas species approaches the surface it experiences an attractive potential whose range depends upon the electronic and atomic structures of the gas and surface atoms. A certain fraction of the incident gas molecules is trapped in this attractive potential well with a sticking probability given by the following: S(, T ) = S0 (1 − ) exp(−E a /RT ),

FIGURE 41 Diffusion of rhenium atoms on W(211) at 327 K. Field ion images are taken after 60 sec diffusion intervals. (Courtesy of Lawrence Berkeley Laboratory.)

(55)

where S0 is the initial (zero coverage) sticking coefficient,  the surface coverage (0 <  < 1), and E a the activation energy for adsorption. If this force attraction is due to a van der Waals interaction, the trapping is due to physical adsorption. If the attraction is much stronger, having the character of chemical bonding then we have chemisorption and the process is known as sticking. The boundary between the two types of bonding is usually

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410 set at a binding energy of 15 kcal mol−1 . Sticking by chemisorption is often preceded by trapping into a physisorbed state, in which case the physisorbed state is known as a precursor state for chemisorption. The presence of a precursor state is indicated by a sticking coefficient that remains almost constant as surface coverage increases until a saturation coverage is reached, when it rapidly falls to zero. This behavior arises because molecules in the relatively mobile precursor state diffuse to parts of the surface which are not covered by chemisorbed molecules. In direct chemisorption the sticking coefficient varies strongely with coverage and with ordering of the chemisorbed layer. The adsorbed species may also desorb from the surface if its energy overcomes the attractive surface forces. When a surface reaction occurs a certain proportion of the adsorbed species either decomposes (unimolecular reaction) or reacts with a second adsorbed species (bimolecular reaction) before the product desorbs. During the initial interaction of the gas molecule with the surface as the incoming molecule falls into a potential well the kinetic energy normal to the surface increases. Unless this energy is transferred to some other degree of freedom the molecule will simply bounce off; there will be no trapping or sticking. In the case of physisorption energy transfer via phonons is usually most important while for chemisorption electronic excitation via electron–hole pairs is thought to be important. The exchange of translational energy T with the phonons Vs is called T − Vs energy exchange. The gas molecule may also exchange internal energy, rotation R or vibration V with the vibrating surface atoms. In this case there are also R − Vs and V − Vs energy transfer processes. In order to understand the dynamics of gas–surface interaction, it is necessary to determine how much energy is exchanged between the gas and surface atoms through the various energy-transfer channels. In addition the kinetic parameters (rate constants, activation energies, and preexponential factors) for each elementary surface step of adsorption, diffusion, and desorption are required in order to obtain a complete description of the gas–surface energy transfer process. Most surface reactions take place at high pressures (1– 100 atm) either because of the chemical environment of our planet or to establish optimum reaction rates in chemical processing. Under these conditions, surfaces are usually covered by at least one monolayer of adsorbed species. Since activation energies for adsorption and surface diffusion are generally small (a few kT ), equilibrium among the different surface species, reactants, reaction intermediates, and products, is readily established. In the simplest (but not general and important) case of localized, associative adsorption into a single state, the surface coverage by

Surface Chemistry

adsorbed species is given in terms of the gas pressure P by the Langmuir isotherms:  = KP/(1 + KP),

(56)

where K is an equilibrium constant. Catalyzed surface reactions usually take place between two or more coadsorbed species which compete for adsorption sites on the surface. When j gases adsorb competitively and associatively, the surface coverage by species i is given by the following:    1+ (57) K j Pj .  = K i Pi j

Many catalyzed surface reactions can be treated as a two-step process with an adsorption equilibrium followed by one rate-determining step (diffusion, surface reaction, or desorption). The surface reaction kinetics are usually discussed in terms of two limiting mechanisms, the Langmuir–Hinshelwood (LH) and Eley–Rideal (ER) mechanisms. In the LH mechanism, reaction takes place directly between species which are chemically bonded (chemisorbed) on the surface. For a bimolecular LH surface reaction. Aads + Bads → products, with competitive chemisorption of the reactants, the rate of reaction is given by the following expression: Rate = kR A B = kR K A K B PA PB /(1 + K A PA + K B PB )2 .

(58)

The reaction rate is proportional to the surface coverages A and B and to the reaction rate constant kR . For noncompetitive adsorption, the rate expression becomes the following: kR K A K B PA PB Rate = kR A B = . (59) (1 + K A PA )(1 + K B PB ) General rate expressions of the form given in equations and have been experimentally verified for many types of LH reactions. Similar but more complicated rate expressions are easily derived assuming different (non-Langmuir) isotherms, higher-order reaction steps, or dissociative chemisorption of the reactants. In the ER mechanism, surface reaction takes place between a chemisorbed species and a nonchemisorbed species, e.g., Aads + Bg → products. The nonchemisorbed species may be physisorbed or weakly held in a molecular precursor state. In this case, the rate expression for the surface reaction becomes Rate = kR A PB = kR K A PA PB /(1 + K A PA ).

(60)

Presently no proven examples exist in which surface reaction occurs by the ER mechanism. Surface reaction kinetics determined experimentally are often expressed in the form of a power rate law as follows:

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Rate = kR



Piαi ,

(61)

i

where kR is the apparent rate constant and αi is the experimental order of the reaction (positive, negative, integer, or fraction) with respect to the reactants and products. The apparent rate constant in Eq. (61) is not that of an elementary reaction step (it contains adsorption equilibrium constants), but it can usually be represented by an Arrhenius equation as follows: kR = A exp(−E R /RT ),

(62)

where A is an apparent preexponential factor and E R is the apparent activation energy for the surface reaction. The magnitude of A and E A can provide important information about the rate-determining step of a surface reaction, and very frequently kR and A display a compensation effect. A related quantity is the reaction probability, λi = (2π m K T )1/2 νR /Pi = rate/flux; that is, the probability that an incident reactant molecule will undergo reaction. The simplified isotherms and rate expressions developed in this section are extremely useful despite the implicit assumption that a single state exists for the adsorbed species. Real surfaces are heterogeneous on an atomic scale with a variety of distinguishable adsorption sites. Gas molecules adsorbed at each type of site may display a wide distribution of excited rotational, vibrational, and electronic states. Experimentally, we can measure meaningful rate and adsorption equilibrium constants provided that adsorption and desorption are fast compared with surface reactions so that an adsorption equilibrium exists. In this circumstance the kinetic parameters are an ensemble average over all surface sites and states of the system.

molecules, together with their angular and velocity distributions provide detailed information about the T –Vs energy transfer processes that occur during the gas–surface interaction. A complete dynamical description for this interaction (T –Vs plus R–Vs and V –Vs ) can be determined if the distribution of internal energy states for the product molecules is determined simultaneously with their velocity distributions. This type of detection is known as state selective detection. The angular distribution of scattered molecules is usually displayed by plotting the intensity of detected molecules per unit solid angle versus the angle of scattering r that is measured with respect to the surface normal. Angular distributions in the two limiting cases of gas–surface interaction, cosine and specular scattering, are shown in Fig. 42. The scattered intensity for the cosine distribution decreases as cos  with respect to the surface normal. Cosine scattering is expected when the adsorbed species have long residence times or are strongly coupled to the vibrational states of the surface atoms. It is a necessary criterion for complete thermal accommodation, a situation in which the molecules desorb with a kinetic temperature or velocity distribution that is the same as the temperature of the solid surface. Specular scattering occurs when the scattered intensity is sharply peaked at the angle of incidence (specular angle). In this case the interaction is elastic or quasielastic and little or no energy transfer takes place between the incident gas molecules and the surface. Sharply peaked angular distributions for surface reaction products (I () ∼ cosm , m > 1) indicate that a repulsive barrier exists in the exit channel. Measurements of velocity distributions provide more

1. Molecular-Beam Scattering The most powerful experimental technique for investigating the dynamics of the gas–solid interaction is molecularbeam surface scattering (MBS). The experimental arrangement is similar to that already described for helium atom diffraction. Instead of using an atomic beam of a light molecular weight gas and observing diffraction effects, a well-collimated beam of molecules strikes the oriented, preferably single-crystal, surface, and the species that are scattered at a specific solid angle are detected by a mass spectrometer. The angular distribution of the scattered molecules can be obtained by rotation of the mass spectrometer about the sample. The velocity distribution of the molecules after scattering is deduced by chopping the scattered molecules and thereby measuring their time of flight to the detector. The surface residence times of the

FIGURE 42 Rectilinear plot displaying the (a) specular scattering and (b) cosine angular distribution of scattered beams. The arrow indicates the angle of incidence.

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412 direct information on inelastic scattering than angular distributions alone. Although considerable information can be gained from such studies it has been impossible to get state specific information and indeed it is often unclear whether internal states present more efficient energy transfer channels than phonons or vice versa. The difficulties in studying internal energy transfer in molecular collisions with surfaces can be resolved by the application of state-specific detection techniques. Laser-induced fluorescence, multiphonon ionization, IR excitation with bolometric detection, and IR emission techniques have all been used to obtain state-resolved measurements of the internal energy distributions of molecules scattering from surfaces. It has also been possible to separate experimentally direct inelastic and trapping-desorption scattering. In the direct inelastic scattering of diatomics, coupling to rotational energy has been found to be very important and to exhibit several interesting phenomena: rotational rainbows and the production of rotationally aligned molecules in scattering.

Surface Chemistry

Exchange of hydrogen and deuterium to form HD is one of the simplest reactions that can be catalyzed on clean metal surfaces at temperatures as low as 100 K. The same reaction is immeasurably slow in the gas phase due to the very high dissociation energies of the reacting molecules (103 kcal mol−1 ). the H2 –D2 exchange reaction has been studied over flat (111) and stepped (332) single crystal surfaces of platinum. The Pt(332) surface contains high concentrations of periodic surface irregularities (steps) that are one atom in height. Reaction probabilities averaged over the cosine HD angular distributions were 0.07 on the (111) surface and 0.35 on the (332) surface under identical experimental conditions (Ts = 1100 K, Tg = 300 K). The reaction probability on the stepped surface varied markedly with the angle of incidence of the mixed H2 –D2 molecular beam. This is shown in Fig. 43. The reaction probability was highest when the beam was incident on the open edge of the step and lowest when the bottom of the step was shadowed

2. Molecular-Beam Reactive Scattering While molecular-beam scattering has made great advances in our understanding of the energy exchange processes during the gas–surface collision, molecular-beam techniques have also made important contributions to the understanding of the mechanisms of chemical reactions occurring at surfaces in the form of molecular-beam reactive scattering (MBRS). The use of time-of-flight techniques permits measurement of product velocity distributions and the detailed time resolution of fast transient reactions. Also of great value is the use of state-specific detection methods to determine product vibrational and rotational states. Although MBRS can only be utilized at low pressures (≤10−4 torr) its pressure range permits wide variations of surface coverages. The reaction probabilities on a single scattering can be determined together with the surface residence times of adsorbates. The surface kinetic information is obtained by measurements of the intensity and the phase shift of the product molecules with respect to the reactant flux. Residence times in the range 10−6 –1 sec can be monitored with relative ease, and activation energy is determined from the temperature dependences of the intensities and the phase shifts. The phase shift of the product molecules is usually measured at different chopping frequencies of the incident beam. At a given chopping frequency, only those product molecules are detected that are formed in the surface process and desorbed in less time than the chopping period. As an example of an investigation of the dynamics of a catalyzed surface reaction studied by MBRS we will consider the isotope exchange reaction, H2 –D2 .

FIGURE 43 HD production as a function of angle of incidence  of the molecular beam, normalized to the incident D2 intensity. (a) Pt(332) surface with the step edges perpendicular to the incident beam (φ = 90◦ ); (b) Pt(332) where the projection of the beam on the surface is parallel to the step edges (φ = 0◦ ); and (c) Pt(111).

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(curve a). When the H2 –D2 beam was incident parallel to the steps, the rate of HD production was independent of the angle of incident at all angles of crystal rotation (curve b). These results indicate that the atomic step sites are about seven times more active than the (111) terrace sites for the dissociative chemisorption of hydrogen and deuterium molecules. Detailed analysis of the scattering data revealed a barrier height of 4–8 kJ mol−1 for dissociative H2 chemisorption on the (111) surface. On the other hand, this barrier did not exist (E a = 0) on the stepped surface. This difference in activation energy alone accounts for the different reaction probabilities of the step and terrace sites. While the dissociation probability of hydrogen molecules was higher on the stepped surface than on Pt(111), the kinetics and mechanism of HD recombination appear to be identical over both surfaces once dissociation takes place. On both surfaces, HD formation follows a parallel LH mechanism with one of the reaction branches operative over the entire temperature range of 300–1075 K. This branch has an activation energy and pseudo-first-order preexponential factor of E a = 54 kJ mol−1 and A1 = 8 × 104 sec−1 for the stepped surface and E a = 65 kJ mol−1 and A1 = 3 × 105 sec−1 for the Pt(111) surface. A second branch is observed for temperatures above 575 K, but the kinetic parameters for this pathway could not be accurately determined. D. Stoichiometric Surface Reactions Stoichiometric surface reactions are those in which the surface participates directly in the reaction by compound formation. Oxidation and corrosion are the two most important classes of such reactions. Surface oxidation of metals encompasses a series of at least three reaction steps that include (1) dissociative chemisorption of oxygen on the metal surface, (2) rearrangement of the surface atoms with dissolution of oxygen into the near surface region, and (3) nucleation of oxide islands which grow laterally and eventually condense to produce continuous oxide films. The oxide islands appear to precipitate suddenly once a critical oxygen concentration is reached in the near surface region. Nucleation takes place most readily at surface irregularities such as atomic steps, dislocations, and stacking faults. At room temperature, noble metals such as Rh, Ir, Pd, and Pt display little tendency for oxygen incorporation or surface rearrangement. Initial heats of oxygen chemisorption on these metals are much greater than the heats of formation of the corresponding bulk oxides. Other metals such as Cr, Nb, Ta, Mo, W, Re, Ru, Co, and Ni, dissolve surface oxygen by a place exchange mechanism in which oxygen atoms interchange positions with underlying metal atoms. These

metals display heats of adsorption for oxygen that are comparable to the heats of formation of the stable metal oxides. Metals such as Ti, Zr, Mn, Al, Cu, and Fe dissolve oxygen more readily and form stable oxide films even at room temperature. At low oxygen pressures these films often assume a crystalline structure, whereas at higher pressures (>10−3 atm) the films tend to be amorphous. At higher temperatures (400–1000 K), oxide formation occurs readily on the surfaces of nearly all metals. Growth of surface oxide films takes place only if cations, anions, and electrons can diffuse through the oxide layer. The growth kinetics of very thin films ˚ often follow the Mott or Cabrera–Mott (∼10–50 A) mechanisms in which electrons tunnel through the film and associate with oxygen atoms to produce oxide ions at the surface. A large local electric field (106 –107 V/cm) results at the surface which facilitates cation diffusion from the metal–oxide interface to an interstitial site of the oxide. The film thickness Z at time t is given by Z = α1 ln (α2 t + 1)

(63)

or inverse logarithmic 1/Z = α3 − α4 ln t

(64)

law of growth depending on whether electron tunneling or cation diffusion is rate limiting. The constants α1 –α4 are determined by the material, its structure, and the reaction conditions. The electron field strength and rate of growth decrease exponentially as the film thickens, resulting in an effective limiting thickness for the surface oxide layer. In addition to surface oxides, a vast array of surface compounds can be produced from the reactions of halogens, chalcogenides, and carbon-containing molecules with metal surfaces. Chemisorption of chlorine near 300 K on Cu, Ti, W, Mo, Ta, Ni, Pd, and Au, for example, results in the formation of stable surface compounds which often evaporate as molecular chlorides upon heating at elevated temperatures. Chemisorption of chlorine at 300 K on Ag(100) and Ag(111) produces chemisorbed chlorine overlayers which react irreversibly at about 425 K to produce AgCl with an activation energy of 56 kJ mol−1 . Upon heating, AgCl desorbs at about 830 K with a desorption activation energy of 192 kJ mol−1 . MBRS has been used to investigate the dynamics of several surface corrosion reactions at low reactant pressures. Systems studied include the oxidation Si, Ge, Mo, and graphite, and the halogenation of Si, Ge, Ta, and Ni. With the exception of silicon and germanium oxidation, where dissociative chemisorption of oxygen is apparently rate limiting, the kinetics of these surface reactions generally appear to be controlled by surface or bulk diffusion of the reacting species.

