Thermodynamics: Principles Characterizing Physical and Chemical Processes

Thermodynamics: Principles Characterizing Physical and Chemical Processes by J. M. Honig • ISBN: 0123738776 • Publisher...

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Thermodynamics: Principles Characterizing Physical and Chemical Processes by J. M. Honig

• ISBN: 0123738776 • Publisher: Elsevier Science & Technology Books • Pub. Date: March 2007

Preface

The present volume involves several alterations in the presentation of thermodynamic topics covered in the previous editions. Obviously, it is not a trivial exercise to present in a novel fashion any material that covers a period of more than 160 years. However, as best as I can determine the treatment of irreversible phenomena in Sections 1.13, 1.14, and 1.20 appears not to be widely known. Following much indecision, and with encouragement by the editors, I have dropped the various exercises requiring numerical evaluation of formulae developed in the text. After much thought I have also relegated the Carathrodory formulation of the Second Law of Thermodynamics (and a derivation of the Debye-Htickel equation) as a separate chapter to the end of the book. This permitted me to concentrate on a simpler exposition that directly links entropy to the reversible transfer of heat. It also provides a neat parallelism with the First Law that directly connects energy to work performance in an adiabatic process. A more careful discussion of the basic mechanism that forces electrochemical phenomena has been provided. I have also added material on the effects of curved interfaces and self assembly, and presented a more systematic formulation of the basics of irreversible processes. A discussion of critical phenomena is now included as a separate chapter. Lastly, the treatment of binary solutions has been expanded to deal with asymmetric properties of such systems. The aim, as before, has been to avoid as much as possible a presentation that is simply a linear superposition of discussions found in many other textbooks. Again, great stress is placed on the benefits of a systematic development of every topic, starting with modest beginnings, and reaping a whole cornucopia of results through self-contained logical operations and mathematical manipulations. I am greatly indebted to many persons for providing help, advice, and criticism. Where appropriate I have acknowledged in footnotes the sources that I have closely followed in my expositions. In revising the earlier versions I am indebted to Professor Dor Ben-Amotz for many insightful discussions, especially those relating to irreversible phenomena. I also value the editorial help by personnel at Elsevier, Inc. The book is dedicated to my parents who helped instill a love of the arts and sciences, to my late wife Gertrude Claryce Dahlbom Honig, to my present, equally wonderful wife, Josephine Neely Vamos Honig, and to the many children, both grown and young, who constitute the immediate family. All of them have been

vi

PREFACE

very understanding in my complete absorption in the preparation of the current volume. I hope the present volume will be found useful to all who are interested in the fascinating field of classical thermodynamics.

July 2006

J.M. Honig Purdue University West Lafayette, Indiana

vii

Preface to the Second Edition

The present volume is an upgraded version of a reference text published by Elsevier under the same title in 1982. The goals of the presentation have remained unaltered: to provide a self-contained exposition of the main areas of thermodynamics and to demonstrate how from a few fundamental concepts one obtains a whole cornucopia of results through the consistent application of logic and mathematical operations. The book retains the same format. However, Section 1.16 has been completely rewritten, and several new sections have been added to clarify concepts or to add further insights. Principal among these are the full use of thermodynamic information for characterizing the Joule-Thomson effect, a reformulation of the basic principles underlying the operation of electrochemical cells, and a brief derivation of the Onsager reciprocity conditions. Several short sections containing sample calculations have also been inserted at locations deemed to be particularly instructive in illustrating the application of basic principles to actual problems. A special effort has also been made to eliminate the typographical errors of the earlier edition. The author would appreciate comments from readers that pertain to remaining errors or to obscure presentations. It remains to thank those whose diligence and hard work have made it possible to bring this work to fruition: Ms. Virginia Burbrink, who undertook much of the enormous task of converting the typography of the earlier edition to the present word processor format; Ms. Gail Shively, who completed this onerous task and patiently dealt with all of the unexpected formatting problems; and Ms. Sophia Onayo, who compiled the index and the table of contents. Purdue University has provided a very comfortable milieu in which both the writing and the later revision of the book were undertaken. It is a pleasure to express my appreciation to various individuals at Academic Press who encouraged me to prepare the revised text and who were most cooperative in getting the book to press. Last, this task could not have been completed without the support of my beloved wife, Josephine Vamos Honig, who gave me much moral support after the death of my first wife, as well as during the book revision process, and to whom I shall remain ever grateful. J.M. Honig Purdue University

ix

Preface to the First Edition The publication of yet another text on the well-explored topic of thermodynamics requires some commentary: such a venture may be justified on the grounds that as scientists our perceptions of any subject matter continually change; even as traditional and established an area as chemical thermodynamics is not exempt from such a subtle transmutation. Thus, there appears to be merit in a continuing series of expositions of the discipline of thermodynamics that differ perceptibly from linear combinations of discussions found in prior texts and monographs. In the present volume there occur several departures from conventional treatments, among them: (i) the presentation of the Second Law based on a simplified approach to Carath6odory's method; (ii) a reasonably comprehensive treatment of thermodynamics of systems subjected to externally applied fields-special emphasis has been placed on the systematics of electromagnetic fields and on gas adsorption processes, concerning which there has been much confusion; (iii) detailed investigations on the uniqueness of predictions of properties of solutions, in the face of a bewildering array of standard states, of methods for specifying composition, and of equilibrium constants; (iv) a rationalization scheme for the interpretation of phase diagrams; (v) a discussion of the thermodynamics of irreversible processes, centered on the macroscopic equations. Most of the above topics are not covered in detail in existing texts. Throughout, emphasis has been placed on the logical structure of the theory and on the need to correlate every analysis with experimental operating conditions and constraints. This is coupled with an attempt to remove the mystery that seems so often to surround the basic concepts in thermodynamics. Repeatedly, the attention of the reader is directed to the tremendous power inherent in the systematic development of the subject matter. Only the classical aspects of the problem are taken up; no attempt has been made to introduce the statistical approach, since the subject matter of classical thermodynamics is self-consistent and complete, and rests on an independent basis. The course of study is aimed at graduate students who have had prior exposure to the subject matter at a more elementary level. The author has had reasonable success in the presentation of these topics in a two-semester graduate class at Purdue University; in fact, the present book is an outgrowth of lecture notes for this course. No worked numerical examples have been provided, for there exist many excellent books in which different sets of problems have been worked out in detail. However, many problems are included as exercises at various levels of difficulty, which the student can use to become facile in numerical work.

x

PREFACE TO THE FIRST EDITION

The author's indebtedness to other sources should be readily apparent. He profited greatly from fundamental insights offered in two slim volumes: Classical Thermodynamics by H.A. Buchdahl and Methods of Thermodynamics by H. Reiss. Also, he found instructive the perusal of sources, texts, and monographs on classical thermodynamics authored by C.J. Adkins, I.V. Bazarov, H.B. Callen, S. Glasstone, E.A. Guggenheim, G.N. Hatsopoulos and J.H. Keenan, W. Kauzmann, J. Kestin, R. Kubo, P.J. Landsberg, EH. MacDougall, A. Mtinster, A.B. Pippard, I. Prigogine, P.A. Rock, and M.W. Zemansky. Specific sources that have been consulted are acknowledged in appropriate sections in the text. He is greatly indebted to Professor L.L. Van Zandt for assistance in formulating the thermodynamic characterization of electromagnetic fields. Most of all, he has enormously profited from the penetrating insight, unrelenting criticism, and incisive comments of his personal friend and colleague Professor J.W. Richardson. Obviously, the remaining errors are the author's responsibility, concerning which any correspondence from readers would be appreciated. It is a pleasant duty to acknowledge the efforts of several secretaries, Jane Biddie, Cheryl Zachman, Nancy Holder, Susan Baker, and especially Konie Young and Barbara Rosenberg~all of whom cheerfully cooperated in transforming illegible sets of paper scraps into a rough draft. Special thanks go to Hali Myers, who undertook the Herculean task of typing the final version; without her persistence the manuscript could not have been readied for publication. Dr. Madhuri Pai contributed greatly by assisting with the proofreading of the final manuscript. In a matter of personal experience, it is appropriate to acknowledge several meaningful discussions with my father, the late Richard M. Honig, who was an expert in jurisprudence and who readily saw the parallels between scientific methodology and the codification of law. He persisted with questions concerning the nature of thermodynamics that I could not readily answer and was thereby indirectly responsible for the tenor of the present volume. Last, it is important to thank my immediate family, particularly my beloved wife, Trudy, for much patient understanding and for many sacrifices, without which the work could have been neither undertaken nor completed.

Honig July 1981

J.M.

Table of Contents

• Preface, Pages v-vi • Preface to the Second Edition, Page vii • Preface to the First Edition, Pages ix-x • Chapter 1 - Fundamentals, Pages 1-110 • Chapter 2 - Equilibrium in Ideal Systems, Pages 111-158 • Chapter 3 - Characterization of Nonideal Solutions, Pages 159-247 • Chapter 4 - Thermodynamic Properties of Electrolytes, Pages 249-285 • Chapter 5 - Thermodynamic Properties of Materials in Externally Applied Fields, Pages 287-346 • Chapter 6 - Irreversible Thermodynamics, Pages 347-396 • Chapter 7 - Critical Phenomena, Pages 397-423 • Chapter 8 - A Final Speculation About Ultimate Temperatures—A Fourth Law of Thermodynamics?, Pages 425-426 • Chapter 9 - Mathematical Proof of the Carathéodory Theorem and Resulting Interpretations; derivation of the Debye-Hückel Equation, Pages 427-444 • Index, Pages 445-452

Chapter 1 Fundamentals

1.0

Introductory Remarks

Thermodynamics is an overarching discipline in the sense that all physical phenomena can be described and analyzed in terms of a general macroscopic framework that contains parameters which may be determined by experiment. It is truly remarkable that with the minimal input of only four postulates, and by the systematic application of mathematical logic, a whole cornucopia of results can be produced for use in the interpretation of experiments and for predictive purposes in a wide variety of physical disciplines. In this book an attempt will be made to stress both the systematics that provides the cornucopia as well as the need to establish a close link between theory and experiment. The exposition will encompass mostly the areas of physical chemistry and parts of physics, but the principles expounded below should enable the reader to apply the thermodynamic discipline and methodology to other areas of research. The fundamental concepts are introduced in the form of four basic laws. The procedure is reasonably axiomatic, so that one can deal with (i) the concept of temperature without initially having to refer to heat flow; (ii) the definition of energy as a function of state, and the definition of heat flow as a deficit function; (iii) the introduction of the entropy function that does not depend on a generalization of the performance characteristics of heat engines. A comparison of the entropy changes for a given process carried out reversibly and irreversibly is then used to obtain a variety of fundamental results. This in a natural manner leads to the introduction of several functions of state; considerable emphasis is placed on systematically exploiting their mathematical properties. The important concept of homogeneous functions of degree one is then introduced and is used to analyze the properties of open systems. This chapter ends with a study of stability problems. The reader should investigate not only the details of the derivations but also the internal structure of every presentation, and should note the benefits of a systematic approach to the study of thermodynamic principles.

2

1. FUNDAMENTALS

1.1

Introductory Definitions

Before launching into the concepts of thermodynamics it is important to agree on the meaning of several basic terms. These are discussed below:

System. A region in space that is identified as a useful object of study and set apart from the remainder of the cosmos for that purpose.

Surroundings. Regions immediately outside and contiguous to the system. Boundaries. Partitions separating a system from its surroundings. Comments. It is very important to set boundaries properly and to distinguish appropriately between system and surroundings. Failure to do so can lead to erroneous conclusions. Boundaries may be real, such as walls or partitions, or may be conceptual, such as geometric surfaces. Body. The content of a specific system. Comments. To be amenable to thermodynamic treatment an actual body must be of adequate size, such that normal fluctuations in its properties are minute compared to their average values, and such that physical measurements do not significantly perturb the properties of the body. The volume of such a body must generally be at least of the order of 10-15 cm-3. Homogeneous vs. heterogeneous systems. Homogeneous systems are uniform in properties over their entire volume. Otherwise such systems are heterogeneous. Subsystem. A portion of the region of a system singled out for special study. Isolated systems. Systems totally unresponsive to any changes that occur in the surroundings, or that have no surroundings. 1 Closed (open) systems. Systems in which transfer of matter to or from the surroundings cannot (can) occur. A closed system may nevertheless be subject to manipulation through external agents such as electric or magnetic fields. Permeable (semipermeable)boundaries. Boundaries that enclose an open system (that permit passage of certain chemical species while excluding other species). Adiabatic systems. Systems whose properties are unaffected when their surroundings are heated or cooled. Comment. A more appropriate definition for an adiabatic system will be provided in Section 1.7. Phase. A physically and chemically homogeneous macroscopic region in a system. Comment. In a system there may exist several sub-regions in distinct states of aggregation or composition. Diathermic boundaries. Boundaries that do not permit matter to be exchanged between systems and their surroundings but that permit changes to take place in properties of the system by heating or cooling of the surroundings. Thermodynamic properties. Physical or chemical attributes that specify the characteristic properties of a system.

INTRODUCTORY DEFINITIONS

3

Thermodynamic equilibrium. A state of a system where, as a necessary condition, none of the properties under study changes with time.

Comments. It is not a simple matter to determine whether a system is at equilibrium. One method described later involves subjecting the system to some process that takes the system away from its quiescent state under a set of prescribed conditions. If, on release of the constraint, the response is out of proportion to such a process and if the system does not then return to its original state it could not have been at equilibrium. If the system does return to its initial state without incurring any other changes in the universe then it is said to be in equilibrium with respect to the tests that have been conducted. It may be very difficult to decide whether equilibrium prevails in systems prone to very sluggish processes. In such situations one attempts to establish a relaxation time over which significant changes in properties are detectable when the system is externally perturbed. It is generally agreed that equilibrium prevails when no changes can be detected over intervals very large compared to the relaxation time. 2 Reservoir. A source or sink used to exchange matter with, or through appropriate interactions, to alter the properties of an attached system. A reservoir is assumed to be of such immense size that its properties remain essentially unaltered during any interactions with the system. Thermodynamic coordinates, variables, degrees of freedom. All three are used to designate linearly independent experimental macroscopic variables that are required to characterize the state of a system. Comments. A minimum number of such variables is required to describe systems at equilibrium; their values do not depend on the manner in which the equilibrium state is reached. Intensive (extensive)variables. Variables whose values are independent of (depend on) the size and/or quantity of matter contained in a system under study. State space, configuration space, phase space. An abstract space spanned by coordinate axes, one for each thermodynamic coordinate, on which a given point represents the numerical value of that coordinate. A hyperspace is then formed by a mutually orthogonal disposition of these axes about a common origin. Representative point. A point in phase space that corresponds to the state or characteristics of the system for which the state space was constructed. Path. A succession of states traversed by a system in passing from a given initial to a given final state. Quasistatic process. One that involves passage through a large succession of very closely spaced equilibrium states. In this process the surroundings may be altered such that on the return path to the original system configuration the universe ends up in a different state. Reversible process. One whose path may be exactly reversed though a succession of very closely spaced equilibrium states, such that on reversal of the path both the system and its surroundings are restored to their original state.

4

1. FUNDAMENTALS

Comments. The distinction between quasistatic and reversible processes may be illustrated by considering the magnetization of a paramagnetic and of a ferromagnetic material. In a paramagnet the gradual application of a magnetic field slowly magnetizes the sample, which can then be completely demagnetized by slowly eliminating the magnetic field--this process is reversible. A ferromagnet can be slowly magnetized by gradual application of a magnetic field, but on gradual elimination of the field the material remains partially magnetized. Here, a succession of equilibrium states, followed by the reverse, leaves the system in an altered configuration. This is an example of a quasistatic process. Reversible processes are necessarily quasistatic, but the reverse may not hold. Steady state processes. Processes which do not alter the state of a system but do change the surroundings. Comments. At equilibrium no changes in properties occur with time either in the system or in its surroundings. However, under steady state conditions inputs and outputs of the system remain in balance so that the properties of the system are not altered, but changes do occur in the surroundings as a result of such processes. 3 A more scientific characterization is provided in Chapter 6. Number of Independent Components. The least number of chemically distinct species whose mole numbers must be specified to prepare a particular phase. Comments. Due account must be taken of any prevailing chemical equilibria since in such cases the concentrations of the various participating species cannot all be independently altered. The number of independent components may then be determined from the number of distinct chemical compounds present in the system minus the number of chemical equations that specify their interactions. This matter is taken up in Section 2.1. Number of Degrees of Freedom. The number of state variables that can be altered independently and arbitrarily, within limits, without changing the number of phases within the system. Before proceeding it is taken for granted that the reader has an intuitive understanding of the concept of mass and volume. Much of the subsequent discussion will initially based on these concepts. REMARKS AND QUERIES 1.1.1. The universe is an excellent paradigm of an isolated and closed system. Ordinarily, events occurring at astronomical time scales may be ignored in the consideration of physical and chemical processes occurring in the laboratory. 1.1.2. As an example of problems involving long time scales consider the reaction of hydrogen and oxygen in a balloon at room temperature. The fact that there seems to be no detectable change in the concentration of either constituent over many months does not mean that the system is equilibrated: insertion of platinum black as a catalyst leads to a measurable rate of formation of water, and heating the balloon with a torch leads to a violent reaction.

THE ZEROTH LAW OF THERMODYNAMICS

5

1.1.3. As an example consider the passage of current from a battery through a conductor. At a steady state the average electron density in any section of the material remains invariant; also, the initial rise in temperature due to Joule heating stops when the rate of heat generation is exactly compensated for by the radiation of heat into the atmosphere. Thus, under steady state conditions the material properties of the conductor during this process do not change with time, but the surroundings are altered: the battery runs down and the air is heated up. 1.1.4. Is it appropriate to classify a definition as being correct or incorrect? Conventional or unconventional? Complete or incomplete? Consistent or inconsistent? 1.1.5. Cite conditions under which a proffered definition may be rejected.

1.2

The Zeroth Law of Thermodynamics

We are now ready to study of thermodynamic principles by enunciating general laws that govern the operation of all possible processes in the universe. Therein lies the power of Thermodynamics. The first of these principles involves the socalled Zeroth Law of Thermodynamics which asserts that

Two bodies in equilibrium with a third are in equilibrium with each other. The seemingly obvious statement of transitive properties of the Zeroth Law has important ramifications: at the outset consider only the case where the properties of a system can be specified in terms of a prevailing pressure P and volume V. We follow the procedure advocated by Buchdahl. 1 Consider then two systems 1 and 2 that are initially isolated; we use pressures P1 and P2 (forces per unit area) to deform their volumes 1/1 and V2. We may have to make thermal or other adjustments that will permit physically possible pairs of pressure-volume variables (P1, 1/1) and (P2, V2) to be independently established in the two systems. Let these two units now be joined and equilibrated; it is an experience of mankind that under these conditions only three of the four variables can be independently altered. This restriction is expressed by a mathematical relation r 1/1, P2, V2) = 0, where f13 is an appropriate mathematical function that provides the interrelation between the indicated variables; its detailed form is not of interest at this point. We now repeat the process for joining system 1 to a new system 3 characterized by the pressure-volume variables P3, V3. By the same line of argument, after setting up the compound system one encounters a second interrelation of the form flz(P1, V1, P3, V3) = 0. Lastly, on joining systems 2 and 3 one must set up a third mathematical restriction of the form/31 (P2, V2, P3, 1/3) = 0. If equilibrium prevails after each combination, we require for consistency with the Zeroth Law that system 3 remain unaltered in its union with either system 1 or 2; this allows us to solve for P3 in the functions f12 and fll to write: P3 = (Pl (P2, g2, V3) -q52 (P1, 1/1, ~ ) , from which we construct the following difference function:

~l(P2, V2, V3)- q52(P1, V1, V3)~ ~.(P1, V1, P2, V2, V3)=0.

(1.2.1)

6

1. FUNDAMENTALS

This unfortunately generates a glaring inconsistency: the functional dependence of )~ on V3 is absent from the function t3 - 0 ; also, it makes no sense to have to refer to system 3 when combining systems 1 and 2. To resolve this difficulty we introduce a new requirement: namely, we demand that V3 occur in the functions q~l and @2 in such a manner that V3 is eliminated when the difference between q~l and q~2 is constructed. This is achieved in most general terms by requiting that the functions q~ assume the forms q~l = f2(P2, V2)h(V3) + q(V3) and ~2 = fi (P1, Vi)h(V3) + q(V3), where h and q are arbitrary functions of V3. Substitution of the last two equations in Eq. (1.2.1) then leads to the relation fl (Pi, V1)= f2(P2, V2).

(1.2.2a)

Similarly, consistent with the Zeroth Law, we obtain

fl (P1, V1)= f3(P3, V3).

(1.2.2b)

These results are sensible: reference is now made only to variables appropriate to each distinct system. Eqs. (1.2.2a) and (1.2.2b) thus characterize the equilibration condition. Moreover, this process permits us to select system 1 as a reference standard to infer whether system 2 and 3 are in mutual equilibrium, according as system 1 is or is not changed when coming in contact with first with system 2 and then with system 3.

1.2.1

Empirical Temperatures and Equations of State

Clearly, the functional interrelation specified by fl (P1, V1) is of great significance; it therefore makes sense to provide for this function a special symbol, rl, as a short-hand notation: more generally, we write ri = f / ( P i, Vi), where ri is called the empirical temperature (function). The relationship ri = fi (P i, ~ ) is known as an equation of state for system i. We can thus specify the empirical temperature of system i by measuring its pressure and volume, and inserting these parameters into the chosen function j5 (P i, Vi) that obviously will have to be specified before ri can be quantified. Variables other than pressure and volume can be used equally well to construct different sets of empirical temperatures. The selection of such variables depends on the characteristics of the system that is being investigated. Clearly, for each different choice one can anticipate a distinct temperature scale; this then presents a problem of unifying all different possible temperature s c a l e s ~ a matter that we will resolve below. The labeling of ri as a 'temperature' is obviously meant to link the physical properties to human sensory perceptions of 'hotness levels'. Minimally one should ask that the temperature increase monotonically with increasing hotness levels. This requires a quantification scheme that utilizes a convenient equation of state of a suitable material as an indicator of hotness. An enormous multitude of

THE ZEROTH LAW OF THERMODYNAMICS

7

indicators have been used for this purpose, such as: measurements of volume of ideal gases, of resistivity of solids, of viscosity of liquids, of spectral emissivity of solids, of thermoelectric voltages, of sound velocity, and of magnetic susceptibility. The methods of measurement and the experimental precautions required to attain reproducible results are listed in special compendia. 2 Each type of measurement provides a different response to increases in hotness levels. To obtain a reasonable quantification scheme it is sensible to pick from all conceivable temperature measurements one that is of particular simplicity and utility, that is linear in the correlation with, and that can be used over a large range, of hotness levels. One system well suited for present purposes is the so-called ideal gas. It has been known for over three centuries that gases approaching this type of behavior closely obey the relation P V - - c o n s t a n t (Boyle's Law) when the gas is kept at a constant empirical temperature in a range well above the conditions where it can be liquefied. We therefore adopt the product P V as a direct measure of r. Over the years He gas has been chosen as the medium p a r excellence for such measurements; equipment used for this purpose is known as a gas thermometer. 1.2.2

A n A b s o l u t e T e m p e r a t u r e Scale

In many temperature determinations one maintains the gas thermometer at a fixed low pressure. A useful quantification scheme is the so-called Celsius scale that assigns the values r - 0 ~ (this was the original intent, but nowadays the standard value is r - 0 . 0 1 ~ and r - 100~ to the He gas thermometer which is at equilibrium respectively with water containing ice and with water equilibrated with steam maintained at 1 bar. 3 Let V, V0, and V]00 be the volume of He gas at a fixed, low pressure at temperatures r, 0 ~ and 100 ~ respectively; then r is to be specified by "r- 100

V - Vo

Vloo- Vo

= 100

v

V~oo- Vo

-

100

Vo

V~oo- Vo

= T + To.

(1.2.3)

The intercept of the straight line generated by the two fixed points (that is, the value of r at which V would vanish on that straight line if He could be maintained as an ideal gas down to extremely low temperatures 4) is found to be To -- -IOOVo/(V]oo - Vo) - -273.15 ~ This suggests a natural lower limit to temperature, namely, the point where V vanishes. It also suggests a shift of scale whereby the quantity T - 100 V/(V] 00 - Vo) is the fundamental entity of interest. Adoption of this method leads an absolute scale for quantifying hotness levels; we construct a thermodynamic temperature scale T(K) - r ( ~ + 273.15, where K stands for kelvins as the temperature unit. This still maintains the desired proportionality between absolute temperature and measured volumes of He at fixed, low pressures. Clearly, one could have used changes in pressure of an ideal gas as a measure of empirical temperature, so long as the pressure remained in a range where ideality

8

1. FUNDAMENTALS

can be maintained. In that case, at constant volume, one would set up the scale as (with an obvious subscript notation) r-

100

P-

Po

= 100

PlOO- Po

P PlOO- Po

- 100

Po

-- T + To. (1.2.4)

P l o o - Po

Here the intercept of the straight line generated occurs at the value where P would vanish if the ideal gas state could be maintained at all temperatures. Again, setting up a linear absolute temperature scales through pressure measurements at constant volume is an obvious next step. 1.2.3

Use of Triple Point

A difficulty with the above scheme is that measurements carried out with various actual gases that approach ideal behavior will lead to slightly different results. A better absolute standard is provided by the so-called triple point of water. As we shall see later, 3 the coexistence conditions of water in the solid, liquid, and vapor state can occur only under a set of precisely controlled, invariant conditions determined by the physical characteristics of H20. These conditions are completely reproducible all over the world. For consistency with the above absolute temperature scheme the triple point of water is assigned a temperature T (triple point of H20) = 273.16 K = Tt. Then any other absolute temperature is determined through the proportionality T = (P/Pt)" 273.16, where P is the pressure at T and Pt is the pressure measured for He in equilibrium with water at its triple point. The use of gas thermometers tends to be awkward. One can use more convenient methods by calibrating any other thermometer against the He gas thermometer in the range of hotness levels where these two overlap. The new system is so chosen that its range of operation extends over temperatures where use of the gas thermometer is awkward or impossible. Such a calibrated unit may be used in turn to calibrate yet another system over their common range of hotness levels; the third system is selected so as to extend the measurements over another range of hotness levels that remained inaccessible to the original equipment. The process can clearly be systematically extended. Details of the procedure are beyond the purview of the present discussion. Readers are urged to consult the many existing sources of information in the literature. 2 ADDITIONAL INFORMATION 1.2.1. H.A. Buchdahl, The Concepts of Classical Thermodynamics, Cambridge University Press, 1966, Chapter 2. 1.2.2. A very comprehensive account may be found in Temperature, its Measurement and Control in Science and Industry, American Institute of Physics, New York, which is a multiauthor, multivolume compendium.

MATHEMATICAL APPARATUS

9

1.2.3. In Section 2.2 it will be shown that when two phases of a pure material (e.g., water and steam) are maintained in equilibrium at a fixed pressure, the temperature of the system remains fixed. Similarly, three such phases (e.g., ice, water, and steam) can coexist only at one particular pressure and temperature, termed the triple point. 1.2.4. According to the Third Law of Thermodynamics, taken up later, the ideal gas concept fails at lowest achievable temperatures; no material remains in the gaseous state for all possible r. This fact, however, does not deter us from carrying out an extrapolation that indicates at what value of r the volume would vanish if an ideal gas could be maintained at all temperatures.

1.3

Mathematical

Apparatus

In subsequent sections we will continually apply various mathematical procedures that are listed below. These operations must be properly mastered before one can undertake the unified description of thermodynamic principles. 1.3.1

Transformation of Variables

The method of transformation of variables in three dimensions, described here, can readily be generalized to higher dimensions. Let F (x, y, z) be some function of three independent variables (in thermodynamics these usually are not spatial coordinates, but thermodynamic coordinates), each of which may be rewritten in terms of three different independent variables u, v, w that happen to be more convenient for the description of phenomena of interest. We write these interrelations as x - x(u, v, w), y - y(u, v, w), and z - z(u, v, w), so that the original function becomes F ( x , y, z ) -

F [ x ( u , v, w), y(u, v, w), z(u, v, w)]

(1.3.1)

= G(u, v, w) -- F(u, v, w).

In passing from (x, y, z) to (u, v, w), the function F assumes a different functional form, G. However, to avoid profusion of symbols and confusion in interpretation, it is customary to retain the same symbol for both functional dependences; for, the physical interpretation remains unaltered by any transformation in coordinate representation. On differentiation of Eq. (1.3.1) with respect to u one obtains through the chain rule of differentiation: --

+

v,w

y,z

v,w

77. x,z

+

,

v,w

x,y

v,w

(1.3.2) with similar expressions for (OF~Or) and (OF/Ow). We now determine the differential of F as dE--

OF ~x

dx + y,z

dy + x,z

~

de, x,y

(1.3.3)

10

1. FUNDAMENTALS

which we next abbreviate as

dF - X dx -t- Y dy + Z dz,

(1.3.4)

with X - (OF/Ox)y,z, Y - (OF/Oy)x,z, and Z - (OF/Oz)x,y. On replacing the partial F derivatives in Eq. (1.3.2) with X, Y, and Z one obtains OF

--X V,W

(0x)(0 t ~uu

+ Y

U,W

+Z I),W

(0zt ~uu

"

(1.3.5)

U,W

Thus, it appears as if on differentiating F in Eq. (1.3.4) with respect to u to obtain Eq. (1.3.5) we had left the coefficients X, Y, Z unaltered and 'differentiated' solely dx, dy, dz. However, Eq. (1.3.5) is equivalent to Eq. (1.3.2), which resolves the apparent puzzle. 1.3.2

Partial Derivatives with Different Constraints

A special case of the above arises when we set u - x and restrict ourselves to two independent variables, discarding z and w. Eq. (1.3.2) then reduces to

Oy

(~X )v-- (~X )y--[- (~y )x(-~X)v.

(1.3.6)

The above is very useful if the experimental determination of (OF/Ox)v at constant v is complicated, but the specification of (OF/Ox)y can be carried through more conveniently, provided the partial derivatives (OF/Oy)x and (Oy/Ox)v can also be readily determined, as is frequently the case in thermodynamics. Often one deals with situations where a particular function of two variables is a constant, C, so that F(x, y) -- C. This immediately shows that x and y cannot be independent: we may solve for y = y(x) to write dy dx, so that

-- (Oy/Ox)F

d F--

-~x

dx+

y

~y

dy-

x

~x

dx+

-~y

y

x

-~x

dx-O.

F

(1.3.7)

This leads to another result of importance, namely

Oy)

-~X F

__(OF/Ox)Y

~

(OF/ay)x

(1.3 , 8)

Here a partial derivative that may be hard to evaluate with F fixed is rewritten in terms of partial derivatives involving F that may be much easier to determine. Many cases of this type will be encountered later. We next solve F (x, y) -- C for x = x (y); by the same steps this leads to the result

Ox ) _ _ (OF/Oy)x -~Y F (OF/Ox)y

(1.3.9) ~

MATHEMATICAL APPARATUS

11

Comparison of these two expressions yields the

(Ox/Oy)F --

Reciprocal Theorem:

(Oy/OX)F

,

(1.3.10)

which is extremely useful when it is difficult to deal with a function y expressed in terms of x, but when it is easy to handle x expressed in terms of y. Note the requirement that F be held fixed; otherwise the expression may not apply. Matters get more complicated when F is a function of three independent variables and when F(x, y, z) -- C, a constant; now only two of the variables are independent. Let us solve for x = x(y, z) or y = y(x, z), so that

dx -

()-~yOx z,Fdy +

() Oy d y - -~x

z,F

(Oy) dx + -~z

dz, y ,F

(1.3.11a)

dz.

(1.3.1 lb)

x,F

Substitute the second expression into the first and collect terms to find

Ox Ox -~Z Y,F] dz. [1-- (-O--fiy)z,F (~XX)z,F] dX -- [ (-O-fiy)z,F (~Z )x,F -~-(OX)

(1.3.12)

On account of (1.3.10) the left-hand side vanishes, and the right-hand side may be rewritten, such that one obtains the Reciprocity Theorem

(0;) (0;)(0z) z, F

-- - 1 ,

x ,F -~X y,F

(1.3.13)

which is useful in specialty applications encountered later. Yet another relation is found by requiring F (x, y) = C and expressing x and y in terms of two other independent variables, u and v. Set x = x(u, v) and y = y(u, v); by the chain rule of differentiation

v

y

v,F

x

v,F

(1.3.14)

which may be rearranged as

(Oy/OU)v,F (Ox/Ou)v,F

(OF/Ox)y (aF/Oy)x

(1.3.15)

On now introducing (1.3.8) one obtains finally

Oy ) _ (Oy/OU)v,F -~X F (OX/OU)v,F

~

(1.3 16) 9

12

1. FUNDAMENTALS

which is useful in cases where the derivative on the left is not readily evaluatedl but those on the fight are easily determined. The above operations are so frequently used that it is advisable to memorize them. 1.3.3

Euler's Theorem of Homogeneous Functions

A theorem of great importance in thermodynamics is based on a thought experiment: consider a system containing n l moles of species 1, n2 moles of species 2, . . . , n r moles of species r. On doubling all moles numbers at constant pressure and temperature the volume of this system also doubles. In thermodynamics we encounter many quantities with the property that a change in all variables (as opposed to the parameters; see below) by a given factor also changes the particular function by this same factor. We examine the consequences of imposing such a requirement. Given a function F ( x l , x 2 , . . . , Xr), we write ! ! ! d F -- F 1 dxl + F 2 dx2 + . . . + F r dnr, where the primes indicate partial derivatives. Now change all independent variables proportionally to their original values, using a common factor d)~, so that dxi -- xi d)~ for all i, and require a proportional change in F, such that d F -- F d)~. Then F d)~ -- ~--~i F{xi d)~, from which we obtain F

F(xl,x2 . . . . . Xr) -- i ~ l X i ( OF

(1.3.17) Xjr

This relationship is known as Euler's Theorem f o r Homogeneous Functions of Degree One. However, in addition to the dependence on the xi the function F may also display a dependence on parameters such as pressure P or temperature T that, of course, remains unaffected by the above manipulations. 1.3.4

Exact Differentials

For a system characterized by thermodynamic variables x l, have many occasions to examine differentials such as dL =_ Z

Xi(xl,x2,... ,xr)dxi,

X2, ...,

Xr, we will

(1.3.18)

i where the d symbol is used whenever the increment in L and hence, the integral f d L , depends on the specific path, described by the xi, by which the system proceeds from a given initial to a given final state. Functions of this type are awkward and ought, if possible, to be avoided: as the path is altered so is the differential and so is the related integral. In thermodynamics great emphasis is therefore placed on setting up and dealing with a special class of functions that depend solely on the initial and final states of the system and that are independent

MATHEMATICAL APPARATUS

13

of the particular path by which the system proceeds. The differential of such a function R (x l, x2 . . . . . Xr) then becomes

dR-i=1

dxi -= Z Xidxi. xJ~=i i

(1.3.19)

Note in particular that all the coefficients Xi a r e obtained by differentiation of the single function R (x l, x2 . . . . , Xr). Such a mathematical entity is known as a function of state of the system and its differential d R is known as an exact differential. Functions of state R that are useful in thermodynamics are subject to the following requirements: 1. R is a real, single-valued, analytic function of the thermodynamic variables that characterize the state of a system. 1 2. The difference in R for a system in two different states depends solely on these two states. 3. The change in R for a cyclic process is identically zero. 4. The quantity dR is an exact differential which has the form of Eq. (1.3.19).

1.3.5

Elements of Vector Analysis

We briefly review here several elements of vector analysis that are needed later; for a better and more complete description the reader is referred to textbooks of mathematics. Examples of vectors are the position vector r -- ix + j y + kz, where i, j , k are unit vectors that coincide with the mutually orthogonal x, y, z axes of the coordinate system, and x, y, z are the corresponding coordinates. A vector in this space is designated by A - i Ax § j a y § kAz, where the A)~ are the components of the A vector along the three axes. We will also need the gradient vector operator, defined by V ----iO/Ox + jO/Oy + kO/Oz. The following vector manipulations are of relevance. (i) The dot product of two vectors,

A. B-

IAIIB[ sinOAB,

(1.3.20)

where IA] is the magnitude of the vector A, and where OAB is the angle between the vectors in the plane defined by them. Clearly, by definition, the dot product results in the formation of a scalar. Since i, j , k are orthonormal it follows that i 9i - j - j - k . k - 1 and ex. e u - 0, with )~ 7~ # and ex - i, j , k. Thus the dot product of two vectors is given by A . B - (iAx + j A y + kAz) . (iBx + j B y + kBz)

= Ax Bx + AyBy + Az Bz.

(1.3.21)

14

1. FUNDAMENTALS

It is easily checked that the operation is commutative, A 9B = B 9A and distrib-

utive, A . (B + C) = A . B + A . C. Another operation of importance involves the gradient vector dot product:

v .A =

i- x + j- y +

9 (lAx + j A y + kAz)

OAx OAy OA z ~. Ox +--~-y + Oz

(1.3.22)

This operation is called the divergence; it measures the degree to which the vector A spreads out from any given point. For, along a given direction x a change in distance dx entails a change of the vector from lAx(x) to i Ax(x + dx) i[Ax(x) + (OAx/Ox)dx], in which the partial derivative in the last term specifies the rate of increase or decrease of the vector along the positive x direction. Eq. (1.3.22) is then clearly the three-dimensional counterpart. (ii) Another useful entity is the cross product of two vectors as defined by (1.3.23)

A x B - h l A I I B I sin0AS.

Here h is the unit vector perpendicular to the plane defined by A and B. It points in the direction specified by the fight-hand rule. In light of their definitions the orthonormal unit vectors satisfy the relations ex x ex - 0, and (i x j ) - k - ( j x i), ( j x k) - i - - ( k x j ) , (k x i) - j - - ( i x k). With these rules it is readily checked that A x B - (iAx + jAy + kAz) x (i Bx + j By nt- kBz) -- ( B x A) and that this cross product may be recast in determinantal form as

AxB=

i Ax Bx

j

Ay By

k Az Bz

(1.3.24)

The cross product obeys anticommutation rules. Similar rules apply to the gradient vector; we obtain

VxA=

i O/Ox

Ax

j k O / O y O/Oz

Ay

(1.3.25a)

Az

which may be expanded as

V x A-

i l/ OAz \ Oy

OAy) (OAx Oz + J Oz

OAz) (OAy Ox + k Ox

~Ax) Oy "

(1.3.25b)

The operation (1.3.25) is termed the curl; the name is appropriate because the curl of a vector that points in a fixed direction vanishes, whereas a vector that curves around a fixed axis has a large associated value of the curl.

MATHEMATICAL APPARATUS

15

Table 1.3.1 Selected vector operations A . (B x C) = B . (C x A ) = C . (A x B ) A x (B x C) = B ( A . C) - C ( A . B )

V(fg) = fVg + gV f V . ( f a ) = f ( V . A) + a . ( V / ) V x(fa)--f(VxA)-ax(Vf)

V.(AxB)=B.(VxA)-A.(V V • (V f ) = 0 V• xA)=VV.A-V.VA V.(V •

xB)

(a) (b) (c) (d) (e) (f) (g) (h) (i)

Selected integral operations if

V~

.

dr - O ( f ) - r

Gradient Theorem

f f f v v ' a d 3 r - - - ffsA~'d2r

Gauss'Theorem

f fs(V X A) . d2r = f A . dr Stokes'Theorem

O) (k)

(1)

Remarks. A, B, C are vector quantities, h is the outer unit normal to a surface element, g, f , r are scalar quantities. The integral in (j) is a line integral connecting an initial to a final state; the ones to the left and right of (k) extend over the volume and over the surface of a body; the ones to the left and right of (1) extend over the surface and form a closed loop on the surface of a body.

Lastly, we cite the relation for the volume of a parallelepiped which generally may be nonorthogonal. If A, B, C represent the vectors of length A, B, C along the tilted x, y, z axes of the parallelepiped, then the volume within this figure is given by

V=(AxB).C.

(1.3.25c)

We call attention to several more involved vector operations listed in Table 1.3.1. These may be verified by writing out both sides of each equation in component form. 1.3.6

First Order Differential Equations

We shall have occasion to deal with a first order differential equation of the form

dy(x) -F R(x)y(x)= X(x). dx

(1.3.26)

16

1. FUNDAMENTALS

Its solution, as may be checked by direct substitution, is given by

+el,

y(x)-•215

(1.3.27a)

where

exp[fdxR(x)],

•

(1.3.27b)

and where C is an arbitrary constant. 1.3.7

Integrals with Variable Limits

On occasion we will encounter cases where differentials or derivatives must be taken of integrals with variable limits. Three such situations are of importance: (i) The first of these is of the type (V is the volume of the system)

P-

-- - ~

OV

dx dy dz f (x, y, z; V, Y)

d d J-L~2

r

,

(1.3.28)

in which the x, y, z coordinates of physical length remain within the limits - L / 2 <~x, y, z <~L/2, where L is the length of a cube of volume V -- L 3, equivalent to the volume V of the actual system. 3 Y is a parameter in the problem, such as temperature, pressure, or magnetic field. Thus, the limits of the integral as well as the integrand itself are variable. 4 Therefore, one cannot simply interchange the integral and differential operators. We handle this case by introducing fractional coordinates ~ - x / L , r/-- y / L , ( -- z / L , such that - 1/2 ~< ~, r/, ( ~< 1/2 and d x dy dz - L3d~ do dr, whence 0

= (3V2/3)_ 1 0

00L

OV = OL OV

1

0

OL = 3L 2 OL"

(1.3.29)

Thus, we obtain

1 [ rrr,,2 P =

3L 2 OL a a ,!-1/2

d~ d~ d f L 3 f (~, rl, (; L, Y)

] y

.

(1.3.30)

Since the limits on the integral are now fixed we may carry out the differentiation under the integral sign to find

P -- -

fff

l/2

d~ d~Td( f ( ~ , JT, (; L, Y)

d d ,I-1/2

L ~f[1/2 "3 ] J] ,!-1/2

d~ drl d (

Of(~, rl, (; L Y) OL

(1.3.31) Y

MATHEMATICAL APPARATUS

17

Now transform back to the original variables:

P=

dx dy dz f (x y z" V, Y)

L3

'

l

fff

L/2

3L 2 a a a - L ~ 2

,,

dx dy dz ( O f ) -~

dV v dL'

(1.3 32) "

which may be put in the final form

l fv dx dy dz f (x, y, z; V, Y) -

P =

V

fv

dx dy dz-::-:_. o vOf Y"

. (1.333)

(ii) Another operation of interest is of the type 81

8

[ i.

=_ - ~ Z

dx dy dz g(x, y, z; Y, Z) (Y,Z)

J,

(1.3.34)

Z

in which the integration limits themselves depend on the variable of the differentiation. Again, this is handled by the method of fractional coordinates: we write

(,,) -~

,;;r l,.