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E. Catalytic Surface Reactions A major goal of basic surface chemistry is in trying to understand heterogeneous catalysis on an atomic scale. Virtually all chemical technologies and many technologies in other fields use catalysis as an essential part of the process. The most important catalytic processes are summarized in Table V. These processes are listed together with the pertinent chemical reactions, widely used catalysts, and typical reaction conditions. There are several definitions of a catalyst, one general definition being that it is a substance that accelerates a chemical reaction without visibly undergoing chemical change. Indeed, a major role of a catalyst is in accelerating the rate of approach to chemical equilibrium. However, a catalyst cannot change the ultimate equilibrium determined by thermodynamics. Another major function of a catalyst is to provide reaction selectivity. Under the conditions in which the reaction is to be carried out, there may be many reaction channels, each thermodynamically feasible, that lead to the formation of different products. The selective catalyst will accelerate the rate of only one of these reactions so that only the desired product molecules form with near-theoretical or 100% efficiency. One example is the dehydrocyclization of n-heptane to toluene: CH3 CH3

(CH2)5

CH3

 4H2

This is a highly desirable reaction that converts aliphatic molecules to aromatic compounds. The larger concentration aromatic component in gasoline, for example, greatly improves its octane number. However, n-heptane may participate in several competing simpler reactions. These include hydrogenolysis, which involves C C bond scission to form smaller molecular weight fragments (methane, ethane, and propane); partial dehydrogenation, which produces various olefins; and isomerization, which yields branched chains. All of these reactions are thermodynamically feasible, and since they appear to be less complex than dehydrocyclization, they compete effectively. A properly prepared platinum catalyst surface catalyzes the selective conversion of n-heptane to toluene without permitting the formation of other products. The catalyst selectivity is equally important for the reactions of small molecules (such as the hydrogenation of CO to produce a desired hydrocarbon) or very large molecules of biological importance, where enzyme catalysts provide the desired selectivity. Catalysis is a kinetic phenomenon; we would like to carry out the same reaction at an optimum rate over and over again using the same catalyst. In most cases such a steady-state operation is desirable and aimed for. In

the sequence of elementary reactions that include adsorption, surface migration, chemical rearrangements, and reactions in the adsorbed state, and desorption of the products, the rate of each step must be of steady state. The rate of the overall catalytic reaction per unit area catalyst surface can be expressed as (moles of product/catalyst area × time). Another expression for catalytic rate is the turnover number or turnover frequency. This is the (number of molecules of product/number of catalyst sites × time). For most heterogeneous catalyzed small molecule reactions the turnover number varies between 10−2 and 102 sec−1 . The calculation of the turnover number is limited by the difficulty of determining the true number of active sites. The reaction probability reveals the overall efficiency of a catalyst. It is defined as follows: reaction probability rate of formation of product molecules rate of incidence of reactant molecules. The determination of the rates of the net catalytic reactions and how the rates change with temperature and pressure is of great practical importance. Although there are many excellent catalysts that permit the achievement of chemical equilibria (for example, Pt for oxidation of CO and hydrocarbons to CO2 and H2 O), most catalyzed reactions are still controlled by the kinetics of one of the surface processes. From the knowledge of the activation energy and the pressure dependencies of the overall reaction, the catalytic process can be modeled and the optimum reaction conditions can be calculated. Such kinetic analysis, based on the macroscopic rate parameters, is vital for developing chemical technologies based on catalytic reactions. The rates of reactions are extremely sensitive to small changes of chemical bonding of the surface species that participate in the surface reaction. Since the energy necessary to form or break the surface bonds appears in the exponent of the Arrhenius expression for the rate constant for the overall reaction, it can increase or decrease the rate exponentially. For example, a change of 3 kcal in the activation energy alters the reaction rate by over an order of magnitude at 500 K. Small variations of chemical bonding at different surface irregularities, steps, and kinks, as compared to atomic terraces, can give rise to a very strong structure sensitivity of the reaction rates and the product distribution. Rate measurements exponentially magnify the energetic alterations that occur on the surface and could provide a very sensitive probe of structural and electronic changes at the surface and changes of surface bonding on the molecular scale. One of the most important considerations in catalysis is the need to provide a large contact area between the =

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Surface Chemistry TABLE V Chemical Processes Based on Heterogeneous Catalysis Processes Ammonia synthesis

Typical reactions

Catalyst

N2 + 3H2 → 2NH3

Triply promoted iron (Fe K2 O Al2 O3 CaO);

Dehydrogenation

Fe2 O3 Cr2 O3 K2 O mixed metal oxides

H2 

Reaction conditions

720–800 K 40–100 atm 800–900 K 10–50 atm

Epoxidation Fischer–Tropsch synthesis of hydrocarbons Fischer–Tropsch synthesis of oxygenates

C2 H4 + 12 O2 → C2 H4 O CO + H2 → alkanes olefins aromatics

AgCl K2 O/Al2 O3 Fe3 O4 K2 O/Al2 O3 supported Co, Ru, Ni, Rh

CO + H2 → aldehydes acids alcohols

Rh2 O3 ·H2 O K2 O LaRhO4 supported Pd, Pt

520–600 K 500–700 K 10–50 atm 500–700 K 10–50 atm

Hydrotreating (desulfurization and denitrification)

R S R + H2 → 2RH + H2 S R N—R + 32 H2 → 2RHH + NH3

Co Mo, Ni Mo,

570–770 K

Ni Co Mo/Al2 O3 Ni W/Al2 O3 , MoS2 , WS2

30–200 atm

Olefins

Solid acids, zeolites Group VIII metals

270–470 K 1–5 atm

Xylenes

ZSM-5-zeolites

480–580 K 2–5 atm

Alkanes

Zeolites, Pt/Al2 O3

570–770 K 5–50 atm

Isomerization

Methanol synthesis

CO + 2H2 → CH3 OH

ZnCrO3 ZnO Cu2 O Cr2 O3 ZnO Cu2 O Al2 O3

570–670 K

Methanol to gasoline

CH3 OH → aromatics olefins, H2 O

ZSM-5-zeolites

480–540 K 2–15 atm

NOx Reduction

NO + 52 H2 → NH3 + H2 O 2NO + 2H2 → N2 + 2H2 O

Ru, Rh, Pd, Pt/SiO2

370–520 K 1–10 atm 450–650 K 1–10 atm

Ru, Rh, metal oxides

2CO + 2NO → 2CO2 + N2 Oxidation

Olefins + Alkanes 2NH3 + CO +

+ O2 → CO2 + H2 O

5 2 O2

1 2 O2

100–600 atm

Group VIII metals

370–670 K

Ag, Fe2 (MoO4 )3

550–570 K

V2 O5

1–10 atm

→ NO + 3H2 O

→ CO2

Partial oxidations Alcohols

CH3 OH + 12 O2 → H2 CO + H2 O

O

o-Xylene

 3O2

O  3H2O O

Olefins

C2 H4 + 12 O2 → CH3 CHO



Reforming Dehydrogenation

1 2

O2

V2 O5 SnO2 ·MoO3

O

 O2  H2O

O

R

R

Bi2 O3 ·MoO3

Pt, Pt–Re, Pt–Ge

700–800 K

 3H2 continues

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TABLE V (Continued ) Processes

Typical reactions

Catalyst

Dehydrocyclization

Pt Au, Pt Re Cu

Reaction conditions 5–50 atm

 3H2 Dehydroisomerization

Ir Au/Al2 O3

 3H2 Isomerization

 H2

Hydrogenolysis

2C3H8

Hydrogenation

 H2 Selective Hydrogenation Olefins Alkynes

H2 

NiS

R C CH + H2 → RHC CH2

Pt/Al2 O3

Steam reforming

CH4 + H2 O → CO + 3H2

Ni K2 O/Al2 O3

Water gas

CO + H2 O → CO2 + H2

Fe2 O3 ·Cr2 O3 ZnO Cu2 O

reactants and the surface. The total rate (moles of product per time) is proportional to the surface area. As a consequence, a lot of effort is expended to prepare large surfaces area catalysts and to measure the surface area accurately. One example of high-surface-area catalysts is the group of catalysts known as zeolites, which are aluminosilicates used for the cracking of hydrocarbons. They have crystal ˚ in size. The structure of one structures full of pores 8–20 A of the many zeolites used for catalysis, faujasite, is shown in Fig. 44. Since the catalytic reactions occur inside the pores, an enormous inner surface area, of several hundred

FIGURE 44 Line drawing of the structure of the zeolite faujasite.

420–500 K 1–10 atm 220–250 K 1–10 atm 850–1100 K 30–100 atm 650–800 J 20–50 atm

square meters per gram of catalyst, is available in these catalyst systems. Transition-metal catalysts are generally ˚ employed in a small, 10- to 100-A-diameter particle form dispersed on large-surface-area supports. The support can be a specially prepared alumina or silica framework (or a zeolite) that can be produced with surface areas in the 102 -m2 /g range. These supported metal catalysts are often available with near-unity dispersion (dispersion is defined as the number of surface atoms per total number of atoms in the particle) of the metal particles and are usually very stable in this configuration during the catalytic reaction. The metal is frequently deposited from solution as a salt and then reduced under controlled conditions. Alloy catalysts and other multicomponent catalyst systems can also be prepared in such a way that small alloy clusters are formed on the large-surface-area oxide supports. Most catalytic reactions take place via the formation of intermediate compounds between the reactants or products and the surface. The surface atoms of the catalyst form strong chemical bonds with the incident molecules, and it is this strong chemical surface–adsorbate interaction which provides the driving force for breaking highbinding-energy chemical bonds (C C, C H, H H, N N, and C O bonds), which are often an important part of the catalytic reaction. A good catalyst will also permit rapid bond breaking between the adsorbed intermediates and the surface and

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the speedy release or desorption of the products. If the surface bonds are too strong, the reaction intermediates block the adsorption of new reactant molecules, and the reaction stops. For too-weak adsorbate-catalyst bonds, the necessary bond-scission processes may be absent. Hence, the catalytic reaction will not occur. A good catalyst is thought to be able to form chemical bonds of intermediate strength. These bonds should be strong enough to induce bond scission in the reactant molecules. However, the bond should not be too strong to ensure only short residence times for the surface intermediates and rapid desorption of the product molecules, so that the reaction can proceed with a large turnover number. Of course, activity is only one of many parameters that are important in catalysis. The selectivity of the catalyst, its thermal and chemical stability, and dispersion, are among the other factors that govern our choices. While macroscopic chemical-bonding arguments can explain catalytic activity in some cases, atomic-scale scrutiny of the surface intermediates, catalyst structure, and composition, and an understanding of the elementary rate processes are necessary to develop the optimum selective catalyst for any chemical reaction. One of the important directions of research in catalysis is the identification of the reaction intermediates. The surface residence times of many of these species are longer than 10−5 sec under most catalytic reaction conditions (as inferred from the turnover frequency). They may be detected by suitable spectroscopic techniques either during the steady-state reaction or when isolated by interrupting the catalytic process. The concept of active sites is an important one in catalysis. A surface generally possesses active sites in numbers that are smaller than the total number of surface atoms. The presence of unique atomic sites of low coordination and different valency that are very active in chemical reactions has been clearly demonstrated by atomic-scale studies of metal and oxide surfaces. A catalytic reaction is defined to be structure sensitive if the rate changes markedly as the particle size of the catalyst is changed. Conversely, the reaction is structure insensitive on a given catalyst if its rate is not influenced appreciably by changing the dispersion of the particles under the usual experimental conditions. In Table VI we list several reactions that belong to these two classes. Clearly, variations of particle size give rise to changes of atomic surface structure. The relative concentrations of atoms in steps, kinks, and terraces are altered. Nevertheless, no clear correlation has been made to date between variations of macroscopic particle size and the atomic surface structure. Most surface reactions and the formation of surface intermediates involve charge transfer, either an electron transfer or a proton transfer. These processes are often

TABLE VI Structure-Sensitive and StructureInsensitive Catalytic Reactions Structure sensitive

Structure insensitive

Hydrogenolysis Ethane: Ni Methylcyclopentane: Pt Hydrogenation Benzene: Ni Isomerization Isobutane: Pt Hexane: Pt Cyclization Hexane: Pt Heptane: Pt

Ring opening Cyclopropane: Pt Hydrogenation Benzene: Pt Dehydrogenation Cyclohexane: Pt

viewed as modified acid–base reactions. It is common to refer to an oxide catalyst as acidic or basic according to its ability to donate or accept electrons or protons. The electron transfer capability of a catalyst is expressed according to the Lewis definition. A Lewis acid is a surface site capable of receiving a pair of electrons from the adsorbate. A Lewis base is a site having a free pair of electrons that can be transferred to the adsorbate. The proton-transfer capability of a catalyst is expressed according to the Brønsted definition. A Brønsted acid is a surface site capable of losing a proton to the adsorbate while a Brønsted base is a site that can accept a proton from the adsorbed species. Perhaps the most widely used catalysts, the zeolites, best represent the group of oxides that exhibit acid–base catalysis. Zeolites are alumina silicates, some of which are among the more common minerals in nature. Modern synthesis techniques permit the preparation of families of zeolite compounds with different Si/Al ratios. Since the Al3+ ions lack one positive charge in the tetrahedrally coordinated silica, Si4+ , framework, they are sites of proton or alkali–metal affinity. Variation of the Si/Al ratio gives rise to a series of substances of controlled but different acidity. By using various organic molecules during the preparation of these compunds that build into the structure, subsequent decomposition leaves an open pore structure, where the pore size is controlled by the skeletal structure of the organic deposit. Very high internal surface area catalysts (102 m2 /g) can be obtained this way ˚ and controlled acidity with controlled pore sizes of 8–20 A [(Si/Al) ratio]. These catalysts are utilized in the cracking and isomerization of hydrocarbons that occur in a shape selective manner as a result of the uniform pore structure and are the largest volume catalysts in petroleum refining. They are also the first of the high-technology catalysts in which the chemical activity is tailored by atomic-scale

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418 study and control of the internal surface structure and composition. A catalyst used in industry is very rarely a pure element or compound. Most catalysts contain a complex mixture of chemical additives or modifiers that are essential ingredients for high activity and selectivity. Promoters are beneficial additives that increase activity, selectivity, or useful catalyst lifetime (stability). Structural promoters inhibit sintering of the active catalyst phase or present compound formation between the active component and the support. The most frequently used chemical promoters are electron donors such as the alkali metals or electron acceptors such as oxygen and chlorine. For example, in the petroleum industry, chlorine and oxygen are often added to commercial platinum catalysts used for reforming reactions by which aliphatic straight-chain hydrocarbons are converted to aromatic molecules (dehydrocyclization) and branched isomers (isomerization). These additives accomplish several tasks during the reaction. By changing the chemical bonding of some of the surface intermediates, the steady-state concentration of these intermediates may be altered, and thus a somewhat higher concentration of the catalytically active species is obtained. In this way the rate of the reaction is increased and the selectivity may be improved. Often multicomponent catalyst systems are utilized to carry out reactions consisting of two or more active metal components or both oxide and metal constituents. For example, a Pt–Rh catalyst facilitates the removal of pollutants from car exhausts. Platinum is very effective for oxidizing unburned hydrocarbons and CO to H2 O and CO2 , and rhodium is very efficient in reducing NO to N2 , even in the same oxidizing environment. Dual functional or multifunctional catalysts are frequently used to carry out complex chemical reactions. In this circumstance the various catalyst components should not be thought of as additives, since they are independently responsible for different catalytic activity. Often there are synergistic effects, however, whereby the various components beneficially influence each other’s catalytic activity to provide a combined additive and multifunctional catalytic effects. It should be clear from this discussion that the working, active, and selective catalyst is a complex, multicomponent chemical system. This system is finely tuned and buffered to carry out desirable chemical reactions with high turnover frequency and to block the reaction paths for other thermodynamically equally feasible but unwanted reactions. Thus, an iron catalyst or a platinum catalyst is composed not only of iron or platinum but of several other constituents as well to ensure the necessary surface structure and oxidation state of surface atoms for optimum catalytic behavior. Additives are often used to block sites,

Surface Chemistry

prevent side reactions, and alter the reaction paths in a variety of ways. While industrial catalytic systems are complex and are not readily suited to basic science studies to understand how they work on an atomic scale, one approach to their understanding is the synthetic approach. In this approach we begin with a very simple system then synthesize complexity from this. The catalyst particle is viewed as composed of single crystal surfaces, as shown in Fig. 45. Each surface has different reactivity and the product distribution reflects the chemistry of the different surface sites. We may start with the simplest single crystal surface [for example, the (111) crystal face of platinum] and examine its reactivity. It is expected that much of the chemistry of the dispersed catalyst system would be absent on such a homogeneous crystal surface. Then high-Miller-index crystal faces are prepared to expose surface irregularities, steps, and kinks of known structure and concentration, and their catalytic behavior is tested and compared with the activity of the dispersed supported catalyst under identical experimental conditions. If there are still differences, the surface composition is changed systematically or other variables are introduced until the chemistries of the model system and the working catalyst become identical. This approach is described by the following sequence:

FIGURE 45 Catalyst particle viewed as a crystallite, composed of well-defined atomic planes. (Courtesy of Lawrence Berkeley Laboratory.)

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structure of crystal surfaces and adsorbed gases ↓↑ surface reactions on crystals at low pressures (≤10−4 torr) ↓↑ surface reactions on crystals at high pressures (10+3 −10+5 torr) ↓↑ reactions on dispersed catalysts Investigations in the first step define the surface structure and composition on the atomic scale and the chemical bonding of adsorbates. Studies in the second step, which are carried out at low pressures, reveal many of the elementary surface reaction steps and the dynamics of surface reactions. Studies in the third and fourth steps establish the similarities and differences between the model system and the dispersed catalyst under practical reaction conditions. The advantage of using small-area catalyst samples is that their surface structure and composition can be prepared with uniformity and can be characterized by the many available surface diagnostic techniques. In this approach to catalytic reaction studies the surface composition and structure are determined in the same chamber where the reactions are performed, without exposing the crystal surface to the ambient atmosphere. This necessitates the combined use of an ultrahigh vacuum enclosure, where the surface characterization is carried out, and a high-pressure isolation cell, where the catalytic studies are performed. Such an apparatus is shown in Fig. 46. The small-surface-area (approximately 1-cm2 ) catalyst is placed in the middle of the chamber, which can be evacuated to 10−9 torr. The surface is characterized by LEED and AES and by other surface diagnostic techniques. The

lower part of the high-pressure isolation cell is then lifted to enclose the sample in a 30-cm3 volume. The isolation chamber can be pressurized to 100 atm if desired and is connected to a gas chromatograph that detects the product distribution as a function of time and surface temperature. The sample may be heated resistively both at high pressure or in ultrahigh vacuum. After the reaction study the isolation chamber is evacuated, opened, and the catalytic surface is again analyzed by the various surface-diagnostic techniques. Ion bombardment cleaning of the surface or means to introduce controlled amounts of surface additives by vaporization are also available. The reaction at high pressures may be studied in the batch or the flow mode. Typical catalytic reactions that have been investigated, in some detail, using this approach include hydrocarbon conversion on platinum and modified platinum surfaces (isomerization, hydrogenolysis, hydrogenation, dehydrogenation and cyclization), dehydrosulfurization on molybdenum, ammonia synthesis on iron, and carbon monoxide hydrogenation on iron. F. Photochemical Surface Reactions Photochemical surface reactions form their own class due to the fact that a thermodynamically uphill reaction ( G > 0) may be carried out with the aid of an external source of energy, light. In fact, one of the most important chemical reactions of our planet, photosynthesis, requires the input of 720 kcal/mol of energy to convert carbon dioxide and water to one mole of sugar: light

6CO2 → 6H2 O−−−−−−→6H12 O6 + 6O2 chlorophyll

FIGURE 46 Schematic representation of the experimental apparatus to carry out catalytic reaction-rate studies on single-crystal surfaces of low surface area at low and high pressures in the range 10−7 to 104 torr.

(65)

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It is useful to consider light as one of the reactants in photosynthesis. By adding the light energy to Eq. (65), the reaction becomes athermic or even exothermic if excess light energy is utilized, hv + H2 O + CO2 = CH2 + 32 O2 We may consider photon-assisted or photochemical reactions of many types that lead to the formation of lower molecular weight hydrocarbons and of other products. One of the simplest of these important new classes of reactions leads to the dissociation of water: hv + H2 O = H2 + 12 O2

(66)

Another leads to the formation of methane: hv + CO2 + 2H2 O = CH4 + 32 O2

(67)

or to the fixation of nitrogen: hv + 3H2 O + N2 = 2NH3 + 32 O2

(68)

Light as a reactant may be employed in two ways. The adsorbed molecules can be excited directly by photons of suitable energy to a higher vibrational or electronic states. The excited species then may undergo chemical rearrangements or interactions that are different from those in the ground vibrational or electronic states. Alternatively, the solid can be excited by light in the near-surface region. Photons of band-gap or greater energy may excite electron–hole pairs at the surface. As long as these charge carriers have a relatively long lifetime (i.e., they are trapped at the surface, so that their recombination is not an efficient process), there is a high probability of their capture by the adsorbed reactants. These, in turn, can undergo reduction or oxidation processes using the photogenerated electrons and holes, respectively. The photographic process is one example of this type of surface photochemical reaction. However, we would like the photogenerated electrons and holes to be captured by the adsorbed molecules in order to carry out photochemical surface reactions of the adsorbates instead of the photodecomposition of the solid at the surface. The cross sections for adsorption of band-gap or higher-than-band-gap energy photons are so large that the photogeneration of electron–hole pairs is a most efficient process. At present, this cannot be readily matched by the efficiency of direct photoexcitation of vibrational or electronic energy states of the adsorbed molecules. Many solid surfaces efficiently convert light to longlived electron–hole pairs that can induce the chemical changes leading to the reactions in Eqs. (66)–(68). In fact, inorganic photoreaction is one of the exciting new fields of surface science and heterogeneous catalysis. It is important to distinguish between thermodynamically uphill photochemical reactions and thermodynam-

ically allowed photon-assisted reactions. The latter reactions are thermodynamically feasible without any external energy input, but light is used to obtain certain product selectively. Excitation of selected vibrations, rotations, or electronic states of the incident or adsorbed molecules by light permits us to change the reaction path or increase the reaction rate. For example, the hydrogenation of acetylene or the oxidation of ammonia can be photon-assisted, leading to different reaction rates than in the absence of light. As an example of a photocatalyzed surface reaction we discuss the photoelectrochemical dissociation of water. It was shown in 1972 that upon illumination of reduced titanium oxide (TiO2 ), which served as the anode in basic electrolyte solution, oxygen evolution was detectable at the anode, and hydrogen evolved at a metal (platinum) cathode. This reaction requires an energy of 1.23 V/electron (a two-electron process per dissociated water molecule). In the presence of light of energy equal to or greater than the band-gap energy of titanium oxide (3.1 eV), an external voltage as low as 0.2 V was sufficient to dissociate water. The process stopped as soon as the light was turned off, and started again upon reillumination. Shortly after, several other systems showed the ability to carry out photon-assisted dissociation of water. When p-type gallium phosphide, GaP, was used as a cathode instead of platinum upon illumination of the TiO2 anode, O2 and H2 could be generated at the semiconductor anode and cathode, respectively, without the need of applying any external potential. When strontium titanate, SrTiO3 , was substituted for TiO2 as the anode, H2 O photodissociation was found to take place without external potential even when a platinum cathode was employed.

FIGURE 47 Energy conditions needed to reduce B+ to B and oxidize A− to A at a semiconductor surface. Electrons that are excited by photons into the conduction band ECB must be able to reduce B+ , and electron vacancies (holes) in the valence band E VB must be able to oxidize A− .

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Figure 47 shows a schematic energy diagram to indicate the conditions necessary to carry out photoelectrochemical reactions efficiently. If the band-gap energy is greater than the free energies for the reduction and oxidation reactions, the photoelectron that is excited into the conduction band by light could reduce B+ to B by electron transfer from the surface to the molecule. The photogenerated electron vacancies (holes) could also oxidize the A− anions to A by capturing the electron. For the photodissociation of water, the conduction band must be above the H+ /H2 potential and the valence band below the O2 /OH− potential to be able to carry out the photoreaction without an external potential. The band gap must be greater than 1.23 V and the flat-band potential of the conduction and valence bands energetically well placed with respect to the (H+ /H2 ) and O2 /OH− couples. The flat-band potentials can be obtained by capacitance measurements as a function of external potential. There is, of course, considerable band bending of the conduction and valence bands of any semiconductor at the surface. This is due to the presence of localized electronic surface states and to charge transfer between the adsorbates and semiconductor. Potential-energy diagrams that show the band positions schematically at an n-type or ptype semiconductor liquid interface are shown in Fig. 48. The band bending provides an efficient means of separating electron–hole pairs, since the potential gradient as shown for the n-type semiconductor drives the electrons away from the semiconductor surface while it attracts the holes in the valence band toward the semiconductor electrolyte interface. As a result, the oxidation reaction takes place at the oxide anode while the reduction reaction takes place at the cathode to which the photoelectron migrates along the external circuit. The magnitude of the band bending at the surface depends primarily on the carrier concentration in the semiconductor and on the electron-donating or -accepting abilities of the adsorbates at the surface. Semiconductors that are not likely to carry out the photodissociation of water, according to the location of their

FIGURE 48 Band bending at the n-type and p-type semiconductor interfaces.

flat-band potential, may become photochemically active as a result of strong band bending at the surface. Often the oxidation or reduction photoreactions lead to the decomposition of the semiconductor electrode material. Instead of the photoreactions of adsorbate ions or molecules, a solid-state photoreaction occurs. This is particularly noticeable at the surfaces of illuminated CdS, Si, and GaP. Much of the research is therefore directed toward stabilizing these photoelectrode materials by suitable adsorbates that could prevent the occurrence of photodecomposition by providing an alternative chemical route for the photoreduction or photooxidation.

ACKNOWLEDGMENT This work was supported by the Assistant Secretary for Energy Research, Office of Basic Energy Sciences, Materials Sciences Division of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.

SEE ALSO THE FOLLOWING ARTICLES ADHESION AND ADHESIVES • ADSORPTION • AUGER ELECTRON SPECTROSCOPY • BONDING AND STRUCTURE IN SOLIDS • CATALYSIS, INDUSTRIAL • CATALYST CHARACTERIZATION • CHEMICAL THERMODYNAMICS • CRYSTALLOGRAPHY • PHOTOCHEMISTRY, MOLECULAR • PHOTOELECTRON SPECTROSCOPY • SOLID-STATE ELECTROCHEMISTRY • TRIBOLOGY

BIBLIOGRAPHY Adamson, A. W. (1982). “Physical Chemistry of Surfaces,” 4th ed., Wiley, New York. Anderson, J. R., and Boudart, M. (1981). “Catalysis Science and Technology,” Vols. 1–7, Springer-Verlag, Berlin/New York. Ertl, G., and Gomer, R., eds. (1983). “Springer Series in Surface Sciences,” Vols. 1–4, Springer-Verlag, Berlin/New York. Ertl, G., and Kuppers, J. (1979). “Low Energy Electrons and Surface Chemistry,” Verlag Chemie, Weinheim. Feuerbacher, B., Fitton, B., and Willis, R. F. (1979). “Photoemission and the Electronic Properties of Surfaces,” Wiley, New York. King, D. A., and Woodruff, W. P., eds. (1983). “The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis,” Vols. 1–4, Elsevier, New York. Morrison, S. R. (1977). “The Chemical Physics of Surfaces,” Plenum, New York. Roberts, M. W., and McKee, C. S. (1978). “Chemistry of the Metal-Gas Interface,” Oxford Univ. Press, London. Somorjai, G. A. (1981). “Chemistry in Two Dimensions: Surfaces,” Cornell Univ. Press, Ithaca, NY. Tompkins, F. C. (1978). “Chemisorption of Gases on Metals,” Academic Press, NY. Vanselow, R., and Howe, R., eds. (1979). “Chemistry and Physics of Solid Surfaces,” Vols. 1–6, Springer-Verlag, Berlin/New York.