--

z

d~ dr/d(L 3 g(~, r/, (; Y, Z)

O~Y a J a-l12 --

fff

z

l/2

O

d~ d q d (

J J ,I--1/2

--

fv

[Vg(~ ri (; Y, Z)] -~

'

'

Z

Og(x, y, z; Y, Z) OY z

dxdydz

+ fvdxdydz[g(x'Y'z;Y'Z)](OV(Y'Z)) V(Y,Z) OY

. (1.3.35) z

(iii) Lastly, in later discussion it will be necessary to consider the differential of the integral f v d3r(M" H), that is encountered in the study of magnetic effects. We treat this quantity as shown below:

d fvdxdydz(M 9 H) --

d f i r 1/2 d~

d~ d((M.

H) V

d d 3-1/2

fff

l/2

d~ dJTd( d [ V ( M . H)]

d d 3-1/2

-

dx dy dz -~ d[V (M. H)]

f M.n_ d3r

~/

dV+

+ f d3r M . d H. Jv

d3rH.dM (1.3.36)

18

1. FUNDAMENTALS

REMARKS 1.3.1. At isolated singular points such functions are allowed to be nonanalytic. 1.3.2. A physical rationalization of line (k) runs as follows. Consider the one dimensional flow along x and compare the mass m per unit volume of fluid at x and at x 4- dx. Then set m(x 4- dx) = m(x) 4- (Om/Ox)dx, where the second term indicates the net accumulation or depletion of m as the material traverses the distance dx. In three dimensions this generalizes to (Om/Ox)dx + (Om/Oy)dy + (Om/Oz)dz = V m 9d 3r. The integral on the left of line (k) thus represents the net accumulation of material within the volume, generated through the balance between influx and outflow. The right side represents the additive effect of summing the transfer of mass patchwise across the entire surface. The operation h 9dZr ensures that only the transfer of m in the direction orthogonal to each patch of size dZr is credited to the total accumulation. The minus sign is needed since ~ points in the direction away from the interior of the body. Stokes' theorem, line (1) is not as readily visualized: basically, the curl operation corresponds to a 'swirl' of material generated by rotating matter flows. In the interior of the body all the swirls cancel out, but the parts of the swirls on the surface can be added up to form any continuous closed path that one wishes to draw. 1.3.3. A simple method for checking the results shown below is to assume that f is independent of spatial coordinates. In that case the triple integral of Eq. (1.3.28) may be replaced by V f ( V , Y), so that we obtain P = - ( O ( V f ) / O V)r; the remaining equations simplify correspondingly. 1.3.4. Similar strategies are employed when the solid body is not a simple cube though the mathematical manipulations become more cumbersome. Alternatively, one approximates the actual shape through a juxtaposition of cubes of appropriate sizes that fit into the body, and one then sums the contributions. In any event, the final outcome is not affected by the shape of the body.

1.4

Thermodynamic Forces

Before discussing the concept of work we briefly take up the concept of thermodynamic force, as presented by Redlich. 1 Consider again two systems 1 and 2 that are initially isolated, whose deformation coordinates are represented respectively by the vectors X l and x2. In this state these coordinates can be varied independently and at will. We now allow the systems to interact, so that certain of the variables can no longer be altered independently; this defines the interaction. Suppose that coordinates Xr and Xs are changed simultaneously as a result of the interaction. This may be formally expressed by the function g (Xr, Xs) -- O. Nothing prevents us from replacing Xr and Xs by monotonically varying functions yr(Xr) and ys(Xs) respectively; this merely alters the mathematical description of the interaction. This freedom permits us to select coordinates Xr and Xs such that their changes obey the relation

dxr + dxs = O.

(1.4.1)

ELEMENTS OF WORK

19

When previously isolated systems interact in this manner there are three possible outcomes: Xr increases while Xs diminishes, Xs increases while Xr decreases, or xr and Xs remain unchanged; in the latter case the systems are at equilibrium with respect to the postulated interaction. We now consider a property fr (Xr) for system r and fs (Xs) for system s, for which three possible outcomes are anticipated: fs > fr

corresponds to

dxr > 0, dxs < 0,

(1.4.2a)

f~ < fr

corresponds to

dxr < 0, dxs > 0,

(1.4.2b)

fs = fr

corresponds to

dxr = dxs = 0.

(1.4.2c)

The functions f are arbitrary in the sense that all classes of functions that change monotonically with a particular deformation coordinate x are equally acceptable; we select those that are convenient in their application. These selected functions are known as generalized forces. In this terminology Xr and Xs are altered because the systems interact through the operation of forces that change the state of the systems. In general, body r may be regarded as the surroundings and body s, as the system. Quantitative measurements require the calibration of interactions against an accepted standard force appropriate to the measurement under study, in consonance with the physical laws that prescribe the interactions. When fr and f~ differ, processes take place in each system until the equilibrium state f~ -- fr is reached, at which point xr and Xs become stationary. The fact that processes come to a halt does not mean that all forces have ceased to exist; rather, they balance each other out. The interplay between forces and their effects will be explored in the next section. At this point we may introduce a better definition for quasi-static or reversible processes. These are changes in a system that result from an imbalance of only those forces that maintain a system at equilibrium. REFERENCE 1.4.1. O. Redlich, Rev. Mod. Phys. 40 (1968) 556.

1.5 1.5.1

Elements of Work General Statements

We now begin the mathematical description of performance of work by the surroundings on a system. As will be seen later, work performance can usually be measured or calculated, and represents a useful means for specifying (portions of) several functions of state. Work is performed by application of a thermodynamic force fi (see preceding section) of type i that results in an infinitesimal change dxi of the relevant thermodynamic coordinate in the system. In general, fi depends

20

1. FUNDAMENTALS

on Xi and is not necessarily collinear with it. Both these matters are handled by writing the element of work performance in vectorial form as d W = f ( x i ) 9 d x i , where the d symbol serves as a reminder that this increment in general is pathdependent, so that the summation of these increments generally changes as the path is altered. Where more than one type of work is performed the contributions must be summed. Matters are simplified by rendering the applied force small enough to incur reversible displacements in x i , so as to avoid the turbulent or dissipative conditions. Otherwise a whole host of extrathermodynamic variables are needed to describe the process. However, this is not a necessary requirement; we will occasionally consider cases where the applied force exceeds that needed for guaranteeing reversibility. The total work of type i performed on the system in taking it from state X l to x2 is specified by Wi --

fxx2f i " d x i ,

(1.5.1)

1

which quantity is path-dependent, unless, as explained below, a function q9 can be found such that one may set f = -Vqg. 1.5.2

Pressure-Volume Work

We now examine several different types of work performance under quasistatic conditions. We begin with the compression of gas in a cylinder of constant cross section A by a piston (cf. Fig. 1.5.1), for which the applied pressure at each stage is only incrementally larger than the opposing gas pressure within the cylinder. With a displacement - d x along the direction of the external force the element of

Pgas

Pa

Fig. 1.5.1. Compression of a gas in a cylinder of constant cross sectional area A. The volume of the gas is diminished by application of an external pressure.

ELEMENTS OF WORK

21

compressive work is then specified by f d W -- f ( - d x ) - - = A dx - - P d V. A

(1.5.2)

As a specific simple example let the contents of the cylinder be an ideal gas, so that we may write P - n R T / V . Then W - - f v V f ( n R T / V ) d V ; assuming infinitesimally slow compression, during which a constant temperature may be maintained, we find

W -- - ( n R T )

din V - n R T l n ( ~ / V f ) ,

(1.5.3a)

which is positive since Vi > VU. For an irregularly shaped, isotropic object we let P d a represent the force acting uniformly on cross section element d a of the boundary (see Fig. 1.5.2) which recedes inward by a distance - d x relative to the applied pressure. The element of work is given by

dW - - f

P d a dx - - P (

f

dcr) dx - - P dV,

(1.5.3b)

as before. However, if the mechanical characteristics of the body are anisotropic the procedure becomes more complicated, as demonstrated in a later section. One may generalize on the above by considering the sudden application of an externally applied constant pressure P0 in excess of the internal pressure P. The do

\; Fig. 1.5.2. Compression of an irregularly shaped object by application of an external pressure that diminishes the volume.

22

1. FUNDAMENTALS

rf

10

dl

Fig. 1.5.3. Stretching of a rod by an externally applied force. The length of the bar is increased in this process.

rapid contraction of the system is accompanied by all sorts of turbulent phenomena that cannot be described in terms of simple thermodynamic principles. However, this is immaterial: when equilibrium is established the internal and external pressures balance out, and the work that has been performed is given by the quantity - P o ( V f - Vi) that involves the difference in final vs initial volumes of the system. The point to be emphasized is that one must formulate W in terms of the externally applied pressure. This is a nice illustration of a case where problems arising from the occurrence of irreversible processes have been circumvented. Other situations will be taken up later. When a thin rod of cross section A and length l, initially at equilibrium, is stretched or compressed by application of a force f (Fig. 1.5.3) it is conventional to introduce a stress cr =_ f / A that induces a strain e =_ d l / l and to set cr > 0, d l > 0 when the rod is under tension and er < 0, d l < 0 under compresion. Note that in either case work is performed on the rod. In the elastic limit when the work element is carried out reversibly, d W -- Voo" d e ,

(1.5.4)

where V0 is the volume of the unstretched rod. Again, this result must be generalized, as is done later, for anisotropic materials. 1.5.3

Work in Gravitational Fields

Work done against gravity may be handled in two distinct ways. The first is to consider a material with n moles of gram atomic mass M being lifted through a height d z ; the element of work done on the system is dW - Mgn dz,

(1.5.5a)

where g is the gravitational constant. The nature of the material is not altered in this procedure; only its potential energy is changed. This should be contrasted with a process of incrementing the mass of the body by adding d n moles to it at height z (relative to a standard height, such as sea level). The work element is now given by dW - Mgz dn,

(1.5.5b)

ELEMENTS OF WORK

23

whereby both the constitution of the system, as well as its potential energy, have been changed. Which of these two descriptions is used depends on the process under consideration. 1.5.4

Work in Electrostatic Fields

A similar situation prevails for a charge q subjected to an electrostatic potential q~. If the latter is changed in the amount dq~ the work involved is dW = q d ~ ; no change is encountered in the thermodynamic deformation coordinates. However, if an element of charge dq is taken from infinity and placed in a location at the electrostatic potential q~, then dW = ~ dq. Here the thermodynamic coordinates as well as the potential energies are altered. This latter formulation requires generalization. We consider the potential to be a function of position: q~ = q~(r), at location r, where the charge density is given by p(r). We now increment the charge d3rp(r) in volume element d3r at that location by the amount dq - d3r dp(r). Then the element of work for bringing the charge increment from infinity to location r is given by d 3 r ~ ( r ) d p ( r ) . The work increment involving a body of volume V then reads dW - f v d3r ~ ( r ) d p ( r ) . Next, we introduce Maxwell's equation V 97~ = 47rp(r), where p(r) is now the free charge density at location r, i.e., a charge adjustable by external means and not intrinsic to the medium under study. Here ~9 is the electric displacement vector, which is related to the electric field vector E, by ~9 -- E + 47r79, in which 79 is the polarization density vector. Thus, dp -- (4zr)-IV 9dZ~. Introduce line (d) of Table 1.3.1, so that we may set

dW - f d3r (4zr)-' (V. diD)~(r) - (4yr)-'

f

d3r [ V . ~ d'V -- d ' V . V~b],

where the integration must be carried out over the entire region of space over which 7~ is present. Gauss' theorem, line (k) of Table 1.3.1, may now be applied to the first of these two integrals to obtain -(47r) -1 f dZr 9h ~ d~9, where h is the outer unit normal vector perpendicular to a given surface element, and where the integration extends over any arbitrary surface enveloping regions in which the electric displacement field exists. This boundary may therefore be extended to enormously remote regions where the surface flux vanishes. The first integral may thus be neglected. In the second we introduce the electric field vector in terms of the electrostatic potential as E = - V q~, so that finally the work element takes the form

dW - (4re)- 1 f d 3r E . d~9.

(1.5.6a)

While appropriate, this expression is not really useful because the integration must be extended over all space. In Section 1.6 we provide an alternative, more

24

1. FUNDAMENTALS

useful formulation, namely, d W - -(47r) -1 f v d3r 7:'. dEo,

(1.5.6b)

in which E0 is the applied electric field before introduction of the sample. The polarization vector 7:' vanishes outside the sample; hence, the integration may be restricted to the sample volume V. Conventionally one writes 7:' = oeg, where oe is the electric susceptibility; alternatively, we set 7:' --ot0g0. Here we have not taken into account the work required to energize the vacuum in setting up the field g0. This quantity, which enters the derivations of Section 1.6 is normally of no interest. 1.5.5

Work in Magnetic Fields

The work incurred in subjecting material to a magnetic field 7-r may be handled through the expression d W = .Ad . d7-r where .A4 is the magnetic moment vector per unit volume. In this formulation the thermodynamic coordinates remain unchanged; only the potential energy of the system is altered. This expression, analogous to the lifting of a given mass of material against a gravitational field, is normally not of interest. As an alternative formulation we introduce a local current density, J (r) that responds to a steady vector potential ,A(r). The associated work element is d W - ( l / c ) f v d3r J " d.A, where c is the velocity of light. We next introduce Maxwell's relation V x 7-r -- (47r/c)J, which applies when the electric displacement vector is independent of time and when J is the free current density. Thus, d W - (47r) -1 f d3r d . A . (V x 7-r On using line (g) of Table 1.3.1, the work increment reads d W -- (47c) -1

f

d3r [7-r (V x d.A) + V . (7-r x d.A)].

The integration extends over the entire region where 7-r is present. The second integral, involving a divergence, is rewritten by application of Gauss' theorem, wherein 7-r x d,A is evaluated over an arbitrary surface that may be drawn infinitely far away. This portion of the integral may be discarded, leaving us with the first part that involves V x d,A --- dB, where/3 is the magnetic induction vector. We thus obtain d W - (4rr) -1 f d 3r 7-[,. d113.

(1.5.7a)

Again, this formulation is awkward because the integral extends over all space and because the independent variable is 13, which includes the response of the material to the applied magnetic field 7-r which alone is subject to experimental control.

ELEMENTS OF WORK

25

For most practical applications it is more convenient to use the expression, derived in Section 1.6. (1.5.7b)

dW - f v d3r 7-Lo . d a d ,

where A,4 = (47r) -1 (B - 7-~), and 7"r is the applied magnetic field prior to the introduction of the sample. The integration involves only the volume of the specimen, since .A/I vanishes outside its confines. We have not included the work required to set up the magnetic field in vacuum prior to insertion of the sample. Also, one conventionally writes .AA --- X 7-r where X is the magnetic susceptibility; alternatively, .A4 = X07"r One should note the sign difference in comparing Eq. (1.5.6b) and (1.5.7b). 1.5.6

Two-Dimensional

Films

Consider a thin film that is mounted on a wire frame (Fig. 1.5.4) and that is stretched by length dx through the application of a force f ; the work element is given by dW = f dx. The area increase is d A = 21 dx, where 1 is the length of the frame perpendicular to the applied force; the factor 2 arises because there is a film on both sides of the wire frame. Thus, dW = ( f / 2 1 ) d A ; conventionally one defines the surface tension by Y =- f /21, whence

dW--ydA.

1.5.7

Elongation

(1.5.8)

of S p r i n g s

We furnish one example to illustrate further the concept of work. Consider a mass M attached to the end of a weightless spring, as shown in Fig. 1.5.5. Initially the system of (mass + spring) is at equilibrium when no net force acts of the

() O I I I I i I I I I

rf

() O dx

Fig. 1.5.4. Setup for expanding a film stretched over a wire frame.

26

1. FUNDAMENTALS

Fig. 1.5.5. Schematic diagram depicting a massless spring with an attached mass point.

system: the spring is extended to its normal length x0 and the mass point is stationary. Now apply a force F that elastically extends the spring to length x > x0; assuming that H o o k e ' s L a w holds, the applied force is specified by F -- k ( x - xo), where k represents the force constant of the spring. Under contraction F points in the direction opposite to that shown, so that x < x0. Let the force F -- Fs be applied quasistatically, so that it barely exceeds the internal force of the spring at each stage of the process; then the velocity v of the mass point is essentially zero. The work carried out on the system in this process is given by W --

fxxl Fs d x

- k

ixxl(x -

0

xo) d x ,

(1.5.9a)

0

where x l is the extension of the spring at the conclusion of the process. The work performed on the system is thus specified by

w-5l k ( x l

- x0) 2

(1.5.9b)

if k is independent of x. The energy increment is stored as potential energy within the spring. Fs thus represents the minimum force needed to extend the spring. Now apply a force F in excess of the force Fs needed to extend the spring quasistatically. Then W -

fxxl Fs d x + ixxl( F 0

Fs) d x ,

(1.5.10a)

0

in which we treat the first term as before. The second contribution describes an acceleration of the mass point M arising from use of the excess applied force. The velocity of the mass point is v = v0 - 0 initially and reaches Vl when the spring is extended to length Xl. According to Newton's First Law, F - Fs -- M d Z x / d t 2 ; with v - d x / d t , ( d Z x / d t 2) d x - ( d v / d t ) v d t - v d r . Hence, the second term in Eq. (1.5.10a) becomes 89(v 2 - 0) and Eq. (1.5.10a) may be written as 1

w - - ~ l k ( x l - x0) 2 + ~ M v 2,

(1 ..5 10b)

ELEMENTS OF WORK

27

where the energy change now involves not just the potential energy contribution but a kinetic energy term as well. This example clearly illustrates that only in the case of quasistatic processes can one replace the applied external force by the opposing internal force of the system. In closing, one notes that in all cases the element of work is specified by the (dot) product of an intensive variable with the differential of an extensive variable. Where forces are superposed one must sum over distinct elements of work. This important conclusion is summarized by the mathematical formulation dW - Z

(1.5.11 a)

Yi d yi , i

in which the Yi represent appropriate generalized forces, and the Yi represent the corresponding displacements. Where appropriate, these quantities must be present in vectorial form.

1.5.8

Path-independent Line Integrals

We have noted that work, as expressed in the form Wi --

ix x2 f

(1.5.lib)

i "dxi,

1

depends on the manner in which the force varies with displacement, when x is changed from X l to x2; thus, generally, W is path-dependent. For obvious reasons it is not easy to develop general statements governing the properties of such quantities. One is thus led to seek out integrals that are actually independent of the path taken between the end points, quantities that were described in Section 1.3 as functions o f state. In the present context we now consider integrals (1.5.1 l b) which can be shown to depend solely on the end points. Such integrals vanish identically when x l = x2. For this purpose choose as a reference state the quantity x 1 -- 0 that is arbitrary but, once selected, not changeable. We may then introduce a function q~(x2) defined by

q5(X2) -- -- f0 x2 f(x)

(1.5.12)

.dx,

called a potentialfunction. It depends solely on the variable X2; the minus sign is introduced as a convention. Then, we can determine the potential difference as qO(x2) - qo ( x l ) -

-

fx X2f ( x ) 1

.dx -

fx X2[ V . 1

~(x)]dx.

(1.5.13)

28

1. FUNDAMENTALS

The fight hand is obtained through line (j) of Table 1.3.1; it involves the definition of the divergence symbol. Since the integrands must match we find that f -- - V ~ .

(1.5.14)

That is to say, the vector f and its three spatial components may actually be specified in terms of the gradient of a single scalar potential function 45. It is therefore simplest to determine the potential function first and then take the gradient to find the components of the force vector. This extraordinary property invites further exploration that illuminates the properties of functions of state. For this purpose we invoke Stokes' Theorem, line (1) of Table 1.3.1 in the form

f F . dx - f fs(V X F) . d2x,

(1.5.15)

where the left-hand side involves any closed boundary path that is located on a surface region. Any function of state F taken around a closed loop must vanish; the left-hand side is then equal to zero, so that the integrand on the fight vanishes as well; we then find that V

x F

-

0.

(1.5.16)

Thus, the three components of the function of state F are not independent: when Eq. (1.5.16) is expanded as shown in Eq. (1.3.25) one finds

OFx 3y

=

3 Fy 3x

,

OFz 3y

=

3 Fy 0z

,

3 Fz 0x

=

OFx 0z

,

(1.5.17)

showing that under the assumed conditions the function F is subject to stringent requirements. On finally setting F - - V q O taking F x - - O ~ / O x , F y - - O ~ / O y , Fz -- O ~ / 3 z , and invoking Eq. (1.5.17) we find that 02~/OxOy - 02~/3yOx - O , and similarly for the other components. Thus, the second order cross derivatives of the potential function taken in either order are the s a m e ~ a comforting result. One should note that the potential function specified by Eq. (1.5.12) is actually a mathematical construct reminiscent of the concept of potential energy, commonly encountered in elementary treatments. However, the two ideas should not be confused; the potential energy is often invoked because, as shown below, the integrand does arise in the development of the concept of energy. The potential qo, however, can be invoked only for line integrals that are independent of the path, i.e., that vanish when taken around a loop. QUERIES 1.5.1. It is stated that the element of work always involves an intensive and the differential of an extensive variable. Yet, in Eqs. (1.5.6) and (1.5.7) only intensive quantities seem to be present. How is this apparent contradiction resolved? 1.5.2. What changes in the analysis of the spring performance must be introduced when the spring itself has a nonzero mass?

THE ELEMENT OF WORK FOR A SYSTEM SUBJECTED TO ELECTROMAGNETIC FIELDS

1.6

29

The Element of Work for a System Subjected to Electromagnetic Fields

We now provide the derivation leading to Eqs. (1.5.6b) and (1.5.7b), beginning with the Maxwell equations 10B

V • E +

= 0,

c Ot 1 01)

V • 7"r

c Ot

=

(1.6.1a)

47rJf

.

c

(1.6.1b)

Here E, "/9, 7"r J f represent respectively the electric field, electric displacement, magnetic field, magnetic induction, and current density; c is the speed of light. Form the scalar product of Eq. (1.6. l a) with 7-r and of Eq. (1.6.1b) with E and subtract, to obtain 7-r

1 x E) - E . (V x H ) + -7-r

OB

c

~

019

1 -4- - g c

~t

4re = ---E. c

Jf.

(1.6.2)

Next, introduce the vector identity 7-r (V x E) - g . (V • 7"r - V 9 (g • 7-s line (f), Table 1.3.1, to obtain V . (E x 7-r

47r c

s

1 Jf +-"~ c

0/3

1 +-g.

"-~

0/)

c

~t

= 0.

(1.6.3)

f

Now integrate over all space. By Gauss' theorem, d3r V . (E • 7-s may be transformed into a surface integral enveloping the fields; the surface integration can be extended to infinitely remote boundaries where the fields ultimately vanish. This term therefore drops out, leaving -

f

l fd3r(E.dg+7-t.dt3).

d3r E,. J f dt - 4re

(1.6.4)

We rewrite the left-hand integrand in the form E . ] s - E . p i ~ s - [ p I E + ~ - ~ p I ~ I • t3] 9 ~ i - f L . ~ I ,

(1.6.5)

in which pf and v f are the free charge carrier density and velocity, respectively. On the fight we inserted the vector identity v f 9 (v f • 1~) - B . (v f • v f ) = O, which holds because the cross product v f • v f vanishes identically. Physically, this corresponds to the fact that a charged particle moving in a magnetic field does no work. The quantity in square brackets represents the Lorentz force density, designated by f L . We finally obtain ctw--fd3rE

9J f d t - - f d 3 r f L

.vfdt

I

1

i

_

_

[ d3r (E, d~D + 7-s dl3) 47r J 9

o

(1.6.6)

30

1. FUNDAMENTALS

as the increment of work. 1 We now follow Heine (1956) in taking note of the mathematical identity

E, . dZ) + 7-L . d13 - E,o . dl~o + 7"r d13o + (8,. d~Do - 1). dE, o) + ('k-Co. d B - 13o. dT-~) + (1)o. ds +

(13o. d7"r

- 8o" d ~ o )

- 7-~o. d B o ) + E . ( d l ) -

+ ('D - / ) o ) " dCo + B o . (dT'L - d7"r

dl)o) + (7-r - 7"r

d13. (1.6.7)

The subscripted vectors represent field variables arising from fixed charge and current distributions existing prior to insertion of the sample, and the remaining vectors represent corresponding quantities when the sample is inserted into the field. We then integrate over all space. In the absence of the sample we can also set B0 = 7-s and Z~0 = E0. The third term then becomes (E - ~9) 9dE0 = - 4 7 r 7 9 . dE0, and the integration may be restricted to the volume of the sample, since 79 vanishes elsewhere. The fourth term reads 7-r ( d B - d7-r = 47r(7-r 9d.A,4); here, the integration may be restricted to the sample volume, since elsewhere ,A,4 = 0. The fifth and sixth terms vanish automatically. When the integration is carried out each of the last four terms vanishes as well, on account of a vector theorem derived by Stratton. 2 This leaves

1 f d3r(E,o.dE, o + B o . d B o ) dW - 4re

f v d 3 r 7 9 . d E , o + fvd3rT-Co.d.AA.

(1.6.8) Here the first two integrals extend over all space and specify the work increment required to set up the electromagnetic field in free space; these quantities normally are of no interest. We thus concentrate on the remainder, which relate to the additional work involved in immersing the sample into the preexisting field. One should note the difference in sign between the third and fourth terms, and the nonsymmetric disposition of the field and response variables in these two integrals. REMARK AND REFERENCE 1.6.1. The sign convention requires a comment. Positive current proceeds in the direction of the electric field; hence, the dot product of the integrand is positive. On the other hand, the charges, flowing from a region of higher to lower potential, have lost energy, which is transferred (as work under isothermal conditions) to the surroundings. This process thus necessitates the minus sign in specifying the work performance. If the carriers are negatively charged then the sign of pf and that of v f are both reversed, so that the conclusion is still the same. 1.6.2. J.A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941, p. 111.

THE FIRST LAW OF THERMODYNAMICS

1.7 1.7.1

31

The First Law of Thermodynamics The First Law of Thermodynamics

We had noted earlier the advantage of adopting functions of state in describing thermodynamic processes, so as to avoid having to deal with path-dependent properties. For present purposes we achieve this aim by invoking an experience of mankind: the performance of work under adiabatic conditions, Wa, in taking a given system from a particular initial state {xi} = (Xl, x2 . . . . . X n ) i to a particular final state {x f } = (X 1, X2, . . . , X n ) f is independent of the type of work performed and is independent of the path taken in proceeding from {x i} to {x f }. We therefore assign this function of state a special name, the (internal) energy E, whose difference, A E = E f -- Ei = Wa, is specified by the amount of work of any kind performed while taking the system from state x i to state x f under adiabatic conditions. If now such a process is carried out under nonadiabatic conditions this situation no longer holds, which seems to impair the usefulness of the energy concept. However, other changes now also take place that were absent in the adiabatic process, so as to maintain the viability of the concept of energy. This experience of mankind gives rise to a new assertion, namely, the First Law of Thermodynamics:

For any process whatsoever there exists a function of state, called the internal energy, E, which depends solely on the initial and final thermodynamic coordinates of the system. Differences in this quantity may be determined by calculating or measuring the work performed adiabatically on the system for the process under study. In other words, corresponding to a given transformation of the system from state {xi } t o {Xf} there exists a unique energy difference A E = EU - - Ei that characterizes this change. Obviously, this assertion is not very useful until one knows how to determine the energy difference for all conceivable processes. To clarify the above concept a reader interested in a more mathematical introduction to the First Law is referred to Section 9.4. There the existence of such a function is derived as a necessary consequence from the existence of the linear representation of work performance as dW - Z i Yi dyi, Eq. (1.5.11 a). The above Law has important ramifications. In any nonadiabatic process the work performed, W, differs from Wa. Since now A E - W 7~ 0 it is expedient introduce a deficit function Q that restores the equality: we write Q - ( A E W) = 0, where Q is called the heat transfer. One must recognize that changes in energy of a system can only be brought about through interactions of the system with its surroundings; internal reconfigurations without changes in surroundings leave the energy function unaltered. Also, for all processes that connect the same initial and final states of the system

32

1. FUNDAMENTALS

A E remains unchanged, but it is the interactions between the system and its surroundings, specified by W and by Q, that depend on the process under consideration. Clearly, for an adiabatic process Q = 0; we henceforth take this to be a sufficient criterion for characterizing adiabatic conditions. Methods for determining Q experimentally will be described in later sections. Here we adopt the measurements of Wa as a method of preference to determine A E. If no adiabatic path exists to proceed from state i to state f one can generally find an intermediate state r such that by performance of work Wa~ and Wa" one can execute the adiabatic transformations i --+ r and r --+ f respectively. Let AEir, AErf, and AEif be the corresponding energy changes; then clearly A E i f = A E i r nt- A E r f -- Wta -+- W~a~ allows the desired energy change to be determined via separate measurements of Wa~ and Wa". We now write the First Law in the following mathematical form: A E = Q + W.

(1.7.1)

As noted earlier, Q and W represent two different manifestations of energy in transit across the boundaries of the system. They incur changes in the energy of the system but do not represent energies per se. Clearly, the energy of the system is increased by rendering Q > 0 and W > 0, in which case one states, by convention, that heat flows into the system and that work is performed on the system. Conversely, the energy is diminished by having Q < 0 and W < 0, in which case heat flows out of the system and work is done by the system. 1 Thus, though Q and W generally are not functions of state, their algebraic sum is. Also, it makes no sense to ask how much heat or work a given system contains, since only the internal energy properly characterizes the state of a given system (a nice analogy that clarifies this idea will be presented in the next section). Moreover, only a difference in energy can be uniquely determined; there is no such thing as 'the energy' of a system. This reflects the well-known fact that energies can be specified only to within an arbitrary constant that has no fundamental significance. 1.7.2

The First Law of Thermodynamics in Analytic Form

We now present the differential form of Eq. (1.7.1) as

dE = dQ + dW,

(1.7.2)

wherein the d symbol is used to emphasize that these infinitesimals are path dependent, whereas the infinitesimal d E is not. We can then broaden the concept of adiabatic processes; in these dQ = 0 for every infinitesimal stage of the process. Lastly, since energy is a function of state involving the thermodynamic coordinates x l, x2, . . . , Xn we can write its differential in the analytic form

n OE d E - - j ~ 1 OXj dxj. 9

(1.7.3)

THE FIRST LAW OF THERMODYNAMICS

33

A similar expression for Q and W is clearly inappropriate. Since these latter quantities represent energies in transit any process taking place totally within a given system does not change its energy. This immediately leads to the Law of Conservation of Energy as a corollary to the First Law. In an isolated system the energy is constant, no matter what processes occur within it. Such processes change the internal configuration of the isolated system but not its energy. It is of interest to note that the First Law may be set up on a more mathematical basis by adapting the so-called Carath6odory theorem to the differential Pfaffian form for work: dW = Z i Yidyi, Eq. (1.5.1 l a). As explicitly developed in Chapter 9, when adiabatic conditions are imposed the theorem, under the constraint dW -- O, necessitates the existence of an associated function of state that is constant under these conditions. 1.7.3

Examples Illustrating Problems in Defining Surroundings During Performance of Work

The foregoing calls for a more protracted discussion of the interchange of work and energy between a system and its surroundings. It is extraordinarily important in all changes of state brought about through the execution of work (i) to distinguish clearly between what constitutes the system, the surroundings, and the connecting link between them; (ii) to note that for the determination of work it is not necessary to be informed about the internal changes in the system when work is performed on it or by it; and (iii) to recognize that work performed on or by the system can be determined only after setting up a "work reservoir", completely external to the system under study, and operating in such a manner that the performance of work can be either readily measured or calculated. We illustrate the problems which arise by reference to Fig. 1.7.1, which depicts an enclosure S equipped with a movable piston P of mass Mp. The latter rests on release pins rl and r2; the container is also furnished with stops s l and s2 that arrest the downward motion of the piston under the action of the earth's gravitational field. Let the space in the enclosure be totally evacuated and let the release pins then be retracted; the piston is then accelerated through a vertical distance h, until arrested by the lower stops, s l and s2. The volume of the enclosure is thereby reduced from Vi to VU. Consider first the case where the system comprises the empty space and piston but excludes the walls. The work performed by the descent of the piston P through the height h is - M p g h , where g is the gravitational constant. The minus sign arises because on completion of the process the compound system has a lower potential energy than at the start. If, on the other hand, the system is restricted to the empty space no work has been done because the internal configuration of the space has remained unchanged. However, as regards the compound system, heat in the form of radiation crosses the boundaries of the system; for, the surrounding walls are heated through the friction of the moving cylinder and the stops are

34

1. FUNDAMENTALS

P ~r~

r2~

h

S

~]s1

.

.

$2C

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Fig. 1.7.1. Schematic diagram depicting a system S in an adiabatic enclosure that contains a movable piston P; r 1 a n d r 2 are retractable release pins, and Sl and s2 are stops that arrest the motion of the piston under the influence of a gravitational field.

deformed by its sudden arrest. Thus, the work performance shows up in the form of a heat transfer into the surrounding space. Next, let the container be filled with a gas at a pressure Pi sufficiently low that the released piston compresses the gas to pressure Pf when ultimately arrested by the stops; i.e., the pressure of the gas satisfies the relations Pi <~ P <~ Pf < Mpg/A, A being the cross-section of the piston. Let the system again exclude the walls. Contrary to popular misconceptions, the work performed on the compound system is still Mpgh, even though there now exists an opposing gas pressure P in the range Pi <~ P <~ PU. However, in the present situation the acceleration of piston P is lower than that prevailing in vacuum; at the same time the gas is now subject to convection, turbulence, and heating in the compression stage; also, the walls are heated, though to a lesser extent than before, by the friction of the moving piston and by its ultimate sudden deceleration. Clearly, even if the system is restricted to the gas alone, its internal coordinates are altered, and the energy of the gas is therefore altered. In this more restricted case the piston is part of the surroundings. Without detailed information it is impossible to ascertain how much of the work, during the descent of R is converted into an increase in energy of the gas, and how much appears as increased energy of the walls. For nonequilibrium situations this type of analysis would require the specification of a host of nonthermodynamic variables. However, as far as the ultimate equilibrium state is concerned, none of this matters; after the system has come to equilibrium,

THE FIRST LAW OF THERMODYNAMICS

35

the work performed by the compound system (P + G) is again A E = -Mpgh, regardless of how A E is partitioned between gas and walls. It is only in adiabatic processes that all of the work shows up as an increase in energy of the system that is comprised solely of the gas space. The potential energy change has been partially or fully converted as a higher energy state of the compressed gas. As the next experiment, let enough gas be admitted to the system to support the piston by virtue of the balancing relation Pg = Mpg/A. Now, add one grain of sand at a time to the top of the piston, thereby forcing P to move downward very slightly at each step against the increasing opposing gas pressure. In this case all frictional effects are negligible. Continue this process until the piston is gently nudged against the lower arrests. Here it is appropriate to consider the piston as part of the surroundings in equilibrium with the system that is comprised solely of the gas phase. Work in the amount g fo d(mz) has been performed on the system that does not include the cylinder; re(z) represents the mass of the piston-plus-sand which is present at the height z above the arrests. A new feature appears in this series of quasistatic steps: the applied external and resistive internal pressures essentially balance, and at each stage the gas is only infinitesimally removed from equilibrium. One may therefore introduce the equation of state of the gas P(T, V) and thereby evaluate both the work f P(T, V)dV and the energy increase of the system arising from this work performance; appropriate equations of state for this process are provided in later sections. On the other hand, if one takes the compound system to comprise both piston and gas the internal coordinates have been altered: energy in one form (as potential energy) has been transformed into energy of another type (described by the greater mass of the piston and by changed properties of the gas, as will be explained in Section 1.13). Further discussion of this matter will be deferred until the effects of external fields are explored in Chapter 5. Another instructive exercise consists in examining the free expansion of a gas, depicted in Fig. 1.7.2. The gas is initially confined to a space of volume VA, volume VB being totally evacuated. A hole of macroscopic dimensions is now opened in the diaphragm separating the two volumes; the gas ultimately occupies the total volume VA + VB. What is the work involved in this process? The answer is not absolutely straightforward: If the system is taken with boundaries encompassing only the volume VA, complications arise because in the expansion process the internal coordinates of the system so defined are altered by the motion of gas through the hole. With respect to volume VA one now deals with an open system, with new ground rules that will be taken up in Section 1.20. If, on the other hand, the system encompasses the total volume VA + VB plus the inside portion of the walls, no work is executed by the process. To understand this point it is important to restrict considerations to the system in its original equilibrium state and in the final equilibrium state of the adiabatic enclosure. There has been no change in surroundings, nor has any work reservoir been needed

36

1. FUNDAMENTALS

Fig. 1.7.2. Schematic diagram illustrating the free expansion of a gas initially confined to compartment A. On opening the partition the gas then expands into compartment B.

to run the process. Put differently, the process of expansion has taken place totally within the completely isolated compound system A + B; no heat or work crosses the boundaries. Once again, there is no need to worry about the complex interactions as the jets of expanding gases exert forces on the container walls first in one direction as they emerge from chamber A and then in other directions as they strike the various walls of chamber B. The final state of the adiabatic system is that in which it has reached a totally quiescent state; note that A E for the process of reaching this final state from the initial state, vanishes. Nevertheless, clearly 'something has happened' that calls for a description which the energy function is unable to provide. The requisite analysis is furnished by the entropy state function that will be introduced below in due course. The adiabatic enclosures mentioned above are ideal partitions which separate regions of thermodynamic interest from the remainder of the universe; in particular, no heat transfer of any type can occur across such boundaries. The walls of the container are in intimate contact with the gas which is being compressed but changes in its internal coordinates cannot be communicated to the remainder of the universe. Thus, quite generally, in calculating work performance, one must start with the initial equilibration state and end with the final equilibration state. The fact that the system passes through intermediate nonequilibrium stages, wherein not all of the alterations in potential energy of the piston are communicated to the gas, does not matter: in determining functions of state of a system the details of the intervening processes are irrelevant in thermodynamic analyses based on equilibrium states.

THE FIRST LAW OF THERMODYNAMICS AS A PARABLE

37

The above examples illustrate the necessity of dealing appropriately with the concept of work and with the placement of boundaries of a system. Concerning the former, one must distinguish carefully between work and other perturbations that affect the system or the state of the surroundings, one must deal with a system at equilibrium before a process is started and after it is completed, and the work involved must be calculable or measurable by standard methods. The placement of boundaries must be such that no frictional or other dissipative effects occur in its immediate vicinity or across the boundary; regions in which such phenomena take place must be clearly included either in the system or else in the surroundings. One should note again that changes in the specification of the boundaries may totally alter the specification of work that accompanies a given process. REMARKS AND QUERIES 1.7.1. In many texts and papers, following historic precedent, the First Law has been written in the form A E -- Q - W. Readers are cautioned to note carefully which formulation is used by any given author. Obviously, the convention for W employed in this text must everywhere be replaced by - W if the above convention is employed, wherein W < 0 and W > 0 represent work done on or by the system respectively. In modern presentations the usage in the present text is the preferred one. 1.7.2. Discuss the relevance of the Law of Conservation of Energy to the operation of perpetual motion machines. 1.7.3. Consider the explosive reaction when a mixture of hydrogen and oxygen is ignited by a spark, developed by equipment located internally in the system. If the system is totally isolated there is supposedly no change in its energy in such a violent process. Yet 'something' clearly has happened. Aside from the spark, what is the source for this process (i.e., where did the explosive 'energy' come from?), and what are the results of this process (i.e., what has happened to the explosive 'energy' liberated in this process)? What can you conclude from the fact that there seems to be a problem in the thermodynamic description of some processes: whereas a remarkable change in constitution of the system has taken place, the energy function of state of the system presumably has remained invariant.

1.8

The First Law of Thermodynamics as a Parable

Callen 1 has provided an interpretation of the First Law of Thermodynamics in terms of a parable that bears repeating. A certain person owns a rectangular swimming pool; he allows water to enter or leave through an inlet and outlet and daily compares the height of the water level with the amount of water that has flowed through the water meters attached to the inlet and outlet. In general the owner notes no detailed correlation between the water meter readings and the change in water level in the pool, but he notes that on rainy days

38

1. FUNDAMENTALS

the pool contains more than the expected amount, and on very dry days less water is present than anticipated. So the owner is led to cover the pool with a tarpaulin. At that point the owner finds that indeed the change in water level can be balanced precisely with the rates of water passage through the inlet and outlet. The analogy with the energy of a system should be clear. The covered pool is the analogue of an adiabatic system; changes in water levels correspond to energy changes. Water flowing through the inlet and outlet corresponds to positive and negative work, respectively. When the tarpaulin is removed the water added through rainfall or diminished by evaporation corresponds to heat influx and outflow. In these circumstances the bookkeeping requires that water gained through rainfall or lost through evaporation be taken into consideration when tracking changes in water level. Notice finally, that the liquid in the pool is just water; one cannot differentiate between liquid that entered or left through the faucets, as compared to water that was accumulated or lost through rainfall or evaporation. This should render more understandable earlier statements that heat and work transfers represent energy in transit across boundaries, and that they cannot be identified as pieces of energy within the system. All parables, being analogies, are to some extent misleading. Chief among the problems is the fact that, unlike water, heat is not a conserved quantity. However, this should not deter readers from pondering the lessons of this very nice tale. REFERENCE

1.8.1. H.B. Callen, Thermodynamics, Wiley, New York, 1960. 1.9

The Second Law of Thermodynamics

We begin here the discussion of the Second Law of Thermodynamics. This law has been enunciated in many different forms, the most prominent being the formulations by Kelvin and by Planck. These will be presented later as consequences of the approach derived below. 1 Undoubtedly, the most elegant statement of this Law was provided by Carathdodory in the following form:

In every neighborhood of a representative point of system in phase space there exist neighboring points that cannot be reached through any adiabatic process. The linking of this statement to consequences involving physical, chemical, cosmological, biological, or related processes requires a certain degree of mathematical sophistication and skill. Interested readers are encouraged to refer to a variety of sources, where these matters are discussed in detail. 2 This approach is important in the mathematically rigorous development of the subject matter. However, its presentation at this stage unnecessarily complicates the fundamental issues that should be emphasized. Hence, the material has been relegated to a

THE SECOND LAW OF THERMODYNAMICS

39

separate chapter at the end of the book. Those interested in the formal approach should examine the derivations after reading through the end of the current section. We develop instead a shortcut for the Second Law along the following lines: Just as we were able to link the performance of work under adiabatic conditions to the existence of a function of state, the energy, so we will postulate that the transfer of heat under reversible conditions is related to another functions of state, termed the empirical entropy, symbolized by s (or, later, the metrical entropy, S), which will then found to be useful to characterize various processes. In short, it is claimed as part of the Second Law of Thermodynamics that

A reversible incremental transfer of heat, dr Q, between a given system and its surroundings is related to a new incremental function of state, termed the empirical entropy, ds, through an integrating denominator, )~, whose physical significance is to be established later The above statement permits the Second Law to be expressed mathematically as

dr Q /)~ = ds.