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Ligand Field Concept Gunter Gliemann

Yu Wang

University of Regensburg

National Taiwan University

I. II. III. IV. V.

Basic Experimental Findings Model and Theory Optical Properties of Complexes Magnetic Properties of Complexes Stabilization of Complexes

GLOSSARY Charge transfer transition Change of the electronic state of a complex ion by the transfer of an electron from mainly the central ion state to mainly the ligand system state or vice versa. Coordination number Number of ligands bound at the central ion. Electron spin Intrinsic angular momentum vector of the electron. It is a quantum phenomenon, which has no analog in classical mechanics. Jahn-Teller effect For a nonlinear molecule in an electronically degenerate ground state, distortion must occur to lower the symmetry and to lower the energy to a more stable nondegenerate ground state. Ligand Atomic ion, molecular ion, or molecule coordinated at the central ion of a complex. Magnetic moment A property associated with a magnetic domain. It is an experimental measure of the magnetism of a compound, generally measured in units of magnetons. Orbital angular momentum Mechanical vector quantity perpendicular to the orbit of a particle. Its magni-

tude depends on the orbit diameter and the mass and velocity of the particle. Spin crossover complex Complex undergoes a spin transition induced by certain external factor such as temperature, pressure, light, etc. Tanabe-Sugano diagram Term splitting as a function of ligand field strength for 3d-transition metal ions in octahedral field, originated by Tanabe and Sugano. Term Entity of states of equal energy. Transition metals Elements with incompletely filled d orbitals: scandium, titanium, through copper (3d series); yttrium, zirconium through silver (4d series); lanthanum, hafnium through gold (5d series).

THE LIGAND FIELD CONCEPT is the basis of a quantum theoretical model developed in the 1950s for describing the electron systems of transition metal complexes. A transition metal complex is composed of a transition metal ion (central ion) surrounded by a system of ligands (atomic ions, molecular ions, or molecules). The ligands produce the electrical field (the ligand field) acting on the electron system of the central ion. As the ligand field theory shows,

523

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the optical, magnetic, and stability properties of transition metal complexes strongly depend on the symmetry and strength of the ligand field.

I. BASIC EXPERIMENTAL FINDINGS Crystallized salts of metal complexes have an amazing variety of brilliant colors, which are expressed by such prefixes as violeo-, praseo-, luteo-, purpureo-, and roseo(Table I). Investigations of the underlying chemical structures reveal a close relationship between color and composition, an immediate challenge for the spectroscopist. Therefore, after the foundation of coordination theory by A. Werner (1907) extended studies on the absorption spectra of complexes were investigated. The aim of these studies was to derive relations between the number and the position of absorption bands and the nature of the central ion and the ligands. Because the techniques and apparatus for measuring absorption spectra were still rather undeveloped, endeavors to interpret the spectra according to the theory of electrons remained unsatisfactory until the 1940s. But it was soon recognized that the existence of d electrons is significant for the color of transition metal ion complexes. The outstanding work of M. Linhard and coworkers, starting in 1944 with the investigation of Co3+ and Cr3+ complexes, set the standard for the precision of absorption spectroscopy of dissolved complexes. One of the main results of Linhard’s work is indicated schematically in Fig. 1. The absorption spectra of transition metal complexes can be divided into two spectral regions. In the longwavelength region (λ ≥ 350–400 nm) one finds one or more weak bands (extinction coefficient ε = 1–102 liters mol−1 cm−1 ). These bands do not appear when the central ion is not a transition metal ion (e.g., Al3+ instead of Cr3+ ; see Fig. 2). Therefore, these weak bands were assigned to transitions involving d electrons of the central ion (central ion bands, d-d bands). In the short-wavelength region (λ ≤ 350–400 nm) strong absorption bands (ε ∼ 103 – 106 liters mol−1 cm−1 ) are observed. These strong bands (ligand bands) are usually charge transfer bands, due to electron transfer between the central ion and the ligand system, or intraligand bands, caused by excitation of the electron system of the ligands.

FIGURE 1 A schematic absorption spectrum of a transition metal complex ion in the UV-VIS region. The central ion bands correspond to d →d transitions; the ligand bands correspond to charge transfer transitions and/or to intraligand transitions.

Stimulated by Linhard’s work, in 1946 F. E. Ilse and H. Hartmann formulated the ligand field concept, based on the classical ionic model of transition metal complexes (W. Kossel and A. Magnus) and on appropriate group theoretical methods (H. Bethe).

TABLE I Nomenclature Describing the Colors of Some Transition Metal Complexes TM complexes

Color

Prefix

cis-[Co(NH3 )4 Cl2 ]+ trans-[Co(NH3 )4 Cl2 ]+

Violet

Violeo-

Green

Praseo-

[Cr(NH3 )6 ]3+

Yellow

Luteo-

[Co(NH3 )5 H2 O]3+ [Co(NH3 )5 Cl]2+

Rose Purple red

RoseoPurpureo-

FIGURE 2 Absorption spectra of [Cr(C2 O4 )3 ]3− (full line) and ¨ H. L. (1957). Z. Phys. [Al(C2 O4 )3 ]3− (dotted line). [From Schlafer, Chem. 11, 65.]

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FIGURE 3 L-edge absorption spectra of Fe2+ in Fe(phen)2 (NCS)2 at 298 K (solid line) and at 15 K (dotted line). [From Lee, J. J., Sheu, H. S., Lee, C. R., Chen, J. M., Liu, R. G., Lee, J. F., Wang, C. C., Huang, C. H., and Wang, Y. (2000). J. Am. Chem. Soc. 122, 5742.]

In addition to the absorption spectra at UV-VIS range, the absorption in the much higher energy range has also been observed in recent years. Here the electron transition is between the inner core orbitals and the valence orbitals of the central ions. An example of Fe L-edge absorption is given in Fig. 3 to display the electron transition between 2 p and 3d orbitals. The magnetic properties of the transition metal complexes are known to exhibit quite a variety even with the same metal ion, for example, diamagnetic, and paramagnetic are found in Fe2+ , Co3+ complexes of various ligands (low-spin, high-spin complexes). The uneven distribution of electron density around the metal ion in a complex is demonstrated by the deformation density at the metal center shown in Fig. 4, where the spherical electron density is subtracted from the observed molecular electron density. Since the spherical density means that even populations are among five degenerated d-orbitals, the deformation density may give the direct observation on the difference among d-orbital population. All these recent experimental findings can be rationalized by the ligand field concept.

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525 tron pairs. Neither the ionic model nor the covalent model provided a satisfactory interpretation of several optical properties, since these models were concerned primarily with the electronic ground state of the complex ions. For a large number of complexes certain spectral regions of the absorption spectra can be assigned alternatively to the different components of the complex (central ion, ligand system). On this account a model that starts with the assumption of separated electron systems for the central ion and for the ligands will be appropriate for a theoretical treatment. If one is interested primarily in the electronic states of the central ion bound in the complex, one has to consider the electric field generated by the ligand system, socalled the ligand field. The effect of this ligand field on the electronic states of the central ion is then taken into consideration. The theory treating this concept of the ligand field is the ligand field theory, where the electronic structure of the ligand system is taken explicitly into account. In other words, the complete molecular orbital treatment of the complex is undertaken. To a first approximation the charge distribution of the ligands is represented by point charges and/or point dipoles in their centers, the ligand field treatment can be reduced to an atomic orbital treatment of the central ion, this extended ionic model is designated as crystal field theory. The starting point of the crystal field theory is the description of the electronic states of the free, isolated transition metal ion (the central ion) in a complex. Information on the electronic ground state and the excited states

II. MODEL AND THEORY In the 1920s and 1930s two apparently contrary models, the ionic model and the covalent model, were developed to explain the binding between the central ion and the ligands, which were both represented by point charges and point dipoles. In the ionic model of Kossel and Magnus, the binding between the central ion and the ligands is due to the electrostatic forces between the components. In the covalent model of Sidgwick and Pauling, the binding between the central ion and the ligands is accomplished by elec-

FIGURE 4 Deformation density of Ni(disn)2 plane around Ni, solid line positive, dotted line negative, contour interval is 0.1 eA˚ −3 . [From Lee, C. S., Hwang, T. S., Wang, Y., Peng, S. M., and Hwang, C. S. (1996). J. Phys. Chem. 100, 2934–2941.]

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of the free ions is available from the analysis of the corresponding absorption or emission spectra and/or from quantum mechanical calculations. A. Electronic States of a Free Ion The electronic states of a free ion can be characterized by their energy and by their angular momentum, which includes the orbital angular momenta and the spins of the electrons involved. For atoms with atomic number Z ≤ 30, the Russell-Saunders coupling is a good approximation. In this approximation the orbital angular momenta of electrons are added vectorically to the total orbital angular momentum L and the spins of the electrons are coupled to the total spin S:   Ij = L; sj = S j

j

where lj and sj are the orbital and the spin angular momentum of the jth electron, respectively. As shown by quantum mechanics the absolute values of the vectors L and S are restricted to certain discrete amounts  |L| = h L(L + 1), L = 0, 1, 2, 3, . . .  |S| = h S(S + 1),  0, 1, 2, 3, . . . for even number of electrons S= 1/2, 3/2, 5/2, 7/2, . . . for odd number of electrons where h is h/2π (h is Planck’s constant); L and S is the quantum number of the total orbital angular momentum and the total spin of the system of electrons, respectively. For the numerical values L = 0, 1, 2, 3, . . . it is conventional to use the letters S, P, D, F, . . . By quantum mechanical rules spin S with quantum number S can take 2S + 1 to different special orientations with respect to a given direction shown in Fig. 5. The 2S + 1 is denoted as multiplicity M. Correspondingly, an orbital angular momentum L with quantum number L can assume 2L + 1 different spatial orientations. Therefore, a given set of quantum numbers L and S can assume a total of (2L + 1)(2S + 1) states with different orientations of orbital and/or spin angular momentum. These states form a Russell-Saunders term, symbolized by 2S+1 L. All (2L + 1)(2S + 1) states of the same term have equal energies, if the coupling between L and S is ignored. They are energetically “degenerate.” Usually, different terms have different energies. By Hund’s rule the term with the highest multiplicity M and the highest L value is the ground-state term (energetically most stable term). The energy difference between the terms is expressed in terms of B, an energy parameter of electronelectron repulsion. For example, a d 2 ion (two d electrons) will result in various states with multiplicity 1 (singlet; spin configu-

FIGURE 5 Spatial orientations of a spin vector S with S = 32 . There are 2S + 12 = 4 different allowed values of projection on the z axis.

ration with the two electron spins antiparallel, ↑↓) and 3 (triplet, ↑↑). There are, in total, two triplet terms and three singlet terms: 3 P, 3 F, 1 S, 1 D, and 1 G. The 3 F [composed of (2 × 3 + 1)(2 × 1 + 1) = 21 states] is the groundstate term. The five Russell-Saunders term of a d 2 ion are shown in the energy-level diagram of Fig. 6. The complete sets of Russell-Saunders terms for the d N ions with N = 1, 2, . . . , 9 are given in Table II. TABLE II Russell-Saunders Terms for Free dN Ionsa Occupation of the d shell d1, d9 d2,

d8

Russell-saunders term 2D 3 F, 3 P 1 G, 1 D, 1 S

d3, d7

4 F, 4 P 2 H, 2 G, 2 F, a2 D, 2 P

d4, d6

5D 3 H, 3 G, a3 F,

b3 F, 3 D, a3 P, b3 P

1 I, a1 G, b1 G, 1 F, a1 D, b1 D, a1 S, b1 S

d5

6S 4 G, 4 F, 4 D, 4 P 2 I, 2 H, a2 G, b2 G, a2 F, b2 F, a2 D, b2 D, c2 D, 2 P, 2 S

a If for a d N ion (N = 1, . . . , 9) several terms with the same L value and the same S value exist, they are distinguished by prefixes a, b, c. Terms with underlined L symbols are the ground terms for the first d N configuration.

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FIGURE 6 Russell-Saunders terms of a d 2 free ion with and without spin-orbital coupling.

If the coupling between the total orbital angular momentum, L, and the total spin angular momentum, S, has to be taken into account, the total angular momentum J should be used. Where J is the vector sum of L and S. The quantum number of the vector J is again restricted to certain discrete amounts. J = L + S, J = |L + S|, |L + S − 1|, . . . . . . . |L − S|  |J| = h J (J + 1) The energy term is symbolized as 2S+1 L J , for example, the ground state of d 2 (3 F) is split into 3 F4 , 3 F3 , and 3 F2 , where the order of energy is such that 3 F4 > 3 F3 > 3 F2 as shown in Fig. 6. Each term is in 2J + 1 degeneracy and can be separated by applying magnetic field, i.e., the Zeemann effect. However, the splitting due to the L − S coupling is much smaller than the splitting due to the electron-electron repulsion designated the energy difference between the Russell-Saunders terms.

B. Crystal-Field Theory In the course of forming a complex, the central ion is bound to be affected by the ligand field (electrical field of the ligands), and thereby the motion modes of the electrons of the central ion will be perturbed. Accordingly, the term system of the central ion will be changed. Some terms of the free ion are energetically merely shifted, while others are split into progeny terms 2S+1 i , i = 1, 2, . . . with different energies (intracomplex Stark effect). The symbol i describes the orbital state of the ith progeny term. The number of the progeny terms 2S+1 i can be exactly determined by the methods of group theory. It depends on the L value of the (parent) term 2S+1 L and on the symmetry of the ligand system: Number of progeny terms 2S+1 i = f (L, symmetry) 1. Term Splitting Under Oh Symmetry The general group theoretical result can be illustrated by a simple model system. First we consider a system of six

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FIGURE 7 Schematic probability distributions and energy states of a p electron (L = 1) in an octahedral and square-planar ligand field induced by negative point charges.

ligands (represented by electrical negative point charges) at the corners of a regular octahedron. That is, the central ion is subjected to an octahedral environment, Oh symmetry. The central ion within this system is to be varied with regard to its orbital angular momentum (quantum number L). To simplify the problem the central ion may contain only one electron, either a p electron (L = 1) or a d electron (L = 2). The corresponding probability distributions of the electron charge known from the theory of atoms are shown in Fig. 7 and Fig. 8. The p electron can occupy three different orbital states (2L + 1 = 3). All three distributions are in equivalent positions with regard to the ligands (Fig. 7). Therefore, the energies of the three

Ligand Field Concept

states will be shifted by the same amount when the ligand field is acting on the central ion. No splitting will be observed. For a d electron (2L + 1 = 5), however, there are distributions that are obviously not equivalent with regard to the ligands (Fig. 8). The maxima of the distributions dx y , dx z , and d yz are directed equivalently into the angular bisectors between the bonds. Therefore, these three states will have the same energy. The distribution dz 2 and dx 2 −y 2 have maxima along the bonds to the ligands located on the x, y, and z axis. It follows that the dx 2 −y 2 and dz 2 states will have higher energy than the dx y , dx z , and d yz states because of the stronger repulsion between the electron of central ion and the point charges of the ligands. Quantitative calculations show that the dz 2 state is energetically degenerate with the dx 2 −y 2 state. Therefore, the fivefold D term of a d 1 ion is split into a threefold state t2 and a twofold state e (Fig. 8). From these two examples we see that for the same symmetry of the ligand field the number of progeny terms depends on the quantum number L of the orbital angular momentum of the electron system. Under the influence of an octahedral symmetry of the ligand field, P terms (L = 1) are merely shifted, whereas D terms (L = 2) are split into a twofold (e) and a threefold ( t2 ) degenerate term. Table III summarizes the resulting term-splitting under the octahedral field for L values up to 4. The numbers in parentheses give the orbital degeneracy of the terms. A, B, E, and T symbolize different orbital symmetries of the terms, i.e., the progeny term 2S+1 i mentioned above, where A and B represent nondegenerate, E represents twofold degenerate, and T represents threefold degenerate terms. The term-splitting of the ground states of d N complexes with N = 1 to 5 in an octahedral ligand field are shown schematically in Fig. 9. The d N and d 10−N ions exhibit equivalent splitting diagrams, for example, the ground state of d 8 (3 F) exhibit the same splitting diagram as that of d 2 (T1 + T2 + A1 ) but the order of energy is inverted. Thereby the order of energy is T1 < T2 < A1 for d 2 , but is A2 < T2 < T1 for d 8 . This is a consequence of the so-called electron-hole correlation. 2. Term-Splitting Under D4h Symmetry

FIGURE 8 Schematic probability distributions and energy states of a d electron (L = 2) in an octahedral and square-planar ligand field induced by negative point charges.

For the same quantum number L of the orbital angular momentum, the number of progeny terms depends on the symmetry of the ligand system. When the octahedral symmetry of the ligand system is reduced to a square-planar (D4h ) symmetry by canceling the ligands on the z axis, the pz distribution is subjected to a weaker field than the px and p y distributions. Thus lowering of the symmetry from Oh to D4h yields a term-splitting of T1 to A1 + E. In the same way, the D terms (L = 2) are split further into A1 + B1 + B2 + E terms (Fig. 8). Such splitting is demonstrated by the uneven population of B1 (dx 2 −y 2 ) and B2

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Ligand Field Concept TABLE III Splitting of Orbital States with L = 0, 1, 2, 3, 4 in an Octahedral (Oh ) and a Square-Planar (D4h ) Ligand Field Orbital state of the free ion Quantum number L

State symbola

0 1 2

S(1) P(3) D(5)

→A1 (1) →T1 (3) →E(2) + T2 (3)

→A1 (1) →A2 (1) + E(2)

3

F(7)

4

G(9)

→A2 (1) + T1 (3) + T2 (3) →A1 (1) + E(2) + T1 (3) + T2 (3)

→A2 (1) + B1 (1) + B2 (1) + 2E(2) →2A1 (1) + A2 (1) + B1 (1) + B2 (1) + 2E(2)

a

Orbital states in an Oh ligand field

Orbital states in a D4h ligand field

→A1 (1) + B1 (1) + B2 (1) + E(2)

The numbers in parentheses give the orbital degeneracy of the states.

(dx y ) terms of Ni2+ in roughly D4h symmetry as shown in Fig. 4. As for the splitting of the other terms, they are listed in Table III. It is worth noticing that the lower the symmetry is, the more splitting of the terms occurs. In D4h , the highest degeneracy is E, whereas in Oh , it is T. 3. Weak- and Strong-Field Methods, Term Diagrams The amount of the energetic splitting or shifting can be (approximately) determined by the methods of quantum mechanical perturbation theory. Since the perturbation comes from the electrical interaction between the electrons of the central ion and the charge distribution within the ligand system, the magnitude of energetic splitting or shifting will depend on the central ion-ligand distance R and on the charges q and electrical dipole moments µ of the ligands: Magnitude of energetic splitting and/or shifting = F(R, q, µ) Since the charge distributions within the ligands are not known exactly, an absolute calculation of these energetic effects is very tedious and time-consuming. In practice these magnitudes (as functions of R, q, and µ) are taken

as parameters in the calculation and are fit to experimental data. Normally a value in Dq is represented. Since both the electron-electron interaction and the influence of the ligand field have to be treated, there are two methods for finding the term system of a transition metal complex when the central ion contains two or more d electrons. Both methods start with the free d N ion of which the electron-electron repulsion is not yet taken into account. They differ only in the order of which one is treated first. According to the expected relative amounts of these energetic quantities, the weak-field method is employed when the effect of electron-electron interaction dominates, whereas the strong-field method is appropriate when the influence of the ligand field is dominant. Complete treatments of a complex ion by both methods will ultimately yield the same results. The weak-field method is described by the following scheme: d N −−−−−−→ 2S+1 L −−−−−→ 2S+1 electr on−electr on interaction

ligand f ield

The first step considers the electron-electron interaction. This step gives result in the Russell-Saunders terms 2S+1 L of the free ion following the procedure described in section II.A. In the second step the splitting of these terms by the ligand field is determined, following the procedure given in Section II.B.1 or B.2. The two steps of the strong-field method have the opposite order: d N −−−−−→ t2n e N −n −−−−−→ 2S+1 ligand f ield(Oh )

FIGURE 9 Term-splitting of the ground states of d 1 to d 5 and d 8 in an octahedral ligand field. Term-splitting of the ground states of d N , N > 5 is the same as that in d 10−N , but with the energy order inverted, see d 8 versus d 2 .

electr on interaction

In the first step the splitting of one-electron states of the d shell in the ligand field is considered. Under the octahedral ligand field, the five orbitals of the d shell are split into a family of two degenerate states e(dx 2 −y 2 , dz 2 ) and a family of three degenerate states t2 (dx y , dx z , d yz ) shown in Fig. 8. By quantum mechanical perturbation theory the energy difference between e and t2 is calculated as ≡ 10 Dq. The N d electrons can occupy the levels e and t2 following Pauli’s principle, such as t2n e N −n . In the second step of the

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increasing Dq value the terms are either shifted (L ≤ 1) or split (L > 1). This Tanabe-Sugano diagram will provide nice interpretation to the optical properties of Cr3+ in ruby (Section III). However, when central ion is d N with N = 4 ∼ 7, not only the shifting and splitting of the terms occurred as indicated in d 3 ion. There also appears to be an apparent change at certain Dq/B value (VT ). An example of Fe2+ complex in an Oh environment, a Tanabe-Sugano diagram of d 6 , is shown in Fig. 12. The ground state of the strongfield (Dq/B > VT ) is 1 A1 , a low-spin state, but that of weak-field (Dq/B < VT ) is 5 T2 , a high-spin state. When the Dq/B value is very close to VT , the spin crossover phenomenon occurs, where the spin state of the central ion can be fine tuned by varying temperature or pressure. The light-induced excitation of the spin state was also observed.

FIGURE 10 Term correlation diagram of d 2 in weak- and strong ligand fields.

strong-field method, the electron-electron interaction is taken into account. Take d 2 as an example. The weak- and strong-field methods are illustrated on the left and right side of Fig. 10, respectively. The solid and dotted lines in the middle indicate the correlation between two methods. This means the two methods will ultimately yield exactly the same result in term splitting. The influence of the ligand field can be conveniently presented in the form of diagrams showing the term energies as functions of the strength of the ligand field. For systems with octahedral symmetry the energies of the terms 2S +1 depend on the single parameter 10 Dq defined above. An example of the term diagram for Cr3+ (d 3 ) in an octahedral environment is shown in Fig. 11, the Tanabe-Sugano diagram of d 3 . When Dq = 0, the termsplitting of the free Cr3+ ion are as in Table III. With

FIGURE 11 Term diagram of a Cr 3+ ion in an octahedral ligand field. Term energies as functions of the ligand field strength Dq, Tanabe-Sugano diagram of d 3 . [From Tanabe, Y., and Sugano, S. (1954). J. Phys. Soc. Jpn. 9, 753.]

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tor is a characteristic of the central ion. The influences of the central ion and of the ligands on Dq value are as follows: 1. Influence of the central ion on Dq. For the same ligand system, (a) The Dq values are very similar for transition metals of the same series and the same charge. For example, the Dq value of the complexes [Mn(H2 O)6 ]2+ , [Fe(H2 O)6 ]2+ , and [Ni(H2 O)6 ]2+ is 8500, 10,400 and 8500 cm−1 , respectively. This indicates that divalent 3d transition metal ions have roughly the same Dq value. (b) The Dq values increase when going from the first to the second and to third series of transition metal ions, i.e., for the analogous complexes, the trend being 3d < 4d < 5d. As an example, for [M(NH3 )6 ]3+ complexes, the Dq value is 2287, 3400 and 4120 cm−1 for M = Co3+ , Rh3+ , and Ir3+ , respectively. TABLE IV Dq Values of Six-Coordinate Transition Metal Complexes

FIGURE 12 Tanabe-Sugano diagram of d 6 . [From Tanabe, Y., and Sugano, S. (1954). J. Phys. Soc. Jpn. 9, 753.]

4. Dq Values and Spectrochemical Series For a given central ion the field strength parameter Dq depends on the central ion-ligand distance (R) and on the charge distribution within the ligands (q, µ). The value of Dq is determined from experimental data. By comparison of the optical absorption data of octahedral complex ions with the Tanabe-Sugano term diagram, the 10 Dq (or ) values for several transition metal complexes are given in Table IV. Jørgensen developed a means of estimating the value of 10 Dq for an octahedral complex by treating it as the product of two independent factors. 10 Dq ≈ f (ligand) × g(central ion) Where the factor f describes the field strength of ligand relative to water, which is assigned to 1.0. The g fac-

f

g (cm−1 )a

(cm−1 )b

[CrF6 ]3− [Cr(H2 O)6 ]3+ [Cr(en)3 ]3+ [Cr(CN)6 ]3−

0.9 1

17,400 17,400

15,060 17,400

1.28

17,400

22,300

1.7

17,400

26,600

[Mo(H2 O)6 ]3+ [MnF6 ]2−

1 0.9

24,600 23,000

26,000 21,800

[TcF6 ]2−

0.9

30,000

28,400

[Fe(H2 O)6 ]3+

1

14,000

14,000

[Fe(ox)3 ]3−

0.99

14,000

14,140

[Fe(CN)6 ]3−

1.7

14,000

35,000

[Ru(H2 O)6 ]2+

1

20,000

19,800

[Ru(CN)6 ]4−

1.7

20,000

33,800

[CoF6 ]3−

0.9

18,200

13,100

[Co(H2 O)6 ]3+

1

18,200

20,760

[Co(NH3 )6 ]3+

1.25

18,200

22,870

[Co(en)3 ]3+

1.28

18,200

23,160

[Co(H2 O)6 ]2+

1

9,000

92,00

[Co(NH3 )6 ]2+

1.25

9,000

10,200

[Rh(H2 O)6 ]3+

1

27,000

27,200

[Rh(NH3 )6 ]3+

1.25

27,000

34,100

[Ir(NH3 )6 ]3+

1.25

32,000

41,200

Note: ox = oxalate = C2 O2− 4 ; en = ethylenedia-mine = HN2 CH2 CH2 NH2 . a Jørgensen, C. K. (1971). “Modern Aspects of Ligand Field Theory,” Chap. 26, Elsevier, New York. b Experimentaln data from Lever, A. B. P. (1986). “Inorganic Electronic Spectroscopy,” 2nd ed., Chaps. 6 and 9, Elservier, New York.

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(c) The Dq values increase with increasing ionic charge on the central ion, for example, the Dq value of [CrF6 ]3− and [CrF6 ]2− is 1506 and 2200 cm−1 , respectively. 2. Influence of the ligands on Dq. For the same central ion, (a) The Dq values increase with the number of the ligands, for example, the Dq value of [Co(NH3 )4 ]2+ and [Co(NH3 )6 ]2+ is 590 and 1020 cm−1 , respectively. (b) The Dq value increases in the order of the spectrochemical series: − I− < Br− < C1− ∼ SCN− ∼ N− 3 < (C2 H5 O)2 PS2 − − − < F < (C2 H5 )2 NCS2 < (NH2 )2 CO < OH − − < (COO)2− 2 ∼ H2 O < NCS < NH2 CH2 COO + < NCSHg ∼ NH3 ∼ C5 H5 N < NH2 CH2 CH2 NH2 − − − − ∼ SO2− 3 < NH2 OH < NO2 < H ∼ CH3 < CN

An underlined atomic symbol indicates that the ligand is coordinated with that atom.

C. Ligand-Field theory In the crystal-field model, it is assumed that the electrons of central ion are perturbed by the crystal field in the form of point charges located at the coordinated atoms of the ligand. Basically the crystal-field approach is still limited at the atomic orbital level, nevertheless, it did interpret successfully on many properties of the transition metal complex (see the following sections). However, purely based on the point charge model, it is hard to rationalize why in the spectrochemical series CN− is such a strong field, but F− is such a weak field. Apparently there is a need of improving the crystal-field model in order to reason the order of spectrochemical series. To improve the model of the crystal-field theory, the electrons are allowed to move over the whole complex ion in molecular orbitals. In other words, we have to consider the complex at a molecular orbital level. In a common approximation, the molecular orbitals are described by suitable linear combination of atomic orbitals (LCAO approximation). In this approach, in addition to d, p, and s orbitals of the central ion, the orbitals of the ligand system will also be included. In recent years, due to the great improvement of both hardware and software in computation, an ab initio quantum calculation of such complexity becomes feasible. However, in this content we will keep it as conceptual as possible. Namely, the group orbitals of the ligand will be included only in the form of σ and π bond. As shown by quantum mechanics and group theory only certain linear combinations yield energetic effects:

FIGURE 13 Combinations of a central ion d : (a) dx2 −y2 (b) dxy orbital, and (c) the σ group orbitals of the ligand σx + σ−x − σ y − σ−y .

1. The combination orbitals must have the same symmetry. For example, in Fig. 13 the dx 2 −y 2 orbital (Fig. 13a) of the central ion has the same symmetry as the σ group orbitals of the ligand system shown in Fig. 13c, therefore a combination between these two can be made. But the dx y orbital (Fig. 13b) has different symmetry from the σ group orbitals; thus no combination can be formed between the dx y orbital and the σ group orbitals. 2. The energies of the combination orbitals must be of comparable magnitude for significant interaction to occur. 3. The combining orbitals must overlap. Every pair of orbitals suitable of forming linear combination yields a stabilized (bonding) and destabilized (anti-bonding) states shown in Fig. 14. The anti-bonding state is usually labeled by an asterisk. When the energy difference between the combining orbitals is small, the energy splitting becomes large and the interaction between two orbitals is strong (Fig. 14a). The energy-level diagram of the molecular orbitals of an octahedral complex is presented in Fig. 15a, where only σ bonds between the central ion and the ligand are considered. According to the group theoretical treatment, the group orbitals of the ligand in the form of σ bond consist of a1 + e + t1 orbitals. The corresponding orbitals of the central ion

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FIGURE 14 Two atomic orbitals form a bonding and an antibonding state: (a) with small difference in energy of A and B and (b) with large difference in energy of A and B.

include a1 (s), t1(u) ( px , p y , pz ) and e(dx 2 −y 2 , dz 2 ), i.e., an sp 3 d 2 hybrid. These orbitals combine with the group orbitals of ligand and yield six σ bonding and six σ ∗ anti-bonding states, each contains a1 , e, t1 , and a1∗ , e∗ , t1∗ , respectively. The central ion orbitals t2(g) (dx y , d yz , dzx ) are “nonbonding” since there are no σ orbitals of the ligands with suitable symmetry as indicated in Fig. 13. Each coordinated ligand atom contributes two σ -electrons (σ -donor) to fill up the σ -bonding orbitals a1 , t1 , e. Therefore, the d electrons of the central ion will occupy the t2(g) and e∗ orbitals. The energy gap between t2(g) and e∗ corresponds to the crystal field parameter 10 Dq. If a π -bond interaction between the central ion and the ligand is taken into account, the group orbitals of the ligand in the form of π , π ∗ consist of t1(g) , t2(g) , t1(u) , t2(u) orbitals. The t2(g) orbitals of the central ion (dx y , d yz , dzx ) are no

FIGURE 15 Molecular orbital diagram of an octahedral complex considering. (a) σ -bonding only. (b) σ - and π-bonding between the central ion and the ligand.