(1.9.1)

The important point here is that under the above conditions one can dispense with path-dependent quantities. The identification of )~ is among the very compelling logical presentations of abstract reasoning. Namely, we examine the process of combining two different systems A and B into a composite system C. Let the heats transferred reversibly from the surroundings to the system in an infinitesimal process be designated by dr Q a and dr Qb for the component parts, and dr Qc for the isolated and internally equilibrated compound system; dr Q a -k- dr Qb = dr Qc. According to the Second Law, then d s c _ )~a dsa +

-s

dsb.

(1.9.2)

To proceed with the identification of )~ let the deformation coordinates of A and B be given by x l, x2 . . . . . Xn-1, t and Yl, Y2, . . . , Ym-1, t respectively, where t is the common empirical temperature of the compound system. Mathematically, we can then 'solve' for Xn-1 in terms of Xl, x2, . . . , Xn-2, t, Sa and for Ym-1 in terms of Yl, Y2 . . . . , Ym-2, t, Sb. We now assert the following: [1] Sc must be independent of X l , X 2 . . . . . X n - 2 , Yl, Y 2 , . . . , Ym-2, t. If this were not so Eq. (1.9.2) would then necessarily contain terms involving dxl, d x 2 , . . . , dym-2, dt, analogous to Eq. (1.3.9) or (1.7.3) for the differential forms of the energy E expressed as an analytic function of the coordinates {xi }. [2] The ratios )~a/)~c and )~b/)~c cannot depend on these variables; otherwise statement [ 1] would be contradicted.

40

1. FUNDAMENTALS

[3] )~c cannot depend on Y l , Y2 . . . . , Ym-2; for, if it did, then )~a in Eq. (1.9.2) would have to depend on these same coordinates in such a manner as to cancel out from the ratio ~.a/)~c, for consistency with statement [2] and [1]. However, )~a cannot possibly depend on the y coordinates appropriate to system B. By the same reasoning ~.c must be independent of x l, x2, . . . , Xn-2. [4] Furthermore, )~a cannot depend on x l, X2,..., X n - 2 , and )~b cannot depend on yl, y2, . . . , Ym-2. For, according to [3] these variables are missing from ~.c whence the ratios )~a/)~c and )~b/)~c must also be free of these variables, to remain consistent with [1 ], [2], [3]. However, this argument does not apply to t, which is common to A, B, and C. [5] As a consequence of [1]-[4] the functions )~a, ~+, and )~c can involve at most the variables (Sa, t), (sb, t), and (Sa, sb, t), respectively. [6] The functional dependences mentioned in [5] must have the form ~ - a - qga(sa)T(t), )~b -- qgb(sb)T(t), )~c -- gOc(Sa, s b ) T ( t ) , in which T ( t ) is a common though perfectly arbitrary function of the empirical temperature t; this function cancels from the ratios )~a/)~c and ~+/)~c in Eq. (1.9.2), in consonance with requirement [2]. [7] In summary, we find that dr Qa - T(t)qga(Sa) dsa - T ( t ) dSa,

dr Qb - T (t)qgb(sb) dsb -- T (t) dSb, dr Qc - T (t)qgc(Sa, Sb) dsc -- T (t) dSc,

(1.9.3)

where S in the differential form d S = go(s)ds is termed the metrical entropy. As described earlier, one can select any arbitrary function for T(t); we thus take T to be identical with the thermodynamic temperature scale introduced in Section 1.2. We show in Section 1.13 that it is this particular choice that corresponds to all possible empirical temperature scales that are monotonic in hotness levels. Moreover, equilibrium conditions have been assumed throughout, witness the fact that a limited number of well defined, equilibrium thermodynamic variables xi or Yi were introduced to characterize the properties of each subsystem. To preserve these all processes must be executed reversibly, as indicated by the subscript r. We are thus led to the fundamental result d S - dr Q / T.

(1.9.4)

The Second Law contains one other fundamental assertion: The entropy o f the final state f o r a process occurring in an isolated system is never less than that o f the initial state.

Implied in the above statement is the concept that processes in isolated systems are not subject to experimental control and therefore are spontaneous. Such

THE SECOND LAW OF THERMODYNAMICS

41

processes continue until equilibrium is established, at which point the entropy of the system is at a maximum, consistent with the properties of the system in its final state. The above statement also applies to adiabatically isolated systems, in which case one deals with processes for which the entropy of the system increases in the absence of any heat transfer to or from the surroundings. One should carefully note that it is possible to decrease the entropy of a system through appropriate interactions with the surroundings, but in that event there must occur elsewhere in the universe a compensating process such that the total entropy of the system plus the surroundings does increase. An important feature, already implicit in earlier discussions, is the fact that in an irreversible change one need to demand only that the system remain equilibrated at the beginning and end of the process. As far as functions of state are concerned it is immaterial that the changes of the system during an irreversible process cannot be described by a limited number of thermodynamic variables. As long as one deals with functions of state their change in an irreversible process may always be calculated by setting up a reversible process that brings the system from the same initial to the same final state. 1.9.1

Final Comments

We call attention to the mathematical construction in Chapter 9: when the transfer of heat assumes the linear form d Q = ~-~i Xi dxi Carath6odory's theorem necessitates the existence of a function of state that is fixed under adiabatic conditions, and whose change is tied to the transfer of heat under reversible conditions. We can now combine the First and Second Laws as follows: when only mechanical work is incurred a n d all p r o c e s s e s are c a r r i e d out reversibly we may write d E = dr Q + dr W - T d S - P d V .

(1.9.5)

This is the fundamental starting point for many subsequent thermodynamic operations. We will later deal with cases where processes are executed irreversibly. The concept of increasing entropy of a system may be linked to increasing disorder within the system. This important topic is beyond the purview of classical thermodynamics; readers are urged to consult an extensive literature dealing with this subject. NOTES AND QUERIES 1.9.1 The Kelvin and Planck statements (see Section 1.10) deal with the impossibility of operating thermal engines under certain prescribed conditions; from these assumptions the Second Law may then be deduced. There is no logical objection to this procedure, but it does seem somewhat unsatisfactory to base a universally applicable Law on principles pertaining to the operation of heat engines. The reverse

42

1. FUNDAMENTALS

procedure, outlined in Section 1.10 does provide what appears to be a better alternative; here the cyclic operation of heat engines is derived as a consequence of the Second Law. 1.9.2. We follow here the presentation provided by H.A. Buchdahl, The Concepts of Classical Thermodynamics, Cambridge University Press, 1966, Chapters 5, 6. Readers interested in a somewhat more detailed exposition of Carathrodory's approach and in the introduction of the thermodynamic temperature concept may also consult Chapter 9, where it was placed so as not to interrupt the current derivation. In retrospect it seems to be more direct to start with the formulation of the Second Law adopted here rather than working with the more elegant theory developed by Carathrodory. 1.9.3. An adiabatic enclosure is filled with supercooled water and then allowed to stand; in due time ice is observed to form. Obviously, the process is spontaneous, yet there is a decrease in overall order, hence, a reduction in entropy. How can this be reconciled with the statement that the entropy of an isolated system can only increase? 1.9.4. Is a process for which dS = 0 necessarily adiabatic? Explain your reasoning.

1.10

1.10.1

Cyclic Processes in Relation to Reversibility and Irreversibility. C a r n o t Efficiency Cyclic Processes

Considerations of cyclic processes constitute a very powerful tool in final deductions based on the Second Law. Consider two points in configuration space that are infinitesimally close to one another, as represented by 1 and 2 in Fig. 1.10.1. Choose a particular quasistatic process that takes a given system from state 1 2 I

r

Xl

o

\ i

~'

~..~

u

....

1

X2

Fig. 1.10.1. Illustration of a reversible path (solid curve) connecting states 1 and 2 and of an irreversible path (dashed curve) connecting states 2 and 1. The dashed path involves coordinates that fall outside the domain of the indicated configuration space.

CYCLIC PROCESSES IN RELATION TO REVERSIBILITY AND IRREVERSIBILITY

43

to state 2; let the heat exchange between system and surroundings be given as dr Q 1~ 2 (in Exercise 1.10.2 we ask whether it matters if this quantity is negative or positive). Select a second, irreversible path that effects the same 1 --+ 2 change and that involves a heat exchange di Q l~2. This latter path is dashed on the diagram: being an irreversible process, the path lies outside the phase space appropriate to quasistatic processes. By the First Law, dr Q 1--+2 _ d E 1-->2 _ dr W 1--->2, and di Q l ~ 2 _ dE1-->2 _ di W 1-->2, so that di Q 1--->2 - d r Q 1--->2 - d r W 1--->2 - di W 1--->2.

(1.10.1)

Note that the energy differentials have canceled out; being functions of state, they are the same for both processes. The differences in Eq. (1.10.1) cannot vanish. If they did the reversible and irreversible paths would have to coincide, because d Q and d W depend on the chosen paths. This, however, would lead to a contradiction of terms. Thus, the algebraic sums in (1.10.1) must be either positive or negative. Suppose first that di Q 1-->2 _ dr Q 1-->2 ~ di Q 1~ 2 + dr Q2--->1 > 0; then, by Eq. (1.10.1), the work involved in completing the cycle by going from 1 to 2 and back to 1 is given by dr W 1-+2 - di W 1--+2 - - - ( d i W 1--+2 + dr W 2--> 1) > 0. That is to say: in executing a cycle we have expended (i.e., put into the system) an amount of heat di Q1--->2 -Jr-dr Q2--+ 1 > 0 and have obtained from the system (i.e., the system has performed) an exactly equal amount of work di W 1~2 -Jr- dr W 2--+ 1 < O. No other changes have occurred in the universe. Suppose next that di Ql-+2 _ dr Q1--->2 ~ di Q1--->2 _+_dr Q2-+ 1 < o, so that by Eq. (1.10.1), dr W 1-+2 - di W 1~2 = - ( d i W ~-->2 + dr W 2-> 1) < 0. A similar line of reasoning now shows that in going around the cycle once we have expended (i.e., put into the system) an amount of work di W 1~ 2 _jr_dr W 2--> 1 > 0 and we have obtained from the system (i.e., the system has transferred to its surroundings) an exactly equal amount of heat di Q1--->2 -k- dr Q2---> 1 < 0. No other changes have occurred in the universe. Which of these alternatives do we choose? Let us simplify the notation by setting di Q 1~ 2 ~ di Q and dr Q 1-->2 _~ T d S 1~2 =_ T dS. One must now consider the possibilities (a) di Q - T d S > O, i.e., T d S < di Q, or (b) di Q - T d S < 0, i.e., T d S > di Q. The present discussion must apply to all situations, including the special case where the irreversible process is carried out adiabatically. Under alternative (b) one finds that T dS > 0 for the adiabatically isolated system, and no contradictions are uncovered. Under alternative (a) one would require T d S < 0 for an irreversible, adiabatic process, and by the corollary to the Second Law, discussed in Section 1.9, this possibility must be ruled out. We thus claim that diQ < TdS-drQ.

(1.10.2a)

This result is eminently sensible: if we take the system from state 1 to state 2 by a reversible sequence of steps the entropy change must be matched by a heat

44

1. FUNDAMENTALS

transfer of magnitude dr Q~ T. If the same change is achieved by an irreversible operation, part of the overall entropy change is generated by processes that cannot be controlled experimentally through manipulation of the surroundings. The amount of heat required to match the remaining entropy change will be correspondingly smaller. Also, from alternative (b) we find that dr

W 1---~2<

di W 1--+2,

(1 .10.2b)

di Q~ T.

(1.10.3)

and for any finite changes 82 -- S1 >

fl

We shall discuss these inequalities in much greater detail in Section 1.12. Equation (1.10.3) must be used with care, for it leaves open the question as to what temperature is to be employed in conjunction with any irreversible process that takes the system from state 1 to state 2. In the present case, as in most treatments, Eq. (1.10.3) is used solely in situations where the system under study is intimately thermally anchored to a reservoir under conditions where T is maintained at the fixed temperature To of the surroundings. Where this is not the case one must resort to the approach taken in Section 1.12, or adopt the procedures outlined in Chapter 6. 1.10.2

The Clausius Inequality

Consider next a different process wherein a system is taken reversibly from some state through a macroscopic cycle back to the same initial state. Denoting such a path by the integral sign fo, we find that the state function S is specified by

fodS-fo dFQ-~

(1.10.4)

On the other hand, Eq. (1.10.3) shows that in this cyclic process, for which

S1 --$2,

fo di Q T

< 0.

(1.10.5)

0,

(1.10.6)

Both statements may be combined to read

fo -T

which is known as the Clausius inequality. Note again the very nature of an irreversible process: one can never reverse the process and return to the starting point without incurring other changes in

CYCLIC PROCESSES IN RELATION TO REVERSIBILITY AND IRREVERSIBILITY

45

the universe; also, one cannot influence any event in an isolated system. If an irreversible process proceeds at all, it must go in the spontaneous direction that increases the entropy; when quiescence sets in the entropy will be at a maximum consistent with the imposed constraints. Note once more that the state function S serves as a means of monitoring whether a given process in an (adiabatically) isolated system is indeed possible. No process for which S decreases can occur in an (adiabatically) isolated system; conversely, any process for which S increases in such a system will be spontaneous. For a system A which exchanges heat with surroundings B we consider an enlarged system in which the original system and all of its surroundings form a composite isolated system for which d Stot = d SA -Jr- d SB ~> O. 1.10.3

Operation of a Heat Engine

Consider now the operation of a heat engine. To be useful as a continuous source of power it must be capable of being operated in cycles, so that at the end of every cycle the engine gets back to the same state from which it started. All net changes accompanying the operation of the engine thus occur in the universe. We operate the engine between a hot and a cold reservoir kept at temperatures Th and Tc. The reservoirs are assumed so large that heat transfers do not appreciably alter the temperatures of each. Let Qh be the heat transfer per cycle across the boundary between the reservoir at temperature Th and the engine (Qh > 0 or Qh < 0 according as heat flows into or is rejected by the engine at the hot junction), and let Qc be the heat transfer per cycle across the boundary separating the engine and the reservoir at temperature Tc(Qc > 0 or Qc < 0 according as heat is caused to flow into or is rejected by the engine at the cold junction). If n o w Qh > Qc, o r Qh -- Qc > 0, then in the course of one cycle we must have W < 0 in order that the First Law, as applied to the cyclic process, ( Q h -Qc) + W --0, may hold. That is, by arranging matters so that heat is transferred into the engine from the hot reservoir and heat is rejected from the engine to the cold reservoir, the difference Qh -- Qc is transformed into work W < 0 performed by the engine. We now ask how efficiently this energy conversion process can be carried out in a heat engine operating in cycles. The efficiency is measured by a quantity r/ defined as ~

Work out Heat in

.

(1.10.7)

To determine this quantity recall the Clausius inequality (1.10.6). In the present case the integration in (1.10.6) reduces to the two processes occurring during heat transfers Qh from the hot reservoir into the engine and - Q c from the engine into

46

1. FUNDAMENTALS

the cold reservoir. 2 Thus, according to (1.10.6),

y ~ Oi _ Oh Th i

Qc <
(1.10.8)

or

(1.10.9)

Qh <~ Q_.f_c Th Tc '

where Qh, Qc are both positive quantities, since a negative sign precedes Qc whenever we consider the heat removed from the engine. Hence, (1.10.9) may be rewritten a s Oc/Oh >/Tc/Th, o r

Qc <~ Tc Qh Th'

(1.10.10)

thus, we rewrite Eq. (1.10.7) (the right side is the Carnot efficiency)

W Q h - Qc Qc = 1<~ 1 ~7- Qh -Qh Qh

Tc T h - Tc = Th Th

=

r/Carno t.

(1.10.11)

Note the following points: (i) The generality of our approach. We have not restricted ourselves as to types of engines, reversibility, or types of processes, other than the restriction to cyclic events and to engines operating with two reservoirs. (ii) The efficiency relates to the isothermal heat transfer across the hot junction. (iii) The equality sign holds only for reversible processes, so that for any irreversible process the efficiency is inevitably less than that for a reversible one. (iv) Even a reversible process is never 100 % efficient, except in the inaccessible limits Tc = 0 o r Th ~ ~ ; it becomes the more efficient the larger Th/Tc is. (v) q depends only on the temperatures of the 'boiler' and 'condenser'. Statements (v) and (iii) are sometimes called the First and Second Theorems of Carnot. We can now recast the Second Law in a variety of ways. Statement (iv) above provides the basis for Kelvin's Formulation of the Second Law:

It is impossible to devise an engine which, working in cycles, shall produce no effect other than to extract heat from a reservoir and to perform an equal amount

of work. By working the derivations given here in reverse we can see that this statement implies the validity of the Second Law.

AN ENTROPY ANALOGY

47

Related to the above is Planck's Statement of the Second Law:

It is impossible to devise a machine which, operating in cycles, transfers heat from a colder to a hotter body without producing any other effects in the universe. This may be demonstrated by noting what would happen if the Planck statement were incorrect. If no other changes occurred in the universe then the heat extracted from the cold reservoir must be transferred without loss to the heat reservoir. Equation (1.10.8) would then have to be altered to read - Q h / T h + Qc/Tc <, 0, with the requirement that Qc be the heat flow into the engine and that Oh = Qc. In these circumstances one finds that Th/Tc ~< 1, which contradicts the definition, Th > Tc.

QUERIES AND COMMENTS 1.10.1. Examine whether the arguments relating to cyclic processes depend on the direction of the heat transfer between the system and its surroundings. 1.10.2. Here again the question arises as to what temperature is involved if the hot and cold reservoirs differ significantly in temperature from those of the engine at the respective junctions. Such a difference must exist to sustain the heat transfers. As shown later, the temperatures are those of the reservoirs. 1.10.3. Does the Kelvin statement apply to rocket performance? Or to engines operating on a single stroke?

1.11 An Entropy Analogy Just as for the First Law, one may provide a parable that illuminates the Second Law of Thermodynamics and its consequences; we follow the presentation by Spalding and Cole. 1 Entropy change may be likened to a difference ASAB in elevation between two points A and B in rough, mountainous terrain. A mountaineer elects to measure ASAB in terms of the number of steps, n =_ nr, each one-foot in height, he must take to get from level A to level B. On a firm path the difference in height is then given by ASAB = nr. But if he chooses a slippery path through the underbrush filled with dirt and gravel, slippage occurs at every step. Now, given a particular height difference, he fails to surmount this difference with the number of one-foot incremental steps, n =_ n i, that would have sufficed had he stayed on the firm path: here ASAB > n i . Considerable effort has been wasted in the form of heat supplied to the countryside when the climber, dirt, and gravel resettle at every step. Furthermore, rather than by counting n r, the value of A SAB may be ascertained by consulting geodetic maps on which altitude contours are entered. It should be clear that A SAB corresponds to an entropy change and that n r and n i simulate f dr Q~ T and f di Q~ T respectively, while contour lines on a map correspond to tabulations of entropy values.

48

1. FUNDAMENTALS

COMMENTS AND QUERIES

1.11.1. D.B. Spalding and E.H. Cole, Engineering Thermodynamics, Arnold, London, and McGraw-Hill, New York, 1959. 1.11.2. Explore the shortcomings of this parable.

1.12

Constraints, Equilibrium, Functions of State I

In this section we study among other topics the processes that involve a difference in temperature between a system and the surroundings with which it interacts. This matter seems not to have received sufficient attention in the literature, even though such a difference is needed in any process that involves a transfer of heat. The discussion is much facilitated by the introduction of a deficit function at various stages of the derivation. These deliberations then lead quite naturally to the generation of a variety of useful functions of state and to a study of their extremal properties.

1.12.1

Entropy Deficit Function

We introduce several concepts that are directly linked to the following properties of energy and entropy:

d E + dEo = O,

(1.12.1a)

dS + dSo >~O,

(1.12.1b)

where the subscript zero represents the surroundings to the system of interest. We also introduce a generally accepted assumption: we take the reservoirs to be so huge and inert that all processes within them occur reversibly at a constant temperature To. Without use of this assumption further progress is difficult. Surroundings of this type are referred to as baths. Consider first a reversible process that changes the entropy of a system through interactions with its surrounding bath but that can produce no net change of entropy in the universe: during an infinitesimal step

dSu =-drS + dlSo = O,

(1.12.2a)

where an obvious subscript notation was used. Let the same process now be executed irreversibly (1.12.2), so that the entropy of the universe increases:

d Su - di S + d2 So > O.

(1.12.2b)

Since S is a function of state of the system, di S = drS - dS; the entropy change is identical for both infinitesimal steps. Comparison of (1.12.2a) and (1.12.2b) shows that d2 So > dl So.

CONSTRAINTS, EQUILIBRIUM, FUNCTIONS OF STATE

49

The use of inequalities is very awkward in any subsequent mathematical manipulations. As a remedy it is apposite to introduce a positive entropy deficit function dO > 0 that converts Eq. (1.12.2b) into an equality: d~ S + d2So - dO = O.

(1.12.3a)

dSu ==_di S + d2So = dO.

(1.12.3b)

It follows that

Thus, dO may be interpreted as representing the infinitesimal differential entropy increase in the universe arising from irreversible interactions between the system and its surroundings. Alternative interpretations are furnished below. Moreover, on subtracting (1.12.2a) from (1.12.3b) one obtains d2 So - dl So =dO,

(1.12.4)

which shows clearly that any excess entropy generated in the irreversible process must end up in the bath, as is entirely consistent with the requirement that S be a function of state of the system. The entropy exchanged between the system and its surrounding bath is greater by the amount dO for an irreversible process than for the same process executed reversibly. Next, we note that in the closed compound unit of (system + bath) any heat that leaves the surroundings (system) must show up in the system (bath). Hence, 2 in an irreversible process, dzQo = - d i Q, so that dzSo = dzQo/ To = - d i Q~ To. Also, for the reversible case dl So = - d S, whence (1.12.4) becomes 3 dS=

~Q

+dO.

(1.12.5)

To

This equation shows that in an infinitesimal irreversible process the entropy change of the system is related but not equal to the heat transfer, that involves the inverse of the temperature of the reservoir, To, which is a well characterized quantity. Eq. (1.12.5) replaces the standard inequality d S > di Q~ T and provides a second interpretation of the deficit function: dO specifies the difference between the entropy change dS and the experimentally determined value of di Q~ To in an infinitesimal step. It is only for a reversible process that these quantities match and that dO drops out; in the irreversible case the heat transfer is numerically smaller. 1.12.2

Two Examples

It may not be out of place to illustrate the above discussion by citing two examples. Consider the magnetization of a paramagnetic sample by an external magnetic field. During a very slow, essentially reversible, process at constant temperature the gradual resulting alignment of electronic spins reduces the entropy of

50

1. FUNDAMENTALS

the sample (see also Section 5.7). Hence, at every infinitesimal step d S < 0; entropy is expelled into the surroundings, and, in accordance with Eq. (1.12.2a), we find that dl So = - d S > 0. If instead the same magnetic field is imposed in one large pulse at constant temperature, the same entropy change takes place in the system for the same degree of spin alignment. However, now, in conformity with Eq. (1.12.3a), dzSo = - d S + dO > 0 incurs a larger entropy flow into the surroundings than before. Thus, dzSo > dl So, as claimed earlier. To maintain constant temperature in both cases we set d S = dr Q~ To in Eq. (1.12.5) to find that di Q = dr Q - TodO < 0 and di Q < dr Q. Thus, with d S < 0, more heat flows out of the system in an infinitesimal step carried out irreversibly than in the reversible execution; di Q is more negative than dr Q. As a second example consider the slow expansion of a gas through a very tiny pinhole from an initial volume to a larger volume at constant temperature. As is intuitively obvious, and will be demonstrated in Section 2.4, its entropy is increased at the expense of a diminution of the entropy of the surroundings. Hence, consistent with Eq. (1.12.2a), dl So = - d S < 0. If the opening is increased to macroscopic dimensions the process runs irreversibly, such that, by Eq. (1.12.3a) d2 So -- - d S -+- dO > dl So still holds, but here in the sense that d2 So is less negative than dl So. Thus, in the irreversible process less entropy is shifted from the surroundings into the system than during the reversible operation. Note that dr Q / T matches the positive dS but di Q / T is less than dS, so that in this example again di Q < dr Q, but in the sense that both differentials are positive. 1.12.3

Characterization of Systems whose Surroundings are at Different Temperatures

We now return to Eq. (1.12.4) and replace - d l S o by d r S - dr Q / T ; it is at this point that the temperature of the system enters. 4 We next again set d2 So - d i Q~ To and thereby obtain a fundamental relation (that appears not to be widely known) for the difference in infinitesimal heat transfers under reversible and irreversible conditions drQ

T

diQ = ~ + dO.

(1.12.6a)

To

One should note how the temperatures of the system and surroundings occur in the above expression. Eq. (1.12.6a) can then be trivially rearranged to solve for the heat transfer for any given infinitesimal process under irreversible conditions" di Q - --~- dr Q - To dO - dr Q + = dr Q + ( To - T ) d S - To dO .

- 1 dr Q - To dO

(1.12.6b)

Obviously, the above relation becomes useful only after a recipe is provided to show how the quantity dO may be determined; this matter is addressed below.

CONSTRAINTS, EQUILIBRIUM, FUNCTIONS OF STATE

51

Under the more restrictive conditions where the temperatures of the system and reservoir are essentially identical, that is, when the system is thermally firmly anchored to the reservoir, one obtains the simpler form 5 di Q - dr Q - To dO - dr Q - de,

(1.12.6c)

where we have introduced a new deficit quantity with units of energy, de =- To dO. As shown in Eq. (1.12.6c), it specifies the difference in heat transfer in a given infinitesimal step between the system and its surroundings at almost the same temperature, when carried out reversibly as opposed to irreversibly. We rearrange (1.12.6a) as diQ

drQ

To

T

(1.12.6d)

=dO,

and observe an important consequence" this relation allows for the possibility that the heat transfer during the irreversible process may be the same as for the reversible process. This can happen when T ~ To and when the deficit function satisfies the relation T-To dO - dr Q ~ TTo

T-To -- ~ To

dS.

(1.12.6e)

In other words, the equality di Q - dr Q obtains not only when d O - 0 (as in Eq. (1.12.6c)), but also when matters are so arranged that dO satisfies Eq. (1.12.6e). 1.12.4

Performance of Work

At this point we invoke the First Law to write d E -- di Q + di W - dr Q + dr W; by substituting for di Q in Eq. (1.12.6b) we find that di W -- dr W - ( To - 1 ) T

dr Q + de -- dr W - ( To - T ) d S + To dO

(1.12.7a)

We now recognize that in comparing a given reversible and irreversible process with the same dQ, any work performance may be mapped onto the equivalent process of raising a weight against the force of gravity, regardless of how the other type of work is performed (i.e., whether it is electric, mechanical, magnetic, etc., in origin). It is intuitively evident that the least amount of work required to raise a weight is expended in a reversible process. Under irreversible conditions extra work is needed to overcome friction, turbulence, etc., that is then lost to the surroundings. Hence, we impose the condition that in comparing a particular process that is performed on the system reversibly as opposed to irreversibly we set di W > dr W. Conversely, the greatest amount of work is available to the

52

1. FUNDAMENTALS

surroundings when a system is operated in a reversible manner; i.e., dr W is more negative than di W. Thus, the last two terms in (1.12.7a) must be positive. By far the most interesting feature of Eq. (1.12.7a) is the fact that it permits a direct determination of dO as

dO - --~ol [di W - dr W -t (To - T) d S ] .

(1.12.7b)

This specifies the infinitesimal entropy increase of the universe for any irreversible practical process in terms of the measurable or calculable quantities di W, dr W, and d S ~ a n apparently unappreciated result. Note that for cyclic processes the dS term drops out, and the difference di W - dr W can grow without bounds if the cycle is endlessly repeated. By contrast, the above arguments impose a lower bound on the sum of the last two terms in (1.12.7a), namely,

TodO=de>

~-1

drQ--(To-T)dS>O.

(1.12.7c)

This inequality should be compared with the equality (1.12.6e). The righthand side reflects the fact that for To > T, heat flows into the system, so that ( T o / T - 1)dr Q is positive. When To < T both factors are negative. Hence, the lower bound, being positive, is more restrictive than the original inequality dO>O. If the system remains thermally well anchored to the reservoir such that T = To throughout any given process, Eq. (1.12.7a) reduces to d~ W = d~ W + d~,

(1.12.7d)

so that de then relates to the work performed in executing at essentially constant T an infinitesimal irreversible, as compared to a reversible, step. We finally return to Eq. (1.12.6b) via (1.12.7c) and note that when we set (To~ T - 1) dr Q - de < 0 we obtain di Q < dr Q, which is a reformulation of the basic thermodynamic inequality that rests on the Second Law. The present analysis could, of course, be run backwards: starting with the basic inequality di Q < dr Q one may use the above derivation to show that di W > dr W.

1.12.5

Basic Formulation of Equilibrium Thermodynamics

At this point we can begin a study of the very basic relations that underlie the remaining thermodynamic developments, namely the construction of appropriate functions of state. For this purpose we now rewrite Eq. (1.12.1 a) as di Q + di W + dEo = 0; then, using Eq. (1.12.5), we obtain the fundamental expression

TodS - de + di W - d E = O,

(1.12.8)

CONSTRAINTS, EQUILIBRIUM, FUNCTIONS OF STATE

53

which makes no explicit reference to heat transfers but involves only the performance of work and irreversibility effects. Note that it is the well-established temperature of the surroundings that enters the equation. 4 1.12.6

Construction of Functions of State and their Extremal Properties

In what follows we apply Eq. (1.12.8) to several special situations and examine the consequences. In some situations we imagine the system at temperature T to be enveloped by surroundings at a temperature To, and to be subjected to a variety of virtual displacements away from the equilibrium state, while maintaining the initially specified constraints. To emphasize this point we introduce the symbol 6 whenever constrained variations are considered. We then examine the response to such a displacement. 1.12.7

Isolated or Isoenergetic Systems

As a first example, consider the isolated system; here ~i Q = ~i W = ~ E -- 0; also note that in the absence of surroundings we simply introduce the temperature of the system, T. Then Eq. (1.12.8) reduces to

T ~ S = 6 e = _ T~O ~ 0 .

(1.12.9a)

Here, 3S = 60 >~ 0 is the entropy change arising from irreversible processes occurring within a completely closed system. As Eq. (1.12.9a) shows, S can then only increase. As soon as these processes have ceased, 30 = 6 S = 0, so that S has assumed an extremal value which is a maximum under the present constraints. For example, the entropy change in the free expansion of a gas can be determined by finding AS under quasistatic conditions, as specified later in Section 2.3. Since S is a function of state the same entropy change takes place in a free expansion under the same conditions. All this, of course, merely repeats what has been well established in earlier sections. Consider next an isoenergetic process for which 6 E -- 0, but for which interactions between system and surroundings are permitted, as long as one requires that the work exchange at each infinitesimal step be precisely balanced by the heat exchange. This can only be done under quasi-equilibrium conditions that exclude any irreversible processes over which the experimenter has no control. As a necessary condition the temperatures of the system and surroundings must converge to the same limiting value. We therefore set 3r W = --6r Q and 6e = 0. Then Eq. (1.12.8) reduces to

T6S=6rQ,

(1.12.9b)

which merely repeats the well-established fact that the entropy change is tracked by the reversible heat transfer.

54

1. FUNDAMENTALS

Alternatively, we write T 3S -- - ( ~ r W ,

(1.12.9C)

which signals the fact that the entropy of a system can be reduced, namely by doing work on the system, ~r W > 0, while maintaining constant energy. This should dispel the idea that in any process whatever the entropy of a system always remains either constant or can only increase. Obviously, the entropy diminution in the system must be compensated for elsewhere in the universe by processes that increase the total entropy of the system plus surroundings. 1.12.8

State Functions for Isothermal Processes

New concepts are established by considering processes that occur isothermally, that is, in the limit: 3T - - 0 and To -- T. We may now rewrite Eq. (1.12.8) as 3 ( E - T S ) -- 3A -- 3 i W -(~B -- (~rW.

(1.12.10a)

In the above we have invented a new function of state, A = E - T S, involving state functions that had been previously introduced. A is called the Helmholtz (free) energy function. The fight-hand side follows from Eq. (1.12.7d). As is seen, changes in A are tracked by the reversible performance of work at constant T. If no work is involved, but irreversible (and therefore, uncontrollable) processes are allowed to occur at constant temperature, we find from the above that 3A = - 3 e

~< 0.

(1.12.10b)

Thus, irreversible processes within the system at constant temperature lead to a decrease in its Helmholtz free energy. When quiescence sets in, 6A -- 0, and A is minimized; this characterizes the equilibrium constraint for a system at constant temperature. In what follows we distinguish again between mechanical and nonmechanical work by setting di W - di Wm + d W n and de - d6m + d6:n. In the Exercises 6 the reader is asked to show that dSm and d6n must separately be nonnegative. If, in addition to T, the volume V is held fixed (isothermal-isochoric conditions, for which no mechanical work is possible) then the above expressions reduce to A = ~i Wn - ~6n "-- ~r Wn.

(1.12.11)

Parenthetically, Eq. (1.12.11) is a restricted version of (1.12.10); contrary to some assertions in the literature, there is no need to require constant volume conditions to prove the assertion that A is a minimum at equilibrium. Suppose, instead, that the system is subjected to processes at constant pressure (isothermal-isobaric conditions). It is then expedient to rewrite Eq. (1.12.8) as d E - d ( T S ) + d ( P V) - di Wn + den = O, or, 3(E - TS + PV) -3G

-- 3iWn - 3en = ~ r W n .

(1.12.12a)

CONSTRAINTS, EQUILIBRIUM, FUNCTIONS OF STATE

55

By arguments now familiar we have introduced a new function of state, G -= E - T S + P V, the Gibbs (free) energy, involving the indicated combination of other functions of state. As is evident, changes in this quantity are tracked by the reversible performance of work at constant T and P other than mechanical. Moreover, if irreversible phenomena occur under isothermal-isobaric conditions in the absence of any work performance then 3G = - 3 e n <. O,

(1.12.12b)

showing that in such circumstances the Gibbs free energy of the system diminishes and reaches a minimum at equilibrium. 1.12.9

Isentropic Processes

We next consider isentropic processes that do not explicitly involve temperature as a variable. Here S is held fixed, and all processes involving Eq. (1.12.8) are subject to the restriction d E + de - di W -- 0. Thus, 7 E = - 3 e + ~i W -- ~r W,

(1.12.13a)

showing that the energy change under such conditions is tracked by reversible performance of work under adiabatic conditions. This, of course, simply repeats what was originally stated while setting up the First Law. The right-hand side follows from the use of Eq. (1.12.7a), with the central term missing. However, if irreversible conditions prevail within the system in the absence of any work, we find 6E = - 6 s

~< 0,

(1.12.13b)

so that under the stipulated conditions the energy of the system diminishes while the process lasts, and E is minimized at equilibrium. A special case arises if the volume is also held fixed; under such isentropicisochoric conditions E = - 6 e n + ~i Wn -- ~r W n ,

(1.12.14a)

E = -3Sn <. O.

(1.12.14b)

and

On the other hand, if pressure is kept fixed at constant entropy, we find for isentropic-isobaric conditions that d E + d (P V) + dgn - di Wn - O, SO that 6(E + P V) =_ 6H = ~i Wn - ~gn = ~r W n .

(1.12.15a)

We thus encounter yet another function of state, the enthalpy, H =_ E + P V, whose change under adiabatic conditions is tracked by reversible performance of

56

1. FUNDAMENTALS

nonmechanical work. When irreversible effects enter and no work is performed we find that (1.12.15b)

6 H = - 3 S n <~ O,

showing that this function too diminishes when the system is subjected to isentropic irreversible processes in the absence of work performance. At equilibrium the enthalpy is a minimum. 1.12.10

Summary

The full utility of the functions of state listed above can become apparent only later. For now we note that changes in A and G may be specified by reversible performance of work under isothermal conditions, while changes in E and H are linked to performance of work under conditions of constant entropy. Variations in S may be similarly monitored under conditions of no energy change. As emphasized earlier, the characterization of processes is greatly simplified by t~tilizing such path-independent functions. Moreover, since work can always be measured empirically or calculated, these functions of state can be directly evaluated or determined experimentally. Except for S the above functions of state are all at a minimum when equilibrium prevails under the postulated constraints, while S is at a maximum. The above results are summarized in Table 1.12.1. The table shows that different functions of state are useful in characterizing equilibrium conditions under a variety of conditions. The table should also put at rest a common misconception that the performance of work is invariably linked to energy changes. Obviously, the relevant function of state depends on the conditions of the experiment. Also, we can illustrate the meaning of 'virtual processes'. These refer to slight displacements of the system from equilibrium while maintaining the indicated constraints. If the function of state does not change in such a displacement the system is at equilibrium; if not, changes take place until a new equilibrium point is reached. Table 1.12.1 Characterization of equilibrium

Fixed parameters (constraints)

Equilibrium

Relation to reversible work

E S S, V

~S = 0, S maximum ~E -- 0, E minimum 6E = 0, E minimum ~ H -- 0, H minimum ~A = 0, A minimum 6A = 0, A minimum SG = 0, G minimum

~S " ~E =

S, P

T T, V T, P

--Sr W/T Sr

W

6 E = ~r Wn ~H = (~r Wn

~F

- - ~r

W

S F = 8r Wn ~G -- Sr Wn

CONSTRAINTS, EQUILIBRIUM, FUNCTIONS OF STATE

57

REMARKS AND QUERIES 1.12.1. The author is greatly indebted to Professor Dor Ben Amotz of Purdue University for his insightful participation that greatly clarified the fundamental concepts on which the present section is based. See also: Dor Ben Amotz and J.M. Honig, J. Chem. Phys. B 118 (2003) 5932. 1.12.2. The notation is a bit awkward at this point: di Q really refers to the heat released by the system to the surroundings or absorbed from it in an irreversible manner. The negative of this quantity is absorbed by or furnished from processes taking place reversibly in the reservoir. Since no heat escapes the compound system, we may set di Q = -d2 Qo. 1.12.3. Strictly speaking, the differential d (in dX) should only be used to indicate an infinitesimal variation of X along a thermodynamically describable path, and should not be applied to extrathermodynamic, irreversible phenomena. In our case, however, we demand only that the initial and final points of the process be representable as equilibrium states. The use of yet another symbol for a nonequilibrium differential process seems excessive. 1.12.4. The question of how to deal with the concept of temperature in a system undergoing irreversible processes will be discussed below and is taken up in Chapter 6. 1.12.5. Of course, this statement should be more carefully formulated. Since no heat transfer can occur without a temperature difference between the system and the surroundings Eq. (1.12.6c) is to be interpreted as a limiting law wherein dQ signifies the heat flow for the case where T closely approaches To. 1.12.6. To show that the individual de are nonnegative consider some spontaneous process for which d8m < 0; and examine the consequences by comparing work performance in reversible and irreversible processes. 1.12.7. At this point it is vital to distinguish between adiabatic and isenthalpic processes. In the former dQ = 0, so that by the First Law di W = dr W; for any given infinitesimal process all types of work performance lead to the same energy differential, dE. In the isenthalpic case (or if T = To) the central term in Eq. (1.12.7a) drops out and Eq. (1.12.7d) is applicable; now the energy change does depend on how the work is executed in the process. If done irreversibly the entropy dO thereby generated must be transferred out of the system by removal of heat to the surroundings. Incidentally, one way of maintaining the T -- To requirement is through the contrived step of ramping the reservoir temperature up or down in step with the temperature of the system at each infinitesimal change of conditions, but it is best to focus attention on the requirement that dS -- 0. 1.12.8. The so-called Grand Potential function J = A - G has occasionally been used to characterize thermodynamic states. Show under what conditions this quantity can be related to reversible performance of work and characterize the properties of this function when irreversible processes occur. 1.12.9. As another example that relates to comparisons between processes executed reversibly or irreversibly, consider at constant volume the conversion of hydrogen and oxygen into steam over a platinum catalyst, and compare this with the explosive conversion by an electric spark. Are any changes needed in the derivations?

58

1.13 1.13.1

1. FUNDAMENTALS

Systematics of Thermodynamic Functions of State Thermodynamic Functions of State

We begin here a study of the properties of functions of state that are useful in characterizing physical processes. One should note the amount of information that becomes available through a systematic approach. We begin with the general formulation for the energy change of a system by noting the corresponding expression for the surroundings, as indicated by the subscript zero. Here we replace the heat transferred into the bath by dr Qo = To dS0, and the mechanical work performed, by dr W = - P d VO, so that d E o = To dSo - Po dVo,

(1.13.1a)

which we adopt because it is assumed at the outset that all processes in the surroundings always take place reversibly. In a closed unit comprised of system + surroundings maintained at constant volume 1 we then set d E o - - d E and d V0 = - d V. We also adopt Eq. (1.12.4), in which we set dl So = - d S; it will be recalled that dl So was defined to be the entropy change in the surroundings when the process in the system was carried out reversibly. Then (1.13.1b)

d2 So = - d S + dO.

When the above substitution is applied to (1.13.1a) one obtains the fundamental result for the differential energy of the system d E -- To d S - Po d V - To dO.

(1.13.1c)

This expression shows that volume and entropy serve as thermodynamic variables, or as control variables, for the internal energy function of the system: E = E ( S , V). Nevertheless, the intensive variables are those of the surroundings, and are therefore well defined, even when the processes in the system proper are far removed from equilibrium. For the present we exclude other types of work that are treated in Chapter 5. We defer the generalization of the present treatment to the case of open systems to Section 1.20. It is now expedient to introduce explicitly the difference in intensive variables between the system and surroundings, by rewriting the above in the form d E = (To - T ) d S -

(Po - P ) d V

if- T d S -

P d V - TodO.

(1.13.1d)

This, however, again raises the problem of defining intensive variables such as temperature for a system that undergoes irreversible processes. One way to approach this difficulty is presented in Chapter 6, where T is allowed to depend on the position within the sample. An alternative is shown in Fig. 1.13.1 which

SYSTEMATICS OF THERMODYNAMIC FUNCTIONS OF STATE SYSTEM

59

SURROUNDINGS

1

T0

Fig. 1.13.1. Temperature profile near the junction between a system (left) at temperature T and a reservoir (right) at temperature T O under QSI operating conditions. The change in temperature occurs over a narrow boundary region.

shows a partition between the system and its surroundings that permits the transfer of heat. If this barrier is a poor thermal conductor the rate of heat transfer will be sufficiently small, such that most of the volume of the surroundings (the sample) will be at temperature To (T). The changeover from T to To is then confined to the immediate vicinity of the thin barrier. Thermal processes carried out under such conditions will be termed quasi-static irreversible (QSI). More generally, we define the QSI processes as those in which the characteristics of the system and, separately, of the surroundings do not deviate seriously from internal equilibrium. However, the QSI state of the system and surroundings may differ to any arbitrary extent. The reader may construct other scenarios in which T represents some suitably averaged value over the volume of the system; this represents no major difficulty when the variation in T over the sample volume is small relative to its average. However, for severe departures from equilibrium the concept of temperature may no longer be viable. Similar considerations apply to all other intensive variables, such as pressure. As a special case consider removing the system from its equilibrium state by external intervention and then allowing the system to respond, all the while maintaining conditions of constant entropy and volume. In that event one finds that d E -- - T o d O .

(1.13.1e)

This shows that in the absence of external driving forces To d S and Po d V any ongoing process will take place intemally, and will not be subject to experimental control. Thus, with dO > 0, the energy of the system spontaneously diminishes until these irreversible phenomena have run their course, at which point E assumes a minimum value consistent with the imposed constraints. Eq. (1.13.1e) coincides with Eq. (1.12.13b) that holds when no work is performed. By contrast, if the conditions of the surroundings and of the system are in near balance, so that T ~ To and P ~ P0, then any process becomes essentially reversible, and the differential energy function (1.13.1 d) assumes the standard form dE = T dS - P dV.

(1.13.If)

60

1. FUNDAMENTALS

Thus, entropy and volume represent the appropriate control variables. Since E is a function of state Eq. (1.13.If) may be subtracted from Eq. (1.13.1d), thus allowing us to solve for (1.13.1g)

TodO -- (To - T) dS - (Po - P) dV.

This relation is of considerable interest: for, it allows us to determine the infinitesimal contribution to the entropy that is attributable to irreversibility effects. Note that this determination is based on changes in volume or entropy that can be measured. Before Eq. (1.13.1g) can be integrated one must specify how the intensive variables To - T and P0 - P are altered in a given irreversible process. If this is done one obtains 0 in terms of S and V. This matter will be taken up below in a slightly different context. We next examine the other functions of state that had been introduced in the preceding section. The first of these is the enthalpy, defined by H = E + P V,

(1.13.2a)

which, sensibly, invokes the pressure P that characterizes the system, rather than P0. The corresponding differential d H = d E + P d V + V d P has the generalized form, obtained from Eq. (1.13.1 d) d H - - ( T o - T) d S -

(1.13.2b)

( P o - P) d V + T d S + V d P - TodO.

The difficulty here is that H = H (S, P) should be written out in terms of S and P as control variables. To address this problem we adopt a transcription based on the relations V -- V (T, P) and S -- S (T, P). We then invert the latter in the form T = T (S, P) and substitute in the former to set V = V ((S, P), P) = V (S, P), from which we derive the relation dV-

( ) -~

dS+

-~

sdP'

(1.13.2c)

which is then substituted in (1.13.2b) to yield d H -- (To - T ) d S -

(Po - P)

+ T dS + V d P -

[()

TodO.

ov --~

P

as+

S

de

] (1.13.2d)

If desired, the partial derivatives may be reexpressed as shown in Section 1.3. As an example, one might decide to write (OV/OP)s = (OV/OP)T + (OV/OT)p(OT/OP)s. The quantities (OV/OP)T or (OV/OT)p are specified by - f l V or ot V respectively, where fl and o~ are the isothermal compressibility coefficient and the isobaric thermal expansion coefficient. Also, in a reversible process (OT/O P ) s represents the change in temperature of the system during an adiabatic

SYSTEMATICS OF THERMODYNAMIC FUNCTIONS OF STATE

61

change in pressure. In any event, we have now expressed d H in terms of independent experimental variables that specify enthalpy changes. In the special case dS = d P =- 0 there is no external driving force for generating changes in enthalpy. Any remaining processes are spontaneous, thus, not subject to experimental control, and are therefore expressed by the relation (1.13.2e)

d H = - To dO,

so that the enthalpy diminishes until quiescence sets in, at which point H is at a minimum consistent with the indicated constraints. This relation is identical with Eq. (1.12.15b) which holds when no work is performed. If, instead, conditions prevail in which the system and surroundings are in near-equilibrium, so that processes occur nearly reversibly with P = P0 and T = To, one obtains the standard formulation (1.13.2f)

dH = T dS + V dP,

where entropy and pressure again serve as the control variables. On subtraction from (1.13.2d) one obtains TodO - ( T o - T) d S -

(Po - P)

-~

dS +

-ff--fi

P

dP .

(1.13.2g)

s

Here 0 involves S and P as control variables. To carry out an integration To - T and P0 - P must be specified for a particular process. Proceeding along very similar lines we introduce the Helmholtz (free) energy through the relation A = E-

TS,

(1.13.3a)

with the differential form dA -- d E - T d S - S d T, so that d A = (To - T) d S -

(Po - P) d V - P d V - S d T - TodO.

(1.~3.3b)

The problem here is that A = A ( T , V) should be written out in terms of T and V as control variables. We address this situation by setting S = S ( T , V), so that dS_

( ) _~

dT +

(~

-ff--ff T d V "

(1.13.3c)

At this stage we anticipate the derivations provided below that result in the formulation of Eqs. (1.13.12) and (1.13.15), according to which we may set (0 S/O T) v = C v / T, where C v is the heat capacity at constant volume, and we may introduce the relation (OS/OV)T = ( O P / O T ) v . These results are derived by independent methods, so that circular arguments are avoided. We next introduce the mathematical identity (1.3.9) to write ( O P / O T ) v = - ( O V / O T ) p / ( O V / O P ) T

62

1. FUNDAMENTALS

and then replace the numerator and denominator by ot V and by - f l V, where ot and fl are the isobaric thermal expansion and the isothermal compressibility respectively. On introducing these substitutions we finally obtain

[

]

Ol

d a -- ( To - T ) CV T d T + -fl d V

- ( P o - P ) d V - P d V - S d T - TodO

(1.13.3d) which expresses the Helmholtz differential in the appropriate independent variables and in terms of measurable quantities. Using familiar arguments we next set d T = d V = 0, and find that in spontaneous processes under these conditions d A = - To dO,

(1.13.3e)

showing that the Helmholtz free energy assumes a minimum value under the indicated constraints. This relation agrees with Eq. (1.12.10b), obtained in the absence of any type of work. When system and surroundings are in near equilibrium a reversible process is characterized by the standard relation d A -- - S d T - P d V .

(1.13.30

When subtracted from (1.13.3b) we find TodO--(To-r) --(To-T)

[( ) [ ~ -OS ~

V

dr+

-~

CV d T + dV T -fl

T

dV

-(Po-

]

-(Po-

P)dV

P)dV,

(1 13.3g)

where we have expressed dO in terms of T and V and quantities that may be measured experimentally. Once again (To - T) and (Po - P) must be specified for a particular process before the integration can be performed. Finally, we introduce the Gibbs (flee) energy function via (1.13.4a)

G=E+PV-TS, whose differential form is given by d G = (To - T ) d S -

(Po - P ) d V + V d P - S d T - TodO.

(1.13.4b)

This must be rewritten so as to involve T and P as control variables. Proceeding in the manner shown above we set S = S(T, P) and V = V (T, P) to write d G - ( T o - T)

OS -~

P

dT +

-ff--fi

dT+

r

(OV)

dR

-if-fiT d P ]

+ VdP-

SdT-

TodO

SYSTEMATICS OF THERMODYNAMIC FUNCTIONS OF STATE

--(To-T)

Cp T d T - o~V d P

+ V dP - SdT-

63

J

- (Po - P)[o~V d T - ~6V d P ]

(1.13.4c)

TodO.

In the second part we again used Eqs. (1.13.10) and (1.13.15) that will be derived below on an independent basis: C p is the heat capacity at constant pressure and ( O V / O T ) p and ( O V / O P ) T were replaced by c~V and by - f l V , respectively. When we require that d P = d T = 0 no controllable changes can take place. Any residual processes are spontaneous, and thus, subject to the relation (1.13.4d)

d G = - To dO,

so that the Gibbs free energy assumes a minimum value under the indicated constraints. Once more, this relation coincides with Eq. (1.12.12b) when no work is incurred. When near equilibrium prevails Eq. (1.13.4b) reduces to the standard form dG = -S dT § V dP.

(1.13.4e)

By methods now familiar we subtract Eq. (1.13.4e) from (1.13.4b) to write -

d T - oeV d P J

_

-

(Po -

P)[o~V dT

-

~V dP],

(1.13.4f)

where we are specifying 0 in terms of T and P. 1.13.2

Discussion

This completes the construction of the elementary functions of state for a closed system; their generalization to open system is deferred to Section 1.20. In the formulations (1.13.1g), (1.13.2g), (1.13.3g), (1.13.4f) the deficit function was written out in terms of differentials of functions of state and in terms of (T0 - T) and (P0 - P). As a first approximation one may expand these differences in a Taylor's series up to second powers. The same applies to the generalized functions of state (1.13.1 d), (1.13.2d), (1.13.3d), (1.13.4c). The ramifications of such steps have not been explored, but a version equivalent to the first power expansion is provided in Chapter 6. To integrate Eq. (1.13.4f) is not a trivial matter. At the outset let us assume that the surroundings are essentially infinite, so that whatever the process, the quantities To and P0 remain unaltered. The remaining variables change continuously during the irreversible process, so that the trajectory of each must be specified as

64

1. FUNDAMENTALS

a function of time t. Thus, the integrated form stretches over limits between the initial time ti and the final time t f . We obtain

-

[/,o-,<,> To

~V(T(t))

-57

dt -

To

~V(P(t))

-3T

dt

]

"

(1.13.4g) Clearly, the above expression is very unwieldy. Some simplification is achieved by ignoring the dependence of Cp, ~ V, and/5 V on the indicated variables, so that the first integral reads Cp f ( 1/ T - 1/ To) d T -- Cp [In (Tf / Ti ) - (1 / To) (Tf /~) ]. The last integral may be rewritten in the form - (~ V~ To) [Po ( P f - Pi) ( p 2 _ p2)/2] Thus, these two integrals are path-independent in this approxif 9 . mation. On the other hand, the two central integrals cannot be simplified in this fashion. One possible approach is to assume that in (1 - T~ To), T may sensibly be replaced by its average value 7~ ~ (Tf - / ~ ) / 2 , and similarly, for P(t) in the third integral. If T does not deviate significantly from To, and similarly for P the two central contributions are small and the approximations are tenable. In this particular simplification the two integrals reduce to -or V (1 - T~ To)(Pf - Pi) and (~ V Po/To) (1 - P / Po) (Tf - Ti), respectively. The extent to which such steps may be justified can only be decided by experiment. In any case, an explicit result has been provided for one particular set of approximations. 1.13.3

Summary

In summarizing the above we should note that the First and Second Laws have now been placed on an equal footing: for any irreversible process we write d E = dQ + d W

(1.13.4h)

d S = d Q / To + dO,

(1.13.4i)

and

in which dO is specified by any of the appropriate variations cited above. Since these relations involve measurable quantities and functions of state the differential for the entropy is now properly specified for any process. The resulting reformulation parallels the First Law, in the sense that it facilitates the determination of entropy changes involved in heat and work exchanges of any kind, just as the First Law quantifies energy changes arising from any kind of interchange of heat and work. In this connection it is also important to recall Eq. (1.12.7b) that expressed dO solely in terms of work and entropy increments. This provides an alternative procedure for use in Eq. (1.13.4i).

SYSTEMATICS OF THERMODYNAMIC FUNCTIONS OF STATE

1.13.4

65

Systematization of Results Based on Functions of State

Henceforth we concentrate on the use of Eqs. (1.13.10, (1.13.20, (1.13.3f), (1.13.4e) as the fundamental building blocks (as applied to equilibrium processes) for all subsequent thermodynamic operations. The enormous advantage accruing to their use is that by the First Law all of these functions depend solely on the difference between the initial and the final equilibrium state. We no longer rely on the use of quantities such as heat and work that are individually path dependent. As will be shown shortly and in much of what is to follow, these functions of state may be manipulated to obtain useful information for characterizing experimental observations. One should note that the choice of the functions E, H, A, or G depends on the experimental conditions. For example, in processes where temperature and pressure are under experimental control one would select the Gibbs free energy as the appropriate function of state. Processes carried out under adiabatic and constant pressure conditions are best characterized by the enthalpy state function. When changing these control variables in going from (If) <-+ (2f) <-+ (3f) (4e) by successively adopting (2a), (3a), (4a) one progresses from one function of state to the next. This process is known as executing a set of Legendre Transformations. Thus, no one function is in any sense more fundamental than any other; the choice is dictated solely by what quantity is the most convenient for use in any experimental situation. 1.13.5

Thermodynamic Interrelations; Maxwell Equations, and Equations of State

At this point the systematics kicks in. It will be evident that an enormous amount of information may be generated by a systematic application of elementary steps, thereby illustrating the power of thermodynamic methodology. Consider first the process of taking first partial derivatives of the functions (1.13.If), (1.13.2d), (1.13.3d), (1.13.4e) with respect to the appropriate independent variables. We obtain T = (3E/3S)v

= (OH/OS)p,

(1.13.5)

V = (3H/3P)s

= (3G/3P)T,

(1.13.6)

S = -(3A/3T)v P = -(3E/3

= -(3G/3T)p,

V)s = -(3A/3

V)T.

(1.13.7) (1.13.8)

Eq. (1.13.7) for the entropy is particularly useful if A or G are known as functions of T, V or T, P, respectively. These can be determined, for example, through the use of partition functions in statistical mechanics, which is usually relatively straightforward, at least in commonly used approximation procedures. Alternative

66

1. FUNDAMENTALS

methods for finding A or G via equations of state or other methods are furnished later. Similarly, determining the pressure via Eq. (1.13.8) is useful when direct measurements (e.g., the pressure exerted by electrons in a metal) are difficult. The remaining relations are likely to be less useful. The next step consists in carrying out a cross differentiation in either order; for example, by applying the relation (02E/OSOV) -- (02E/OVOS) to Eq. (1.13.If) one obtains Eq. (1.13.9) below. The other expressions are similarly derived.

(OT/O V)s = - ( 0 P/aS)v,

(1.13.9)

(aSIa P)T = --(O V/aT) p,

(1.13.10)

(OT/OP)s = (OV/OS)p,

(1.13.11)

(OS/OV)T = (OP/OT)v.

(1.13.12)

The above relations are known as Maxwell Equations. Eqs. (1.13.10) and (1.13.12) are particularly useful: if the equations of state is known in the form P = P (V, T) or V = V (P, T) for any given material; then its entropy may be determined by integration of the partial differential equation with respect to P or V. 2 More generally, these expressions are used to eliminate partial derivatives of the entropy in favor of temperature derivatives of pressure or volume, quantities that are directly accessible by experiment.

1.13.6

Caloric Equations of State

Other deductions are found by substituting S = S (T, V) or S = S (T, P) into E = E (S, V) or into H = H (S, P) to obtain E = E (S (V, T), V) = E (T, V) and H = H(S(P, T), P) -- H(T, p),3 so that one may write

aS -~

dE--T

v

OE --

-~

dH-T-~

dT+

dT + V

OS

-~

-~

T

-P

dV

OV,

(1.13.13)

T

P

OH

T -~

[()]

dT + T - ~

aT +

T

+ V dP

dP.

e

(1.13.14)

V

Comparing the coefficients leads to the results

--

OT

-~ Cv,

~

V

-

P

~ Cp,

T

P

(1.13.15)

SYSTEMATICS OF THERMODYNAMIC FUNCTIONS OF STATE

() OE -~

-- T T

-~

T

(0,) -~

- P - T

-~

T -- T

-~-fi

67

- P ( V , T),

(1.13.16)

p + V(P, T),

(1.13.17)

V + V-- - T

-~

T

where the appropriate Maxwell relations were used to obtain the right-hand side of the last two equations. The quantities C v and Cp are defined as shown in (1.13.15), and are known as heat capacities at constant volume or at constant pressure. These can be experimentally determined as a function of T over wide temperature ranges, normally by standard calorimetric methods (see also Section 1.16). Integration of the experimental heat capacities with respect to temperature then yields E, H, or S as a function of T (see Section 1.17), with either V or P as parameters. Alternatively, by inserting the equation of state into (1.13.16) or (1.13.17), followed by integration, one can find the dependence of E on V or H on P, with T as a parameter. Eqs. (1.13.16) and (1.13.17) are known as caloric equations o f state. As an aside, Eqs. (1.13.13) and (1.13.14) illustrate the difference between what is often referred to as natural and unnatural coordinates. It is seen that E, considered as a function of S and V, has a very simple differential form, Eq. (1.13.If); this is obviously the natural coordinate representation. By contrast, E, considered as a function of T and V has a more complicated 'unnatural' differential form, in which the coefficients of d T and of d V are specified by the relation (1.13.13). Similar remarks pertain to the enthalpy, H. Lastly, insertion of Eq. (1.13.7) into A = E - T S leads to

A--E+T

~

v

or

~

v

-~

P

or

-~-

P

~

T

(1 13.18)

T

(1 13 19)

~

Similarly, one finds G--H+T

-----

~

o

o

o

The above relations are known as G i b b s - H e l m h o l t z equations. The solution of these partial differential equations, after insertion for E or H from (1.13.13)(1.13.17), yields A or G. We have thus achieved a complete specification of the various thermodynamic functions of interest in terms of experimental information. We shall later discuss simpler experimental techniques for determining these quantities. 1.13.7

The Equilibrium State Reviewed

We have seen above that after processes have run their course the various functions of state E, H, A, G, and - S have assumed minimal values consistent with

68

1. FUNDAMENTALS

the constraints imposed on the system. To undo the minimization process work must be executed, or, as seen later, material must be transferred across the boundaries of the system. On this basis we may identify the equilibrium state as that for which the appropriate thermodynamic function of state (depending on the various applicable constraints) is at a minimum. Again, any displacement from this state requires a relaxation of the constraints and/or performance of work and/or transfer of matter across boundaries.

1.13.8

The Absolute Temperature Scale; Reprise

We are also now in a position to introduce a very elegant proof set forth by Landsberg, 4 which shows that all classes of different physically sensible temperature scales do in fact correspond uniquely to the absolute temperature scale introduced in Section 1.2. We thereby complete the discussion which was briefly alluded to in Sections 1.2 and 1.9. We attempt to correlate an empirical temperature scale t to the thermodynamic temperature scale T. For this purpose we rewrite Eq. (1.13.16) in the form

OP

-1

1 (1.13.20)

Multiply both sides by ( d T / d t ) d t and integrate from to (corresponding to To) to ti (corresponding to 7~)

Ct fT~ i d T T

1

I i

(0P / O T ) v

Jto P + ( O E / O V ) t

dt =-- Ki

1

ln(Ti/To)

(i

1, 2 . . . . ).

1

(1.13.21) Here to is a standard temperature that has been arbitrarily chosen and the set ti represents a set of other temperature values on the chosen empirical t scale. Note the definition for Ki. From (l. 13.21) one immediately deduces that T2 -- eK2 To ~ e K2

TO e K2 T1 - To (T1 - T o ) - ~ K1 ( T l-e 1

- To).

(1.13.22)

Let T1 be deliberately chosen such that tl - to corresponds to the difference between the boiling and freezing points of water maintained at one bar. Then we may set T1 - To = 100 K. Suppose now another empirical temperature scale is selected in which t il corresponds to tj correspondingly, K~ must match Ki. Thus,

' ftj;P +(OP/Ot')v (0 E/O V)t' dt'- f6

Ki =~

dt = Ki

(i = 1, 2 . . . . ). (1.13.23)

SYSTEMATICS OF THERMODYNAMIC FUNCTIONS OF STATE

69

!

Note that the equality K i - - K i arises because the evaluation of an integral is independent of the selection of a dummy integration variable. It now follows from Eqs. (1.13.22) and (1.13.23) that eK2 T 2 - eK 1 __ l ( r ; !

7'( -- Td Z ; ) - Z2 T1----2--~o.

(1 13.24)

!

Hence, if tl - to - t 1 - to are both so chosen as to correspond to 7'( - Td -- T1 To -- 100 K, then T ~ - T2, which proves the assertion. The absolute temperature is thus independent of the choice of an empirical temperature scale. 1.13.9

Equations

of State

In the foregoing we have emphasized the useful role that an equation of state plays in the determination of the various functions of state displayed above. Hence, we briefly comment on some mathematical relations that have been found useful for this purpose. The simplest of these is the perfect gas law which may be written in the form PV

Z -

RT

= 1.

(1.13.25)

The use of this relation is obviously very restrictive. One may attempt to improve on its usefulness by introducing an expansion of the form B(T) C(T) V~ + 9 2 + " "

Z(T, 9)--I+

(1.13.26)

where the coefficients B, C . . . . can generally be determined from theories of intermolecular forces; they have been tabulated for various materials. Eq. (1.13.26) is known as a virial equation of state. A more ubiquitous relation was first proposed by Johannes van der Waals in 1873 that is applicable to fluids; it has the form RT

a

P :- 9 - b

92"

(1.13.27)

This formulation has been found so useful that compilations of the parameters a and b are widely available. Eq. (1.13.27) leads to the relation 1

a

Z =

~

1 -b/V

~.

(1.13.28)

RTV

In many cases b / V << 1, in which case one may expand the denominator of the first term to obtain a )1 Z-l+

b-

b2 +

+..-

"

(1 13.29)

70

1. FUNDAMENTALS

A comparison of (1.13.29) and (1.13.26) establishes a relation between the virial coefficients B, C . . . . , and the van der Waals coefficients. Other equations of state for fluids have been proposed in the literature, among them the Berthelot relation

(

P + T 9 2 ( 9 - B b ) - RT,

(1.13.30)

P ( V -- b')e a'/RTf' = RT,

(1.13.31)

the Dieterici relation

and the Redlich-Kwong relation P =

RT V-b

a T1/2V(V+b)

.

(1.13.32)

When solids rather than fluids are considered one invokes the so-called Griineisen Law. In differential form (at constant volume) it reads

dP--yG(T)Cv(T)dT,

(1.13.33)

where yG(T) is a temperature dependent parameter that has been specified in the literature for a variety of solids. If the solid is metallic yG(T)Cv(T) must be replaced by yG(T)Cv(T) + 2Cv,el/3. where the correction term involves the electronic contribution to the heat capacity of the solid. This latter term is generally swamped by the first term except at lowest temperatures. Frequently one simply employs an empirical relation of the form Z =

RT

(1 + at - tiP),

(1.13.34)

where Vo, or, fl are parameters that are found in appropriate tables; t is the temperature in ~

1.13.10

Thermodynamics of Anisotropic Media

At this point we take up the thermodynamic principles governing mechanical work in anisotropic media. Here the changes in physical extension depend on the direction of application of nonhydrostatic pressure. Consider a deformation in which s(r) and s(r + dr) are displacement vectors that move element A, originally at r, to r + s(r), and a neighboring element B, originally at r + dr to an adjacent location. For small displacements, s(r + dr) ~ s(r) + d r . Vs(r). Here V s ( r ) is a matrix (a tensor) with columns Osx(r)/Ox, Osx(r)/Oy, Osx(r)/Oz, and similarly with sy(r), Sz(r ). As is evident, any matrix with entries Mij may be rewritten in the form Mij = (1/2)(Mij + Mji) + (1/2)(Mij - Mji), involving

SYSTEMATICS OF THERMODYNAMIC FUNCTIONS OF STATE

71

Table 1.13.1 Matrix entries for deformation of anisotropic materials A. Entries to the symmetric strain tensor

es=

exxexyexz eyxeyyeyz ezxezyezz

exx : (OSx/OX) eyy=(OSy/Oy) ezz : (OSz/OZ)

exy : (1/2)[(OSx/Oy) + (OSy/OX)] : e y x exz=(1/2)[(OSx/OZ)+(OSz/OX)]=ezx ey z : (1/2)[(OSz/Oy) + (OSy/OZ)] = ezy

B. Conventional notation for strain components

el =- exx e4 =- 2exy = 2eyx

e 2 =- eyy e5 =- 2exz = 2ezx

e 3 =_ ezz e6 =- 2eyz =- 2ezy

C. Conventional notation for stress components

o.1 -= o.xx pressure along x o.2 =- o.yy pressure along y o.3 = o.zz pressure along z

~ =- o.xy -- o.yx pressure 2_ x along y or vice versa ~ --o.xz = o.zx pressure _1_x along z or vice versa o6 = o.yz -- o.zy pressure 2- y along z or vice versa

respectively a symmetric (es) and an antisymmetric component (ea). The matrix for es ~ in the overall matrix Vs(r) - es + ea is shown in Table 1.13.1 which involves the entries eij - - ( 1 / 2 ) [ O s i / O r j + O s j / O r i ] . For the ea entries the ' + ' sign is replaced by ' - ' , so that the diagonal elements of ea vanish. ,,.,

,,.,

Consider again a dilatation of a solid in which element A at r is displaced by s(r) to A ~, and a neighboring element B at r + d r is displaced to B ~ by the operation s(r + d r ) . The A ~- B' separation distance is then d r ~ - d r + [s(r + d r ) - s ( r ) ] - d r + d r . Vs(r) - d r + d r . ( e s + e a ) . Note that e a represents a pure rotation; on rotating the coordinate system by - e a , we obtain d r ' - d r + d r 9 es in the rotated system. Here d r is the row vector [dx, dy, dz] while the es matrix is shown in Table 1.13.1. A linear dilatation along x is then specified by d x ' - (1 + e x x ) d x + eyx dy + ezx dz, with similar expressions for dy' and dz ~. Thus, the diagonal element exx of Table 1.13.1 indicates the fractional lengthening of the solid along the x-axis, and similarly, for y and z. Angular dilatations, on the other hand, involve a displacement dl whereby idl originally perpendicular to ) dl (i and ) are orthogonal unit vectors along x and y) now change into Ax - i dl + i dl . es - i(1 + exx) dl + ~

) e x y d l -q- ~:exz d l , a n d Z y =_ ) d l

q-)dl.es-

i e y x d l nt- ) ( 1 -+- e y y ) d l + ~:eyz d l .

The angle between Ax and A y in the x - y plane is specified by

COSq~xy =

('i dl + "idl . e s ) . () dl + ) dl . e s) ~ ^ ~ . li dl + i dl . es I x Ij dl + ) dl . es [

(1.13.35)

When the appropriate entries for es are introduced and an expansion is carried out to first order in eij o n e obtains (after a large number of elementary steps)

72

1. FUNDAMENTALS

Table 1.13.2 Maxwell relations for anisotropic solids; constant V

1 (0eS/) (Off/) W T -- - ~ ei

I(OS) V

~i

ei = -

(Oei) OTis

(T1.13.2.1)

V

(T1.13.2.3)

l ( O~ie)i O"i = - ( -oai ) S (T1.13.2.4) W ~

T

\ a T , ] ei

COS ~xy

"~ 2exy. Now introduce the deviation from the right angle by ~'2xy dPxy, so that c o s dl)xy,~ ff2xy ~ 2exy. Similarly F2xz ~ 2exz, F2yz ,~ 2ey z.

--

re~2-

It is conventional to introduce a more compact notation for the strain components, as shown in Part B of Table 1.13.1; also, one sets up six components of a stress tensor ff that induces the strain. Here ~Yi - - [ O ( U / V ) / O e i ] S , e j # i ; these are listed in Part C of Table 1.13.1. The element of work is then given by V Z i r dei. With these definitions the above concepts are embedded in thermodynamics by writing the differential of the energy (of the strained relative to the unstrained body) as 6

dU -- TdS + V ~ criei.

(1.13.36a)

i=1

One should note the positive sign ahead of the second term: work is done by the system as the strain is diminished. We next introduce the enthalpy by H U - ~ 6 = 1 o"i ( V e i ) . The enthalpy differential is then given by 6

6

dH - dU - V Z

ei dai - V ~

i=1

6

cri dei - Z

i=1 6

= T dS - V Z

aiei d V

i=1

6

ei doi - Z

i=1

oiei dV.

(1.13.36b)

i=1

From (1.13.36a) and (1.13.36b) Maxwell relations may be constructed in the usual manner; these are listed in Table 1.13.2. Integration of those involving the entropy differentials then specify the entropy of the solid in its dependence or ai or on ei under different conditions. Moreover, from the definition of the stress tensor it follows that

(O0"i/Oej)S,ejr i ~Cijls--(Ocrj/Oei)S,ej#i ~ c j i l s , (Ocri/Oej)T,ej#i ~

cijlT -

(Ocrj/Oei)T,ej_r ~

cjilw,

(1.13.37a)

(1.13.37b)

which are know as isentropic and isothermal elastic stiffness coefficients. The inverses (Oai/Oej)s,aj#i or (Ooi /Oej)T,aj#i represent isentropic and isothermal compliance coefficients. There also exist six thermal strain coefficients Oti

INTERRELATIONS INVOLVING HEAT CAPACITIES

73

(Oei/OT)aj and six thermal stress coefficients (~i ~ (Oai/OT)ej that deal with the temperature variations of the respective coefficients. This completes the brief survey of commonly used coefficients that characterize the deformation of anisotropic elastic solids. QUERIES AND COMMENTS 1.13.1. The imposition of a constant volume condition is actually too restrictive. It is only necessary to demand that the change of volume in the system be exactly compensated for by a corresponding change in the surroundings. This does not rule out other volume changes in the surroundings that do not affect the properties of the system. 1.13.2. Can you think of reasons why Eq. (1.13.10) is generally more readily applicable to experimental measurements than Eq. (1.13.12)? 1.13.3. Strictly speaking, one should employ different symbols to distinguish between energy as a function of S, V and as a function of T, V, but such proliferation of symbols is undesirable. What is meant should be clear from the indication of the relevant independent variables. 1.13.4. ET. Landsberg, Thermodynamics and Statistical Mechanics, Oxford University Press, 1978, Chapter 5. 1.13.5. Determine the dependence of E on V and of H on P for isentropic processes using an ideal gas. Note the resulting expressions. 1.13.6. Repeat Exercise 1.13.5, using the van der Waals gas as a working substance. In carrying out the mathematical manipulations do not attempt to solve the van der Waals equation for V(T, P), but use appropriate mathematical 'tricks' of Section 1.3, so that you can always employ the van der Waals equation in the form P = P ( T , V).

1.14

Interrelations Involving Heat Capacities

We turn to relations characterizing heat capacities. Experimentally, it is much easier to measure heat capacities at constant pressure than at constant volume; the latter inevitably changes with temperature. However, the quantity relevant to theoretical interpretation is Cv rather than C p. A relation between these may be established, beginning with the definition H = E § P V, so that

Cp = ( O H / O T ) p = (OE/OT)p + P ( O V / O T ) p .

(1.14.1)

Next, we write E = E ( T , V), so that

dE-

()

- ~E

dT +

-ff-ff T d V - Cv T §

-ff-ff T

We now differentiate with respect to T at constant P; by Eq. (1.3.5) the coefficients remain unaffected. Thus,

( ) OE 0-T

P

-- Cv +

~ T

. P

(1.14.3)

74

1. FUNDAMENTALS

Insert Eq. (1.14.3) into ( 1.14.1 ) to obtain Cp-Cvn

t-

[

P+

(1.14.4) T

P

On substitution, using Eq. (1.13.16), we find

(1.14.5) V

P

It is very difficult to measure (OP/OT)v because of the problem of keeping V fixed while varying T. Accordingly, we use Eq. (1.3.8) to write

OP)_ v

(OV/OT)p -(av/aP)v

(1.14.6)

Insertion into Eq. (1.14.5) leads to

ce - Cv - - T (aV/aT)~ (OV/OP)T

_- o t 2 V T . [3

(1.14.7)

Here ot -- V -1 (0 V/OT)p is the isobaric expansion coefficient of the material, and - - V -1 (0 V/OP)T is its isothermal compressibility. These two parameters are usually available in tabulations, so that the difference Cp - Cv can be determined. It is readily checked that for an ideal gas,

Cp - C v =nR,

(1.14.8)

where n is the number of moles of gas and R is the gas constant. Heat capacities are also useful in determining the variation of temperature with pressure or volume in isentropic processes. For this purpose introduce Eq. (1.3.8) to write

( O T ) _----(OS/OP)T--T_ ( 0 V ) , -ff-fi s (a S / a T ) p C p - ~ p

(1.14.9)

where we had invoked Eqs. (1.13.15) and (1.13.10) to obtain the final result. Once again, knowledge of the equation of state of the material and of the heat capacity at constant pressure suffice to determine the quantity of interest. It is not hard to prove in the same manner that

(OT)

_

s

(OS/OV)T_

T (OP)

-(as/ar)v-----~

(1 14.10)

-~ v"

Eq. (1.14.6) may then be introduced to substitute for the troublesome partial derivative on the right. These matters are left as exercises for the reader.

THE JOULE-THOMSON EXPERIMENT

75

The variation of heat capacity with pressure or volume is readily found by inserting Eqs. (1.13.15) or (1.13.10) into the expression o Z S / O T O P = (1/T)(OCp/OP)T --(02V/OT2)p, whence OCp OP

- -T~oT2

,

T

(1.14.11)

P

and similarly OCv OV

- T ~

-f~

.

(1.14.12)

v

Once more, the equation of state may be used to explore the indicated partial derivatives. EXERCISES

1.14.1. Determine the variation of Cp with P and of Cv with V for an ideal gas and for a van der Waals gas. Comment on your findings. 1.14.2. Determine the dependence of T on P and on V for isentropic processes using an ideal gas. Note the resulting expressions. 1.14.3. Repeat Exercise 1.14.2, using the van der Waals gas as a working substance. In carrying out the mathematical manipulations do not attempt to solve the van der Waals equation for V(T, P) but use appropriate mathematical 'tricks' of Section 1.3, so that you can always employ the van der Waals equation in the form P=P(T,V).

1.14.4. Is it not a contradiction to differentiate Cp, the heat capacity at constant pressure, with respect to pressure? How do you resolve this apparent problem?

1.15 1.15.1

The Joule-Thomson Experiment Analysis of the Joule-Thomson Effect

One of the good illustrations of thermodynamic methodology is based on the Joule-Thomson porous plug experiment; the results were also of considerable practical interest in the latter half of the 19th century, at a time when the liquefaction of gases was of great importance. Consider the passage of a gas through a porous plug in an adiabatically insulated enclosure, as shown schematically in Fig. 1.15.1. Let the gas initially be in volume V1 and at pressure P1 on the left and let it be completely transported quasistatically at constant pressure through the porous plug. Let it emerge on the right where the pressure is maintained at the constant value P2 and where the final volume is V2. This process is isenthalpic: for, the work involved in removing the gas entirely from the left is given by W1 - - f o P1 d V1 - P1 V1, while the work involved on the right is specified

76

1. FUNDAMENTALS

Fig. 1.15.1. Porous plug experiment of Joule and Thomson. (a) Initial state: the gas is on the left in volume V1 and at pressure P 1. A pressure P slightly greater than P 1 is applied to the piston to drive the gas through the porous plug. (b) Final state: the gas has been reversibly and under isenthalpic conditions forced through the plug and appears on the right-hand side in volume V2 and at pressure P2 < P 1.

by W2 - - j o ~ P2 d V2 - - P2 V2. The net work is then W - W1 -q- W2 - P1 V1 P2 V2 - - A E -

E2 - E l , since Q -

0. T h u s , E 2 + P2 V2 -

H2 -

E1 + P l V1 -

H1.

Therefore, the process, known as the Joule-Thomson porous plug experiment is isenthalpic, as claimed. It is characterized by the Joule-Thomson coefficient

rij =-- (OT/OP)H,

(1.15.1)

which correlates the change in temperature of the gas as it passes through a pressure difference across the porous plug. Depending on whether 0J > 0, r/j - - 0 , or r/j < 0, a gas will either cool, remain at constant temperature, or heat up as it passes through the plug (N.B. d P is negative). We examine the magnitude and sign of 0J using the van der Waals gas as an illustrative example. We first adopt Eq. (1.3.8) to rewrite (1.15.1) in the form rlj - - --

(OH/OP)T (OH/OT)p

= --

(OH/OP)T Cp

.

(1.15.2)

Introduce Eq. (1.13.17) into the expression (OH/OP)T -- - r I j C p to obtain (on a molar basis; V =- V/n; C =- C / n ) T

(0~)) --],)--17jCP. ~-~ p

(1.15.3)

We now adopt the van der Waals equation of state: P =

RT

a

V- b

V 2"

(1.15.4)

THE JOULE-THOMSON EXPERIMENT

77

Unfortunately, its inversion to obtain V(T, P) for insertion in (1.15.3) is algebraically very cumbersome. We thus resort to approximations. Multiply both sides of (1.15.4) by (V - b ) / P to obtain

RT P

V=

a ab Fbq-~ PV P V 2"

(1.15.5)

We regard the last three terms as 'corrections' to the perfect gas law, and in this spirit replace V on the right-hand side by R T / P to find

RT P

V-

a +b+ RT

abP (RT) 2"

(1.15.6)

Then

OV )

R

a

2abP RZT3.

P -- --~ -4 R T 2

--~

(1.15.7)

Now solve Eq. (1.15.6) for R~ P and substitute the result in (1.15.7) to obtain

OV) -~

_ V - b p

T

2a

3abP

-1 R T 2

R2T3.

(1.15.8)

When (1.15.6) and (1.15.8) are inserted in (1.15.3) one finds, on neglect of the term involving 1/T 3"

1 ~ J -- -~p

2a RT

b

3abP ) R ZT 2 "

(1 15.9) "

Under conditions where the third term does not dominate/TJ is rendered positive for 'large' a and for 'small' b, i.e., for large interatomic forces and for small effective volumes. In addition, the heat capacity should be as small as possible. These requirements must be optimized to achieve the liquefaction of gases by cooling. However, as the pressure is raised the third term in (1.15.9) becomes increasingly important, and ultimately causes rlj to pass through zero and then turn negative. Hence, the operating pressure drop across the porous plug must be chosen carefully to ensure conditions of gas liquefaction.

1.15.2

Inversion Temperature

A practical problem of interest is to determine the inversion temperature, Ti, at which r/j - 0 . According to Eq. (1.15.9) this happens when Ti satisfies the quadratic equation Ti2

2a 3a P Rb Ti -I- R---T- = 0.

(1.15.10)

78

1. FUNDAMENTALS

500 ~

400 ~

300 ~ "1 t.,.

-

~ ~ ~ ' X

E 200~ 0

9~ loo ~ 0~ -

t

" ~" I S

j

j

l

~

-100

.... _:! ......

20

._ t

60

_

I ................

100

1..= .

200 Pressure

I

..........

300 Atm.

Fig. 1 . 1 5 . 2 . J o u l e - T h o m s o n inversion curve for N2 gas; actual data as compared to predictions based on the van der Waals equation of state.

A plot of T/ vs. P, as applied to N2 gas, is shown in Fig. 1.15.2. The calculated curve is in fair agreement with experiment. Better agreement is achieved by use of more realistic approximations to the equation of state. Somewhere below 350 bar there exist two inversion temperatures that are specified by Eq. (1.15.10). For all temperatures within the curve r/j remains positive; this maps out the conditions that must be satisfied for effective cooling of a gas via adiabatic expansion through a porous plug.

1.15.3 IsenthalpsI Isenthalps can be used to establish the enthalpy of a van der Waals gas at any point in its adiabatic expansion through the porous plug. We begin with the expression (for the remainder of this section all extensive quantities will be specified as being molar variables) dH = (OH/OT)p dT + (OH/OP)T dP, whence

dH

-- Cp(T,

P)dT

-

Cprlj

dP

3abP ] =Cp(T,P)dT+ -~--~ + b + R2T2 dP.

(1.15.11)

THE JOULE-THOMSON EXPERIMENT

79

Since d H is an exact differential the cross derivatives taken in either order must match:

OCp ) _ 2a OP T RT2

6abP R2T3"

(1 15.12)

Next, carry out an indefinite integration over pressure at constant temperature"

2a Cp - R T 2 P

3ab R2T 3 p2 + L(T),

(1.15.13)

where L ( T ) is an arbitrary, unspecified function of the temperature. Then, by combining (1.15.2), (1.15.11), and (1.15.13) the Joule-Thomson coefficient may be written as

2a/RT - 3abP/R2T 2 - b

qj - -

L(T) + (2a/RT2)p - (3ab/R2T3)p 2

CP-~T--

2a/RT - b L(T) + (2a/RT2)p '

(1.15.14)

where the higher order terms have been dropped. Let the van der Waals gas be taken from an initial state T1, P1 to a final state T2, P2 in two steps: (i) Cool the gas reversibly from T1 to T2 at constant pressure P1. (ii) Expand the gas reversibly from P1 to P2 at fixed temperature T1. In executing these steps under these conditions we find. on use of (1.15.11) and (1.15.13): 0 -- AH

-

-

fT T2 Cp d T

--

1

+ f t P2 d H 1

P1

1

-R-T--2

T2

R 2 T 3 .ql_ L ( T )

d T 21-

,

- - - ~ 2 -~ N 2 T 2

Ji- b

dr.

(1.15.15) On dropping out higher order terms and taking a, b, L to be sensibly independent of temperature we can carry out the integration to obtain 0--AH=

R

T2

T1

+ L(T2 - T1) +

b

RT2 (P2 - P1).

(1.15.16)

This may be rearranged to read

LT1 + bP1

2a P1

R T~

= LT2 + bP2

2a P2

R T2

= H(T1, P1) -- H(T2, P2) = H

(a constant). (1.15.17)

80

1. FUNDAMENTALS

Thus, the quantity H ( T , P) =

2a P RT

+ L T + bP

(1.15.18)

is a constant under the assumed conditions and approximations and represents the desired isenthalp. Eq. (1.15.17) can be used to determine the variation of temperature with pressure, given a, b, H, and L. Eq. (1.15.13) may be used to fix L experimentally, and to check whether the assumption that this quantity is independent of T actually holds. If this is not the case the L ( T ) dependence must be introduced in Eq. (1.15.15) before the integration is performed. In the present case H is found by insertion of one pair of P and T values. Having thus fixed H, Eq. (1.15.17) is then employed again to trace out the general dependence of T on P. Note that, on account of Eq. (1.15.1), the slope of this curve furnishes a value of 17j for each specified set of (P, T) values. The maximum of the slope for each H curve thus determines the largest accessible value of the Joule-Thomson coefficient for a given set of conditions. In numerical work one should recall the approximations that have been introduced. More accurate results are obtained by use of fewer restrictive assumptions and of more realistic equations of state. Aside from its intrinsic interest the above discussion furnishes a nice illustration of how the methodology of thermodynamics can be used to obtain results of practical value. ACKNOWLEDGMENT

1.15.1. The author is greatly indebted to Professor James W. Richardson at Purdue University for useful discussion that formed the basis for formulating the present section.

1.16

Heat Measurements and Calorimetry

Up to now heat has been treated as a somewhat aetherial quantity, having been introduced as a deficit function that restores the balance between energy changes and work performance in a system. We complement this earlier presentation with a more meaningful description by introducing a set of units and a method for measuring heat transfers. The heat transfer was originally measured in units of calories, where one calorie was defined as the quantity of energy required to raise one gram of pure water from 14.5 to 15.5 ~ at one atmosphere. This definition has been supplanted by the introduction of the joule, which represents the energy specified by the conversion factor: 1 cal - 4.184 joules. One joule is also equivalent to the energy developed in a circuit by an electric current of one ampere flowing through a resistance of one ohm (driven by a potential difference of one volt) in one second.