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FIGURE 16 Comparison of the ligand field strength, value of ligands with σ -donor, π -donor, and π-acceptor character.

longer nonbonding orbitals, they can combine with πL or πL∗ group orbitals of the ligand shown in Fig. 15b. This π -interaction can either increase or decrease the 10 Dq value (of which only σ -bond is considered) depending on whether the ligand is a π -acceptor or a π -donor. When the ligand is served as a π -acceptor, for example CN− , where a low-lying empty πL∗ is available, the t2(g) orbitals of the central ion are stabilized by the π -interaction, therefore increasing the 10 Dq value. On the other hand, when the ligand is served as a π -donor, for example, fluoride F− ion, the t2(g) orbitals of the central ion are destabilized and thus decrease the 10 Dq value. The effect of this πinteraction on the 10 Dq value is illustrated in Fig. 16. With this π -interaction in mind, it is not too hard to understand the order of ligand field strength in the spectrochemical series given above. The energetic order within the system of progeny terms 2S+1 i (A1 , T1 , E, etc.) determines the important properties of the complex ions: 1. Optical properties. The energetic differences between the ground state and the few lowest excited states, mainly d-d transition, correspond to the absorption bands in the spectral region with λ ≥ 300–700 nm, with extinction coefficient of 100 –102 . There are also bands with much higher extinction coefficient (102 –106 ) which normally correspond to charge transfer band or intraligand transition. These bands are responsible for the color of the compound (Section III). 2. Magnetism. The spin multiplicity M = 2S + 1 of the ground term determines roughly the magnetic behavior of the complex ion (Section IV). 3. Stability. The stabilization of the ground term by the ligand field stabilization energy represents an

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increase in the binding energy of the complex ion, in addition to that obtained by the ionic model of Kossel and Magnus (Section V).

III. OPTICAL PROPERTIES OF COMPLEXES The main features of the optical absorption and emission spectra of transition metal complexes can be interpreted on the basis of crystal-field or ligand-field theory. Generally, the energies of the absorption and the emission bands correspond to energetic differences between electronic states. Therefore, an interpretation of the optical spectra will start with a comparison between the experimental spectra and the term diagram of the complex ion, according to the crystal-field theory. This procedure will be demonstrated by two informative examples. The absorption spectrum of the [Ti(H2 O)6 ]3+ ion consisting of one d-d band with its maximum at 492 nm is shown in Fig. 17. The ground state of this Ti3+ (d 1 ) free ion is 2 D (Table III). In an octahedral ligand field the 2 D term is split into the low-lying 2 T2 ground state and a 2 E state at higher energies (Fig. 9). By the absorption of a photon with energy of E(2 E−2 T2 ), the complex ion will be excited from its ground state 2 T2 into the state 2 E. The excitation energy in an octahedral d 1 ion is by definition equal to 10 Dq. From the wavelength of the absorption maximum at 492 nm, it follows that Dq has a value of ∼ 2030 cm−1 . In the molecular orbital theory the 492-nm band corresponds to the electron transition between the nonbonding t2 state and the anti-bonding e∗ state (see Fig. 15). The

FIGURE 17 Absorption spectrum of [Ti(H2 O)6 ]3+ . [From Hartman, H., Schlafer, H. L., and Hansen, K. H. (1957). Z. Anorg. Chem. 40, 289.]

FIGURE 18 Absorption (a) and emission (b) spectra of Cr3+ ion in ruby.

strong increase in absorption below ∼ 350 nm belongs to the charge transfer band, possibly from t2 to empty ligand excited states of suitable symmetry and spin states. The second example is the absorption spectrum of Cr3+ in ruby shown in Fig. 18. Two relatively strong bands I and II and three very weak absorption J1 , J2 , and J3 can be seen. At wavelengths below 300 nm a very strong increase in the absorption due to charge transfer transitions is observed (not shown in Fig. 18). Ruby is an Al2 O3 crystal wherein some Al3+ ions are substituted by Cr3+ ions. The absorption spectrum (Fig. 18) is due to the Cr3+ ions (d 3 ions) since Al3+ does not have any d electrons. This can be confirmed by the absorption spectra of Cr(ox)3− 3 and 3+ Al(ox)3− shown in Fig. 2. Every Cr ion is surrounded 3 by six oxygen ions forming an octahedron. The three lowest terms of the free Cr3+ ions are 4 F (ground state), 4 P, and 2 G (Table II). In the presence of the Oh ligand field, the terms 4 F and 2 G are split into three (4 A2 , 4 T2 , 4 T1 ) and four (2 E1 , 2 T1 , 2 T2 , 2 A1 ) progeny terms, respectively. A comparison between this term diagram (Fig. 11) of Cr3+ in ruby and the absorption spectrum yields the following assignment. The relatively intense bands I and II belong to the spin-allowed transitions 4 A2 → 4 T2 and 4 A2 → 4 T1 , respectively, whereas J1 , J2 , and J3 are due to quartet-doublet spin-forbidden transitions (4 A2 → 2 E, 4 A2 → 2 T1 , 4 A2 →2 A1 ). On the basis of this assignment and the corresponding Tanabe-Sugano diagram (Fig. 11), one can determine the Dq and B values of the complex. The significant intensity difference between the highenergy charge transfer bands (λ < 300 nm), the bands I and II, and the weak bands J1 , J2 , and J3 originates in the different nature of the corresponding transitions. As shown by quantum theory, the absorption of photons by

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Ligand Field Concept

molecules is regulated by some general rules or the socalled spectra-selection rules: 1. The total spin of the electron system remains constant (spin-allowed). 2. The direct product of the symmetries corresponding to ground state, excited state, and the optical transition moment should be totally symmetric (symmetry allowed or LaPorte allowed) These two criteria are fulfilled by most of charge transfer bands. The d → d transitions, however, are distinctly weaker than the charge transfer transitions, since they are LaPorte forbidden. Among the d → d transitions the spinallowed transitions show a significantly stronger absorption (bands I and II) than the spin-forbidden transitions (bands J1 , J2 , and J3 ). Ruby was the first crystalline compound to exhibit optical laser properties. The ruby laser works as follows. The excited quartet terms 4 T2 and 4 T1 (ref. to Fig. 11) are populated by irradiation (optical pumping) using the broad absorption bands at I and II from the ground term 4 A2 . Then radiationless transitions into the 2 E state take place within a period of <10−10 sec. The electron at 2 E state is stable for a relatively long time ( ∼5 × 10−3 sec) before the photon emission occurs. Therefore, the 2 E state is populated more and more by optical pumping, and finally the number of Cr3+ ions in 2 E excited state is larger than the number in the 4 A2 ground state (population inversion). When this “laser condition” is fulfilled, the accumulated energy can be emitted as an intense beam of photons with energy ∼14000 cm−1 ( E(2 E−4 A2 )), the brilliant red beam of the ruby laser. The X-ray absorption spectra can probe the atomspecific transition between inner core orbitals and valence orbitals. Take L-edge absorption as an example, it is the transition between 2s, 2 p orbitals and 3 p, 3d orbitals: LI -edge absorption is assigned to the transition between 2s and 3 p. LII -, LIII -edge absorption spectra are assigned to the transition between 2P1/2 , 2P3/2 , and 3d transition. The LII -, LIII -edge absorption spectra of Fe2+ in high-spin (solid line) and in low-spin state (dotted line) are displayed in Fig. 3. It clearly demonstrates that the difference in d orbital populations is distinctively reflected by these L-edge absorption spectra. The multiplet calculation according to the crystal-field theory, including the L-S coupling, can be utilized to reproduce the experimental measurement and estimate the Dq value. The observed and calculated LII -, LIII -edge absorption spectra of the low-spin Fe2+ in Fe(phen)2 (NCS)2 complex, are shown in Fig. 19. The corresponding 10 Dq values are 7420 and 17,750 cm−1 for high- and low-spin complex, respectively.

FIGURE 19 The observed (top) and calculated (buttom) LIII,II -edge absorption spectra of Fe2+ in low-spin state. [From Lee, J. J., Sheu, H. S., Lee, C. R., Chen, J. M., Liu, R. G., Lee, J. F., Wang, C. C., Huang, C. H., and Wang, Y. (2000). J. Am. Chem. Soc. 122, 5742.]

IV. MAGNETIC PROPERTIES OF COMPLEXES Every electron has a permanent magnetic moment of amount µB = 9.27 × 10−24 J T−1 (Bohr’s magneton, BM), which is parallel to its spin s. For an electron system with total angular momentum quantum number J , the magnetic moment is µ = g[J (J + 1)]1/2 BM where g is the Land´e splitting factor, and g = 1 + {[J (J + 1) + S(S + 1) − L(L + 1)]/2J (J + 1)}. For complexes in which the L-S coupling √ is negligible, the magnetic moment becomes µ = 4S(S + 1) + L(L + 1) BM. However, in most 3d transition metal complexes, the orbital contribution is insignificant, so the effective magnetic moment becomes √ √ µef f = 4S(S + 1) = n(n + 2) BM, where n designates the number of unpaired electrons. This is a spin-only model. The magnetic properties of some 3d transition metal complexes are listed in Table V, the calculated

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Ligand Field Concept TABLE V Magnetic Properties of Some Complexes of the First-Row Transition Metalsa High-spin complexes Central metal

No. of d electrons

No. of unpaired electrons

Ti3+ V4+

1 1

V3+ V2+ Cr3+

Low-spin complexes

µ(expt) BM

µ(calc)b BM

No. of unpaired electrons

1

1.73

1.73

—

—

—

1

1.68–1.78

1.73

—

—

—

2 3

2 3

2.75–2.85 3.80–3.90

2.83 3.88

— —

— —

— — —

µ(expt) BM

µ(calc)b BM

3

3

3.70–3.90

3.88

—

—

Mn4+

3

3

3.80–4.00

3.88

—

—

—

Cr2+

4 4 5

4 4 5

4.75–4.90 4.90–5.00 5.65–6.10

4.90 4.90 5.92

2 2 1

3.20–3.30 3.18 1.80–2.10

2.83 2.83 1.73 1.73

Mn3+ Mn2+ Fe3+

5

5

5.70–6.00

5.92

1

2.00–2.50

Fe2+

6

4

5.10–5.70

4.90

—

—

—

Co3+

6 7

4 3

— 4.30–5.20

4.90 3.88

— 1

— 1.8

— 1.73

Co2+ Ni3+

7

3

—

3.88

1

1.80–2.00

1.73

Ni2+

8

2

2.80–3.50

2.83

—

—

—

Cu2+

9

1

1.70–2.20

1.73

—

—

—

a b

Burger, K. (1973). “Coordination Chemistry: Experimental Methods,” Butterworth, London. µ is spin-only value.

µef f values based on the spin-only model are in adequate agreement with the corresponding experimental ones. As mentioned above, the magnetic properties are directly related to the spin multiplicity of the ground state, or the number of unpaired electrons at the ground state. In the case of d 1 ∼ d 3 and d 8 ∼ d 10 of octahedral complexes, the spin multiplicity of the ground state keeps the same (high-spin state) even at very strong ligand field. But in the case of d 4 ∼ d 7 system in Oh , the spin multiplicity of the ground state drops beyond certain Dq value (VT ) called a low-spin state, therefore an apparent change of magnetic moment is observed. For example, the µef f of high-spin state of Fe2+ (d 6 ) complex is ∼ 5.0 BM, whereas that of low spin is ∼ 0 BM. For the spin crossover system, two different spin multiplicities can be detected on the same complex at different temperatures. For example, the magnetic moment of Fe(phen)2 (NCS)2 measured at various temperatures is displayed in Fig. 20. The spin multiplicity below 176 K is a singlet, but the spin multiplicity above 176 K becomes a quintet. The corresponding effective magnetic moment is 0.5 and 4.9 BM, respectively. The deviations between the theoretical (spin only) and experimental values of the magnetic moment in Table V are mainly due to the fact that besides the spins the orbital angular momentum can also contribute to the effective magnetic moments.

V. STABILIZATION OF COMPLEXES In the ionic model, the formation of a complex ion from the free central ion and the ligands is combined with a stabilization by the electrostatic binding energy Eclass . The resulting state corresponds to the electronic ground state of the complex. From the crystal-field theory, however,

TABLE VI Ligand Field Stabilization Energy for Octahedral Transition Metal Complexesa High-spin configuration

Low-spin configuration

Number of d electrons

Ground term

LFSE (Dq)

1 2 3

t12 [2 T2 ] t22 [3 T1 ] t32 [4 A2 ] t32 e1 [5 E]

−4 −8

4 5 6 7 8 9 a

t32 e2 [6 T1 ] t42 e2 [5 T2 ] t52 e2 [4 T1 ] t62 e2 [3 A2 ] t62 e3 [2 E]

Ground term

LFSE (Dq)

−6 0

t42 [3 T1 ] t52 [2 T2 ]

−16 −20

−4

t62 [1 A1 ]

−24

−8 −12

t62 e1 [2 E]

−18

−12

−6

Calculated in the strong-field approximation.