HEAT MEASUREMENTS AND CALORIMETRY

81

Heat transfers are conveniently measured by calorimetric techniques. Citing Eq. (1.13.13), we write

dE-- -~

dT-+- -~ TdV"

(1.16.1)

At constant volume the second term drops out and the partial derivative in the first term may be replaced by the heat capacity at constant volume, Cv (see Eq. (1.13.15)). Then, in the absence of any other work for an infinitesimal step we may write

dE--ctQlv -CvdT,

(1.16.2a)

and in a finite step,

AE-- Q[V-- fT72CvdT= (Cv)(T2- T1),

(1.16.2b)

where the quantity in angular brackets represents an averaged value. This equation shows that the heat generated by a given process in an adiabatically insulated calorimeter may be determined by the temperature rise, once the heat capacity of the apparatus is known. This may be found by supplying a known amount of electrical energy to the system and recording the temperature increase of the calorimeter. Naturally, the actual procedures of calibration and operation are much more involved than indicated here. In fact, one cannot easily measure temperature increases at constant volume. Hence, we now consider processes at constant pressure. For this purpose one invokes Eqs. (1.13.15) and (1.13.17) to establish that (for fixed P and in the absence of any work)

dn--dale --fedT

(1.16.3a)

and

zXn- Qlp- fv : CpdT= (Cp)(T2- T1).

(1.16.3b)

The earlier remarks apply here as well, except that temperature changes are monitored at constant pressure, so that temperature changes cause no difficulties in experimental measurements. There exist many other techniques for determining heat generation and transfer, which readers are invited to explore by reference to appropriate texts and monographs. The principal purpose of the present discussion is only to outline one method by which Q or changes in energy and enthalpy may be measured.

82

1.17 1.17.1

1. FUNDAMENTALS

Determination of Enthalpies and Entropies of Materials Entropy Determinations

One of the important problems in thermodynamics involves the determination of the entropy of any material. Before considering this matter we must examine the characteristics of phase transitions. 1.17.2

Heat Transfer in Phase Transformations

Consider a material that undergoes a first order phase transition such as fusion, vaporization, allotropic transformation, and the like. In each case two phases remain in equilibrium at a fixed temperature while heat flows in or out of the system during the transition. The theoretical background for characterizing this process will be provided in Chapter 2. The energy change of the system on completion of the transition is specified by (in the absence of any work other than mechanical)

AEt-

Qt + wt - Qt - f P dV.

(1.17.1)

Transitions are generally carried out at constant pressure; we thus replace the integral by P A Vt and then obtain

AH lp - Q, Ip

(1.17.2)

The corresponding entropy change on completion of the transition is found by invoking A H -- T AS at equilibrium, 1 so that for the change in phase

Ast Ip - AHI Ip/ Tt

(1.17.3)

Henceforth we drop the P subscript. We also reintroduce Eq. (1.13.15), (OS/OT)p = C p / T , to write

$2 - S1 - fTy 2CP T dT 1.17.3

CpdlnT.

(1.17.4)

Determination of Entropies

For purposes of illustration consider a material whose entropy is to be determined as a function of increasing temperature. Let the sample undergo an otfl transition at temperature Tc~t~, melt at a temperature Tm > T,~, and boil at a temperature Tb > Tm. We wish to determine the entropy at a temperature T > Tm. For convenience we divide the temperature interval into segments 0 --+ Ta --+ Tot~ --+ Tm --+ Tb ~ T; here Ta is a temperature somewhere in the range 0-10 K, below which

DETERMINATION OF ENTHALPIES AND ENTROPIES OF MATERIALS

83

it is difficult to determine Cp accurately. We proceed by considering the various ranges separately. Between 0 and Ta one frequently resorts to the Debye theory for the heat capacity of a nonconducting solids, and extended to metals by Sommerfeld. As a first approximation one uses the relation

Cp - a T 3 + FT,

(1.17.5)

where a and Y are parameters that in principle can be specified by microscopic theories. In practice their values are usually determined empirically by plotting Cp vs. T2; the straight line has a slope a and intercept F. Fig. 1.17.1 shows three

1.6

1.2 0

E 0.8 E

Copper

~ o.4

o

1

,

2

I

,

i

4

,

6

[

l

8

1

,,,a

10

I

l ...... J~ . . . . l

!

12

14

16

L ....... I . . . .

18

4.0 93"}

f

"O

3.0 O

E E b

2.0 1.0 !

0

i

I,

2

i

[,,

4

I

6

I

..

8

1...,.~.

t

10

.

12

J.....

14

,

I..

~.

16

!

18

10 ~,

8

S

"o m O

6

E E O

4 2 I J

2

,

!

4

i

i

6

. L

l

8

.

l

10

,

!

12

,

I

14

-~ . . . . |

16

i

J,

18

T 2, deg 2

Fig. 1.17.1. Heat capacity measurements on elemental metals at low temperature. After W.C. Corak, M.P. Garfunkel, C.B. Satterthwaite, and A. Wexler, Phys. Rev. B 98 (1955) 1699.

84

1. FUNDAMENTALS

such plots for the indicated elemental metals; these show the degree to which the present analysis is applicable, and the extent of extrapolation required to determine the intercept. Inserting Eq. (1.17.5) into (1.17.4) we write

S(Ta)- S(O)- fo Ta (aT 2 + y ) d T - aT3 3 + yTa.

(1 17.6a)

As an aside, S(0) vanishes only under the conditions prescribed in Section 1.18. In the range Ta to Ta~ we obtain

S(Ta~)- S(Ta)- f r~ C~pdr, d Ta T

(1.17.6b)

that requires an empirical determination of C~ as a function of T for phase or, so that the integration may be performed. At the or/3 phase transition we write Q~t~ (Sfl -- Sot)T~# =

T~

AH~ -

-

-

Ta~

~

(1 17.6c) ~

The entropy change in the range T~ to Tm is given by

S(Tm)- s ( r ~ ) - fTf m C~P T dT "

(1.17.6d)

It is evident that the remaining contributions are as follows:

Qm (Sl -- SS)Vm =

Tm

=

A Hm

(at the melting point),

Tm

S(Tb)- S(Tm)-- f~~b Cl T dT Qb ( S v - Sl)Tb - -

Tb

S ( T ) - S(Tb)-

A Hb

Tb

T dr

(for the liquid phase),

(1.17.6e) (1.17.6f)

(at the boiling point),

(1.17.6g)

(for the gas phase).

(1.17.6h)

We then find S ( T ) - S(O) by addition of Eqs. (1.17.6a)-(1.17.6h). While this is straightforward the difference S(T) - S(O) depends on the pressure to the same extent that Cp, Q, and the various temperature ranges do. It is therefore conventional to measure, or at least report, all results under standard conditions of one bar and to let users of such information introduce corrections such as Eq. (1.13.10) to determine S(T) at other pressures. Conventionally, entropies cited for standard conditions are denoted by S O(T).

DETERMINATION OF ENTHALPIES AND ENTROPIES OF MATERIALS

85

It should be evident that a similar approach is used to find the enthalpy. Beginning with Eq. (1.17.2) and (1.13.15), we obtain under standard conditions

H~

o +. frf~ Cp,o dT 4F yT2a . + [ r . e. C~0 dT . + Qu#

H ~ 4 a T .a 4

.

J Ta

(1.17.7) We may combine the above to determine the standard Gibbs free energy according to

G~

(H~

H~

H~

- T(S~

S~

(1.17.8)

Tabulations of [G~ - H~ [H~ - H~ and of S~ - S~ are available in numerous reference works in the literature and play a large role in industrial applications. By way of illustrations we display in Fig. 1.17.2 a plot of the molar heat capacity of oxygen under standard conditions. The plot of Cp vs. In T is then used to determine the entropy of oxygen from the area under the curves. Note that the element in the solid state exists in three distinct allotropic modifications, with transition temperatures close to 23.6 and 43.8 K; the melting point occurs at 54.4 K, and the boiling point is at 90.1 K. All the enthalpies of transition at the various phase transformations are accurately known. An extrapolation procedure was employed below 14 K, which in 1929 was about the lower limit that could conveniently be reached in calorimetric measurements. 14 ~ 12

T

Liquid

t

02

10 0

E

6, 8 "13 n t~

o

&

Gas_____........

6

0

__.i17t II 1 1

2 Log T

Fig. 1.17.2. The molar heat capacity of oxygen. After W.F. Giauque and H.L. Johnston, J. Amer. Chem. Soc. 51 (1929) 2300.

86

1. FUNDAMENTALS

REMARK

1.17.1. Here we are getting slightly ahead of the logical presentation. In Chapter 2 we learn that two phases, 1 and 2, at equilibrium are characterized by the same Gibbs free energy: G1 = H1 - Tt S1 = G 2 = H 2 - Tt $2. These findings are not based on arguments developed in the present section; hence, the reasoning is not circular. 1.18

The Third Law of Thermodynamics

The Third Law deals with processes taking place close to T - - 0 ; clearly, problems arise since the integrating factor 1/ T begins to diverge at that point. We first inquire whether the state corresponding to T -- 0 can be reached. Consider a system characterized by a deformation coordinate z with a conjugate variable Z such that the element of work is given by dW = - Z d z . Then the energy of the system is expressed functionally by E = E(S, z) -- E ( S ( T , z), z); thus,

dS-

T-I(dE + Zdz)-

T - l { OEoTd T +

Z + ~

dz .

(1.18.1)

On taking second derivatives of S with respect to z and T in either order one obtains the expression

O---z

OT

-- ~

Z+ ~

.

(1.18.2)

After carrying out the indicated differentiations one obtains

OE

Z + ~

Oz

OZ

= T~.

(1.18.3)

OT

We now consider an adiabatically reversible process, since it is only under conditions of such isolation that one can hope to attain ultralow temperatures. On setting d S = 0, Eq. (1.18.1) becomes

dT = -

Z+OE/Oz

dz,

(1.18.4)

whence, by (1.18.3),

dT =-T

(OZ/OT)z dz (OE/OT)z

T(OZ/OT)z dz. Cz

(1.18.5)

The above expression carries an important general message: any adiabatic reversible process resulting in a change of thermodynamic coordinates z necessarily alters the temperature of the system. If now the heat capacity at constant z, Cz, were to remain constant at low T and if (OZ/OT)z were to be positive in this range, then it would indeed be

THE THIRD LAW OF THERMODYNAMICS

87

possible to attain the absolute zero of temperature. However, it is an experience of mankind that for all materials Cz varies with temperature a s T -a, a >7 1, as T --+ 0. Everything thus hinges on the question whether (OZ/OT)z > 0 approaches zero faster than does the quantity T 1-a which diverges at the limit T -- 0. As an experience of mankind the answer is found to be in the affirmative. It is therefore impossible to reach the limit T - - 0 . To examine the implications we note that the differential of the Helmholtz free energy A - E - TS is given by d A - - Z d z S dT. On cross differentiation with respect to z and T we obtain a Maxwell relation of the form (OZ/OT)z = (OS/Oz)~, so that Eq. (1.18.5) may be rewritten as

d T --

T(OS) dz. C z -~z T

(1.18.6)

In other words, since one cannot attain the limit T - - 0 one must require that in every conceivable situation (OS/Oz)TT/Cz --+ 0 as T ~ 0. Thus, (OS/Oz)T not only approaches zero but with Cz ~ T -a does so faster than 1 / T a - 1 . This gives rise to the principle of unattainability of the absolute zero of temperature. The statement (OS/Oz)T --+ 0 as T --+ 0 is incorporated in another Law: The Third Law of Thermodynamics asserts:

As the temperature of any system approaches the lowest possible temperature of O K the entropy of the system assumes a particular, least value when the system is in its lowest energy state. The entropy reaches this value with zero slope taken with respect to all thermodynamic deformation coordinates. One should carefully note that we do not claim that S itself vanishes at T = 0. The statement about lowest energy attends to the fact that in the cooling process to lowest temperatures excited energy states may accidentally be frozen in. This then keeps the system from attaining equilibrium, so that it cannot be properly characterized in terms of deformation coordinates. Also, any equilibrium state that remains intrinsically disordered as T ~ 0 will have a nonzero entropy; examples are furnished in the Remarks section. 1 Thus, at equilibrium in its lowest energy state the system is in its most stable configuration, for which the entropy at absolute zero, So, has the lowest possible value, whatever the coordinate z under consideration. Moreover, the lowest possible entropy is attained in the limit of vanishing slope: (OS/Oz)~ --+ 0 as T --+ 0. Despite the above disclaimer one often does set So = 0, namely when So is not altered during a given process, in which case the actual entropy change does not depend on the value assigned to So. As a simple example one may consider processes that do not involve nuclear transformations. Here the entropy at T - - 0 associated with the mixture of different isotopic species does not change. Hence, for practical purposes, we may ignore this contribution, thus allowing us to set the effective entropy at the absolute zero to zero. However, one must obviously

88

1. FUNDAMENTALS

be very careful in determining whether setting So - - 0 is justified; some counterexamples are offered in the Remarks section. 1 An important consequence of the Third Law is that it denies the existence of an ideal gas. For, as we establish in Chapter 2, its entropy is given by

S =CvlnT + RlnV,

(1.18.7)

whose derivative, (OS/OV)r = R / V does not vanish at T = 0. This fact, of course, does not prevent us from using the ideal gas law at elevated temperatures as an approximation to characterize properties of actual gases. However, it is clear that the approximation fails at low T. 2 REMARKS AND QUERIES 1.18.1. Exceptions to setting So = 0 arise whenever configurational disorder must be taken into account, as in cases where a material may be disordered or be in several states of equal energy that are frozen in at T = 0. Several instances come to mind: (a) Helium, which remains a liquid at T -- 0, unless subjected to external pressure. (b) Solid CO, H20, N20 and the like: neighboring pairs may be encountered in configurations such as CO-CO or CO-OC of nearly equal energy. At ultralow temperatures even that small energy difference may become important, and a disordered state is frozen in. On heating the disorder is annihilated. (c) Glasses or solid mixtures such as AgC1 + AgBr that may be regarded as frozen liquids or solutions, with a residual entropy of mixing. (d) Paramagnetic materials whose electronic spins remain disordered down to lowest attainable temperatures. (e) Materials in which it is important to take account of isotopic distributions or nuclear spin degeneracies, if these quantities change in a given process. 1.18.2. There is also a logical problem in the elementary derivations of the Second Law that are quoted in many textbooks. These depend on the use of an ideal gas as a working substance in Carnot cycles, that are then used in setting up the Second Law. Clearly, it is awkward to have to acknowledge at a later stage that the very existence of such a working substance is denied by the Third Law. 1.18.3. Does the van der Waals or Berthelot equation of state satisfy the requirements of the Third Law? Discuss the implications of your answer.

1.19

The Gibbs-Duhem Relation and Its Analogs

So far we have not taken into account the chemical constitution of m a t t e r ~ a subject of central importance in chemical thermodynamics. We now discuss several fundamental issues, and in subsequent sections provide a systematic thermodynamic analysis of compositional changes.

1.19.1

Partial Molal Volumes

We begin by showing how the volume of a system depends on its chemical composition in a mixture for which the mole numbers of the constituent components

THE GIBBS-DUHEM RELATION AND ITS ANALOGS

89

are specified. Consider as an example an aqueous solution containing sulfuric acid and sodium chloride at constant temperature and pressure. If we double the mole numbers of each of H 2 S O 4 , NaC1, and H20 we double the volume of the system. Thus, more generally, at fixed T and P the volume of a system should be a homogeneous function of the mole numbers n i of all species in the system, independent of the state of aggregation of each constituent. Then according to Euler's Theorem, Section 1.3,

r

(0nV/) (1.19.1)

V (T, P, n l , n2 . . . . . nr) -- i~l ni 9

T,P,nj#i

We now introduce the short-hand notation Vi =- (0 V / O n i ) T , P , n j # i to write F

V(T, P, nl, n2, ..., t / r ) - Z

ni Vi(T, P).

(1.19.2)

i=1 n

Here V/ is known as the partial molal volume of component i. It is the effective volume of one mole of that component in an infinite copy of the solution. Alternatively, this quantity may be regarded as the incremental change in volume of the solution in which temperature, pressure, and all other components are held at fixed values. In general, the volume of the mixture is not equal to the sum of volumes of the individual components. Thus, for Eq. (1.19.2) to be useful we need to find out how the individual Vi are to be determined experimentally. We shall address that problem below. A distinction must be made between the functional dependence of Vi on the mole numbers n i, and the parametric dependence of Vi on T and P; for, doubling T and P obviously does not double the volume of any mixture. These two quantities are held fixed in specifying the partial derivatives in Eq. (1.19.1). However, as T and P are assigned different sets of values (P1, T1), (P2, T2), and so forth, Vi and V change in a manner prescribed by experiment. Thus, the fact that these parameters are held fixed in a given set of partial differentiations does not preclude the partial molal volumes from changing as one passes from one set of experimental conditions to a different set. 1.19.2

A Variant on the G i b b s - D u h e m Relation

We examine a consequence of setting V -- V(T, P, n l, lows that

dV -

(I

OV dT + - ~ P,ni

~

n2, . . . ,

dP + ~r f'i dni, T,ni

i--1

n r). It then fol-

(1.19.3)

90

1. FUNDAMENTALS

which is to be compared with the differential form derived from Eq. (1.19.2): r

r

(1.19.4)

d V - Z ~" drti -F y'~rti dVl'. i=1 i=1

Consistency then demands that we set r

.idf i=1

dP,

-

(1.19.5a)

T,ni

P,ni

which for fixed T and P reduces to r

(1.19.5b)

Znid(-'z i --0. i=1

The above relations represent variants on the Gibbs-Duhem relation. Note the important point that Eqs. (1.19.5) represent constraints. One cannot arbitrarily change T, P, and the summation y~r=l n id Vi independently. Rather, the formulation (1.19.3) has consequences that are not obvious, and that must be individually examined; see below. 1.19.3

Determination of Partial Molal Volumes

We begin with the definition of the molar volume of a mixture by writing I7' = V~ ~ i ni. On introducing mole fractions we then convert (1.19.2) to the form -- ~

Xi Vi.

(1.19.6)

i

It is evident that for a one-component system 17'1- 17'k. For a binary system we differentiate V - (1 - x2)1/1 + x2 V2: -- - V 1 -~- (1 - x2) ~,~x2

(o 2)

T,P

q- f2 q- X2 ~ ~ X 2

T,P

m

-

(OV2~

-- - g l + X l -~x 2 ,] T, p --]- V2 --k-x 2 --~X2 ,] T, p

- V 2 - ~71,

(1.19.7)

where we noted that xI(OV1/OX2)T,p -Jr-x2(OV2/OX2)T,p - - 0 on account of Eq. (1.19.5b). Next, set V - xl I7'1 + x2132 and eliminate V2 from Eq. (1.19.7). This yields (1.19.8a)

gl -- V - x 2 T,P

THE GIBBS-DUHEM RELATION AND ITS ANALOGS

91

V2

i

J J J

/ ~ //......- O i

,z

O

~'''~

i I I i ! b

1 x2

Fig. 1.19.1. Plot illustrating how the partial molal volumes of components in a binary mixture may be determined by extrapolation.

Similarly, it may be shown that

(1.19.8b) This provides a means of finding the partial molal volumes: as illustrated in Fig. 1.19.1, one measures the molar volume of the solution at a set of x2 values. At the particular value x2 = b a tangent to the curve is drawn. The points of intersection of this tangent at x2 - - 0 , 1 yields the desired quantities V1 and V2 respectively. 1.19.4

A l t e r n a t i v e M e t h o d for D e t e r m i n i n g Partial M o l a l V o l u m e s

An alternative procedure draws on the definition of an apparent molal volume in a binary mixture, based on the relation V - nl ~"~ + n2d#,

(1.19.9)

where f,o is the molar volume of pure component 1. This is converted to the mass of the solution by switching to molalities m2 as the concentration unit for the solute, in terms of 1000 g of pure solvent. The total mass of the solution of density p is specified by Vp - 1000 + m zM2, where M2 is the gram molecular mass of solute. Correspondingly, for the pure solvent V~ 1P0 - 1000. Now solve

92

1. FUNDAMENTALS

(1.19.9) for ~b and substitute for V and for IP~ in the resultant expression. This yields ~b -

m2

[

n2p(m2)

M2

lO00 p(m2) - Po 1 . m2 Po

(1.19.10)

Measurements of the various quantities on the right of (1.19.10) then specifies ~. This, in turn, allows one to find V2 via Eq. (1.19.9) as

V2--

~

= ~b -+- n 2

r,P,nl

= ~b -+- m 2

T,P,nl

.

(1.19.11)

T,P

The relation V -- n 1I7'1 + n2 V2 - n117,0 + n2~b may be rewritten as gl -

g 0 nt- - - ( , -

f/r2)- g 0 -m2--

171 ~

tll

m2M1

T,P

(1.19.12)

Eqs. (1.19.11) and (1.19.12), along with (1.19.10), are now available to determine the desired partial molal volumes. 1.19.5

The Gibbs-Duhem Relation

The above treatment with respect to volume may be carried over to other thermodynamic functions in which T and P appear as parameters. Anticipating the next section, we introduce the Gibbs free energy in the form G = G (T, P, n 1, n2, . . . , nr). As in the case of the volume we assert that the Gibbs function is to be homogeneous in the mole numbers n i. We proceed by strict analogy to the above treatment of effective volumes. For a one component system we write the Gibbs free energy in the form G - n lG1. For a set of components this generalizes to G - Y~.i G i n i , where Gi represents the effective Gibbs free energy of component i in the mixture. The fact that G is to depend on the n i in the indicated form means that we must satisfy the Euler criterion by writing r

G ( T , P, n l , n2, . . . , nr) -- E

nilzi(T, P),

(1.19.13a)

i=1

where

#i=--(O~ni)r,P,ny#i is known as the

(1.19.13b)

chemical potential; the nomenclature is explained in Section 2.2.

THERMODYNAMICS OF OPEN SYSTEMS

93

The differential of (1.19.13a) may be written out as

dG - Z

ni dlzi @ Z i

lzi dni

(1.19.14a)

i

and is to be compared to the differential of G ( T , P, n l, n2, . . . , nr), namely,

dG -

dT +

lg~idni.

d P -at-

P,ni

T,ni

(1.19.14b)

i--1

We thus establish that

Z

nid~i i

dP = -S dT + V dP,

-~

(1.19.15a)

P,ni

which at constant T and P specializes to Z

ni d # i - O.

(1.19.15b)

i Both expressions are known as the G i b b s - D u h e m relation. Again, these relations impose important constraints, this time on the chemical potentials encountered in a mixture of different components. An example of such a restriction will be furnished in Section 3.14. Since G is not as readily measured as V, methods other than those discussed above for specifying partial molal volumes must be introduced to determine the chemical potentials. These procedures will be taken up at a later stage. QUERY 1.19.1. Cite an example of a system for which the following statement is incorrect: consider two systems at the same temperature and pressure containing the same chemical materials in identical amounts. When these are combined the total energy is twice that of each subsystem. What does this teach you? Explain in detail what conditions must be met so that the statement is corrected.

1.20

Thermodynamics of Open Systems

1.20.1 Thermodynamic Functions of State At this point we extend the earlier discussions to open systems, in which the mole numbers n i of the different components of a systems are allowed to change through the exchange of material with the surroundings. Thus, the various thermodynamic functions of state, V, E, H, S, A, G are now functions of these mole numbers in the manner already displayed for V and G in the preceding section.

94

1. FUNDAMENTALS

An alternative, more systematic formulation will be provided at the end of the present section. We begin by adopting Eq. (1.19.14b):

dG--SdT

+ V dP + E lZi dni.

(1.20.1a)

i

On applying the usual Legendre transformations which are designed to hold for both closed and open systems we set up the remaining functions of state

dA--S

d T - P dV + ~_~ lZi dni,

(1.20.1b)

1

i

dH -- T dS + V dP + ~

dni,

(1.20.1c)

lZi dni.

(1.20.1d)

~i

I

i

d E - T dS - P dV 4- ~ i Here we have set

T,P,njr

S, V,njr (1.20.2) In Exercise 1.20.1 it is to be shown that all four definitions for the chemical potential are identical; hence, one symbol suffices. Experimentally it is simplest to realize the process of adding dni moles of material to a system, at constant pressure, temperature and remaining mole numbers; for that reason the definition ~i = (OG/Oni)T,P,nj#i is ordinarily used. However, under different constraints another of the above definitions will need to be employed. 1.20.2

T, V,njr

S,P,njr

~

First Derivatives and Maxwell Relations

We now initiate the ordinary procedure: we carry out a single differentiation to obtain n - -

s--

Uv

() OA

-~

s,x j

v,~

-

-~

__()_ OG -~

v,x j '

, P,xj

T

-~

S,xj

--~

V,xj

-~

_(o,) -~

T,xj' n,xj

The above are obvious generalizations of Eqs. (1.13.5)-(1.13.8), which, incidentally, justifies the adoption of Eq. (1.20. l a). As before, the most useful formulations are the ones that specify S and P. Also, E and H are the appropriate functions of state under adiabatic constraints, whereas A and G are appropriate for characterizing isothermal processes. The next step involves the set of double differentiations in either order: we set (02X/OxiOxj) -- (02X/OxjOxi), with X -- E, H, A, G; x - P, V, S, T, n. This

9

9

~~

~

g

~

~~,,~oa g ~

=r

~

::r

=

~.=r"

~

~.

~"

~'~ - ~

~

~'~

9

~

~

.

~,.

9 9

< ~

b~

~"

~

0

~-~.

.~.

~

,.<

9

9

0

t~

o

',<

9

t~

t~

::r

~.~~

cr

::r

9

t~

c~

I

II

II

~1 ~

I

II

c~

c~

c~

II

II

II

~

I

II

II

,.,.

I

I

II

II

~

I

~

II

II

~..~ I ~ D

k,N.

r~

r~

r~

@

r~

@

BI

~

~

m

m

-<

m

0

o

Z

0

I m :D

96

1. FUNDAMENTALS

1.20.3

Further Generalizations

Several additional points should be noted: from G -- H - T S we deduce that /Zi

--

/-ti

--

T Si,

(1.20.5)

where Hi =--- (OH/Oni)T,P,nj#i and Si =-- ( O S / O n i ) r , P , n j # i . We also generalize the definition of heat Q as the deficit function needed so that dQ - ( d E - d W - Z i [zi dni) vanishes identically, in order that E remains a function of state. We then view the entropy change as being given by

drQ

dS = ~, T

(1.20.6)

as before; also, when only mechanical work is involved, the element of work may be generalized to read dr W - - P d V + Z

(1.20.7)

lzi dni. i

We may also rewrite the First Law in the form dE

T

OS "'-

dT +

-~

V,xi

-- P d V + Z

dV + T,xi

i

T, V,nj#i

dni] (1.20.8)

~ i dni, i

which should be compared with

dE--

OE -~

dT+ V,xi

dni. T,xi

i

(1.20.9)

T,V,nj#i

On matching coefficients we generalize Section 1.13 as -- T

=---Cv,xi,

V,xi

V,xi

T,xi

V,xi

(1.20.10a)

(1.20.10c) T,V,nj#I

Here we had introduced the appropriate Maxwell relation to arrive at (1.20.10b). Thus, we can also write d E - Cv,xi d T +

T

-~

Y,xi -- P

J

d V qL ~

Eidni. i

(1.20.11)

THERMODYNAMICS OF OPEN SYSTEMS

97

Eq. (1.20.10c) involves a short-hand symbol for the differential energy. The reader should be able to construct corresponding arguments for the enthalpy, and to derive the expressions

-- T

-~

(1.20.12a) P,xj

and dH

= Cp,xi

dT +

- T

-~

P ,x i

"Jf-V

d P 4- Zi

tSli d n i ,

(1.20.12b)

wherein fli = (OH/Oni)T,P,ni#j. We may replace (OV/OT)p,xi with otV, where ot is the isobaric coefficient of expansion. 1.20.4

Two Important Relations

We note two relations that will be extensively used in later derivations: According to Table 1.20.1, term [3], we write

( O#i ~

OP ]T,xi

-- Vi.

(1.20.13)

Also, staging with G~ T = H~ T - S we obtain H

dH

d(G/ T) - ---T~ NT ~- ~1

H

-dS-

VdP

T2 dT --1-------~.

(1.20.14)

Thus, at constant P, O(G/T)) OT

_ P,xi

H T2 "

(1.20.15a)

If we now allow G and H to depend on the {Xi } and take partial derivatives of (1.20.15a) with respect to xi at constant T and P, we obtain (on interchanging the order of the partial differentiations on the left-hand side),

( O(lzi / T ) ) OT

ISli P,xi

T 2'

a relation that will be cited frequently in our further development.

(1.20.15b)

98

1.20.5

1. FUNDAMENTALS

Extension to Nonequilibrium Processes

A more systematic presentation may be developed, which is an extension of the procedures developed in Section 1.13. We consider a very general process that couples events in the open system to those of the surroundings; the latter are designated with the subscript zero. By convention, all events in the reservoirs occur reversibly, so that the energy change in an infinitesimal step of any process in the surroundings may be written as d Eo - To d So - Pod Vo + Z

lzOi dnoi ,

(1.20.16)

i

where To and P0 are the prevailing temperature and pressure, So and V0 the entropy and volume,/x0i is the chemical potential of species i and noi, the corresponding mole number. We now invoke Eqs. (1.12.1a) and (1.12.4), so as to refer to properties of the system: for the closed unit (system + surroundings) maintained at a constant volume we find that 2 d Eo = - d E , d So =- d2 So = dl So + dO = - d S + dO, d Vo -- - d V, dnoi = - d n i . After the substitutions we obtain dE-

To d S -

PodV + Z

(1.20.17a)

lzOi dni - To dO,

i

which we rewrite in the equivalent form d E -- (To - T ) d S -

+ TdS-

(Po - P ) d V + Z(ll~oi i

PdV + Zlzidni i

- TodO.

- ll~i) dni

(1.20.17b)

This expression holds for any irreversibly executed step in which the intensive variables for the system differ from those of the surroundings. One of the limitations is the need to specify the intensive variables of the system executing processes that depart substantially from equilibrium. One possible way of addressing this problem is sketched in Fig. 1.20.1, which pertains to the temperature of a system exchanging heat with its surroundings. At sufficiently slow exchange rates most of the reservoir and most of the system are at well defined, different temperatures, while the temperature change occurs in a poor conductor of heat over a distance that is small compared to the extension of system and reservoir. Other schemes may be conceived, such as replacing the slowly varying temperature within the system by a suitably averaged value. Alternatively, T, P, and/z i may be regarded as functions of position within the system, as is done in Chapter 6. Eq. (1.20.17) then becomes a local function in which contiguous regions form the surroundings. However, as always, when the system departs

THERMODYNAMICS OF OPEN SYSTEMS

99

System

> / <

Reservoir

To

Fig. 1.20.1. Sketch of a temperature profile for the combined system and reservoir at different temperatures T and T 0. The temperature in each phase remains essentially constant over almost the entire region; the gradient in temperature develops over only a small region 1 at the interface.

extensively from its equilibrium configuration the problem of assigning a temperature becomes acute. Similar considerations apply to the other intensive variables. Lastly, if work other than mechanical is carried out, the appropriate conjugate intensive-extensive variable pairs must be included in the above formulation. This problem is taken up in Chapters 5 and 6. Clearly, under quasi-equilibrium conditions T -- To, P -- P0, /Zi = / Z 0 i , and dO = 0, so that the standard form dE-

T dS-

P dV + ~

lZi dni

(1.20.17c)

i

is recovered. The fact that this relation applies only under limiting conditions is frequently not sufficiently emphasized. Moreover, since E is a function of state we may subtract this relation from Eq. (1.20.17b) to obtain TodO - (To - T) d S -

(Po - P) d V + ~ ( ~ o i

- lzi) dni,

(1.20.17d)

i

which shows how the deficit function may be specified when entropy, volume and composition represent the relevant control variables. We next introduce the Helmholtz free energy by the relation A - E - T S, which leads to the differential form dA

-

(To -

T)dS-

(Po -

P)dV

+ ~--~(lzoi - ~i) dni i

- S dT-

P dV + ~

ll~i dni - To dO.

(1.20.18a)

i

However, the appropriate control variables for the function A - A ( T , V, {ni }) are temperature, volume, and composition. Accordingly, it is necessary to introduce these quantities as independent variables for the entropy" S - S ( T , V, {ni}). We

100

1. FUNDAMENTALS

thus write

dS -

( ) OS -~

dT +

-~

V,ni

dV +

T,ni

z(01) i

dni.

(1.20.18b)

T, V,nicj

In the above we next introduce the same substitutions that led to setting up Eq. (1.13.3d), and we also set Si ~ (OS/Orti)T,V,nj:/:i to write dA -

(To - T )

[ Cv'ni

dT +

T - (Po - P ) d V

-Jr-E

~

dV

-[- E

fl

Si d n i

]

i

+ E ( l ~ o i - l~i)dni - S d T - P d V i (1.20.18c)

~ i d n i - To dO ,

which is the desired expression for an infinitesimal change in Helmholtz free energy under nonequilibrium conditions. Here ot and fl are the isobaric thermal expansion and the isothermal compressibility coefficients respectively. The above correctly reduces to the standard form when the infinitesimal step is executed reversibly, namely, (1.20.18d)

l~i d n i .

dA - -S dT - P dV + ~

i When this is subtracted from (1.20.18c) one obtains

T - (Po - P ) d V

i

+ E(~o i

(1.20.18e)

i - lzi)dni,

which specifies the deficit function when temperature, volume, and composition are the appropriate control variables; their coefficients are experimentally accessible. Similarly, the Gibbs free energy is introduced by G - E + P V - T S, whose differential form is dG -

(To - T ) d S -

- SdT

+ VdP

(Po - P ) d V

+ ~#idni i

+ E(~o i

- TodO.

i - tzi) dni (1.20.19a)

THERMODYNAMICS OF OPEN SYSTEMS

101

Since temperature, pressure, and composition are the appropriate independent variables for the Gibbs free energy we must now write out the entropy and volume in the form S = S(T, P, {ni}) and V = V(T, P, {ni}), take their differential forms, and substitute these in Eq. (1.20.19a). Following the method used in setting up Eq. (1.13.4c), we next introduce the heat capacity at constant pressure, the appropriate Maxwell relation, as well as o~ and/3. We also introduce the partial molal entropy Si and volume Vi to obtain

dG-(To-T)ICp'n~dT-etVdP+ESidni] T i

i

i

- SdT + V dP + ~--~,#i dni - TodO,

(1.20.19b)

i

which applies to any process with the indicated control variables, and which properly reduces to the standard form when the infinitesimal step is executed reversibly, namely,

dG--SdT

+ V dP + E ll.i dni.

(1.20.19c)

Subtraction from (1.20.19b) yields

TodO--(To-T) ICp'ni d T - o t V d P + E S i d n i ] T i -(Po- P)I~VdT-flVdP

+ K ~'dni] + E ( # o i - # i ) d n i , i

i

(1.20.19d) which specifies the deficit function in terms of T, P, and ni and variables that can be measured. Lastly, we turn to the enthalpy H - E + P V. By methods now familiar we obtain

dH - (To- T ) d S - (Po - P)dV + E ( # o i - #i) dni i

+

T d S + VdP + E t z i d n i -

TdO.

(1.20.20a)

i

Since H - H(S, P, {ni}), S, P, and ni are regarded as control variables. We therefore consider the volume first in the form V - V(P, T, {ni}), then introduce the entropy as a function of the same variables" S - S(P, T, {ni }), which function

102

1. FUNDAMENTALS

we invert to read T - T ( S , P, {ni}). Lastly, we insert this expression into the equation of state: V -- V ( P , T ( S , P, {ni}), {ni}) = V ( S , P, {ni}). On taking the differential of this latter relation and substituting for d V we obtain d H -- (To - T) d S

- (Po - P)

3V -~

dS +

-ff--fi

P,ni

dP +

S,ni

dni

i

S,P,ni#j

+ y~(ld, Oi -- lzi)dni + T d S + V d e + Z[dl, i dni - TodO. i i

(1.20.20b)

This specifies the enthalpy change in an infinitesimal step under nonequilibrium conditions. When the above infinitesimal step is executed reversibly the above reduces to the standard form (1.20.20c)

lZi dni.

d H - T dS + V d P + ~

i When this is subtracted from (1.20.20b) one obtains To dO - ( To - T ) d S

- (Po-- e ) [ ( ~ S ) p , n + Z ( l d , Oi -- lzi)dni, i

idS +

( ) OV

-~

dP+~-~ S,ni

~ i

dni S,P,ni#j

]

(1.20.20d)

which completes the specification of the deficit function in terms of S, P, and n i. The integration of the various differential forms dO is not a simple matter. The subject was extensively discussed in Section 1.13 for a closed system by means of a specific example; the procedure may readily be extended to open systems, though the final formulation is then much more complicated. As in Sections 1.12, 1.13 we recognize that the state functions E, H, A, G allow us to analyze any process in terms of quantities that depend only on the initial and final configuration of the system, thus dispensing with such path-dependent quantities as heat and work. Which of the four functions are found to be useful depends on the conditions of the experiment: as an example, if a process is carried out under conditions where temperature and pressure are the relevant experimental variables then the Gibbs free energy is the quantity on which all deductions are based. EXERCISES

1.20.1. Verify that the various definitions for chemical potential cited in Eq. (1.20.2) are equivalent. Is it appropriate to define a chemical potential by the relations (OE/Oni)T,V,ni or (OH/Oni)T,V,ni .9 Document your answer fully.

EFFECT OF CHEMICAL CHANGES ON COMPOSITION

103

1.20.2. The above conditions are somewhat restrictive. We need demand only that the change in volume of the system be attributed to a compensating change in volume of the surroundings, which does not preclude other volume changes in the reservoirs from occurring. Also, for the time being, we deal solely with changes in composition that arise from a transfer of material across the boundaries of the system. Internal compositional changes are treated in the next section and in Chapter 2. 1.20.3. Could you have adopted the view that the term - Z i # i dni should have been grouped with the deficit function dQ rather than with dW ? What would be the consequences of adopting such a stance? 1.20.4. Expand on the analysis of the present section by considering types of work other than mechanical to be performed. 1.20.5. The quantity Z i ~ i dni is sometimes referred to as 'chemical work'. Do you consider this designation to be appropriate? Or should such a quantity be considered as part of the heat transfer, as suggested in Exercise 2? 1.20.6. Would it be feasible to replace the commonly used expression for the differential of the Gibbs free energy by dG -- - T dS + V dP + Z i # i dxi, where X i designates a mole fraction, and where the summation now runs over the c - 1 components? 1.20.7. Derive the following relation that is commonly used in statistical mechanics for a one-component system: (d#/dn)T,V = - ( V / n Z ) ( d P / d V ) T , n . You will need to consult several of the theorems developed in Section 1.3 and recognize that an equation of state interrelates T, P, V, n, whereas # involves T, P, and n. 1.20.8. The grand potential is defined by the relation Y2 = E - T S - Z i Izi dni, Derive expression for S, P, and n i in terms of partial derivatives involving ~ . Then discuss the utility of this function. 1.20.9. Establish the Gibbs-Duhem relation in the following form: E d ( 1 / T ) + V d ( P / T) ~ i ni d(lzi / T) -- O. 1.20.10. Would it be appropriate to introduce a Legendre transform that interchanges # and n, so that the various functions of state would involve terms of the type Y~i ni d#i ? -

1.21 1.21.1

-

Effect of C h e m i c a l C h a n g e s on C o m p o s i t i o n The Isolated System as a Black Box I

So far we have made no distinction between changes in mole numbers of constituent i that are brought about by transport of i across the system boundaries (ni) and those arising from occurrence of chemical reactions within the system ( N i). We now attend to this by writing the First Law in the form

dE-

T dS-

P dV + ~

~i

i

d(ni + Ni).

(1.21.1)

104

1. FUNDAMENTALS

Consider the special case of a closed system for which dni = 0 for all i. The resulting expression

d E - T d S - P dV

+

~-~l~i dNi

(1.21.2)

i

must be handled with care. Consider first the case of a set of reagents in the isolated system whose mole numbers Ni are kept fixed by constraints. Now let the constraints be lifted so that interactions may take place reversibly, adiabatically, and at fixed volume, without transfer of material across the boundaries. Under these conditions we must require that d S = 0 and that d E = 0, which in turn requires that

Z l ~ i dNi - 0 .

(1.21.3)

i

This result appears to be counterintuitive, especially since we normally allow the energy to depend on mole numbers, as specified by the relation E = E(S, V, {Ni }). However, this problem is apparent rather than real 2" from the viewpoint of chemistry the fundamental species in any chemical reaction are the participating atoms whose numbers are strictly conserved~witness the process of balancing any chemical equation. Thus, while the arrangement or configuration of the atoms changes in a chemical process their numbers are not altered in this process. Under conditions of strict isolation the system behaves as a 'black box'; no indication of the internal processes is communicated to the outside. 3 One should not attempt to describe processes to which one has no direct access. However, under conditions illustrated in Remark 1.21.2, even an isochoric reaction carried out very slowly in strict isolation, produces an entropy change dS - dO = Z i ~i dNi > 0. See also Eq. (2.9.3) which proves Eq. (1.21.3) under equilibrium conditions. 1.21.2

Counteracting the Isolation Problem

There is nothing to prevent us from setting up schemes that allow us to investigate the internal equilibrium state of the black box, but only by at least temporarily destroying its isolation. For example, we could lift the initial constraints to allow for an infinitesimal advancement of the reaction at constant volume, and transfer the concomitant energy in the form of heat, by an amount equivalent to the change in entropy - T d S , to the outside world. We could then reintroduce this amount of heat back into the system, corresponding to the transfer +TdS. The net process is adiabatic, but gives rise to a temperature change specified by Cv d T, where the heat capacity is that of the system. As an alternative, we could slowly and reversibly introduce an infinitesimal jet containing chemical species of appropriate composition, whose constituents are poised to interact. We then allow the reactions to proceed during the transfer stage.

EFFECT OF CHEMICAL CHANGES ON COMPOSITION

105

At the conclusion of this process the mole numbers dni of all species inside the system are now altered; also, the 'heat of reaction' has been transported across the boundaries. These events lead to a change in the energy coordinates of the system that is specified by Eq. (1.20.8). Let us specialize to the case where the volume is held fixed. That relation then reduces to

dE - T[dtS + dnS] + E

ll~i dni

i

=7"

[()OS

dT + v, i

l z

dni Jrr,v, j i

,.

= CvdT + T E Si dni + E # i dni (V fixed), i

l~i

dni

i

(1.21.4)

i

which shows the general response of the system (i.e., the energy change) in terms of an entropy change dt S triggered by the energetics of the reaction, and an entropy change dnS associated with the change in the amounts of material in the system. The former effect produces a change in temperature measured by the heat capacity at constant volume. If we now demand that the energy of the system remain constant during the above process we impose the restriction E

(lzi + r g i ) d n i

-- - E

i

Ei d n i -

-Cvdr

-

-r

diS;

(1.21.5)

i

that is, heat must be transferred across the boundaries as required to satisfy Eq. (1.21.5). This latest derivation is an illustration of the fact, previously mentioned, that there do exist processes during which the energy of a system can be held fixed, but only if the work performance (in this case, in the form of mass transfers) is compensated for by a heat exchange. As an alternative, we may contemplate the reversible jet transfer under the adiabatic conditions mentioned above. In such a situation the energy of the system is altered but in a reversible process we now demand that the entropy remain fixed. This imposes the restriction

dS-

(as) -~

dT+ E (O~ni) V,xi

i

dni - CVdT-F E S i d n i -O. T,V,nj=/:i T i

(1.21.6) Thus, the temperature of the system rises as prescribed by Eq. (1.21.6), which should be carefully contrasted with Eq. (1.21.5) and with the earlier discussion. REMARKS 1.21.1. The author thanks Professor Dor Ben Amotz of Purdue University for very insightful discussions concerning this topic.

106

1. FUNDAMENTALS

1.21.2. The problem that arises is illustrated by the following example: Two moles of hydrogen gas and one mole of oxygen gas are placed in a totally isolated container and then exposed to a platinum catalyst. The reaction to form water as a condensed phase is allowed to proceed exceedingly slowly. After a very long time interval one encounters all three species. It is difficult to argue that, as might perhaps be expected, the entropy change in this totally isolated reaction should be zero. The resolution of this paradox hinges on the definition of what is meant by a reversible reaction. Even though the reaction itself proceeds very gradually, no provision has been made to bring the system back to its original state, let alone without incurring any other changes in the universe. Thus, the process must be regarded as being spontaneous, with d S = dO > 0. This indicates that the execution of a very slow reaction is a necessary but by no means sufficient condition to guarantee reversibility. To achieve the latter one may, for example, include in the isolated system an interactive electrochemical cell (see Chapter 4) that is very slowly charged by the reaction. When the process is terminated the initial conditions may be restored by slowly running the cell backwards. In this enlarged system the entropy changes are represented by dSr + dSb = 0, where the two symbols stand for the entropy change in an infinitesimal step of the reaction and of the battery operation. More generally, to ensure reversibility of a chemical reaction one must include in the isolated system some type of storage unit to which the entropy changes may be transferred reversibly and from which they may be recovered in the reverse process. It is in this sense that the quantity d S (=- d Sr + d Sb) used in the text must be understood when dealing with isolated systems, since the ordinary differential relations for functions of state only apply under reversibility conditions. In the absence of an internal entropy reservoir the reaction within an isolated system always proceeds spontaneously. 1.21.3. In other words, the quantity d S in Eq. (1.21.2) is determined by heat transfers across the boundaries, and, as indicated in the previous footnote, is not an entropy change brought about by a purely reversible, internal rearrangement of atoms in the absence of external influences. One must recall that Eq. (1.21.2) only applies to reversible processes, for which dSr + dSb (see Remark 1.21.2) must vanish.

1.22 1.22.1

Legendre Transforms and Stability of a System Generalized Legendre Transforms

For later use we investigate some characteristics of Legendre transforms. Consider a function of state that depends solely on a set of t 4- 1 extensive variables: Y -- Y({xi}). Then t

d Y -- Z i=0

Pi dxi,

(1.22.1a)

LEGENDRE TRANSFORMS AND STABILITY OF A SYSTEM

107

with (1.22.1b)

Pi ~ (O~iXi)xjr

Now introduce a partial Legendre transform as follows: n Z(p0,

9 9 9

Pn, Xn+l .....

Xt) -- d Y ( x o , . . . , xt) - ~

n

Pi dxi - Z

i =0

t --

Xi

dpi

i =0

n

Z pi d x i - Z x i i=n+ l i=0

dpi,

(1.22.2)

(k ~< n)

(1.22.3)

(k > n).

(1.22.4)

where Eq. (1.22.1 a) had been introduced. Then, OZ(po, . . . , pn,Xn+l . . . . , x t ) ) OPs

= -x~ pj~k

and O Z ( p o , . . . , Pn, Xn+l , . . . , Xt) ) -O-Xk

-- p~ x j T~k

It follows from Eq. (1.22.1 b) that

( Op~~

02 Y

(1.22.5) x j s~k

On carrying out a second partial differentiation of Z with respect to p~ in Eq. (1.22.3) we find that

p-7) 02Z

1

(k ~< n).

(1.22.6)

In cumbersome language: the second partial derivative of the function of state Y with respect to the extensive variable x~ has a sign opposite to the second partial derivative of the partially inverted Legendre transform Z taken with respect to its conjugate variable p~. 1.22.2

Stability Conditions Based on Fluctuations

We now apply the above results to the energy function E ( S , V, n l . . . . . n t ) . To explore the effect of entropy fluctuations giving rise to slight departures from

108

1. FUNDAMENTALS

equilibration while keeping all other variables fixed we may expand E about its equilibrium value E0:

E - Eo--

-~ V,x~dS + -~( O2E ) V,x~d 2S + . . . .

(1.22.7)

The first derivative vanishes and the second must be positive to guarantee that at equilibrium E be at a minimum value consistent with the fixed constraints; we exclude here and below the trivial case d S --d2S - - 0 . Thus, on account of (1.13.1c), (1.13.15),

> O.

v,x

(I .22.8a)

Cv, i

This shows that the entropy at constant volume and composition rises with temperature, as is intuitively evident; also, it shows that the heat capacity at constant volume and composition cannot be negative. In similar fashion one may show that 02E

- -

> 0,

(1.22.8b)

S,xk

which indicates that (0 P/O V)s,x~ < 0; under adiabatic and constant composition conditions any pressure rise produces a shrinkage of v o l u m e ~ a common sense experience. In addition, > 0.

(1.22.8c)

S, V,nj:/:k

In other words, the chemical potential of species i rises as more i is added to the system at constant S, V, and constant nj~i. This, of course, is a common feature of all potential functions. The above results also show that, whatever the extensive variable, E remains convex. We now apply the same procedure to the enthalpy H = H(T, P, {ni ]): first we write [cf. (1.13.2d), (1.13.15)] > 0. OS 2

P,xk

-~

P,xk

(1.22.9a)

C P,xi

Thus, as above, a rise in temperature raises the entropy of the system during a process at constant pressure and composition. Also, the heat capacity at constant pressure and composition cannot be negative.

LEGENDRE TRANSFORMS AND STABILITY OF A SYSTEM

109

When we consider changes of H with P we must invoke Eq. (1.22.6) which addresses the change from the extensive variable V to its intensive conjugate, P. Thus, we write --

~

] S,xi

0P 2

S,xi

< O,

(1.22.9b)

> 0,

(1.22.9c)

S,xi

which is identical with Eq. (1.22.8b). We finally cite the relation

02H

an2i ) s,p,~j~

Ort i ,,l S , P , n j:/:k

with the same message as Eq. (1.22.8c), except that constant pressure conditions prevail. In a similar vein we derive the relations .

.

showing that

.

(+,)

.

v,x,

~,5-~]v,x,

Cv,xi >

U

v,x,

=

T

< 0

'

"

0, in conformity with (1.22.8a). Furthermore,

( 2A) 0V2

(1.22 10a)

/22.0b. T,xi

--

-~

r,xi

showing that the isothermal compressibility fi is positive. This relation is the analogue of Eq. (1.22.8b). Finally,

(02A~

--

~-~2i]t,V,xj~i

(Ol.ti) ~ Ol'li

> 0,

(1.22.10c)

T,V,xj~i

which is the analogue of (1.22.8c); again, the chemical potential increases with mole number of species i under the indicated constraints. Lastly, we consider the Gibbs free energy . showing

.

.

.

=

0S2 P,xi ~ P,xi - ~ P,xi that Cp,x~ is positive, as already established -

-

\5~

r,x,

T

< 0,

~

'

(1.22.11a)

in (1.22.9a). Also,

-

ap2 ~,x,

< 0

(1.22.11

b)

(1.22.11

c)

~,xi

as already established in (1.22.10b). Also, --

T,P,nj#i

> 0,

Oni ] T,P,nj#i

which is the pressure analogue of (1.22.10c).

110

1. FUNDAMENTALS

1.22.3

Virtual P r o c e s s e s in C l o s e d S y s t e m s

We next consider virtual energy changes in a closed system. Here the second order term in the expansion of E -- E(S, V) must be positive to guarantee stability and to avoid the trivial case d S - d E - O. We now introduce the symbols E ss = (02 E/O S2)v and similarly for the other partial derivatives. The stability condition then reads 1

f!

a2E -- '~[E~s(aS) 2 -Jr-2E~vaSaV -+- E v v ( a V ) 2] > O.

(1.22.12)

It will be shown in Section 2.2 that the following conditions must apply to satisfy the requirement 32E > 0" " > 0, ESS

" > 0, EVV

" " EssEvv

-

" 2> (Esv)

0.

(1.22.13)

The first two conditions duplicate Eqs. (1.22.8a) and (1.22.8b). The third condition assumes the form

(Esv)2 " < - T(OP/OV)s .

(1.22.14)

Cv

The positive fight-hand side furnishes both an upper and a negative lower bound ff for the cross derivative Esv. If desired the quantity (OP/O V)s may be rewritten via Eq. (1.3.8) in terms of partial derivatives that involve the entropy. Similar arguments may be advanced for virtual changes involving the other thermodynamic functions of state. However, one must be careful: for example, virtual changes in the Helmholtz function assume the form ,, aT aV + Avv ,, a 2 A - ~1 [ - A T,,T ( a T ) 2 + 2ATv

(aV)2] > 0 '

(1.22.15)

where we recognized that the Legendre transform interchange of S with T requires a sign change in the indicated second partial derivative. In fact, we write

--ART --

v

= ~ >0 T

Avv - - '

- f l V > 0,

(1.22.16)

T

in conformity with (1.22.8a) and (1.22.10b). Also,

(A}v)2

<

,,

(aS)

3P) P --~ r

_

Cp

flVT'

(1.22.17)

which establishes an upper and negative lower bound for the cross derivative. The reader should take the time and trouble to use the above arguments to find the second derivatives of the enthalpy and of the Gibbs free energy functions so as to establish analogues of Eqs. (1.22.13), (1.22.14) and (1.22.16), (1.22.17).

111

Chapter 2 Equilibrium in Ideal Systems

2.0

Thermodynamics of Ideal Systems with Several Components and Phases

In the preceding chapter we have considered in general terms the fundamental principles of thermodynamics that govern any physical process. We now specialize to ideal systems in which several components coexist in one or more phases. We proceed gradually from the simple equilibrium conditions to more complex cases, noting the underlying thread that if equilibrium is to prevail the system must be subject to constraints, all of which are ultimately based on Gibb's criterion: at equilibrium the chemical potential of a given species must be the same in all phases.

2.1

Coexistence of Phases: The Gibbs Phase Rule

We examine here the very stringent constraints which arise when two or more distinct phases are to be maintained in equilibrium. We note that equilibrium between two or more phases, considered as coexisting open systems with no rigid partitions, requires minimally the uniformity of temperature and of pressure throughout the entire system. This makes it apposite to deal with the Gibbs free energy as the appropriate function of state. We also restrict ourselves to mechanical work; the generalization to other types of work is taken up later. We finally assume that each of the c components is encountered in every one of the p phases; removal of this constraint is considered in Exercise 2.1.1. Thus, let the total Gibbs free energy be written as G - G ~+ G tl + . . . + G (p),

(2.1.1)

in which G ~ G ~ G (p) a r e the corresponding quantities in phase 1 2, ,p As discussed in Section 1.13, the condition for equilibrium is specified by setting ~

9

9

9

~

~

.

.

.

.

112

2. EQUILIBRIUM IN IDEAL SYSTEMS

the Gibbs free energy at a minimum, with 6G -- 0; hence,

6G - 3G I + 3G" + . . . + 3G (p) --O,

(2.1.2)

which is, however, is minimally subject to the following restrictions: T -- 0,

8 P -- 0.

(2.1.3a,b)

Consider initially the various phases in isolation; in particular, for phase 1, identified here by the prime symbol, C

8G O' -- _so'STO' + vO'spO' + Z l Z ~

0'.

(2.1.4)

i=1

The superscript 0 serves as a reminder that we deal with isolated phases, each at its own temperature T ~ pressure p0, and with mole numbers n o that generally differ in each of the phases. Observe that 2 + c - 1 - c + 1 independent variables are present in Eq. (2.1.4), namely T, P, and the c - 1 mole fractions constructed from the c mole numbers. The total number of variables associated with the p isolated phases, involved in the sum 3G ~ + . . . § 3 G O(p)t, is thus p(c + 1). We now combine all phases and allow thermal, mechanical, and chemical equilibrium to take place throughout the composite system. The temperatures T, pressures P, and mole numbers n i now differ from the values that obtained when the phases were isolated. From Eqs. (2.1.2) and (2.1.4) we find for the new set of variables:

3G--(SI3T

I + S ' 3 T " + . . . + S(P)3T (p))

+ (V'3P'+ V"3P"+... +

+

+ V(P)6P (p))

+... +

+

+""

+""

+ (lZ'cgn'c + lz~gn~ + . . . + lz~P)3n~p)) --O.

(2.1.5)

One must now consider the constraints. On account of the uniformity of temperature and pressure we set

T t- T'=...=

T (p) =-- T

or

3T ~ - 3T" . . . .

- - 6 T (p) - - a T -

O, (2.1.6a)

P~-- P ' = - . . =

P(P) =~ P

or

8pI _ 8 p , = ... = 3p(p ) =~ 3P -- O. (2.1.6b)

and

COEXISTENCE OF PHASES: THE GIBBS PHASE RULE

113

This set of requirements guarantees constancy of T and P for every phase. On applying this constraint the first two bracketed terms in the first two lines in Eq. (2.1.5) are found to vanish. A second set of constraints arises by imposing the conservation of mole numbers for every component in the overall closed system: n l - - n tl + n tlt + - - . + n

(lp) or

3n t1 + 6 n ttl + . . . + 6 n

or

nc - n c + n c + . . . + n

(~nc + 6n c + . . . +

(lp) = - - 6 n 1 - - 0 ,

=- (~nc - O. (2.1.6c)

From (2.1.6c) it is seen that if we wish to set 6G - 0 , with 8T -- 6 P --0, then in Eq. (2.1.5) we must also require that /zlt -- #1tt = " " = / z (lp) = / z l , /z I

2~

/z II

2~'"~

/z~p)

~ k 62,

9

~

~

o

#c

I

-- #c

,

--'"--

/z~p)

=

#c.

(2.1.7)

For, in these circumstances Eq. (2.1.6c) applies, so that the conditions 3nl 6n2 . . . . . 6nc - 0 hold. Thus, when Eq. (2.1.7) is imposed as a requirement, Eq. (2.1.5) is identically satisfied: 6 G - 0 for the entire system at equilibrium. We have thus arrived at a very important necessary and sufficient condition which characterizes the equilibrium conditions among phases: A s i d e f r o m uniform i t y o f t e m p e r a t u r e a n d pressure, one requires that the c h e m i c a l p o t e n t i a l lzi f o r e a c h one o f the c c o m p o n e n t s be the s a m e t h r o u g h o u t all p p h a s e s .

We can also calculate the number of degrees of freedom f for the assembly of phases. As stated earlier, if the phases were all separate systems, p ( c + 1) independent variables of state would have to be specified. However, after establishing equilibrium among the phases one must take account of the 2(p - 1) constraints of Eq. (2.1.6a) and (2.1.6b) to ensure uniformity in T and P, and one must note the c ( p - 1) interrelations in Eq. (2.1.7). The totality of constraints therefore is (c + 2)(p - 1). The number of degrees of freedom remaining is then f - p ( c + 1) - (c + 2)(p - 1) - 2 4- c - p.

(2.1.8)

Equation (2.1.8) specifies the famous p h a s e rule o f G i b b s (1875-1878). Knowing the number of components and phases in a given system, and assuming that T and P for the system as a whole are uniformly variable, Eq. (2.1.8) indicates how many state variables may be independently adjusted over limited ranges without altering the number of phases of the system. The ramifications of the phase rule will be discussed in Section 2.3.

114

2. EQUILIBRIUM IN IDEAL SYSTEMS

Further insight regarding the concept of the chemical potential may be obtained by considering a two-phase, one-component system at fixed temperature and prest t tt t/ t tt sure, for which G - n 1# 1 + n2/z2" Suppose now that at some instant # 1 > # 1" The system can then not be at equilibrium; instead, spontaneous processes will occur that move the system closer to equilibrium, which ultimately results in the equalization of #1t and/z t1/ . At constant T and P this can occur only by a transfer of matter from one phase to the other, Let there be a transfer of - d n ' l - +dnl" > 0 moles from phase ~ to phase "; then d G - (#~( - / Z l )t d n l , where we have set tt dn 1 =- dnt[. Since we assumed #1t > #1, the preceding relation shows that d G < 0 in this case; i.e., matter is transferred spontaneously from the phase of higher chemical potential to the phase of lower chemical potential. Thus, a difference in chemical potential represents a 'driving force' for transfer of chemical species, rather analogous to the difference of electrical potential that is a 'driving force' for electrically charged species. As for an electrical potential, equilibrium is achieved only by an equality of the chemical potential for the species in question throughout the entire system. Just as the relative magnitudes of electrical potentials determine the direction of current flow between the two conductors, so the relative magnitudes of chemical potentials of a given component in two phases in contact determine the direction of transfer of the component between the phases. These considerations will be put on a firmer footing in later discussions. EXERCISES

2.1.1. How must the Gibbs phase rule be modified to take account of the following cases: (a) A multiphase system is placed between two charged parallel condenser plates? (b) One or more of the components are absent from one or more of the phases present? (c) Several distinct regions of the system are maintained at different pressures by means of semipermeable membranes? Document your answers fully. 2.1.2. How must the derivation of the Gibbs phase rule be modified if work other than mechanical P - V work is performed on or by the system? (Hint: classify these degrees of freedom with P and V and proceed with an expanded derivation.) 2.1.3. What tacit assumption has been made in proceeding from Eq. (2.1.1) to Eq. (2.1.2)?

2.2

Achievement of Equilibrium

A basic problem in thermodynamics consists in determining the final equilibrium state that an isolated system reaches after starting out from a given set of initial conditions and constraints. In this matter we are guided by two corollaries of the First and Second Laws; namely, that in an isolated system subjected to any change the entropy cannot decrease, and that its energy must remain constant. These requirements may not be sufficient to determine the final equilibrium state, in which case other experimental data or additional constraints must be inserted to provide a unique solution to the problem.

ACHIEVEMENT OF EQUILIBRIUM

2.2.1

115

C h a r a c t e r i z a t i o n of H e a t F l o w

We now examine several equilibration processes in detail. The first relates to thermal conditions which prevail when two adjacent isolated systems, designated as ' and ", initially at temperature T' and T ' , are equilibrated, after allowing their rigid adiabatic partition to become slightly diathermic (see Fig. 2.2.1). The restriction that the compound system remain isolated and that the energies and entropies be additive yields the relations (ignoring interfacial contributions) or

E'+E"=Et

dE '+dE'=0,

(2.2.1)

and

S--S'(E',V')+S"(E",V")

(2.2.2)

If the walls are rigid we also require d V ' = d V " = O. In an infinitesimal exchange of heat between the two subsystems,

)

( OS~ dE' + d S -- \ OE, v'

(0,,,) OE"

v"

d E " >~O.

(2.2.3)

On account of Eq. (2.2.1) and the relation (OE/OS)v = T, Eq. (2.2.3) now reads dS--

(1 l) T'

T"

d E ' >~0,

(2.2.4)

which immediately establishes the requirement T' -- T" as a necessary condition for thermal equilibrium; this confirms what is already known. Also as the system approaches its equilibrium value the entropy will increase towards its final maximum. In now differentiating with respect to time t we find )-

T'

T"

~

~>0.

(2.2.5a)

> /<

Fig. 2.2.1. Temperature profile for the flow of heat between a region at temperature T p and a second region at temperature T", separated by a narrow transition region of cross section A and length 1.

116

2. EQUILIBRIUM IN IDEAL SYSTEMS

Here S = dS/dt represents the rate of entropy production during the energy transfer; this quantity cannot be negative and it vanishes at equilibrium. In the present case no work has been performed; therefore, all energy changes involve solely heat transfers. It is therefore reasonable to equate dE~/dt with the rate of heat flow, .Q, across the internal boundary. We thus rewrite Eq. (2.2.5a) as ~d-- A(1/T)Q. Next, we define a heatflux by the relation Jo - Q/A, where A is the cross-sectional area of the diathermic partition. Note that by Eq. (2.2.5a) and Fig. 2.2.1, when d E~/dt is positive, Ja involves a heat flow into system ~. We also suppose that the temperatures T ~ and T ~ are very nearly constant in both compartments and that the changeover from T t to T" takes place only over a small distance 1 perpendicular to the partition (see Fig. 2.2.1). Then the product A1 roughly defines a volume V over which the temperature changes occur; thus, we may write S - A (1 / T) V JQ / 1. In the limit of small 1 the ratio A (1 / T) / 1 becomes the gradient V ( 1 / T ) ; also, S / V = 0 is the rate of entropy production per unit volume, which turns out, as seen later, to be a quantity of great theoretical interest. Eq. (2.2.5a) has thereby been rewritten in the more fundamental form

0 -- V(1/T)JQ >~O.

(2.2.5b)

The preceding chain of reasoning is obviously very crude; for a proper derivation of Eq. (2.2.5b) the reader is referred to Chapter 6. We can nevertheless proceed with several interesting deductions involving irreversible thermodynamics without having to cope with the full machinery of Chapter 6. An important aspect of Eq. (2.2.5b) is the fact that JQ and V ( 1 / T ) = F r are conjugate variables, in that they occur as the product of a flux JQ and a generalized (thermal)force, or affinity Fr, in the expression 0 -- F r Jo. Note that 0 > 0 means either that F r > 0, JQ > 0 (i.e. T ~ < T") or that F r < 0, JQ < 0 (i.e. T ~ > T ' ) ; in either case heat flows spontaneously from the region of higher to the region of lower temperature. This is not exactly news, but at least shows the consistency of the entire approach. When 0 -- 0, equilibrium prevails; JQ and F r both vanish. These facts give rise to the viewpoint that the force F r 'drives' the heat flux Ja. The question should be raised whether it is meaningful to apply the temperature concept to a nonequilibrium situation. The answer is in the affirmative if the following sufficiency conditions are met: the two portions of the system are very large, and the heat transfer occurs very slowly. Then T ~ and T t~ are sensibly uniform over both regions and most of the temperature variation takes place in the immediate vicinity of the volume Al of the interface, in the manner sketched in the figure. The relation between F r and JQ c a n n o t be determined from classical thermodynamics alone. Further information is needed, such as microscopic transport theory, experimental results, or other postulates. Sufficiently close to equilibrium the flux can be expected to vary linearly with the applied force according to

JQ = L r F r ,

(2.2.6a)

ACHIEVEMENT OF EQUILIBRIUM

117

where L 7" is a parametric function independent of JQ nomenological coefficient. We then find that JQ=LTV(1/T)=

LT -~VT T2

or

F r , known as the phe-

----KVT,

(2.2.6b)

where K = L T / T 2 is the thermal conductivity; Eq. (2.2.6b) is a formulation of Fourier's Law of heat conduction. In the present scheme we can set (2.2.7) Then, according to Eqs. (2.2.6b) and (2.2.7), we require that L T ~> 0 and tc >~ 0, in order that t) remain nonnegative. 2.2.2

System with Energy Transfer at Variable Volume

We next examine an isolated compound system with a fixed total volume containing a sliding partition that is initially locked and that provides for adiabatic insulation of two compartments at pressures P' and P", temperatures T' and T", and individual volumes V' and V". The system is allowed to relax after slowly releasing the lock and slowly rendering the partition diathermic. Entropy changes in both compartments can now occur in accord with the relation dS - T - l [ d E + PdV], no other forms of work being allowed. The constraints are d V ' + d V " - 0 (rather than d V ' = d V " -- 0, as before) and dE' § d E " -- 0. By the procedure adopted before we write

{ OSl -

ds-\

OS" oE"

v,,

dE"

OS

v ..

dV I +

d

..

>~O.

ov"

(2.2.8) With dS = T - l [ d E + P d V ] one obtains (OS/OE)v = 1 / T and (OS/OV)E = P / T ; Eq. (2.2.8) then becomes 1 1 pl p. dS - T' d E ' ~ T" d E " t T' d V f + --T-gd V

"

>~0.

(2.2.9)

Finally, with d E " = - d E ~ and d V " = - d V ~ one obtains

-

-

T'

T"

--~ +

T~

T"

-~

~>0"

(2.2.10)

This expression, in conjunction with the arguments that led to Eq. (2.2.5b), suggests that we introduce the fluxes JE = dE~/A dt and Jw = d V ' / A dt, and that we convert A ( 1 / T ) = 1 / T ' - 1 / T " and A ( P / T ) = P ' / T ' - P " / T " into affinities such that for small l, F r = A (1 / T) / l = V (1 / T) and Fp = A (P / T) / 1 =

118

2. EQUILIBRIUM IN IDEAL SYSTEMS

V ( P / T ) . We further introduce the rate of entropy production per unit volume in the manner discussed above to obtain

0 -- V(1/T)JE + V(P/T)Jw,

(2.2.11)

which identifies V ( 1 / T ) and JE as well as V(P/T) and Jw as conjugate fluxforce variables. Equilibrium is then characterized by the necessary condition FT = Fp = 0, which leads to the requirements that T ~ = T" and P ~ = P" as equilibrium constraints. As a very important new principle, based on Eq. (2.2.11), one now postulates a linear dependence of the fluxes on the forces, such that both forces combine to drive both fluxes. The resulting relations JE = LllFT + L12Fp, Jw = L12FT +

L22Fp,

(2.2.12)

are known as phenomenological equations, and the various L are known as phenomenological coefficients. Clearly, Eq. (2.2.12) displays interference effects, in that the driving force for energy flow (or for work performance) also affects the work (or energy) flux. We shall discuss these matters in much further detail in Chapter 6. For now we only note that the rate of entropy production in the above case is specified by 0 -- FTJE + F p J w -- LllF2T + (L12 + L21)FpFT + L22 F2 >/O.

(2.2.13)

In order to render 0 nonnegative it is necessary and sufficient to require that

L l l / > 0,

4LllL22 - (L12 + L21) 2/> 0,

L22/> 0,

(2.2.14)

for which a derivation is furnished below. 2.2.3

Transfer of Energy and Matter

We next examine the case of two subsystems separated by a rigid partition that is diathermic and permeable to one species present in different amounts in two compartments held at different temperatures. By an extension of earlier reasoning we invoke the relation d S = T -1 [dE - lz dn] to write

(1 S-

T'

1)dE'

(lz"

T"-~--+

T"

lz')dn' T'--~-~>0,

(2.2.15)

from which it follows that at equilibrium T t = T"

and

/z ~= / z " ,

(2.2.16)

ACHIEVEMENT OF EQUILIBRIUM

119

in consonance with earlier findings. According to Eq. (2.2.15) we can further write down a set of fluxes JE = - d E ' / A dt and Jn =-dn'/A dt, and the corresponding generalized forces F r -- V ( 1 / T ) and Fn =-- V(/x/T). One then sets up linear phenomenological equations of the form, analogous to (2.2.12),

JE = L33FT q- L34Fn, (2.2.17)

Jn = L 43 FT -Jr-L44Fn. The rate of entropy production is specified by

0 -- FTJE -}- FnJn -- L33 F2 -Jr-(L34 q- L43)FnFT q- L44 F2 ~ O,

(2.2.18)

together with the requirement that L33 ~ O,

4L33L44 - (L34 q- L43) 2/> O,

L44 >~ O.

(2.2.19)

The generalization to the case of a sliding partition is to be handled as an exercise 1. 2.2.4

Inequalities that Guarantee a Nonnegative Rate of Entropy Production

We consider here the necessary and sufficient conditions that guarantee that Eq. (2.2.18) remains nonnegative. For this purpose introduce the change of variable

G =~2L33FT q- (L34 q- L43)Fn

(2.2.20)

and eliminate FT from Eq. (2.2.18). This yields the relation ~4L33 G2

if-[ L 4 4 - (L34-ffL43)2] 4L33 F2>/O"

(2.2.21)

Clearly, the multipliers of G 2 and of F 2 must remain nonnegative to satisfy Eq. (2.2.21). Accordingly, we first set L33 ~> 0, then require that 4 L 4 4 L 3 3 (L34 -Jr-L43) 2 /> 0, and finally note that one is forced to set L44/> 0. This establishes Eq. (2.2.19). EXERCISE 2.2.1. Generalize the above derivations so as to handle the case of two subsystems at different temperatures, pressures, and mole numbers of a gas, separated by a movable partition that is permeable to the gaseous species.

120

2. EQUILIBRIUM IN IDEAL SYSTEMS

2.3 System of One Component and Several Phases; The Clausius-Clapeyron Equation According to the Gibbs phase rule the number of degrees of freedom of any system is diminished by one for every additional phase of identical composition that is added to it. Thus, for a one-component system, f = 2, 1, 0 when the system consists of one, two, or three phases. Taking liquid water as an example, the two degrees of freedom are temperature and pressure, which may be varied independently over wide limits without changing the state of aggregation of the liquid phase. However, when water and steam are forced to coexist T and P are no longer independently adjustable; the pressure is now determined by the temperature, or vice versa~only one degree of freedom is left. Thus, if at fixed T < 647.2 K the prevailing equilibrium pressure were raised (for example, by application of pressure through a piston) steam would continually condense, at close to the equilibrium pressure until only water is present (when the piston rests on the liquid level). The heat of condensation must be continually withdrawn, so as to maintain the temperature at a fixed value. The pressure on the water will then rise, as the piston continues to be forced against the liquid. Conversely, if the equilibrium pressure is reduced (by withdrawal of the piston) water would continue to evaporate, so long as the temperature of the liquid is maintained by supplying the heat of vaporization from external sources. Ultimately, only steam remains in the system. If one wishes to have ice, water, and steam coexist no degree of freedom is left; this state can only exist if the system is maintained at the so-called triple point T = 273.16 K and at P = 4.58 bar. For this reason the triple point of water serves as a convenient thermometric reference standard, as already mentioned in Section 1.2.

2.3.1 Phase Diagram of Water A pictorial survey of the above discussion is provided in Fig. 2.3.1 (not drawn to scale), as a plot of P vs. T. The three solid curves delineate three distinct regions within which pressure and temperature may be varied independently within limits, corresponding to the presence of only one phase. On the other hand, the curves on the diagram delineate the conditions under which two phases can coexist; the curves represent the loci of experimental (T, P) values compatible with the presence of two phases. In other words, they indicate the dependence of T on P, or vice versa. To have solid and gaseous water coexist one must adjust T and P to fall somewhere on the curve OT, and similarly for the other branches. T in the diagram represents the triple point, and C, the critical point (T - 6 4 7 . 2 K, P = 217.7 bar), beyond which water and steam are indistinguishable. The description of critical phenomena will be taken up in detail in Chapter 7.

SYSTEM OF ONE COMPONENT AND SEVERAL PHASES

121

X 217.7 atm

Solid, I

1 atm _Ii ~ . ~ / "

4.58 atm (nonlinear) (3

i~

'[,

',

Gas

1

273.15 273.16 373.15 T(K) (nonlinear)

647.2

Fig. 2.3.1. Sketch of the phase diagram of water (not drawn to scale).

,3

S n 217.7 (Pc)

~-! .... 675 T(K)

84.8

-

-'-" 647.2

473.2 Vc

Fig. 2.3.2. Pressure-volume relation for water in the vicinity of its critical point. The transformation between water and steam can be characterized as shown in Fig. 2.3.2, which represents a plot of P vs. the molar volume V at various temperatures near the critical point. The shaded region indicates an excluded domain, in the sense that water with molar volume Va at P - 15.3 bar and T -- 473.2 K coexists with steam whose molar volume is Vb; no water with intermediate molar volumes can exist under those specific conditions. As the pressure is raised beyond 15.3 bar at 473.2 K gas at molar volume Vb condenses to form liquid at

122

2. EQUILIBRIUM IN IDEAL SYSTEMS

molar volume Va. Furthermore, as the temperature is raised the two volumes approach each other and ultimately merge at 647.2 K to a common value Vc. Above that temperature the two phases are indistinguishable. So far we have simply described the curves in the phase diagram of Fig. 2.3.1 as loci of conditions under which two phases coexist. No analytical relations have been specified for such curves; we now attend to this matter.

2.3.2 The Clausius-Clapeyron Equation Equilibrium between two phases A and B is characterized by the equality between chemical potentials,/,a and d/za d/zB and by the uniformity of pressure and temperature; then, by Eq. (1.13.4), --

//,B

--

--SA d T + VA d P -- --SB d T + (TB d P ,

(2.3.1)

which may be rewritten in the form dP

SB -- SA

dT

~zB--V A "

(2.3.2)

This expression, known as the Clausius-Clapeyron Equation, is of great historic significance, being a very early derivation that links seemingly unrelated variables. This was considered to be a noteworthy example of the power of thermodynamic theory and may be considered a precursor to later theoretical developments. The above relation shows up in many different guises. At equilibrium we may set lZB - l Z a - - 0 H B - - H A - - T ( S B - S A ) , w h e n c e we find dP

I-IB--I-7tA

dT

T(VB-

(2.3.3)

VA)'

where T is the coexistence temperature for the two phases. When liquid-gas equilibria are considered one frequently employs the approximations ~ << Vg and Vg - R T / P , which apply under conditions well below the critical point but at temperatures sufficiently high for the perfect gas approximation to hold. On now defining Lv - Hg - Ht as the molar latent heat of vaporization of the liquid Eq. (2.3.3) assumes the approximate form dP dT

=

LvP R T 2'

(2.3.4a)

which may be recast as dln P Lv = RT 2 R T 2'

(2.3.4b)

SYSTEM OF ONE COMPONENT AND SEVERAL PHASES

123

or as dln P

Lv R

d(1/T)

(2.3.4c)

Thus, a plot of In P vs. 1/T should yield a curve whose slope at any point is proportional to Lv. As a final approximation it is frequently assumed that Lv is independent of temperature; this is a poor approximation at best. In that case Eq. (2.3.4b) may be integrated to yield

ln(P2/P1)-

L,V(1R T2

T1 '1)

(2.3.5)

a relation that should be used with some trepidation. In any case, Eqs. (2.3.4a)(2.3.4c) may be employed to determine the molar heat of vaporization where direct determinations might be difficult. One should note that the Clausius-Clapeyron equation is sometimes used in reverse. Rather than employing temperature to control the equilibrium vapor pressure, experimentalists adjust the vapor pressure of a helium bath to control its temperature in the range 0.3 to 4.2 K, where the vapor pressures fall in a conveniently manageable range. By way of contrast to Fig. 2.3.1 we briefly consider the phase diagram for CO2, shown in Fig. 2.3.3. Here the solid-liquid coexistence curve has a positive slope, as contrasted with the case of water, where the slope is negative. This reflects the

i

72.9

X

002

. . . . . . .

C i

60.6

t

P(]tm)

So id

(nonlinear) 5.11

/

I

. . . . . . . . . . .

', a

i

I

vapor .

I

1.0 0

194.7

216.6 T(K) ~

296.2 304.2 (nonlinear)

Fig. 2.3.3. Sketch of the phase diagram for carbon dioxide (not drawn to scale).

124

2. EQUILIBRIUM IN IDEAL SYSTEMS

unusual case for H20 for which ~ < Vs, corresponding to the teleological case of ice floating on water. In almost all other substances the reverse inequality holds, as in Fig. 2.3.3. Note further that the triple point of CO2 occurs at roughly 5.11 bar. Hence under ordinary conditions solid CO2 does not melt but vaporizes directly, a process frequently referred to as s u b l i m a t i o n . 2.3.3

The van der Waals Equation of State

P - V curves of the type shown in Fig. 2.3.2 are very common and are well simulated by the v a n d e r W a a l s equation of state for n moles of fluid (already introduced earlier) nRT

P =

an 2

V - nb

(2.3.6)

V2 '

for which the variation of pressure with volume is sketched for a series of temperatures in Fig. 2.3.4. The symbols a and b represent parameters that must be determined experimentally. The dashed curve shows the exclusion region corresponding to the hatched region in Fig. 2.3.2. Many other equations of state lead to very similar results, but the van der Waals equation of state is adopted frequently in the theoretical characterization of properties of fluids. We therefore examine its properties in further detail.

T7

2

rl v

Fig. 2.3.4. Schematic presentation of pressure-volume variations generated by the van der Waals equation of state.

SYSTEM OF ONE COMPONENT AND SEVERAL PHASES

125

Clearly, a curve located between T8 and T9 in Fig. 2.3.4 corresponds to the isotherm at the critical point, where its first and second derivatives vanish, as shown by the curve for 647.2 K in Fig. 2.3.1. On imposing the requirements ( O P / O V ) T -- ( 0 2 P / O V 2 ) T -- 0 one obtains the results

2a

RTc t

~

V3

3a ~

v4

=0,

(2.3.7)

=0.

(2.3.8)

(Vc _ b)2 [

~

RTc

(Vc - b ) 3

Division of (2.3.7) by (2.3.8) leads to the relation f'c - 3b.

(2.3.9)

When this is inserted in (2.3.7) one obtains 8a

Tc = 27b-----R"

(2.3.10)

Substitution of (2.3.9) and (2.3.10) in the van der Waals equation of state leads to a

Pc = 27b2.

(2.3.11)

RTc 8 Pc f'c = ~

(2.3.12)

It now follows that the ratio

is a universal constant, whereas the individual critical parameters do depend on the parameters a and b. Moreover, it is not difficult to show that on introducing the reduced variables Pr =- P/Pc, Vr =-- V~ Vc, Tr -- T~ Tc the van der Waals equation of state may be rewritten in the universal form

( r+ tI r 3

3 '

(2.3.13)

which is a very elegant formulation, since it involves solely the critical parameters of fluids. Eq. (2.3.13) is one example of the so-called Law of Corresponding States that illustrates the universality of some physical properties of materials. Unfortunately, the universality implied in Eqs. (2.3.12) and (2.3.13) is not well satisfied experimentally. As we shall see in Chapter 7, this situation arises because the van der Waals equation of state does not reproduce the critical characteristics of fluids satisfactorily. Other equations of state in common use suffer from similar defects. As will be shown later, a completely different approach must be employed to characterize the properties of fluids near criticality.

126

2.4 2.4.1

2. EQUILIBRIUM IN IDEAL SYSTEMS

Properties of Ideal Gases Functions of State for an Ideal Gas

An ideal gas obeys the equation of state P ( / - R T; for a mixture of ideal gases the individual chemical constituents i obey the relation Pi Vi - n iR T, or Pi Vi R T. As stated in Section 1.18, no actual material with such properties exists, but the ideal gas serves as a good model substance for many gases at a sufficiently high temperature and low pressure. On account of the simple equation of state it is worthwhile examining the thermodynamic characteristics of ideal gases. 2.4.2

The Energy and Enthalpy of an Ideal Gas

We begin by inserting the constitutional equation of state in the caloric equation of state, Eq. (1.13.16); this leads to the important finding that ( O E / O V ) T = O, regarded as a second criterion to be imposed on ideal gases. Thus, the energy of an ideal gas depends solely on temperature. As a result we now write out the differential energy in the abbreviated form d E = (0 E / 0 T) v d T, whence dE - Cvd T

(ideal gases).

(2.4.1 a)

Moreover, for ideal gases, OCv

which shows that C v likewise can at most depend on T. We next appeal to experiment: the heat capacity of a gas that approximates ideal behavior at high T and low P is constant. At low temperatures this is no longer true, but at some point the ideal characteristics of the real gas then also no longer hold, in conformity with the corollary of the Third Law. As a third criterion of an ideal gas we therefore adopt the requirement that Cv be a constant. Integration of (2.4.1a) then leads to the result E-

Cv T + Eo,

(2.4.2)

where the constant /~0 represents the molar energy of an ideal gas if it could be maintained down to T = 0. This particular choice is obviously unrealistic, but immaterial, since any energy can only be specified to within a completely arbitrary constant; only energy differences are unique. By insertion of (OE/O V)T = 0 into Eq. (1.14.4) we find that Cp - Cv - R.

(2.4.3)

Accordingly, the molar enthalpy of the ideal gas becomes I2I - E + P ( / -- E + R T -- (Cv + R ) T -- C;pT + Eo.

It is readily shown (Exercise 2.4.1) that (0 Cp / 0 P) T -- 0.

(2.4.4)

PROPERTIES OF IDEAL GASES

2.4.3

127

The Entropy of an Ideal Gas

The entropy of an ideal gas is found via dS-

T-I(dE + P dV)-

R

,,,

~

CV dT + _ dV _ Cv d ln T + R d ln V. T V

(2.4.5)

On setting P d (z + (/d P - R d T and substituting for P d (z in the above equation we obtain the equivalent relation dS-

--+--

dT---dP-CndlnT-RdlnP. P

(2.4.6)

Lastly, introducing d In T -- d In P + d In V into either of the above expressions yields dS-CvdlnP

(2.4.7)

+ Cp dln i?.

We have thus obtained three equivalent formulations for the differential entropy of an ideal gas. We may now carry out an integration between two specific limits; selecting Eq. (2.4.6) we obtain dS-

CpdlnT-

R

dlnP.

(2.4.8)

i

Assuming constant Cp we find S f -- Si + C p

ln(Tf /Ti) - R l n ( P f /Pi).

(2.4.9)

Let us now choose T/ and Pi as reference values and let TU and PU assume any arbitrary value; then we may rewrite the above expression in the form S - S* 4 - C p l n T -

RlnP.

(2.4.10)

Here we seem to run into a problem of dimensional analysis since the logarithmic arguments are not pure numbers; however, this difficulty is only apparent. For, in the above expression the reference entropy is specified as S* - Si - CP In Ti + R In Pi. Hence, if the units selected for of P or T are altered in Eq. (2.4.10) there also occurs a compensating change in S*. Thus, the value of S relative to S* is unaffected by the choice of units for temperature T (though, these should always be in K) and for pressure P. However, the numerical values of S and S* obviously do depend on the energy unit chosen for Cp and for R. An alternative for interpreting Eq. (2.4.10) has often been proposed, whereby it is claimed that T and P 'really' represent the ratios T/1 and P/1 in whatever units are selected (though the temperature units must be equivalent to K). This

128

2. EQUILIBRIUM IN IDEAL SYSTEMS

seems a less desirable approach; rather, Eq. (2.4.10) shows that S* represents the molar entropy of the ideal gas when the final and initial temperatures, as well as the final and initial pressures of the experiment are identical. However, there is nothing in the theoretical development to specify what such an initial condition should be. To remedy this indeterminacy a convention has been established that specifies the standard state for the ideal gas: at whatever temperature prevails in a given experiment, the standard state of an ideal gas is set at a pressure P = 1 bar. We deal in an analogous fashion with the equivalent forms for the entropy of the ideal gas:

2.4.4

- S*' + Cv lnT + Rln IY,

(2.4.1 la)

-- S*" + Cv In P + Cp In I?.