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A. Relative Stability of Low-Spin Complexes As shown in the preceding section octahedral complexes with d 4 , d 5 , d 6 , and d 7 electron systems in principle can form high- or low-spin configurations. Why some of these transition metal ions form low-spin complexes and not the high-spin modification can be understood by comparison of the CFSEs of both spin configurations. From Table VI the gain of CFSE for the transition from the high- to the low-spin complexes amounts to 10 Dq for d 4 and d 7 complexes and 20 Dq for d 5 and d 6 complexes. Therefore, in strong ligand fields the low-spin complex is more stable than the high-spin complex. However, in low-spin complexes the CFSE is partly compensated by an increase in electron-electron repulsion, since two electrons are moving in the same orbital space. B. Heats of Hydration m+ By the dissolving of gaseous metal ions Mgas in water, the m+ corresponding hexa-aquo ions [M(H2 O)6 ]aq are formed and the heat of hydration, H H , is generated: 2O

m m+ Mgas −−−→[M(H2 O)6 ]aq + H H

FIGURE 20 Effective magnetic moment, µeff , of Fe(phen)2 (NCS)2 as a function of temperature. [From Lee, J. J., Sheu, H. S., Lee, C. R., Chen, J. M., Liu, R. G., Lee, J. F., Wang, C. C., Huang, C. H., and Wang, Y. (2000). J. Am. Chem. Soc. 122, 5742.]

it is known that the ground-state term of the central ion can be split into several progeny terms, depending on its orbital angular momentum and the symmetry of the ligand field. The lowest energy term after the splitting becomes the ground state. It is stabilized by the crystal-field stabilization energy (CFSE) in addition to the electrostatic binding energy. For central ions with an octahedral ligand system the five d states are split into t2 and e states. According to crystal-field theory, the t2 state is stabilized by −4 Dq, whereas the e state is destabilized by +6 Dq. Therefore, in the strong-field approximation for an octahedral complex ion with t2n e N −n electron system, the CFSE amounts to CFSE(Oh ) = −[4n − 6(N − n)]Dq, where N is the total number of d electrons, n is the number of t2 electrons, and N − n is the number of e electrons. The CFSE (Oh ) values for high- and low-spin complexes for N = 1, . . . , 9 are summarized in Table VI. They yield important information about the relative stability and several thermodynamic properties of transition metal complexes.

The variation of the experimental values of − H H with the atomic number of the bivalent and trivalent metal ions of the first transition series is plotted in Fig. 21. The solid

FIGURE 21 Heats of hydration of transition metal ions as a function of atomic number, (•) experimental data with standard derivation, (×) corrected from LFSE. [From George, P., and McClure, D. S. (1959). Prog. Inorg. Chem. 1, 418.]

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lines connecting the data points are not monotonic. On the basis of the ionic theory it is expected that with increasing atomic number (within a series) the ionic radii decrease monotonically because the electron cloud contracts under the influence of the increasing nuclear charge. Therefore, for electrostatic reasons the heat of hydration should also increase monotonically with increasing atomic number, as indicated by the dashed lines in Fig. 21. The discrepancy between this expected behavior and the experimental data is due to the fact that the hexaaquo complexes have nonspherical symmetry. The octahedral ligand field splits the ground-state terms, yielding the CFSEs given in Table VI. Therefore, the total heat of hydration H H will be composed of two parts, namely, the “classical” part Hclass , resulting from the ionic theory, and the “ligand field” part HL F ≡ CFSE:

High-spin

      d 3 : 4 A2 t23 ; d 5 : 6 A1 t23 e2 ; d 8 : 3 A2 t26 e2

Low-spin

  d 6 : 1 A1 t26

All the other octahedral complexes are Jahn-Teller unstable. In some of these complexes stability can be achieved by elongation or compression of the coordination octahedron along one of its axes. Often the distortions are expressed in the physical and chemical properties of the complexes. For example, the asymmetry of the absorption band of [Ti(H2 O)6 ]3+ (Fig. 17) is due to the Jahn-Teller effect. The distortion splits the degenerate ground state 2 T2 into a low-lying nondegenerate level and other progeny terms. The distortion may also split the excited level 2 E, thus more absorption bands would be observed. Since the energies of these bands are very close, only one band with asymmetric shape appears.

H H = Hclass + HL F The Hclass values can be obtained by subtraction of the CFSEs from the experimental H H values. If the CFSEs are calculated with Dq values obtained from absorption spectra, the resulting Hclass values are very close to the dashed lines as shown in Fig. 21. By corresponding considerations, apparent irregularities in the lattice energies of metallic halogenides in the heats of formation and heats of reactions and in the kinetic stability of transition metal complexes can be rationalized. The structural stability of transition metal complexes is largely influenced by the Jahn-Teller effect. As Jahn and Teller have proved, the nuclear configuration of a nonlinear polyatomic molecule is unstable when its electronic ground state is orbitally degenerate. A molecule in such a situation will undergo a structural distortion such that the resulting ground state is orbitally nondegenerate. The character of the distortion can be static, yielding a permanent deformation in geometry, or dynamic by oscillating between several structures with nondegenerate ground states. Only the following octahedral transition metal complexes have an orbitally nondegenerate ground state (see Tables V and VI):

SEE ALSO THE FOLLOWING ARTICLES ELECTROMAGNETICS • ELECTRON TRANSFER REACTIONS • GROUP THEORY • MICROWAVE MOLECULAR SPECTROSCOPY

BIBLIOGRAPHY Burger, K. (1973). “Coordination Chemistry: Experimental Methods,” Butterworth, London. Cotton, F. A. (1990). “Chemical Applications of Group Theory,” 3rd ed., Wiley, New York. Douglas, B. E., McDaniel, D. H., and Alexander, J. J. (1994). “Concepts and Models of Inorganic Chemistry,” 3rd ed., Wiley, New York. Drago, R. S. (1992). “Physical Methods for Chemistry,” 2nd ed., Saunders College, Orlando. Huheey, J. E., Keiter, E. A., and Keiter, R. L. (1993). “Inorganic Chemistry: Principles of Structure and Reactivity,” 4th ed., Harper Collins College, New York. Jørgensen, C. K. (1971). “Modern Aspects of Ligand Field Theory,” North-Holland, Amsterdam. Lever, A. B. P. (1984). “Inorganic Electronic Spectroscopy,” Elsevier, Amsterdam. Schl¨afer, H. L., and Gliemann, G. (1969). “Basic Principles of Ligand Field Theory,” Wiley, New York.

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Valence Bond Theory Richard P. Muller William A. Goddard III California Institute of Technology

I. II. III. IV.

Introduction Bonding in H+ 2 Bonding in H2 Bonding of Hydrogen to Carbon

GLOSSARY

to molecules, ground to excited states, reactants to products, etc.

Orbital Function that describes the distribution of one electron; used in approximate wave functions. Schr¨odinger equation Fundamental equation of quantum mechanics; wave functions are solutions of the Schr¨odinger equation. Wave function Function defined in quantum mechanics, which contains all the information about a system.

THE DEVELOPMENT of quantum mechanics (QM) placed a, theoretical foundation under the experimental science of chemistry. One could argue—as did Physics Nobelist Paul Dirac in a famous quote—that understanding chemistry is no more than solving these equations. In fact, this is not correct. Doing chemistry requires a qualitative understanding of how a reaction would change if the solvent, temperature, or pressure is changed or if the concentrations or ingredients are changed. Solving the Schr¨odinger equation for any specified conditions would not necessarily provide any understanding of the chemistry. To gain an understanding would require concepts based on QM that allow us to make comparisons: atoms

.

I. INTRODUCTION The orbital description of atoms and molecules is probably the most powerful unifying concept to provide an understanding of chemistry. At the foundation is the Aufbau principle for atoms, which serves as a semiuniversal ordering of hydrogen-like orbitals that explains the periodic changes in the ground-state character of the atoms and their excited states. Similarly, our most powerful concepts of molecular structures and properties are based on combining these atomic orbitals in ways appropriate for molecules. There are two major paradigms for doing this: 1. Molecular orbital (MO) theory. In MO theory, one starts with a molecular framework and considers combinations of the atomic orbitals optimum for the molecule. These one-electron orbitals are ordered by energy, and the electrons are used to populate the lowest ones (retaining the Pauli principle—no more than two electrons per orbital). This approach is particularly useful

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for predicting excited states and is quite popular in spectroscopy. Also, it is the framework used in rigorous QM calculations identified by names such as Hartree–Fock (HF), self-consistent field (SCF) and density functional theory (DFT). Thus, to describe N2 , we would take the five occupied atomic orbitals (denoted 1s, 2s, 2 px , 2 p y , and 2 pz ) of each N atom and form 10 molecular orbitals (denoted σg 1s, σu 1s, etc.) and occupy the lowest 7 with two electrons each (one with up spin and one with down spin). 2. Valence bond (VB) theory. In VB theory one starts with the occupied atomic orbitals of the atoms and constructs a many-electron wave function to describe bonding directly in terms of these atomic orbitals. Although similar to MO, the differences will become transparent (below). VB theory is most useful for describing reactions and bond dissociations because the many-electron states of the atoms are built into VB. However, VB is computationally much more complicated than MO, and it is much less obvious how to describe excited states in terms of VB. Important chemical concepts such as resonance are based on VB concepts. In this article, we will illustrate the use of VB and MO concepts for the simplest molecules and then apply these ideas to two simple problems that detail how to use them.

II. BONDING IN

H+ 2

Consider first the smallest possible molecule, H+ 2 , consisting of one electron and two protons that are separated by a distance R. This system is sketched in Fig. 1, where the two protons are denoted as a and b. The most important question is whether this system forms a bond (i.e., is the lowest energy for a finite value of R).

χl = N exp(−Ra )

(1)

χr = N exp(−Rb ),

(2)

and respectively, where χl and χr denote hydrogen 1s orbitals centered on the left and right protons, respectively, and N is a normalization factor—a scalar factor included to guarantee that the wave function contains only one electron. For finite R, the exact wave functions no longer have the atomic form, but useful approximate wave functions are obtained by combining the atomic orbitals to form the wave function ϕ = Cl χl + Cr χr .

This simple wave function is often referred to as linear combination of atomic orbitals (LCAO). The optimum linear combination of atomic orbital wave functions is the symmetric combination ϕ g = χl + χ r ,

(4)

where for simplicity we now ignore the normalization factor. The other combination of the orbitals is the antisymmetric combination, ϕu = χl − χr ,

(5)

where again we ignore normalization. The g and u labels denote the symmetry of the wave functions with respect to inversion, and stand respectively for the German words gerade (even) and ungerade (uneven). Although symmetry arguments can be very powerful, we eschew them here, and the reader may assume that these labels represent good and ungood, respectively. The energies for the wave functions ϕg and ϕu in Eqs. (4) and (5) are shown as a function of R in Fig. 2. Here we see that the g state is strongly bonding (the energy

A. Description Based upon Linear Combination of Atomic Orbitals In the case of R = ∞, the two protons are infinitely far apart, and the ground state is obtained by placing the electron in the 1s orbital of one or the other of the two protons. This leads to the two states H H+ and H+ H, which are described by the wave functions

FIGURE 1 Coordinate system for H+ 2.

(3)

FIGURE 2 Bonding of the g and u states for H+ 2.

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drops as the nuclei are brought together), Whereas the u state is strongly antibonding (the energy increases as the nuclei are brought together). Thus, starting with two ˚ (4 bohr), the g state would experience atoms at R = 2 A forces pushing it toward smaller R, whereas the u state would have a force pushing the atoms apart to R = ∞. The objective of this section is to understand both the the ϕg and ϕu state bonding and antibonding character.

B. Contragradience and the Origin of Chemical Bonding Starting with the atomic orbitals and combining them to form MOs, we find that g is bonding and u is antibonding. Now we want to understand the physics that underlies the bonding and antibonding character. One common idea is that the g coherent superposition of atomic orbitals causes the electron density to increase in the bond region and that this increased charge attracts the two positive nuclei, leading to stabilization. The resulting change in the electrostatic energy (denoted Vg ) is shown in Fig. 3, where we see that it increases as the atoms are brought together. The problem is that the increase in the density in the bond region comes at the expense of a decrease in the nuclear region. As indicated in Fig. 4, the potential is most negative near the nucleus, so that the increase of the density in the bond region is antibonding in nature. The only energy term in addition to electrostatics is kinetic energy. In QM, kinetic energy (denoted as T ) has a dramatically different form than that of classical mechanics (CM): TCM =

p2 , 2M

(6)

TQM =

(∇)2 , 2M

(7)

FIGURE 3 Kinetic, T, and potential, V, energies of the bonding g and antibonding u orbitals for H+ 2 . Note that even though Vg is unstable, the Tg effect compensates to create a bonding state.

FIGURE 4 Potential energy for H+ 2 . The potential is −∞ at the nuclei a and b and is negative and finite at the midpoint.

where ∇ is the gradient (derivative) of the wave function. Thus, in CM the lowest T is for p = 0 (electron standing still), whereas in QM it is lowest for a very smooth wave function (∇ = 0). Indeed, as is shown in Fig. 5, adding the two atomic orbitals to get g dramatically decreases the gradient in the bond region; for u the subtraction dramatically increases the gradient. These changes in T are much larger than the changes in V . The result is that bonding is dominated by the decrease in the kinetic energy resulting from the symmetric combination of atomic orbitals. The amount of the bonding depends on how much the gradient decreases. Thus, for the best bonding, we want the case in which both atomic orbitals have large gradients in opposite directions so that adding them leads to a big decrease in gradient. From Figure 3, we see that Tg has a ˚ The reason is clarified in Fig. 6. For minimum at 1 A. large distances, the orbitals in the bond midpoint region have small slopes and hence do not change much in forming ϕu . For R near zero, the atomic orbitals have large gradients but the decrease is only for the very small region between the atoms, which is too small to be significant. Thus, the optimum bonding is created at a distance where both atomic orbitals are significant at the bond midpoint. This is approximately the sum of the atomic radii ˚ = 1.06 A). ˚ (2 × 0.53 A

FIGURE 5 Bonding with g orbitals reduces the electron density at the nuclei.

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left and electron 2 on the right. The wave function for this configuration is a (r1 , r2 ) = χl (r1 )χr (r2 ).

(9)

This wave function a says that the probability of electron 1 being at a particular position is independent of where electron 2 is, and vice versa. Since the atoms are infinitely far apart, the electrons are not influenced by each other. A second wave function that is just as good, or as bad, as Eq. (9), is b (r1 , r2 ) = χr (r1 )χl (r2 ),

FIGURE 6 Overlap of 1s orbitals at (a) bonding and (b) too-close distances, and overlap of 2 p orbitals at (c) bonding and (d) tooclose distances. Note that in both of the too-close examples, the contragradient region extends over a smaller region of space than in the bonding examples, leading to less stabilization.

where the electrons have been interchanged. This wave function b is different from a since electron 1 is now on the opposite side of the universe. However, the energies of b and a must be the same since electrons 1 and 2 have the same properties (they are indistinguishable particles). It will be useful to combine a and b into two new wave functions, VB g (1, 2) = a (1, 2) + b (1, 2)

C. The Nodal Theorem We saw in Section II.B that kinetic energy dominates bonding in H+ 2 . An analysis of kinetic energy allows us to make some general predictions about the shape of wave functions. Consider a one-dimensional potential V (x) and consider the set of all eigenfunctions (all solutions of the Schr¨odinger Equation) Hφj = Eφj .

(10)

(8)

We can show quite generally that, except for very diabolical V (x), the ground state is nodeless. It is always positive except at the boundaries, where it goes to zero. A simple example is seen with H+ 2 , where we found two orbitals g and u. The nodal theorem indicates that the lowest state is g-like, i.e., nodeless. We will use this below to work out the bonding in H2 .