(2.4.1 lb)

The Gibbs Free Energy of an Ideal Gas

The Gibbs free energy for a particular ideal gaseous species is found by writing

CJ(T, P) -- 121 - T S - - C p T + E O - T ( C p l n T - R l n P + S*) = G*(T, 1) + R T l n P ,

(2.4.12a)

wherein (~*(T, 1) _=/~o + Cp(T - T l n T ) - TS*(T, 1)

(2.4.12b)

represents the standard value of the molar Gibbs free energy of the ideal gas at a pressure of P -- 1 bar at the prevailing temperature T. Eq. (2.4.12a) reduces to an identity for P -- 1 bar; also, G(T, P) is identical with the chemical potential #(T, P) of the ideal gas. We generalize the above expression for component i in a mixture of ideal gases. In principle we could simply replace G by /~i and P, by Pi. However, the relation to be established is so important that we proceed by a different route: Eq. (1.13.6), applied to the present case, reads (OG/OP)T -- V - Z j n j R T / P . On partial differentiation with respect to n i we obtain

(Olzi'~ _ ~. = R T _ (Olxi'~ (OPi'~ OP J T P \OPi ] r - ~ / r '

(2.4.13a)

where Pi is the partial pressure of component i, which is related to the total pressure P by the mole fraction xi: Pi -- xi P. We thus obtain

(

Olxi ~ _ R T OP~ J T Pi '

(2.4.13b)

PROPERTIES OF IDEAL GASES

129

which may be integrated between the limits p O and Pi at constant T to yield

lzi(T, Pi) - lzi(T, pO) + RTln (Pi/P~

(2.4.14)

By convention we select for pO the standard value of 1 bar; then lzi(T, 1) _= /z/~ (T) is the standard chemical potential for component i of the ideal gas mixture. Thus, we obtain the final formulation,

lZi( T, Pi) -- ~?P(T) --[-RTlnPi.

(2.4.15)

It is possible to introduce the relations Pi = ci R T = xi P, where Ci is the concentration of species i in the volume of the gas space. This leads to the alternative formulations

t-Zi(T, Ci) --/zOC(T) + RTlnci, in which #/~

_= #/~

(2.4.16)

+ R T l n R T . We may also write

l z i ( T , xi) - - / z ? X ( T , P) + R T lnxi,

(2.4.17)

in which/z/~ P) =_ #~ + R T l n P; thus, this particular reference chemical potential now also involves the total gas pressure. Eq. (2.4.17), while formally correct, is not very useful; for, the reference value #/~ P) depends on the constitution of the entire gas mixture, and thus changes each time the composition of the gas phase is altered, even when the mole number of species i is kept constant. It is best to adopt the quantity #/~ as the reference value, which depends solely on the properties of species i in the mixture. The above relations again raise questions regarding the problem of pressure and concentration units. These matters are resolved in the same manner as was done for the entropy equations, with minor modifications: On account of (2.4.12b) lZ/~ now involves the quantity/~0, and therefore is specified only to within an arbitrary constant, as is appropriate for a quantity that represents a generalized energy. Eqs. (2.4.15)-(2.4.17) serve as prototype expressions for chemical potentials in other types of situations and will be referred to as canonical forms. 2.4.5

Adiabatic Changes in the State of an Ideal Gas

Here we begin with the relation d E = Cv d T = d W -- - P d V , which holds for reversible adiabatic expansion of an ideal gas. By definition Cv is a constant. We immediately find that nRTdV CvdT = - ~ , V

(2.4.18)

130

2. EQUILIBRIUM IN IDEAL SYSTEMS

which may be integrated between the limits T1, V1 and T2, V2 to yield C v ln(T2/T1) -- - n R ln(V2/V1) - n R ln(V1 / V2).

(2.4.19)

We now substitute C p - C v for n R and set C p / C v - y. Then Eq. (2.4.19) assumes the form ln(T2/ T1) -- (y - 1) ln(V1/ V2),

(2.4.20a)

rz/ rl -- (Vl / V2)R/d~ ;

(2.4.20b)

or in alternate form

that is to say, the product T V R/dv for an ideal gas is a constant under adiabatic changes. By substitution of V - n R T / P or of T -- P V / n R the reader should have no difficulty in proving that for an ideal gas under adiabatic conditions the following relations hold: T 2 / T1 -- ( P 2 / P 1 ) R/(JP ,

(2.4.21)

and P 1 / P 2 -- (V2/V1) Y

or

P V • --const.

(2.4.22)

These are standard relations in the literature.

EXERCISES 2.4.1. Establish that (OCp/OP)T --0. 2.4.2. Interrelate the reference entropies S*" and S* ~and S*. 2.4.3. Verify Eqs. (2.4.21) and (2.4.22).

2.5 2.5.1

Properties of Ideal Solutions in Condensed Phases Fundamental Characterization Condensed Phases in Solution

We establish here the characterization of liquid and solid solutions. It would be sensible to try to carry over Eq. (2.4.17), lzi(T, xi) - - / z ? X ( T , P) -Jr-R T lnxi, to condensed phases: in many ways the solvent acts as a volume within which the solute is dispersed, in a manner analogous to the role of the container in the dispersion of a gas. We now show that this relation does apply to the special case of ideal solutions. These are mixtures forming a homogeneous condensed phase that satisfy three criteria.

PROPERTIES OF IDEAL SOLUTIONS IN CONDENSED PHASES

131

(i) There shall be no volume change in preparing the solution from its individual components. Let Vi* represent the molar volume of pure i and Vi, its partial molal volume in solution; then the combined volume of all pure constituents prior to mixing is ~ i rti Vi*' and the volume of the solution after mixing is ~ i ni Vi. If there is to be no change in volume on mixing we must require that ~'i - Vi* for every component i. (ii) There shall be no enthalpy change in the mixing process. Let H/* and Hi represent the molar and partial molal enthalpies of i in pure and in final solution form, respectively. Then the total enthalpy of all constituents prior to mixing is ~--~ini H/*, and the total enthalpy of all constituents in solution is ~ i rtilE]i" For ideal solutions the enthalpy of mixing is thus A H -- ~-~i ni (H* - Hi) -- O, requiring that 14" -- Hi for every i. (iii) R a o u l t ' s L a w shall be obeyed: the partial pressure of species i in the gas phase in equilibrium with the solution is specified by Pi - xi Pi*, where xi is the mole fraction of i in solution, and Pi* is the equilibrium vapor pressure of the pure component. An illustration of Raoult's edict for a binary phase is furnished by Fig. 2.5.1; the asterisks refer to pure phases. We next show that the above criteria are met by postulating the following canonical f o r m for the chemical potential of species i in an ideal solution:

(2.5.1)

#i(T, P, xi) -- ~i (T, P) -Jr-R T lnxi,

p~

0

1

XB ~

XA

Fig. 2.5.1. Raoult's Law for binary solutions.

1

0

132

2. EQUILIBRIUM IN IDEAL SYSTEMS

where/~* (T, P) is the chemical potential of pure i, for which xi -- 1; this quantity depends parametrically on T and P. The above equation is self-consistent; it reduces to an identity for xi = 1. We now verify that Eq. (2.5.1) does satisfy all criteria for an ideal solution: (i) Since ( O G / O P ) T , x i - V and (OG/Orti)T,P,njr = lZi,

0 2G _ Oni OP

0V ~

r,P,njei

=- ~" - 0 POni

0 P ,] T,xj

(2.5.2)

_

r,xj It follows that the molar volumes occupied by i in the pure state and in solution are identical. (ii) From Eqs. (1.20.15b) and (2.5.1) we find that _

OT

OT

P,xj

1 P,xj _ T2 ./_

,,,,

T2

(2.5.3)

Thus, the molar enthalpy of i in solution and in the pure phase are the same, so that the enthalpy of the formation of the solution from its components is zero. (iii) From the constraints for equilibrium between liquid and vapor for each species (2.4.15) and (2.5.1) we obtain

/,t?P(T) +

R T In Pi -- It* (T, P) + R T lnxi.

(2.5.4)

Hence, at constant T and P, In Pi =

R 1T [#?P(T) - / z * ( T , P)] + lnxi.

(2.5.5)

The above is equivalent to the relation Pi = C x i , where C is a parameter that is fixed by requiting that at xi = 1, Pi = Pi*" Eq. (2.5.5) thus becomes ei - xi e ? .

(2.5.6)

In other words, Raoult's law is satisfied. Thus, the canonical form (2.5.1) does satisfy the criteria for the properties of ideal solutions; this is the basic relation on which all future derivations conceming solutions will be built. One should note that in Eq. (2.5.1) the chemical potential of i is referenced to that of pure i, independently of the other component in the solution.

PROPERTIES OF IDEAL SOLUTIONS IN CONDENSED PHASES

2.5.2

133

Thermodynamic Characterization of Ideal Solutions

We first derive several thermodynamic quantities that pertain to the mixing of constituents to form ideal solutions" Let G be the total Gibbs free energy of the solution and let G* be the molar Gibbs free energy of constituent i in pure form. Then the Gibbs free energy of mixing is given by A G -- G -- Z n i O * i

-- ~ - ~ n i [ l z * @ R T l n x i ] i

- Enilz* i

- RT Zni i

lnxi.

(2.5.7) By putting everything on a molar basis via AG =-- A G / E j n j, we obtain the more symmetric relation

AG - RT E

lnxi.

Xi

(2.5.8)

i

This relationship will later be exploited in the discussion of phase diagrams. The entropy of mixing may be found from the relation A S - - A G / T , since A H -- 0. We obtain

AS- -R Z

Xi

lnxi.

(2.5.9)

i

Since all Xi are nonnegative and less than unity the entropy of mixing is indeed positive, this rationalizes the experimental fact that the mixing of components to form an ideal solution is spontaneous. Also, in view of conditions (i) and (ii), ,,,,

,..,

.,,,

AE-

ACp

2.5.3

Use of Molarity and Molality as Concentration Units

-- A C v -

O.

From the theoretical standpoint the use of mole fractions for indicating concentrations is to be preferred. However, in the literature molarity ci and molality m i are frequently employed for the same purpose. By definition these quantities are interrelated as follows"

ci

-- =

xi

lO00ni Y~j nj

V

ni

=

1000 ~-~j nj

V

=

lO00p(T, P) ~_~jnj

y~j nj Mj

'

where V is the volume of the solution in c m 3, n j and M j are the number and the gram molecular mass of species j, and p is the density of the Note that the volume and density depends on temperature and pressure varies with T and P, even when the composition of the solution remains rather awkward feature. For pure i the above reduces to c[ -- 1000 p~(T' P) .

Mi

(2.5. lOa) of moles solution. and thus fixed--a

(2.5.10b)

134

2. EQUILIBRIUM IN IDEAL SYSTEMS

It should be noted that this is also the limit attained for a very dilute solution for which one can sensibly expect ideal behavior. This is so because of the expansion

Ci -- =

xi

1000p(1 + n2/nl + n3/nl + . . . )

M1 4- (nz/nl)M2 4- (n3/nl)M3 -4-...

,

(2.5.10c)

which reduces to (2.5.10b) when ni#l ~ nl, unless Mi:/:l ~ M1, a case the reader is asked to take up as an exercise 1. In a similar vein we write

mi xi

=

1000ni Z j nj nlM1

ni

1000

=

xlM1

,

(2.5.1 la)

where the subscript 1 refers to the solvent. The molality of a pure material is thus given by

m* -- lO00/Mi.

(2.5.11b)

This is also the expression for the molality of i in an extremely dilute mixture as applied to an ideal solution. This is the case because of the expansion

mi lOOO ( n2 n3 = nl 1 + ~ + - - + . . . xi nlml nl nl

)

,

(2.5.11c)

which reduces to (2.5.11b) when ni#l << nl. 2.5.4

Chemical Potentials in Terms of Molarities and Molalities

In principle one may proceed to rewrite the chemical potentials by substituting Eq. (2.5.10a) into (2.5.1) to obtain

lzi(T, P, ci) - lzi(T, P,c r) 4- R T l n c i ( T , P),

(2.5.12a)

in which

~i (T, P, c r) = #* (T, P) - RT In

lO00p(T, P) y-~j nj

Zj.jMj

(2.5.12b)

is then the reference chemical potential of i relative to molarity. The problem here is that ~i(T, P, c r) involves the properties of the entire solution; this quantity therefore changes as the constituents of the solution are altered, even when the molarity of species i remains fixed. We therefore set ci - c* in (2.5.12a) and replace the second term of Eq. (2.5.12b) with (2.5.10b). For the case of pure i we then obtain

lzi(T, P, c*) -- ~i (T, P) - R T l n

1000,o* (T, P)

M/

+ RTlnc*(T, P) -- lz*(T, P). (2.5.12c)

PROPERTIES OF IDEAL SOLUTIONS IN CONDENSED PHASES

135

The last two terms in the central part cancel out on account of (2.5.10b), so that /Zi (T, P, c?) --/z* (T, P). As is physically sensible, the chemical potential of pure i does not depend on how the 'concentration' of pure i is specified. This is a warrant for the self-consistency of the approach. There remains the problem of eliminating the dependence of the reference chemical potential on the properties of the solution. For this purpose we adopt a different procedure: We return to Eq. (2.5.12a), and consider the case of an ideal solution, to which (2.5.10b) also applies, so that ci/xi ~ c*. Then

lzi(T, P, ci)

- - #*(T,

P) - R T l n ~

lO00p*

Mi

+ R T In Ci (T, P),

(2.5.13a)

to which we add and subtract R TIn Ci*. Using (2.5.10b) this leads directly to the final expression

lzi(T, P,

Ci) - -

lz*(T, P) + RTln[ci(T, P)/c*(T, P)].

(2.5.13b)

This relation self-consistent, as it reduces to an identity when we set Ci - - C * . Moreover, the reference value for the chemical potential now involves only the properties of pure i, as specified by Eq. (2.5.10b). In a similar manner we could substitute Eq. (2.5.11 a) into (2.5.1) to obtain r

(2.5.14a)

lzi(T, P, mi) - lzi(T, P, mi) -+- R T l n m i , with

#i(T, P, mri) =--lz*(T, P) -- RTln(lOOO/xlM1).

(2.5.14b)

However, the reference value # i ( T , P , m ri) now depends on the properties of the entire solution through the mole fraction x l, which changes with the constitution of the solution, even when the mole number of species i remains fixed. This problem is addressed in the same manner as used in constructing Eq. (2.5.13), except that we now substitute m i for ci. We first apply (2.5.14a,b) to pure i to find lzi(T, P,m*)

= lZ i* ( T , P ) - R T l n ( I O O O / M i )

+ RTlnm

i* -- tz i* ( T , P ) .

(2.5.14c) As is physically reasonable, on account of (2.5.1 l b), the chemical potentials of pure i, expressed in terms of mole fraction and molality are identical. We then consider a very dilute solution for which (2.5.11b) also holds, and which therefore also applies to ideal solutions. We thus write

lzi(T, P, mi) -- lZi* ( T ,

P) - RTln(IOOO/Mi)

+

RTlnmi,

(2.5.15a)

and add and subtract R T In m i , s o that on account of Eq. (2.5.1 lb), Eq. (2.5.15a) assumes the self consistent form

lzi(T, P,

m i ) - - lzi*

(T, P) + R T In[m/ /m*]

(2.5.15b)

136

2. EQUILIBRIUM IN IDEAL SYSTEMS

as the basic equation for the chemical potential when molality is used as the concentration unit; m* is specified by Eq. (2.5.11b). Equations (2.5.13b) and (2.5.15b) have the added advantage that the problem of units never arises.

2.5.5

Standard Chemical Potentials

There is, however, another unifying feature that is achieved by referring the chemical potentials of solutions to the chemical potentials of the pure constituents under standard conditions, P -- 1 bar, as follows:

lzi(T P ~ X i )

--

lz*(T, 1) + R T l n x i l

(2.5 16a)

I~i(T, P, ci) -- #*(T, 1) + RTln[ci(T, P)/c*(T, 1)]

(2.5.16b)

lzi(T, P, mi) - lz*(T, 1) + R T l n [ m i / m * ] .

(2.5.16c)

Here the point of reference is the chemical potential of i in pure form, at a pressure of one bar: lzi(T, 1,x*) = lzi(T, 1, c*) = lzi(T, 1,m*) - / z ~ ( T , 1), known as the standard chemical potential for the pure material at temperature T. This is adopted regardless of the pressure under which the actual experiments are performed. The above expressions are self consistent. As usual, use of mole fractions for compositional variations offers the simplest formulation for the chemical potential. Also, in all three cases there are no problems with regard to units and dimensions. 2 Eqs. (2.5.16a)-(2.5.16c) are considered to be the canonical forms for specifying the chemical potentials of ideal solutions. QUERY 2.5.1. How is the exposition following Eq. (2.5.10c) altered if M2 >> M1 ? 2.5.2. However, a potential problem does lurk behind the use of Eqs. (2.5.16): If one wishes to satisfy criterion (i) for ideal solutions, a straightforward differentiation of Eq. (2.5.16) gives rise to a vanishing volume. State what has gone wrong and how this matter may be addressed. Hint: The chemical potential 114(T, 1, x i ) may be considered as the limit of #i(T, P, xi) when P --+ 1 bar.

2.6

The Duhem-Margules Equation and its Consequences

We study the consequences of applying the Gibbs-Duhem relation to a twocomponent system. Divide the expression nl d # l + n2 d#2 = 0 by (nl + n2) to obtain Xl d # l + (1 - Xl) d#2 = 0. Since dx2 = - d x l we find that at constant T and P

OlZ 1 dx l + X2 I~,~X2 Xl ~ T,P

( Oq/s )

T,P

dx2

/ Zl 2 ) T,P d x l - 0 , dxl + ( 1 - X l ) \ - ~( 0- X

(2.6.1)

THE DUHEM-MARGULES EQUATION AND ITS CONSEQUENCES

137

which may be rewritten as

( ) 0#1 01nxl

dxl --

E

T,P

0 ln(1 -- xl)

-- I ( 01s ) Olnxl

0t*2

]

T,P

-- ( O['L2 ) T,P

dxl

ldxl--O.

Olnx2

(2.6.2)

T,P

On setting d#i IT, P -- R T d In Pi for both components, we find (O In P 1 ) _ (O In P 2 ) , O lnxl T,P O lnx2 T,P

(2.6.3)

which is the so-called Duhem-Margules equation. From this relation it follows that if Raoult's law applies to one component of a binary solution it applies to the second component as well. For, suppose that P1 - X l Pl*, where, at constant T and P, Pl* is fixed. Then (0 In P1/O lnxa)T,p -- 1 -- (0 In P2/O lnx2)T,p, as follows from (2.6.3). Integration then yields P2 = xzP~, as was to be proved. The composition of the vapor phase, relative to that of the liquid phase in equilibrium with it, is found from

P -- Xl P? --[-x2Pf -- x1 P? -[- (1 = PI* -[- x2 ( / ~

-

x1)Pf -- Pf -]- Xl (P~ - Pf) (2.6.4)

P?),

which indicates a linear dependence of the total pressure on the mole fraction of either component. These dependences are illustrated in Fig. 2.6.1. The composition of the gas phase may be ascertained as follows: let the mole fractions in the gas phase be indicated by primes; then !

X1 X2 1

Eliminating

Xl

=

P1 P2

=

xlP~ x2P 2 ,

!

x1 =

xlP~ f

1 -x 1

=

(1 - x l ) P

i" ,'~ tZ_ ~ ' x

,

.

,~.o._,,

2

between (2.6.4) and (2.6.5) leads to P =

P~P~

,',* -

.

(2.6.6)

X'l (t'1" -

One notes that whereas P is linear in X l this is not the case for x '1. The vapor phase is richer in the more volatile component: for, if PI* > P2*, then, by Eq 9 (2.6.5), x'1 / ( 1 - x '1) > x 1/ ( 1 - x 1), or x 1 > X1. These findings are illustrated in Fig. 2.6.1.

138

2. EQUILIBRIUM IN IDEAL SYSTEMS

p~

P2

X'l

1

Xl

0

Fig. 2.6.1. Equilibrium vapor pressure of a liquid solution as a function of the mole fraction

of component 1 in the liquid phase (Xl) and in the vapor phase (x~).

2.7

Temperature Dependence of Composition of Solutions

We examine the effects of varying the temperature on the composition of a binary solution in equilibrium with the vapor phase. The situation is rather complicated because the composition of the two phases, as well as the total pressure, are altered with temperature changes. This comes about because an increase in T favors the evaporation of the more volatile component, thereby enriching the gas phase and depleting the liquid phase of this component. Thus, both xi and x~ are changed, even though the overall composition of the closed system remains the same. A full analysis becomes rather complicated; we therefore impose the additional restriction that the total pressure remain fixed. This may be done in principle by metering in or withdrawing, as required, (via a semipermeable membrane) a third, inactive gas that does not dissolve in the liquid. Alternatively, a movable piston may be employed. For this somewhat contrived situation we invoke the equilibrium constraint for each species:/Z i (g) = / / ~ i (l). Then, according to Eqs. (2.4.15) and (2.5.1), #~

+ R T In Pi -- r

P) + R T lnxi.

(2.7.1)

Differentiating both sides with respect to T yield

+ .--re- j , -

+(0

lnx/)

(2.7.2)

LOWERING OF THE FREEZING POINT AND ELEVATION OF THE BOILING POINT

139

The partial pressure of i is specified by Pi - x~ P, where the presence of any inert gas must be taken into account in specifying x i' . Insertion into the above leads to l n Xi_ Xi

_

p

/z i

.

RT

_-- -

p

H i

-

ft?

RT 2

_-- -

H i

- H

RT 2

i

'

(2.7.3) where we first utilized Eq. (2.5.3) and then introduced criterion (ii) for the constituents of an ideal mixture: H/* and/_)o for the isolated components are equal respectively to the partial molal enthalpies of the same constituents in solution (/-)i) and in the vapor phase (H/'). The relative changes in composition are linked to changes in the enthalpy, which accords with the qualitative description, furnished earlier, on the effects of changing the temperature. The above derivation is not restricted to just miscible solutions. It holds equally well for any other two-phase mixture, including liquid solutions in equilibrium with solid solutions, or, more importantly, to a liquid solution in equilibrium with a pure solid component (such as ice immersed in a salt solution). In that case x i' - 1 refers to the pure component and we obtain O lnxi ) _ AHf OT p RT 2 '

(2.7.4)

where A H u is the enthalpy of diffusion of the pure solid. This matter will be taken up in much greater detail in the next section.

2.8

Lowering of the Freezing Point and Elevation of the Boiling Point of a Solution

One of the interesting problems in solution thermodynamics is the variation in freezing and boiling points of a solvent to which different quantities a solute have been added. Extensive experimentation has shown that the addition of a material B that dissolves in liquid A causes a lowering of the freezing point of the solution relative to the freezing point of pure A. For example, A and B may represent water and sugar, respectively; at the freezing point the sugar solution is in equilibrium with pure ice. More generally, we look for an analytic relation that shows how much the freezing point of a pure solvent is depressed when a given quantity of solute is dissolved in the solvent.

2.8.1

Depression of the Freezing Point

We consider a two-phase system consisting of pure solid A (e.g., ice) in equilibrium with the solution of B dissolved in liquid A (e.g., sugar in water). This requires the equality of the chemical potential/ZA in both phases: #*A( T, P ) -- #1 (T, P ) + R T lnx~,

(2.8.1)

140

2. EQUILIBRIUM IN IDEAL SYSTEMS

where /ZA and #1 are the chemical potentials of pure A in the solid and in the liquid state, respectively. Note the implication of the equilibrium condition (2.8.1). At constant P, a given temperature corresponds to particular composition, x l, of the solution, and vice versa. Thus, Xl becomes a function of T; Xl = x l(T). To investigate this dependence further it is apropos to carry out a partial differentiation of Eq. (2.8.1) with respect to T:

/

-

OT

_-('In'l)

p

RT 2

RT 2

Y

p

In this expression we have first invoked Eq. (1.20.15b) or, alternatively, (2.7.3) in different notation: H~ a n d / t ~ are the molar enthalpies of pure liquid A and of pure solid A respectively. Next, we have defined L f as the molar heat of fusion of pure A; it is not the molar enthalpy change accompanying the transfer of pure A into the solution A + B. Lastly, we invoked Eq. (2.8.1) to obtain the quantity on the right. Equation (2.8.2) may be rewritten as

-

dlnxl - -

RT 2

dr

(P constant)

(2.8.3)

where the limits on the integrals denote two different situations: T f is the freezing point of pure A (xl = 1), and 7'1 is the freezing point of the solution for which the mole fraction of A is Xl. In Exercise 2.8.1, we ask why, on physical grounds, a change in T should result in a change in composition of the solution. We must take account of the temperature dependence of L f , in first order, via the Kirchhoff relation, - ( d ~ ) t - (d))A - A d e , where A d e represents the difference in molar heat capacity of pure liquid A and pure solid A. If the temperature dependence of this difference is negligible we obtain

(aLf/ar)e

[,f_

[,Of 4- A C p ( T -

(2.8.4)

T f ),

w h e r e / , ~ is the value o f / , f at the freezing point, T f . Then at fixed P

-

f..

dlnxl - -

[ (.e.) ] RT 2

+

RT 2

(T -- T f )

dT.

(2.8.5)

Next, define the lowering of the freezing point by O f = TU - 7"1,

(2.8.6a)

so that

TI - Tf[1 - O f

(2.8.6b)

LOWERING OF THE FREEZING POINT AND ELEVATION OF THE BOILING POINT

141

Equation (2.8.5) may then be integrated to yield (constant P) - l n x l --

Lf RTf

- I firl

A~;p] Tf - 7"1 - AC.p ln(T1/Tf ) R T1 R

aG

][ 1 --o,/7 0 fi/ T f ]

(2.8.7)

R

In general, it is an excellent approximation to expand in small powers of Ou/Tf" We obtain (1 - Of/Tf) -1 ~ 1 + Of/Tf +... and ln(1 - Of/Tf) -[Of~ Tf + O}/2T~ + . . . ] . Further, - l n x l -- - l n ( 1 - x2) ~ x2. Here x2 is the mole fraction of the solute. Equation (2.8.7) then reduces to

R T f Tf

F RTf

2R

(O f / Tf

,2

(2.8.8)

The replacement of - lnxl by X2 does limit the applicability of Eq. (2.8.8) to dilute solutions, but this limitation is clearly essential if the theory of ideal solutions is to be applicable to real cases. Eq. (2.8.8) is a quadratic equation in (~f which may be solved to find (~)f in its dependence on x2. However, in view of all of the above approximations we are generally justified in neglecting the quadratic term in (Of~ TU)2. Equation (2.8.8) may then be inverted to read

R# Of-

7_,0 X2. f

(2.8.9)

Generally, the lowering of the freezing point is expressed in terms of the molality m2 of the solute. Here m2 = lO00n2/nlM1, where M1 is the gram molecular weight of the solvent and n2 is the number of moles of that component: further, x2 -- n2/(nl +n2) ~ n2/nl. Hence, x2/m2 ~ M1/1000 for dilute solutions. Thus, after applying all the above approximations one finds

RT~M1 O f "~ ~ m2 =-- K fm2. IO00L~

(2.8 10)

This relation is remarkable in several respects. First, in the present approximation the lowering of the freezing point is a linear function of x2 or m2. Second, the proportionality factor depends solely on the properties of the pure solvent (i.e., o n M1, Tf, and L~) and not on the properties of the solute. For a fixed solvent K U is thus predetermined and is maximized for solvents with high freezing points, high molecular weights, and low heats of fusion. All of this accords with most experiments; in fact, in the early days of chemical research the determination of the freezing point depression was used to find the molecular weight of

142

2. EQUILIBRIUM IN IDEAL SYSTEMS

many materials. One should recognize, however, that numerical estimates based on Eq. (2.8.10) fail in cases where solutes associate or dissociate. The analysis of this situation is left as an exercise. Also, allowance must be made for cases where the solvent ionizes when going into solution. This problem is addressed in Chapter 4. 2.8.2

Elevation of the Boiling Point

The foregoing procedure requires small changes to become applicable to the elevation of the boiling point Tb for an ideal solution containing nonvolatile solutes and a volatile solvent. The equilibrium constraint for the solvent now reads # *1( T , P) + RTlnxl(1) -- # ~

~

P) + RTlnxl(g)

(2.8.11)

where 1 and g refer to the liquid and gaseous phases. Accordingly [note Eq. (2.7.3)],

[o( ox ,)]

- [

- #1

- -~

RT

__ H 0 - H i ~ _

a ln[xl (g)/xl (1)1

RT 2

-OT

p

]

.

(2.8.12)

p

Consider the case where the solvent vapor alone constitutes the gas phase (e.g. a sugar solution in equilibrium with steam). In that event Xl (g) = 1, and we can then set Xl (1) = Xl. Equation (2.8.12) now reduces to i3 In Xl )

aT

/_~o _ Hi*

Lv

RT 2

RT 2'

p

(2.8.13)

where Lv is the molar heat of vaporization of component A from the pure liquid to the gas phase. If P is set at one bar one deals with the normal boiling point. We now (i) rewrite (2.8.13) as

Lv

-dlnxllp

d T2 = ~RT

(2.8.14) P

(ii) introduce Kirchhoff's law in the approximation -

60

+

-

(2.8.15)

(iii) introduce the elevation of the boiling point by the definition ~)b ~ T1 - Tb so that T1 -- Tb(1 + Ob/Tb), and (iv) expand (1 + Ob/Tb) -1, as well as ln(1 + {~)b/Tb) in powers of Oh~ Tb. This yields

LO Ob - lnxl ~ x2 -- RTb Tb

[ L0 RTb

ACp ] 2 2R ( O b / T b ) 9

(2.8.16)

CHEMICAL EQUILIBRIUM: GENERAL PRINCIPLES AND APPLICATION TO IDEAL GASES

143

On neglect of the last term on the right, the resulting equation may be inverted to read

R# O b --

L0

X2

(2.8.17)

and RTb2M1

Ob-

1000/~ 0 m 2 = K b m 2 .

(2.8.18)

The various remarks made in conjunction with Eq. (2.8.10) also apply here with appropriate modifications. EXERCISE

2.8.1. Provide a physical explanation why the freezing point of a solvent is lowered when solute is added to form a solution.

2.9

2.9.1

Chemical Equilibrium: General Principles and Application to Ideal Gases Representation of Chemical Equations

One of the most important applications of thermodynamics to chemistry occurs in the treatment of chemical equilibrium. We develop some general principles below. Consider the prototype of a chemical reaction, VlrR1 + vzrR2 + " " :rip P1 + 1)2p P2 + ' " , in which, for the reaction as written, R1, R2 . . . . are different chemical reagents, P1, P2 . . . . are different chemical products, and the Vir, rip are stoichiometry coefficients for reagents and products, respectively. It is convenient to abbreviate the above schematic reaction as 1)i Ai -- 0,

(2.9.1 a)

I

in which, by convention, the 1)i are negative or positive according as the chemical species Ai designate reagents or products, f o r the reaction as written. We first take up the case where we have arrested the reaction at a particular stage for which the participating species involve a concomitant set of mole numbers n i, such that the Gibbs free energy in the arrested stage at a specified T and P is given by G = ~--~inil~i. One method of achieving this state is to carry out the reaction in a reversible electrochemical cell, in the manner described in Chapter 4. A special case of particular interest, which requires no outside intervention, is the equilibrium state, with G - Z i nilLileq , where the mole numbers ni and chemical potentials/Li are those of the various species in mutual equilibrium.

144

2. EQUILIBRIUM IN IDEAL SYSTEMS

To initiate our analysis we now undertake a virtual displacement for the above reaction (see Section 1.12) from its steady state or from its equilibrium configuration at a fixed pressure and temperature. The Gibbs free energy change is then given by

6G- Z i

I~i6ni -+- Z

rti61~i - Z i

(T, P constant),

I~i6ni

(2.9.1b)

i

where the term Y ~ i ni61~i vanishes at constant T and P by virtue of the GibbsDuhem relation. An important constraint in carrying out virtual displacements is that the individual mole numbers ni must change in proper synchronization to accord with the stoichiometry requirements of the chemical reaction. This can be guaranteed by introducing an infinitesimal, virtual (which also implies "reversible") unit advancement, 6)~, of the chemical reaction )--~i 1)iAi = 0, such that the change in mole number of every species i is given by 6ni = vi6~.. Then Eq. (2.9.1b) reads

SG--ZlziSni--(ZVilZi)SX~(AGd)6X i

(T, P constant)

(2.9.2a)

7",p .

(2.9.2b)

i

in which AGd --= -~- 7",P - -

i

Vi~ i

One must be very careful about the interpretation of the quantity AG~: as indicated by the defining relation; it is a differential Gibbs free energy change per unit advancement in the reaction 1. This accompanies an infinitesimal reversible change induced by the reaction at constant T and P, which very nearly preserves the number of moles n i of every species i participating in the reaction 2. The quantity A Gd may be calculated, as shown in (2.9.2b), from the chemical potentials /Zi prevailing in the mixture during the infinitesimal, reversible advancement of the reaction. As already indicated, Eq. (2.9.2) applies whether the system is held by external constraints in a steady state condition in which the mole number n i remains invariant at arbitrarily prescribed values, or whether equilibrium prevails, with the n i assuming appropriate equilibrium values. However, according to Section 1.12, in the latter instance the constraint (6G/6)~)~,p = 0 must then also be taken into account; we thereby obtain an important condition characterizing chemical equilibrium, namely,

(z i i) _o i

eq

Note that if Y ~ i vi ~i < 0 ( > 0) the reaction as written occurs spontaneously (in the opposite direction). We now first take up the case of chemical equilibrium.

CHEMICAL EQUILIBRIUM: GENERAL PRINCIPLES AND APPLICATION TO IDEAL GASES

2.9.2

145

Chemical Equilibrium in Gas Phases

For an ideal homogeneous gaseous system, /Zi- /z?P(T)q-RTlnPi Eq. (2.4.15)], so that the equilibrium condition (2.9.3) reads

0

z

[see

+ i

t

eq

Note that at equilibrium both T and P are fixed. It is useful to rewrite Eq. (2.9.4) as follows: - y~,,___.

= In K p ( T ) -

vi In Pi

i"

9

(2.9.5a)

9 eq

Here we have separated terms such that the left-hand side is a parametric function of T alone, while the fight-hand side depends on the characteristics and composition of the gas mixture. It is therefore appropriate to introduce, as in (2.9.5a), a new quantity Kp which shows explicitly that the left-hand side is actually independent of composition variables. K p is termed an equilibrium c o n s t a n t ~ a highly undesirable appellation because this quantity obviously varies with T and thus ought to be designated an "equilibrium parameter". However, the term "equilibrium constant" is so firmly entrenched that we shall continue to use it here. A convenient reformulation of (2.9.5a) is found by taking antilogarithms: Kp(T)-

I-I

(PiVi)eq"

(2.9.5b)

i

The equilibrium constant actually involves a ratio of partial pressures, since the stoichiometry coefficients of the reagents are negative. Eq. (2.9.5b) is in the standard form used to represent equilibrium constants in terms of composition variables. It is possible to provide alternative formulations for Eq. (2.9.3). For an ideal gas one can set [see Eqs. (2.4.16), (2.4.17)] ~ i - - / z / ~ @ R T l n c i or /Z i - /z/~ P) + R T l n x i , to obtain In Kc ( T ) = - Z

ln K x ( T , P) =--

vi In ci

-i

--

(2.9.6)

9

~

RT i

'

eq

--

1)i

.

lnxi

.

(2.9.7)

eq

We have thereby introduced two new equilibrium 'constants', again independent of concentrations, the one in Eq. (2.9.7) also depending parametrically on the total pressure of the equilibrated system. The problem of dimensionality arises once again: This matter is briefly discussed later and fully treated in Section 3.4.

146

2.9.3

2. EQUILIBRIUM IN IDEAL SYSTEMS

Free Energy Changes of Reactions in the Gas Phase

The evaluation of A Gd is of considerable interest. In the general case, when equilibrium does not prevail, the mole numbers n i are arbitrary. If at any fixed stage the reaction E i 1)iA i - - 0 is advanced infinitesimally and reversibly at constant T and P, then

AGd

Vil~OP(T) -+-RT ~ vi lnPi

-- ~

i

(2.9.8a)

i

-- - R T ( ~ i

vi lnPi)e q + RT y~.

1)i In Pi

(2.9.8b)

i =

- R T In K p -+-RT Z

Vi In

Pi,

(2.9.8c)

i

where we have inserted Eq. (2.9.5a) on the right-hand side. Here and later on it is of the utmost importance to distinguish between E i lai In Pi, which merely has the same form as In K e, from (~i vi In Pi)eq, which is identical with In K p. The partial pressures Pi are those appropriate to the arrested stage. Confusion at this point can be disastrous. It is evident that numerical values of Kp and Kc depend on the units chosen for gas pressures and concentrations, but the corresponding AGd/RT is independent of such choices, as is clear from Eq. (2.9.8b). It is convenient at this stage to introduce the standard free energy change A G o for the reaction. This is the value of A Gd when the reaction is theoretically advanced infinitesimally and reversibly under conditions where all gaseous constituents are at unit partial pressure (usually one bar) at the temperature of interest. It may not be possible actually to execute this reaction under such conditions, but this fact does not detract from the assertion taken as a definition. The definition is also consistent with the fact that we had earlier set A Gd = Z i 1)i/s hence AGO =--Y-~-ivi #~ e. Equation (2.9.5a)thus reduces to AG ~

or

t~

K p - - exp

(Aoo) RT

(2.9.9)

More generally, Eq. (2.9.8c) reads AGd

1)i In Pi.

- A G OP + R T Z

(2.9.10)

i

Note that the equilibrium constant K p is directly related by Eq. (2.9.9) to the differential free energy change of a chemical reaction under standard conditions. Further, the A Ga value for the reaction under arbitrary, reversible, isothermal, and isobaric conditions involves the sum of AG o and of a 'correction term'

RT y~.i Vi ln Pi.

CHEMICAL EQUILIBRIUM: GENERAL PRINCIPLES AND APPLICATION TO IDEAL GASES

147

If equilibrium prevails, A Gd --0; in this event Eq. (2.9.10) reduces to

AG~ -- - R T ( Z v i l n P i

)

i

(2.9.1 la)

eq

which is consistent with both (2.9.9) and (2.9.5). The remarks made earlier concerning A Gd apply to the quantity AG o in Eqs. (2.9.9)-(2.9.11), with the additional requirement that all species must be maintained at their standard states during the infinitesimal advancement of the reaction. The definition

AG~

P,

(2.9.11b)

i leaves no ambiguity: one measures, or looks up in appropriate tables, the molar Gibbs free energy/z/~ (specified for the desired standard state, usually P - 1 bar) of all gaseous species i involved in the reaction. These quantities are then to be combined 3 as required by Eq. (2.9.1 l b). There should be no difficulties in constructing the analogues of Eq. (2.9.8c), namely

AGd - - R T l n K c + RT Z l)i lnci -- A G '0 + RT ~ 1)i lnci, i

(2.9.12)

i

AGd -- - R T In Kx + RT ~ Vi lnxi -- AG ''0 + RT ~ Vi lnxi. i

(2.9.13)

i

In these equations AG ~~ and AG "~ are again specified by Eq. (2.9.1 lb), but now #/oP must be replaced by either #/oc or #/ox. Once more one must be at great pains to avoid confusing Z i Vi In ci or Z i t)i ln xi with (~-~i vi In Ci)eq - In Kc or (Y-~i vi In Xi)eq In Kx, because of the close similarity of the expressions. -

2.9.4

Temperature Variation of the Equilibrium Parameter

We now consider the variation of the equilibrium 'constant' with temperature and with total pressure. We base our further discussion on Eq. (1.20.15b) and we consider the definitions (2.9.5)-(2.9.7). Obviously, (OlnKp/OP)T = (0 In Kc/O P)T = 0. Thus,

dlnKp

Vi d (

/~OP\

Zil)i[~?

AM 0

(2.9.14)

i Equation (2.9.14) is one formulation of van't Hoff's equation (1886). The cautionary discussion concerning AGd and AG o also applies to A H ~ this represents the differential enthalpy evolution accompanying unit advancement of the

148

2. EQUILIBRIUM IN IDEAL SYSTEMS

Z i viAi = 0 when the latter is changed hypothetically, reversibly, and infinitesimally while maintaining constant T and P, and all species in their standard states. No ambiguity results when one uses the definition A H ~ - Z i 1)i/_~o. Since ~ o arose through the partial differentiation in Eq. (2.9.14) it represents the partial molal enthalpy of species i at temperature T under standard conditions. This quantity may be determined as prescribed in Section 1.17 or is available from tabulations 3. One may reformulate the results: using Pi --- ci R T and Pi = xi P, one obtains reaction

In K p

--

In Kc + ~

Vi In R T -- In Kx + ~ i

Vi In P.

(2.9.15)

i

Thus, d ln Kc d ln Kp dT = dT - Z

i

d ln R T An 0 Vi d---~ = R T 2

Z i Vi A n 0 -- R T A v T = RT 2 (2.9.16)

With AVIr, p - A v R T / P , this becomes d In Kc A E~ = dT R T 2"

(2.9.17)

Finally, OlnKx) OT p

_

d l n K p

dT

_ ~

AH~ R T 2'

(2.9 18) ,

and in view of Eq. (1.20.13),

tOln xt_ oe

r

_

Rr

5?