III. BONDING IN H2 With this understanding of the fundamental of chemical bonding in H+ 2 , we move our attention to H2 . A. The Valence Bond Description of H2 In the VB description of a molecule, we start with the full wave function for each atom at R = ∞ and combine the wave functions to form the wave function of the molecule, and then bring the atoms together to obtain optimal bonding. For H2 at R = ∞, this configuration consists of two hydrogen atoms infinitely far apart, say electron 1 on the

= χl (1)χr (2) + χr (1)χl (2),

(11)

VB u (1, 2) = a (1, 2) − b (1, 2) = χl (1)χr (2) − χr (1)χl (2),

(12)

without normalization, because at finite R these are the optimum wave functions. Before examining the energies, we need to understand how to think about the relative locations of the electrons in these wave functions. In Fig. 7, we plot the four wave functions a , b , VB g , VB and VB . We see that  has a nodal plane corresponding u u to z 1 = z 2 , whereas g does not. Indeed, along the line between the two peaks in Fig. 7c, we notice that the gradient of the VB g wave function is smaller than that of a or b , whereas the gradient of the VB u wave function is larger. VB This decrease in the gradient of VB g and increase for u are inversely related to R. Thus, based on kinetic energy, VB we would expect that VB g is bonding and u is antibonding, and indeed this is the case: Figure 8 shows the energies E g and E u for the valence bond wave functions of H2 . B. The Molecular Orbital Description of H2 The simplest wave function for H2 is found by starting with an electron in the best molecular orbital of H+ 2 and to place a second electron in this ϕg orbital. This leads to the molecular orbital wave function for H2 , MO gg (r1 , r2 ) = ϕg (r1 )ϕg (r2 ).

(13)

The first step is to examine the meaning of the wave function (13). The total probability for electron 1 to be

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dice are independent so that the probabilities multiply. To summarize: a product wave function, as in Eq. (13), implies that the electrons move independently of each other, i.e., there is no correlation of their motions. In addition to using the ϕg molecular orbital, we may construct wave functions of H2 using the ϕu molecular orbital, leading to wave functions of the form MO ug (1, 2) = ϕu (1)ϕg (2),

(15)

MO gu (1, 2) = ϕg (1)ϕu (2),

(16)

MO uu (1, 2) = ϕu (1)ϕu (2),

(17)

and

FIGURE 7 (a) a (1, 2) = χl (1)χr (2); (b) b (1, 2) = χr (1)χl (2); (c) g = a + b = χl χr + χr χl ; (d) u = a + b = χl χr − χr χl .

at some position r1 when electron 2 is simultaneously at some position r2 is  2 P(r1 , r2 ) = MO (r1 , r2 ) = |ϕg (r1 )|2 |ϕg (r2 )|2 = Pg (r1 )Pg (r2 ).

(14)

This expression is just the product of the independent probabilities for electron 1 to be at position r1 and electron 2 to be at position r2 . Thus, the probability distribution for electron 1 is independent of electron 2. Consider the analogous case of a red die (electron 1) and a green die (electron 2). The probability of rolling a red 3 is 1/6 and the probability of rolling a green 5 is 1/6, so that the total probability of getting both a red 3 and a green 5 is 1/6 × 1/6 = 1/36. The

FIGURE 8 Energies of the g and u states of the VB wave function.

Since the ϕu orbital is antibonding, the above wave functions for H2 lead to much higher energies than Eq. (13), except at large R; we expect an energy level diagram as in Fig. 9. So far we have discussed the molecular orbital wave function assuming that the bonding orbital ϕg is much better than the antibonding orbital ϕu . This is true for shorter internuclear distance R, but does not remain true as the bond is stretched and broken. In general, molecular orbital wave functions lead to a good value for the bond length but a very poor description of the processes of dissociation. The origin of this problem can be seen by substituting the atomic orbital description of the molecular orbital (4) into the molecular orbital wave function (13), leading to MO χgg (1, 2) = χl χl + χr χr + χl χr + χr χl ION = VB g + g ,

(18)

(again ignoring normalization) where VB g = χl χr + χr χl ,

(19)

ION = χl χl + χr χr . g

(20)

At very large R, the exact wave function will have one electron near the left proton and one near the right, as in Eq. (19), which we will refer to as the covalent part of the wave function. The other terms of (18) have both electrons near one proton and none near the other, thus creating an ionic wave function. At R = ∞, these ionic terms lead

FIGURE 9 Simple energy diagram for molecular orbital wave functions of H2 .

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to the energy of H− and H+ 2 rather than the energy of two hydrogen atoms. Since the molecular orbital wave function must have equal covalent and ionic contributions, it yields unfavorable energies for large R. The basic problem with the molecular orbital wave function is that both electrons are in the same ϕg orbital, and hence each electron has an equal probability of being on either center, regardless of the instantaneous location of the other electron. In the exact wave function, the motions of the electrons tend to be correlated so that if one electron is on the left, the other tends to be on the right. This correlation is necessarily ignored in the molecular orbital wave function, and the resulting error is often referred to as the correlation error. For small R, the two centers are close to each other, and this neglect of correlation is not particularly important. At R = ∞, however, the correlation of electrons is of paramount importance and neglect of correlation leads to ludicrously poor wave functions. In the next section, we will discuss a simple wave function, the valence bond wave function, that eliminates this problem of describing large R. C. Comparison of VB and MO Wave Functions The molecular orbital wave function, ignoring normalization, is

VB FIGURE 11 (a) MO gu = φg (1) φg (2), (b) g = χl χr + χr χl .

The wave functions are compared in Fig. 11, showing graphically how the valence bond wave function has smaller probability of having z 1 = z 2 , leading to lower electron repulsion energies. However, the molecular orbital wave function is smoother, leading to smaller kinetic energies. For normal bond distances, the electron repulsion effects dominate, so that the valence bond wave function is better, but for very short R, the kinetic energy becomes dominant, so the molecular orbital and valence bond wave functions lead to nearly identical total energies. Expanding the molecular orbital description of the excited states in terms of atomic orbitals, ignoring normalization, leads to

MO gg (1, 2) = ϕg (1)ϕg (2)

MO gu = (χl + χr ) (χl − χr )

= [χl χr + χr χl ] + [χl χl + χr χr ], (21) whereas the valence bond wave function is VB g (1, 2) = [χl χr + χr χl ].

= χl χl + χr χl − χl χr − χr χr

(23)

and (22)

The energies for these wave functions are compared in Fig. 10, where we see that the valence bond is always better, but that the difference becomes negligible for small R. Figure 10 illustrates the energy of the molecular orbital wave function for the ground state of H2 with comparison to the valence bond and exact energies.

MO ug = (χl − χr ) (χl + χr ) = χl χl − χr χl + χl χr − χr χr

(24)

We may rewrite these equations as VB ION MO gu = u + u

(25)

VB ION MO ug = u − u

(26)

(again ignoring normalization), where VB u = χr χl − χl χr

(27)

ION = χl χl − χr χr u

(28)

That is, the first excited state of the molecular orbital description is identical to the first excited state in the valence bond description. Both describe a covalent repulsive state that separates to two free H atoms. The molecular orbital description of the other excited state is MO uu = (χl − χr ) (χl − χr ) FIGURE 10 Energies of MO and VB wave functions.

= χl χl + χr χr − χl χr + χr χl

(29)

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FIGURE 12 The ground state of the C atom.

FIGURE 14 Bonding an H to a C lobe orbital, resulting in the 4 − state.

Recalling Eq. (21), we see that combining the gg and uu MO wave functions forms the VB description of the bond: MO MO VB g = gg − λuu ,

(30)

where λ is a scalar factor, and we again ignore normalization. Thus, we can fix up the molecular orbital wave function so that it behaves like the valence bond wave MO function by mixing the MO gg and uu wave functions.

IV. BONDING OF HYDROGEN TO CARBON The principles of bonding discussed in simple examples above apply also to chemical bonding in general. In this section, we will apply these simple valence bond ideas about chemical bonding to simple hydrocarbon examples. A. Ground State of Carbon Atom The ground state (3 P) of carbon has the electronic configuration (1s)2 (2s)2 (2 p)2 . Here the 1s electrons are core electrons and stay very close to the nucleus, and thus do not participate in chemical bonding. Thus, when we consider the chemical bonding of the C atom, we normally consider only the behavior of the other, valence electrons. Schematically, we represent the wave function for C atom in Fig. 12. The ovals above and below the atom indicate a lobe orbital: the 2s the pair of electrons that have hybridized with the 2 px orbitals; the line connecting the lobe orbitals indicates singlet pairing of the electrons in the orbitals. The dots without lines connected to them indicate electrons that are not spin paired.

FIGURE 13 Bonding an H to a C p orbital, resulting in the 2  state.

We now consider binding an H atom to the ground state of C. B. Low-Lying States of CHn The lowest states of CH arise from bonding an H to either a carbon p orbital, forming the 2  ground state, as in Fig. 13, or to a lobe orbital forming the 4 − state, shown in Fig. 14. The major difference between the two states is that in order to bond to a lobe orbital, the two lobe orbitals must undo the spin pairing; such rearrangement costs energy and weakens the overall bond. Thus, bonding to the p orbital, forming the 2  state, is more favorable. Now we take the 2  state of CH and bond another H to it to form CH2 . The lowest states of CH2 are obtained by bonding an H either to a p orbital, leading to a singlet state (1 A1 ), shown in Fig. 15, or a lobe orbital, leading to a triplet state (3 B1 ), shown in Fig. 16. For CH2 , the 3 B1 state is favored by 9 kcal/mole. Interestingly, for SiH2 , the 1A1 is more favorable, lying 18 kcal lower than the triplet state. Comparing CH2 with SiH2 , we find two important factors favoring the 3 B1 state relative to the 1A1 . First, the bonds of the 1A1 state involve perpendicular p orbitals, and hence will favor small bond angles, approximately 90◦ . The bonds in the 3 B1 state favor larger angles, approximately 128◦ (see above). For firstrow compounds, the importance of bond–bond repulsions, which are largest at small bond angles, leads to a

FIGURE 15 The 1 A1 state of CH2 .

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FIGURE 18 Two methylene 3 B 1 molecules forming a σ and a π bond in ethylene. FIGURE 16 The 3 B 1 state of CH2 .

significant increase in the energy of states involving small bond angles relative to those with larger angles. This effect increases the energy of the 1A1 state relative to the 3 B1 state. The second factor that account for the favoring of the 3 B1 state is that the lobe orbitals of first-row atoms are closer in energy to p orbitals than in atoms in the Si row. The 1A1 state involves two bonds to p orbitals, whereas the 3 B1 state involves one p bond and one lobe bond. Thus, this second effect also lowers the energy of the 3 B1 state relative to the 1 A1 state for CH2 . The net result is that the ordering of the first two states of CH2 is inverted with respect to the energies of SiH2 . Starting with the 3 B1 ground state of CH2 , there are two possibilities for bonding a third hydrogen, the unpaired lobe orbital or the p π orbital. These two options are shown as B and C, respectively, in Fig. 17. Again, due to the importance of bond–bond interactions, the lower configuration will be the one with larger bond angles. Bonding to the lobe leads to planar CH3 , with 120◦ bond angles, whereas bonding to the p π orbital leads to two 90◦ bond angles. Thus, the lowest energy configuration of CH3 is planar, whereas SiH3 was found to be pyramidal. Because of the ability of 3 B1 CH2 to bond an H to either the lobe or p orbital, the first constant for pyramidal distortion of CH3 is quite small. C. Ethylene VB theory allows us to start with two methylene radicals, CH2 , in the ground state, and, by forming ethylene, H2 C CH2 , we can form both a sigma bond and a pi bond,

FIGURE 17 Options for bonding to the 3 B 1 state of CH2 .

leading to the planar molecule ethylene molecule shown in Fig. 18. In the triplet state of CH2 , the HCH bond angle is 132.3◦ , but forming a bond to the carbon should decrease this angle because of Pauli repulsion from the new bond pair. Thus, in CH3 the bond angle is decreased to 120◦ , and in ethylene the HCH bond angle is decreased further to 117.6◦ . Consider now the case where the plane of one methylene group is rotated about the CC axis by an angle of 90◦ with respect to the other. The structure of the resulting molecule is shown in Fig. 19. Here only a sigma bond is formed. The nonbonding orbitals πl and πr can be combined into singlet and triplet states. The singlet state is usually referred to as N , for normal or ground state, and the triplet state is referred to as T . Based solely on Hund’s rule, we would expect the T state to be slightly lower than the N state for the twisted geometry. However, since these orbitals are localized on different centers, the energy splitting is quite small, approximately 1–2 kcal. In fact, other small effects lead to the N state at 90◦ being about 1 kcal below the T state. Comparison of the energies of twisted and planar ethylene leads to the results in Fig. 20. The singlet state prefers planar geometry since the overlap of the πl and πr orbitals leads to a strong pi bond. However, for the triplet state, the πl and πr orbitals must be orthogonalized to each other, and this state prefers the 90◦ twisted geometry. According to Fig. 20, the energy to twist the N state of ethylene by 90◦ is 65 kcal, breaking the pi bond. This barrier has been observed experimentally from studies of the kinetics for cis–trans isomerization dideuterated ethylene.

FIGURE 19 Twisted ethylene, where only a σ bond may be formed.

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geometry; the adiabatic excitation energy for θ = 0◦ is 91 kcal.

SEE ALSO THE FOLLOWING ARTICLES BONDING AND STRUCTURE IN SOLIDS • HYDROGEN BOND • KINETICS (CHEMISTRY) • QUANTUM CHEMISTRY

BIBLIOGRAPHY FIGURE 20 Energy profile for the singlet N state and the triplet T state of ethylene as the molecule is twisted about the C C bond.

As the molecules are rotated, the strength of the pi bond is weakened, and the CC bond length increases. Thus, for ˚ at θ = 0◦ and RCC = 1.50 A ˚ at the N state, RCC = 1.34 A ◦ θ = 90 . In the T state, the optimum geometry is twisted, θ = 90◦ , since triplet pairing of the orbitals prefers that the pi orbitals not overlap. As the molecule is twisted toward the planar geometry, the CC bond length increases ˚ at θ = 90◦ to 1.57 A ˚ at θ = 0◦ , and the from 1.50 A energy barrier is 25 kcal. Note that Fig. 20 shows the T -to-N energy separation at θ = 0, using the ground-state

Cooper, D. L., Gerratt, J., and Raimondi, M. (1987). Modern valence bond theory, Adv. Chem. Phys. 67, 319. Coulson, C. A. (1961). “Valence,” Oxford Univ. Press, London and New York. Klein, D. J., and Trinajsti´c, N., eds. (1990). “Valence Bond Theory and Chemical Structure,” Elsevier, Amsterdam. Matsen, F. A., and Pauncz, R. (1986). “The Unitary Group in Quantum Chemistry,” Elsevier, Amsterdam. McWeeny, R., and Sutcliffe, B. T. (1969). “Methods of Molecular Quantum Mechanics,” Academic Press, New York. Pauling, L. (1969). “The Nature of the Chemical Bond,” Cornell Univ. Press, Ithaca, New York. Pauncz, R. (1979). “Spin Eigenfunctions,” Plenum, New York. Schaefer, H. F., III, ed. (1977). “Methods of Electronic Structure Theory,” Plenum, New York. Slater, J. C. (1963). “Quantum Theory of Molecules and Solids,” Vol. 1. McGraw-Hill, New York.