Rr

Rr

In practice Eq. (2.9.14) and its analogues are frequently used in reverse. For ideal gas mixtures d ( # ~ and d ( # i / T ) / d T are identical; hence, A H ~ = A Od Z i Vi t2Ii, SO that we may write -

-

AHd - R T 2 d In Kp dT '

(2.9.20)

which shows that if the variation of K p with T is known empirically or from a theoretical analysis, AHd may be found for the gaseous r e a c t i o n }-~i v i A i - O. Note further that by integration of (2.9.20), In Kp(T1)Kp(T2)_-- fr~ 2 AHdRT 2 dT,

(2.9.21)

CHEMICAL EQUILIBRIUM: GENERAL PRINCIPLES AND APPLICATION TO IDEAL GASES

149

which requires that one specify the dependence of A Hd on T. Here Kirchhoff's law may be used if no other detailed information is available. REMARKS 2.9.1. The restrictions should be carefully noted. We assume that the reaction has either run its course (at equilibrium) or has been arrested, so that the existing reagents and products are in balance. We then investigate what would happen if this state were slightly perturbed by adding or taking away a tiny set of participating species, keeping all restrictions in place. This operation is to be contrasted with a later study, in which we allow a chemical reaction to proceed whereby all starting materials are entirely converted into products. The thermodynamic characterization of such a process differs sharply from the analysis offered here. 2.9.2. For example, the reaction 2H2 + O2 -- 2H20 may be carried out reversibly by operating a fuel cell containing nH2 moles of H2 gas and no2 moles of O2 gas in appropriate compartments over the electrodes, and containing nH20 moles of water as the medium into which the electrodes are dipped (see Section 4.6). As long as the cell is operated reversibly by maintaining an appropriate counter emf, so that at the conclusion nil2 - 2~)~ moles of H2 gas and no2 - ~)~ moles of 02 gas remain in the compartments, and nH20 -~- 2~)~ moles of water are present, a measurement of the emf yields AG~(T, P) directly (see Chapter 4 for details; these are not essential to the present argument). However, if the reaction is allowed to proceed to the extent that vi6)~ becomes comparable to the above n i's, the mole numbers are no longer constant; the resulting Gibbs free energy change is then no longer identical with the quantity A Gd in Eq. (2.9.2). An extreme example is the detonation of two moles of H2 and one mole of O2 in a bomb calorimeter: here the mole numbers are altered to the maximum possible degree, T and P are no longer maintained constant, and the reaction obviously is not carried through reversibly. Any measurement of the total free energy change A G for such a process will not even be remotely related to A Gd as defined earlier. These examples should alert the reader to problems that may arise when different authors refer to a free energy change in a reaction; we shall consistently use the subscript d as a reminder of the differential nature of the quantity. Equation (2.9.2b) shows clearly that A Gd involves the chemical potential #i for every species participating in the reaction under the prevailing steady state or equilibrium conditions which remain essentially unaltered in the virtual displacements. Alternatively, one may view A Gd as the change in Gibbs free energy when the reaction is carried out such that 3)~ equals one mole in an essentially infinite copy of the system under study. Obviously, any method by which #i can be determined is satisfactory for use in Eq. (2.9.2b); one need not restrict oneself to measurements carried out during an actual reaction. 2.9.3. One may raise the problem of the reference state that should be used in specifying chemical potentials and enthalpies of individual species. This presents no problem here because we deal with linear summations over all species participating in the reaction; hence, the arbitrarily chosen reference energy, common to all species, cancels out.

150

2.10

2. EQUILIBRIUM IN IDEAL SYSTEMS

Chemical Equilibrium in Homogeneous Condensed Ideal Solutions

The methods of the previous section will now be extended to deal with equilibrium in homogeneous liquid or solid solutions that form ideal phases. 2.10.1

Use of Mole Fractions

We begin by introducing Eq. (2.5.1) in the formulation lzi(T, P ) -- l z * ( T , P ) + R T ln xi. The equilibrium condition in the present case is given by

-- ~-~Vilz*(T, P) W R T ( E v i lnxi ) .

O-- (E1)ill'i) i

eq

i

i

(2.10.1)

eq

This allows us to introduce the corresponding equilibrium 'constant' as l n K x -- - E

i

vil~*(T, P ) RT

-- (~i vi lnXi)eq"

(2.10.2a)

Alternatively, we may write

Kx(T' P ) - H

(x;i)eq '

(2.10.2b)

i

whereby we have separated compositional terms on the fight from quantities that depend parametrically on T and p.1 This equation has the same form as Eqs. (2.9.5a,b), except that now K x also depends parametrically on P. By analogy to Eqs. (2.9.18), (2.9.19) we then find that

(O,n or

p

ON

T

i

RT 2

i

RT

_

RT 2 ,

(2.10.3)

RT

and by analogy with Eq. (2.9.8), AGd

-

-RT

E 1)i l n x i i E 1)il n x i i

In K x + R T

-- AG*d x + R T

(2.10.5a) (2.10.5b)

where AG*ax - Y~i Vil~ i* (T , P). The variables on the fight are those prevailing under nonequilibrium conditions. For reasons mentioned previously this is the preferred manner of dealing with condensed phase equilibria.

CHEMICAL EQUILIBRIUM IN HOMOGENEOUS CONDENSED IDEAL SOLUTIONS

2.10.2

151

Use of Molarity and Molality

As was extensively discussed in the preceding section, alternative formulations are found by introducing molarity, ci, or molality, m i , as the concentration variable. According to Eq. (2.5.10a) we relate the concentration to the mole fraction to write l n K c - - ( ~'~ l)i In Ci) i

In [lO00p(T,P)}-~jnj] , (2.10.6a) Z j nj M j

Vi

--lnKx+~ i

eq

which may be reformulated conventionally as Kc(T'

P)- I--[ (cVi)eq"

(2.10.6b)

i

Similarly, for molalities as concentration units we find

lnKm--(Zvilnmi )

- In Kx

i

eq

+Zi

l)i In

1000 ]

Mix1

(2.10.7a)

along with

Km(T, P)- U (mivi )eq"

(2.10.7b)

i

For solutions that are ideal only in the limit of considerable dilution the logarithmic arguments on the right-hand side of Eqs. (2.10.6a), (2.10.7a) may be replaced by 1000pl/M1 and by 1000/M1 respectively, as was shown in Section 2.5. Furthermore, A G d --

-RT

In Kc + RT ~ Vi In Ci -i

AG*dc + RT Z

vi In Ci,

(2.10.8)

i

AGd---RTlnKm + RTZvilnmi -- AG*dm + RT~vilnmi. i

(2.10.9)

i

Finally, it should be noted that

(OlnKc) _(OlnKx) ff-(~i vi)(O~n?) OT p OT p P (OlnKm) _(OlnKx) OT p OT p 0 l n K c ) _(OlnKx) OP r OP r 3 P )T'

-k-(~ivi)(Olnp

OIn Km

OP

)

r

(Oln ) OP

T

(2.10.10a)

(2.10.10b) (2.10.10c)

(2.10.10d)

152

2. EQUILIBRIUM IN IDEAL SYSTEMS

where Eqs. (2.10.3), (2.10.4) are then to be used for insertion on the right-hand side. 2.10.3

Standard State and Equilibrium Constants

Equilibrium constants cannot be set up in a unique manner. This is evident by studying the fundamental definition that relates these quantities to the reference chemical potentials/z* (T, P). We could equally well, or perhaps preferentially, relate the equilibrium constants to the standard chemical potentials, based on the relation lzi(T, P) -- lzi (T, 1) + RTlnxi. It is easy to verify that this leads to the definition ln/Cx -- - Z

vilz*(T, 1) RT -- (~i vi lnXi)eq ' i

(2.10.11a)

and to the equivalent formulation

1Cx(T, 1 ) -

H

(xVi)eq"

(2.10.1 lb)

i

The right-hand side is formally identical with Eq. (2.10.2b). However, there is a difference in the numerical values of xi in the two cases, because the reference and standard values of the chemical potential of species i are slightly different (see also Query 1.10.1). Moreover, the left-hand side depends only on temperature, with the pressure set equal to 1 bar. One also recovers Eqs. (2.10.3), (2.10.4), except that now P -- 1 bar. These changes demonstrate again that equilibrium constants cannot be uniquely specified. (0 lnl~x/OT)p and (0 lnl~x/OP)T are found as in Eqs. (2.10.3), (2.10.4), on setting P -- 1. Eq. (2.10.5) also holds with the same modification. Lastly, the entire machinery involving the quantities K~c and K~mmay be developed directly from Eqs. (2.10.6)-(2.10.10) after everywhere setting P -- 1 bar. It is customary to work with the equilibrium constants ~x, K~c, and ~m, so that one always refers to free energy changes for the reaction E i 1)iA i - - 0 under the hypothetical constraint of standard conditions. The same applies to the quantities H/* and Vi* for the chemical reaction under the same conditions. Values of the equilibrium constants can be deduced from tabulations of the Gibbs free energies for all the species engaged in the chemical reactions, which are generally provided in terms of standard conditions. As shown above, the equilibrium constants are not fundamental quantities; only the Gibbs free energy changes for a given reaction are unique. This, however, does not detract from the overall utility of equilibrium constants. Consider the relationships /t~q - -

[ AGdq(T, 1)] RT '

exp -

(2.10.12)

CHEMICAL EQUILIBRIUM IN IDEAL HETEROGENEOUS SYSTEMS

153

with q - x, c, m. If/~q is very large and positive t h e n A Gd q must be a very large negative quantity; that is, the reaction as written will proceed spontaneously, such that at in the final equilibrium state one ends up with a preponderance of products over reagents. Conversely, for very small values of the equilibrium constants A Gd q will be a large positive quantity, so that the reaction proceeds spontaneously in the direction opposite that which is written down. Then at quiescence reagents will predominate over products. Equilibrium constants in the neighborhood of unity indicate an equilibration of reagents and products at comparable concentrations. Obviously, Eqs. (2.10.2b), (2.10.6b), (2.10.7b) represent a quantitative formulation of these statements. COMMENTS

2.10.1. To avoid excessive repetition we remind the reader to set P -- 1 (bar) whenever standard conditions are called for. See also the end of the present section, especially Eqs. (2.10.11). 2.10.2. Relate the equilibrium mole fractions cited in Eq. (2.10.2b) to those cited in Eq. (2.10.1 lb). Explain the origin of the difference and the implications of your findings.

2.11

Chemical Equilibrium in Ideal Heterogeneous Systems

We briefly extend the preceding discussion to systems in which one or more pure condensed phases coexist with an ideal homogeneous mixture in gaseous, liquid, or solid form. It is now expedient to distinguish between pure condensed phases, subscript s, and species involved in the solution, subscript i. For the reaction, written as }--~ v~A~ + E i l)i A i - - 0 we write out the equilibrium condition as

AGd--El)s~*nt- (K1)ilzi) s

i

-o,

(2.11.1)

eq

which forms the basis of our subsequent discussion.

2.11.1

Equilibrium with Ideal Gases

Here we apply Eq. (2.4.15) to (2.11.1) to find

Evilz?P(T) + EvslZ*(T, P)-+-RT( EvilnPi) i

s

i

=0.

(2.11.2)

eq

We again separate pressure and temperature variables pertaining to the gaseous mixture from the remainder. This generates an equilibrium constant via In Kp =

'Iz 1)oP i

RT

.

!

(T) +

z S

9

Vs#s ( T P ) '

]

'

(2.11 3) "

154

2. EQUILIBRIUM IN IDEAL SYSTEMS

so that

lnKp -- (~--~vilnPi) i

Kp(T, P) -- H (PiVi)eq,

or eq

(2.11.4a,b)

i

from which the quantities referring to the pure phases are lacking. In some sources one finds statements to the effect that the pure condensed phases are considered as being at "unit activity", whence there is no reference to the s species in Eqs. (4a,b). In fact, this situation arises because of the separation of variables into terms that do and do not involve the composition of the solution. One may also set up expressions using the relations Pi = ci RT = xi P to obtain

lnKc--(Zvilnci

)

i

lnKx--(Zvilnxi)

and eq

i

(2.11.4c,d) eq

The various equilibrium constants are interrelated by

lnKp(T,P)--lnKc(T,P)+

)

vi R T - - l n K x ( T , P ) + 9

vi l n P ,

(zi )

(2.11.5)

whence

OlnKp) _ ( O l n K x ) OT p OT

p

~-,i ViI-I~( T, P) + Y-~s vsHs*(T, P) _ AH,~

RT 2

(2.11.6)

-- RT 2 ,

( 8 1 n K p ) - - ( 8 1 n K c ) - -~QsvsV*(T'P) _-OP T OP T RT RT

, (2.117).

whereas

( OlnKp ) p _ ~ _if _ vi OT

OlnKc) _ OT p

RT 2

,

2118

and

K-"

8 lnKx)

8P

T-

i --fi + \ , l O T

- --/"

P

= -

R-------T- -

RT

=

RT" (2.11.9)

CHEMICAL EQUILIBRIUM IN IDEAL HETEROGENEOUS SYSTEMS

155

Lastly, we obtain

vilz~

AGd - Z

+ ~ VslZ*s(T.P)+ RT ~

i

s

= A G OP + R T Z

1) i

lnPi

i

(2.11.10a)

vi In Pi i

= - R T In Kp + R T Z

vi In Pi

(2.11.10b)

vi In ci

(2.11.10c)

i

= - R T In Kc + R T Z i = -RTlnKx

+ RTEvi

(2.11.10d)

lnxi.

i

2.11.2

Ideal Solutions in Equilibrium with Pure Condensed Phases

We use the same strategy here as in the preceding section. Relative to the reference state we characterize the equilibrium condition by

AGd -- E vi.*(T. P)+ E Vs/Z*s(T. P)+ RT Z ( v i lnxi)eq i

s

= A G d~X -at- RT

i

~-~(1)i lnxi)eq

(2.11.11)

--O.

i

Once more, s refers to liquids or solids in pure form, i, to constituents in the solution. We separate out terms that do not depend on the solution composition from those that do; this leads to the equilibrium constant

ln Kx =

RT

Vs#*(T, P ) + Z i vilz*(T, P)

--

RT '

so that

lnKx

--

RT ~--~(v i lnxi)eq

or

i

Kx -- 17

(x~i)eq"

(2.11.13)

i

The remainder of the analysis proceeds as before. In particular, we obtain

p

OT

RT 2

RT 2 '

(2.11.14)

and

OlnKx) OP

=

r

~i viVi*(T, P)

-t- ~-~s vsV*(T, P) RT

AV~ RT

(2.11.15)

156

2. EQUILIBRIUM IN IDEAL SYSTEMS

Also, it is customary to refer all thermodynamic properties to chemical potentials o f all species, whether in the pure state or in solution, to their values under standard conditions. In that case the equilibrium constant will be designated, as before, by 1Cx and the pressure in the above equations is set at P = 1 bar. Finally, it is possible to specify compositions in terms of molarity ci or molality m i, leading to the specification of Kc and Km or ~c and/Cm. The resulting analysis becomes

somewhat involved and will not be taken up here; interested readers should read Section 3.7 for a full scale analysis of the treatment of nonideal solutions.

2.12

Equilibrium Between Two Ideal Phases

Consider a given species i distributed between two phases ' and " at equilibrium. Equality of the chemical potential leads to the expression lz ~ (T) -t- R T In PS( - / x ~ (T) -t- R T In c i" - - #~9 X(T, P) + R T l n x ~' :

(2.12.1)

11~i * ( T , P) + R T l n X i'

for the equilibration of i in the gas and in the condensed phase. Six additional interrelations could be presented in which the composition of the condensed phase is specified as molarity or molality, but we shall not present these interrelations here. Based on the above one can introduce equilibrium constants of the type Kpx =-- - [ / z ? P ( T ) - # * ( T , P ) ] / R T , etc., so that one can rewrite the equilibrium constraints in the form: Kpx -- .1-". i1 f/ Xl i i ,

Kcx -- C i11/x~ ,

Kxx - x i1/fx. Ii

(2.12.2)

where equilibrium values of Pi, Ci, Xi are to be specified. One may also introduce Eqs. (2.10.7) or (2.10.9) to convert mole fractions to molarity or molality; the appropriate numerical factors or densities may be absorbed in the definitions of equilibrium constants, so that one obtains Kpc

-- P/t/c I ,

gcm

--

ci'/m I ,

Kcc -- r "/c I ' gxm

_ xitt"

Kxc - - x i"/ c' ' i ,

Kpm

-- P/t/m

t/mi"

I,

(2.12.3)

is simply a reformulation of Raoult Law when applied to the solvent. Any of the other relations are equivalent to Henry's L a w when applied to the solute. Kxx -- x i11/ /x iI is also known as the Nernst Distribution Law. ___ relation _ _ Kpx

2.13

-

-

1" i / X ii ,,tl,

Chemical lrreversibility in Chemical Reactions; The Affinity

In Section 1.12 a distinction was made between the entropy change d S - dr Q / T involving reversible processes in a system and the entropy change d S - di Q~ T +

CHEMICAL IRREVERSIBILITY IN CHEMICAL REACTIONS; THE AFFINITY

157

dO corresponding to irreversible changes in the system. We also introduced a deficit function de -- TdO for analyzing irreversible phenomena. The First Law

may then be written in the form d E - d Q + de + d W + Z

#i dni,

(2.13.1)

i

where de vanishes when d Q and d W refer to reversible processes. Consider a process carried out in the absence of any work, for which the last term in (2.13.1) refers solely to a chemical reaction. As explained in Section 1.21, we initiate this reaction by causing a jet of appropriate composition to cross the boundary, at which time it is allowed to execute the reaction in incremental form. This process changes the concentrations of all species and introduces the concomitant heat of chemical reaction into the system. Thus, on the basis of Eq. (1.21.4) we write d E -- T d S + ~

I~i d n i -- T d S -4- ~ i

Villi d~. 7s O.

(2.13.2)

i

This result does not contradict Eq. (2.9.3), because the latter relation, for which Z i l~i dni vanishes, applies only under equilibrium conditions. Here the execution of the reaction changes the internal coordinates of the system, hence, its energy. The differential dS reflects the transport of the heat of reaction across the boundaries, as well as the changes in mole numbers, n i, and is thus associated with the above-mentioned process. The quantity A _= - Z i 19i//~i, was termed the chemical affinity by de Donder (1923). The rate of entropy production generated by the reaction process is thus given by (t is time) ~)=

dS dt

=

A d)~ T dt'

(2.13.3)

which admits of the interpretation (see Section 2.2) that A / T represents a generalized force and that d ~ / d t represents a generalized flux, as applied to chemical reactions. The four fundamental thermodynamic relations may then be written in the general form d E = T d S - P d V - A d)~,

d A = - S d T - P d V - A d)~,

d H = T d S + V d P - A d )~,

d G = - S d T + V d P - A d)~.

(2.13.4)

To guarantee a positive rate of entropy production we require that ( A / T ) ( d ) ~ / d t ) > 0. Thus, with A > O, Z i 1)ilZi < 0 and d k / d t > 0, whereas if A < O, ~ i 1)i~i > 0 and d)~/dt < 0. The chemical reaction as written proceeds to the right or left according to the sign of A. Equilibrium is attained when A = Z i villi = d k / d t = 0, which we have previously characterized as the equilibrium constraint.

158

2. EQUILIBRIUM IN IDEAL SYSTEMS

The generalization to r distinct chemical reactions is simple: every such reaction is specified by its own degree of advancement d)~, and the different chemical reactions are specified by Y~ik VikAik : 0 (k : 1, 2 , . . . , r), for which the affinities are given by Ak = - Z i k 1)iklzi" Note that the subscript k is missing from//~i (why?). The total rate of entropy production due to all the chemical reactions that run concurrently is then given by

0----1 ~ T ~

Ak dXk dt

(2.13.5)

At equilibrium 0 - 0 , which is guaranteed only by requiting that all Ak --0 and that each of the r chemical reactions cease. 2.13.1

Effects of Chemical Reactions on the Entropy Change of a System

As a reprise to Section 1.21 we consider again the dependence of the intemal energy on chemical reactions that are carried out inside a system through the transfer of matter across the boundaries. We begin by citing Eqs. (1.20.9), (1.20.11) in the form

dE - Cv,x dT +

-~

T,x

i

(2.13.6)

Specializing to processes at constant volume we consider two possible cases: (i) T dS - Cv,x dT - - Z i l)i Ei d)~ -- ~ i niCi dT. In this process heat is transferred across the boundaries at a rate commensurate with the heat influx or outflow during the infinitesimal reaction step as it crosses the boundary. In such a case dE - - 0 . (ii) If the above process is carried out adiabatically during a reversible, infinitesimal step in the reaction, dE no longer vanishes; instead, we demand that

dS -- Cv,x dT + ~

Vi Si d~. -- O,

(2.13.7)

i Si :~ (OS/Oni)T,V,nj~i. Measurements of the heat capacity provides a means for determining the rise in temperature of the system that accompanies the process. where

159

Chapter 3 Characterization of Nonideal Solutions

3.0

Introductory Remarks

Much of the material covered in Chapter 2 will now be repeated in a form applicable to nonideal solutions; we concentrate particularly on the proper characterization of the chemical potentials of the constituents. Once this quantity is known all thermodynamic properties of the system may be determined. Particular emphasis is placed on the many alternative concentration units that may be adopted. Pains must be taken to ensure that the various final mathematical formulations uniquely describe a given experimental situation. In what follows two guiding principles will be adopted. (i) The formulations presented in Chapters 1 and 2 are so convenient that it is worth preserving their form in characterizing nonideal systems. (ii) The present analysis must correctly reduce to that of Chapter 2 when systems approach ideal behavior; in particular, for gases at low pressure and high temperature.

3.1

Thermodynamic Treatment of Nonideal Gas Mixtures

Equation (2.4.15) relates the variation of chemical potential of an ideal gas to RT In Pi; in accordance with (i) this suggests that/L i should be specified by an analogous expression, R TIn fi. The quantity fi is known as the fugacity of the ith component of the gas. In accord with (ii) this quantity must approach the pressure Pi at ideality. Since #i is specified only to within an arbitrary constant we can determine uniquely only the difference in chemical potential of the nonideal gas in two states, 1 and 2, given by

#i(T, f2) - lzi(T, 3el)= RT In f2 - RT In fl.

(3.1.1)

Now let 1 represent a specifically chosen reference state in which the gas has fugacity f/0 and let 2 represent any other experimental state of the gaseous system,

160

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

with fugacity fi. Then Eq. (3.1.1) may be rewritten as

/Z i (T, ft') -- "i (Z, # )

nt- R T ln(fi/f/o).

(3.1.2)

This relation is self consistent; it reduces to an identity for fi - fo. It is customary to choose as the reference state the gas species i in ideal form; that is, we select f o _ Pi, and we then adopt Eq. (2.4.15)"

lZi (T, pO) _ ij,?P (T) -+- R T In Pi.

(3.1.3)

When (3.1.3) is substituted in (3.1.2) one obtains the canonical form

lzi(T, fi) -- lz/~

+ RTln3~,

(3.1.4a)

which properly reduces to the ideal gas case. Here the reference state is that of the ideal gas, that is, the state in which all gas interactions are turned off, so that ideality is achieved. The fact that such a state cannot be realized should act as no deterrent, so long as all properties of the actual gas are always taken relative to that same reference state, which, in any event, is completely arbitrary. A particular method for emphasizing the deviation from ideality is to introduce the so-called activity coefficient or fugacity coefficient defined by Yi = f i / Pi, so as to rewrite Eq. (3.1.4a) in the form

# i ( T , fi) - / z ~

+ R T ln yi 4- R T ln Pi,

(3.1.4b)

which shows that lz~ is the standard chemical potential for species i when Yi -- 1 and when Pi - 1 bar. 1 Clearly, we must require that in the limit P --+ 0, Yi ~ 1, so that f/--+ Pi.

3.1.1

Experimental Determination of Fugacities

The above treatment is purely formal until it is established how fugacities may be determined experimentally. This is not the place to provide an exhaustive treatment of this topic; we merely cite one particular technique for accomplishing this aim. Starting with the relation dlzi = R T d i n f/, which holds at a fixed temperature T, we obtain for constant composition of the gas phase, xi

(01nfi) 0P

_ (0/Zi~ _ Vt" . T, xi \ 0 P ~I T,xi RT

(3 15). .

Note that we are invoking here the total pressure variation, since only this quantity can be readily measured. Next, introduce a term

bi =-- Vi - R T / P

(3.1.6)

THERMODYNAMIC TREATMENT OF NONIDEAL GAS MIXTURES

161

that measures deviations of the actual gas from ideal gas behavior. Then

bi 1) bi dln filT,xi~" -- -RT + -P d P - RT dP + d l n P

(3.1.7)

or, on integrating,

dln(fi/p) - ~

bi dp - l n ( f / / P ) - ln(f/l/Pl),

(3.1.8)

at constant T, xi. Here Pl represents some very low pressure where the fugacity is sensibly equal to the partial pressure of the gas, so that we can set f/l = xi P. Then l n ( ~ / P ) -- ~-~

bi dp + lnxi

(T,

Xi constant).

(3.1.9)

In principle this equation may be used to find the fugacity of i in the gas mixture once the dependence of bi o n P has been empirically established. 3.1.2

Fugacity of a Pure Gas

For purposes of illustration we now consider a one-component gas for which b = V - RT/P. It is simplest to integrate Eq. (3.1.5) directly to find

ln(f / Pl) - ~

V (T, p) dp.

(3.1.10)

In principle this accomplishes the task: one measures the (molar) volume of the nonideal gas as a function of the applied pressure at fixed temperature T, beginning at a very low, fixed value of the pressure, Pl, and ending at the pressure P for which the fugacity f is to be found. Alternatively, an equation of state may be employed for insertion in Eq. (3.1.10). If this is not convenient one may integrate by parts to obtain

l n f - lnPl +

= In Pl +

-~[PV-

PlVl]-

[ P V (RT P'T)-I]-1

Pdv ~

f9 ~ P dr.

(3.1.11)

Here one must measure P in its dependence on (molar) volume at the temperature of interest or else determine P(T, V) empirically; also, we set Vl - RT/PI. The reader is invited to work out the fugacity for the special case of a van der Waals gas.

162

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

NOTE 3.1.1. If one wishes to adopt a chemical potential reference standard that is experimentally accessible on could return to Eq. (3.1.4a), set fo = 1 to obtain # i ( T , fi) = tzi(T, 1) nt- R T l n f i . This reference value differs from that in Eqs. (3.1.4), and leads to different fugacity scales. For better or for worse the standard value is usually adopted, because criterion (ii) is then satisfied.

3.2

Temperature and Pressure Dependence of the Fugacity of a Gas

The temperature dependence of the fugacity may be found by rewriting Eq. (3.1.4b) for a one-component system in the form In y =

t.t(T, P) - # ~

- In P,

RT

(3.2.1)

so that, using Eq. (2.5.3), (Olny)

_

I2I(T,P)-I4~ o

OT

p

RT 2

OT

p

(3.2.2)

The fight-hand side obtains because at fixed P, f and ?, = f / P change in the same manner with T. The quantity to be specified here is the enthalpy of the ideal gas, H ~ which is given by ( 5 / 2 ) R T . Alternatively we note that at very low pressures # (T, Pt) --/x oP (T) + R TIn Pl. Then (3.2.3)

lz(T, f ) - lz(T, Pl) = R T In P - R T In Pl + R T In y,

so that (Olng) OT

_-- _ H ( T ' P ) - H ( T ' P I ) _ ( O l n f ) P, PI

.

RT2

OT

(3.2.4)

P, PI

Here we relate the temperature variations to the enthalpy difference of the actual gas at pressure P and at a very low pressure Pl. Methods for evaluating such a difference may be based on Section 1.17; alternatively, this quantity may be evaluated by use of the equation of state, Eq. (1.13.17). The pressure variation is found from Eq. (3.2.1) by writing (Olng)0P r

1 (O/z) R T -ff-fi r

1 _ P

I? RT

1 ~

o

P

(3.2.5)

On the other hand, from y = f ~ P we obtain ,

OP

7-

OP

r

P'

(3.2.6)

THERMODYNAMIC DESCRIPTION OF REAL SOLUTIONS IN THE CONDENSED STATE

163

so that Olnf) OP r

I~'(P, T) RT '

(3.2.7)

requiring the specification of the equation of state of the real gas, or an empirical determination of the actual molar volume.

3.3

Thermodynamic Description of Real Solutions in the Condensed State

We begin our specification of the chemical potential of nonideal solutions in the condensed state that is based on the canonical formulation of ideal solutions, introduced in Section 2.5, t"

~ i -- lZi + R T l n

yi,

(3.3.1)

where Y i - x i , c i , m i and ~ i r is the reference value of the chemical potential #*(T, P) or the quantity that is cited in Eqs. (2.5.12a,b) and (2.5.14a,b) for the concentration units introduced in that section. Standard operating procedures call for the preservation of this formulation for nonideal solutions. This is done by introducing activities ai s u c h that for actual solutions an equation of the general canonical form ~ i -- / -tr --I-- R T In ai

(3.3.2)

is applicable, where the choice of ai and #r depends on what concentration units and standard or reference states have been selected. The specification of ai and #~ requires a rather tedious analysis which will be discussed in detail in Sections 3.4-3.6. A summary is provided at the end for readers not interested in the detailed exposition; in brief it is desirable to employ Eqs. (3.4.23,24) or (3.5.21,22) or (3.6.4) below, as the starting point for further analysis. A very important feature of nonideal solutions is their departure from Raoult's Law; in later sections we shall repeatedly examine and make use of the information provided here. Positive and negative departures from Raoult's Law for a binary solution are schematically illustrated in Fig. 3.3.1. Attention is directed to the following facts: (a) If one component exhibits a positive (negative) departure from Raoult's Law, the other must do likewise; a proof for this statement is to be furnished in Exercise 3.3.2. (b) As the mole fraction xi of component i (i = 1, 2) approaches unity (i.e., as the solution becomes very dilute by virtue of a large excess of component i as solvent), the partial pressure Pi of the solvent closely approaches the value specified by Raoult's Law: as xi ~ 1, Pi = xi Pi*, where Pi* is the partial pressure of pure i. (c) As the mole fraction of component i approaches zero (i.e., when component i as solute is present at close to infinite dilution) the vapor pressure of the solute does not generally follow Raoult's Law,

164

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS p~,[!

P = pA+ PB

"

(a) B

~Raoult's law region

tl

I

/ Henry's

~

o

law region

1

XB (b) A

e8

\ "-PA" N ~ , , ~

r

~., ~-. ~ B

-//

"Raoult's law region

o

z

. . . . . . XB

Heo ', law region 1

Fig. 3.3.1. Diagram showing positive (a) and negative (b) deviations from Raoult's Law. Dashed lines show ideal behavior; dotted lines show Henry's Law.

but is linearly related to the mole fraction: as xi ---->0, one finds that Pi = Kxi. This relationship is known as Henry's Law, and K is known as the Henry's Law constant. (See also Section 2.12.) (d) For ideal solutions Raoult's and Henry's Laws are identical. We shall now proceed with the methodology for determining ai, and we return later to the question of determining the extent of the departure of real solutions from Raoulrs Law. EXERCISES 3.3.1 List reasons why the choice of Eq. (3.3.2) is of great convenience in describing physical properties of nonideal solutions. 3.3.2 On the basis of the Duhem-Margules equation, prove that if one component of a binary mixture exhibits positive (negative) deviations from Raoulrs Law, the second must do likewise. (See, e.g., S. Glasstone, Thermodynamics for Chemists, D. Van Nostrand, New York, 1947, Chapter 14.) 3.3.3 By using Raoult's Law for component 1 and the Gibbs-Duhem relation, show that component 2 must satisfy Henry's Law over the composition range x2 = 1 - Xl for

CHARACTERIZATION OF NONIDEAL SOLUTIONS; PRELIMINARIES

165

which Raoult's Law applies for component 1. (See, e.g., S. Glasstone, Thermodynamics for Chemists, D. Van Nostrand, New York, 1947, Chapter 14.) 3.4

Characterization of Nonideal Solutions; Preliminaries

A proper exposition of the subject rests on the following cardinal principle: Let q represent any composition variable which specifies the makeup of a uniform solution. In what follows we let q stand for mole fraction x, molarity c, or molality m: then the chemical potential of species i in the homogeneous mixture shall be given by the expression

#i(T, P, qi) -- #i(T, P,q?) + RTln[ ai(T' P'qi) ]

(3.4.1)

ai (T, Pi qif i ' in which T is the temperature, P the pressure, and qi+ is any arbitrary reference value of the composition variable qi for the i th species in a uniform mixture; the quantity ai ( T, P, qi ) is termed the activity of species i relative to the composition variable qi at temperature T and pressure P. This quantity is an as yet unknown function of the indicated parameters and variables, whose dependence will be determined later. Eq. (3.4.1) specifies the chemical potential lzi(T, P, qi) relative t o / z i (T, P, qi§ the reference chemical potential, obtained by substituting q+ for qi in the functional dependence. Since Eq. (3.4.1) reduces to an identity when qi -- q+ one may choose for q/+ any value that happens to be convenient. This flexibility is at once a blessing and a curse: it provides enormous freedom of choice, at the same time that it produces a multitude of seemingly different thermodynamic formulations, all of which must ultimately be rendered equivalent. There is the further complication that one also has so many choices for specifying the composition variable q. Readers not wishing to wade through all the intermediate steps or wanting to avoid listings of alternative formulations may proceed directly to the end of this section, where the expressions (3.4.23), (3.4.24) offered below may simply be taken as a sensible starting point for further analysis. 3.4.1

Standard Chemical Potentials

We begin the thermodynamic analysis by specializing Eq. (3.4.1) in two steps: First we desire that the reference value/Z i (T, P, qiif-) be specified at unit pressure P = 1 (usually one bar); thus, we rewrite (3.4.1) as

,

" qi ] '

iTi i i ~ ;

'

(3.4.2)

which again reduces to an identity for P - 1 and qi - - q ? . Next, we select as q+ that particular standard value which renders ai (T, 1, q~) -- 1. These two

q7

166

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

conditions specify the standard state of the system: the standard chemical potential lzi(T, 1, qi~) is taken with respect to species i at unit activity and unit pressure. However, this relation does not suffice to determine ai (T, 1, qi~), since q/O is itself unknown at this point. Therefore, in conformity with the requirement that Raoult's Law must hold for the majority component in exceedingly dilute solutions, we set

ai (T, 1, q*) -- q* (T, 1),

(3.4.3)

which fixes the scale for ai; here q.*,(T, 1) is the 'concentration' of pure i, which is specified by x * - 1, or by Eq. (2.5.10b) with P - 1 when q - c, or by Eq. (2.5.1 lb) when q - m. With the above choices for ai(T, 1, qi~ and q[(T, 1) Eq. (3.4.2) reduces to the

canonical form: lZi( T, P, qi) -- lZi( T, 1, qi~) + RTlnai(T, P, qi),

(3.4.4)

which is clearly patterned after Eqs. (2.5.1), (2.5.12a) and (2.5.14a). With the adoption of q/O = x~ - 1 for a pure material, and by employing standard conditions on setting P - 1, we automatically satisfy the requirement that ai (T, 1, x*) -- 1. For, it is only with this choice that Eq. (3.4.4) reduces to an identity. However, as shown later in Section 3.7, at any other pressure ai(T, P, x*) differs from unity, although under normal experimental conditions the deviations from unity are small. The selection of molarity or molality engenders complications that are addressed later. The problem of dimensionality in Eq. (3.4.4) can obviously be disposed of quite readily in the present approach. However, the specification of the standard chemical potential, requiting more ingenuity, is postponed to later sections. 3.4.2

Specification of Activity Coefficients

In the next step of the procedure we introduce the concept of activity coefficient I] (T, P, qi ) through the expression

Fi(T, P, qi) =-

ai(T,P, qi) , qi

(3.4.5)

so that Eq. (3.4.4) may be rewritten as

lZi (T, P, q i ) - lZi (T, 1, qi~ + RT In Fi (T, P, qi) d- RT ln qi.

(3.4.6)

This equation has the drawback that it specifies lzi(T, P, qi) at pressure P relative to the standard chemical potential at unit pressure. If desired one may keep the

CHARACTERIZATION OF NONIDEAL SOLUTIONS; PRELIMINARIES

167

pressure parameter uniform throughout by returning to Eq. (3.4.1), setting q+ = q/O, and introducing a second activity coefficient via

~/i(T, P, qi) =

ai(T, P, qi) ai(T, P, qi~

--

I~(T, P, qi) , ai(T, P, qi~

(3.4.7)

with which Eq. (3.4.1) may be rewritten as

#i(T,P, q i ) - - # i ( T , P , q i ~

qi)+RTlnqi.

(3.4.8)

The quantity//,i (T, P, q/@) will be designated a reference chemical potential. Alternatively, Eq. (3.4.6) may be reformulated as

lzi(T, P, qi) - lzi(T, P, qi~ + RTln(yia@i (q)qi),

(3.4.9a)

with

A|

=- ai (T, P, qi@).

(3.4.9b)

All three formulations, Eqs. (3.4.4), (3.4.8), and (3.4.9) are encountered in the literature. One should note that we use Yi only in equations involving the reference chemical potential/Z i (T, P, q/@), whereas/-} by itself occurs only where the standard chemical potential//,i (T, 1, q/@), is involved. 3.4.3

Specification of Activities and Activity Coefficients

For q = x and x* = 1, i.e., for pure material, Eqs. (3.4.3) and (3.4.5) reduce to the important relation

ai(T, 1, 1 ) = ~.(T, 1, 1)-- 1.

(3.4.10a)

Equation (3.4.10a), coupled with the requirement (3.4.3), shows at once that x/~ - x + - - x ~ - 1, consistent with the requirement that the standard state of i be the pure substance under a pressure of one bar. Furthermore, Eqs. (3.4.7) and (3.4.10a) lead to yi(T, P, 1) --ai(T, P,x*)/ai(T, P, xi~ -- 1, or

ai(T, P, 1)--ai(T, P, xi@),

(3.4.10b)

which is a self-consistent result. When q = c, or q = m, Eq. (3.4.3) reads :r

ai(T, 1, c * ) - c i (T, 1),

(3.4.1 la)

ai (T, 1, m i ) -- m i ,

(3.4.1 lb)

where the 'concentrations' and 'molalities' of pure materials have earlier been specified by Eqs. (2.5.10b), (2.5.1 l b). Also, by definition,

a/IV, 1,

a/IV, 1,

1

(3.4.12)

168

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

The foregoing analysis shows up the advantages of working with mole fractions as composition variables: here x/e is simply unity. By contrast, c/e and m/e are yet to be determined; they differ for each solution that is made up. 3.4.4

Unit Molarity or Molality as Reference States

To avoid having to determine c/e or m/e one may adopt schemes wherein one sets c/e - 1 mol/liter or m/e at unit molality. This approach may be incorporated here by specializing (3.4.4) to P = 1, qi 1, so that =

lzi(T,l,1)--lzi(T,l, qie)+RTlnai(T,l,1),

q--c,m.

(3.4.13)

On eliminating lzi(T, 1, q/e) between (3.4.13) and (3.4.4), we obtain (q - c, m)

[

J

lzi(T, P qi) -- lzi(T, 1 1) + RT In ai(T, P, qi) ' ' ai(T, 1, 1) '

(3.4.14)

which properly reduces to an identity for P --qi 1. The minor drawback to Eq. (3.4.14) is that it cannot be reduced to a canonical form. In this connection one may introduce the activity coefficient - -

l'i'(T, P, qi) = _

ai(T,P, qi) ai ( T, 1, 1)qi

(3.4.15)

to find

lzi(T, P, qi) -- lZi( T, 1, 1) + RTln~.'(T, P, qi) -k- RTlnqi.

(3.4.16)

Alternatively, one may now solve Eq. (3.4.4) for lzi(T, 1, qi@), and substitute the resultant in Eq. (3.4.13); this yields (with q/* - 1)

lzi(T, 1 1 ) - lzi(T, P 1)-{- RTlnF ai(Z, 1, 1) ] ' ' L ai(T, 19-, 1) .] '

(3.4.17)

and when this expression is introduced in (3.4.14) one obtains

lag(T, P, qi) ] #i(r, P, qi) - lzi(r, P, 1) + RT lnk a i i ~ ~,~ i.)-

(q -- c, m).

(3.4.18)

Then, by use of the definition

y/(T, P, qi) =-

ai(T,P, qi) ai(T, P, 1)q

(3.4.19)

one finds

lzi(T, P, qi) -- #i(T, P, 1) + R T l n v / ( T , P, qi) -+-RTlnqi.

(3.4.20)

CHARACTERIZATION OF NONIDEAL SOLUTIONS; PRELIMINARIES

3.4.5

169

General Commentary

The preceding discussion shows the flexibility available for specifying the chemical potential of component i in a uniform mixture. In every instance it is possible to cast the final relations into the general form ll~i - - lZ 0 +

R T In fii -at- R T In qi ,

(3.4.21)

where #/0 is a standard or reference chemical potential and fli is an appropriate activity coefficient. The various forms assumed by the preceding expressions differ from each other solely in the choice of composition variable and in the desired reference or standard state. The foregoing is sufficiently elastic that one should seek a simplified approach. This is achieved if one uses solely the mole fraction xi as the composition variable and if all thermodynamic characterizations refer only to the standard state at a total pressure of P - 1 bar. In such circumstances the self-consistent equation (3.4.1) reduces to #i(T '

P xi) - # i ( T , 1 1) + R T l n ' '

[

a i ( T , P, xi) a i ( T , 1, 1)

J

(3.4.22)

"

Here again, #i (T, 1, 1) -- #i* (T, 1) is the chemical potential for pure i at P --1 bar. By Raoult's Law, ideal behavior obtains when xi ~ 1, so that we may set a i ( T , 1, l ) - x i* -- 1 , whence we obtain the c a n o n i c a l f o r m # i ( T , P, xi) -- #i*(T, 1) + R T l n a i ( T ,

On now introducing the activity coefficient F i ( T , write l z i ( T , P, xi) -- [z i (T, l) + R T l n x i

P, xi).

P, xi)-

4- R T l n ~ . ( T ,

ai(T,

P, xi).

(3.4.23) P, x i ) / x i

we

(3.4.24)

Note that Ni (T, 1, 1) = 1. However, as shown later, for pressures other than 1 bar, /~ (T, P, 1) and ai ( T , P , 1) deviate from unity. This reflects the fact that a compressed solution experiences greater interactions among its constituents than one at lesser compression. Thus, even though the composition is unaltered the activities do vary with pressure, as does Fi (T, P, x i). The discussion surrounding Eqs. (3.4.3-5) shows that there is only one unique set of conditions, here P - 1 and x i* - 1, for which self-consistency is achieved in Eq. (3.4.22), namely: ai = ~ = 1 only when the mole fraction of the majority component approaches unity and when the solution is under a total pressure of 1 bar. It is this formulation, Eqs. (3.4.23-24), that is found to be most useful. EXERCISE 3.4.1 Prove that yi(T, ai(T, 1, m i) -- 1.

1 , c i*(T, 1)) --

yi(T, 1, m*) - 1, and that ai(T, 1 , c i*(T, 1)) --

170

3.5

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

Standardization of Thermodynamic Analysis for Nonideal Solutions

In Section 3.4 we have displayed many different modes for characterizing the chemical potential of a given species in a nonideal solution. While these various descriptions all look different, surely all physical predictions must be independent of the particular standard or reference state which has been chosen, and surely they cannot be allowed to depend on the choice of concentration units. We now adopt restrictions that guarantee that the chemical potential of any species i in any solution relative to the standard potential shall indeed be unique, i.e., invariant under any change in choice of concentration units. By way of introduction we recall the relations (see Section 2.5) X___~i__

ci

~--:~j n j M j

,

xi _-- M l X l

1000p(r, P) Y-:~j nj

mi

(3.5.1)

1000'

where p ( T , P) is the density of the solution, and the subscript 1 refers to the solvent. From Eq. (3.5.1) we can readily obtain the 'molarity' c* and 'molality' m* for component i present in the isolated, pure state, in which xi -- x i* - 1. Equation (3.5.1) then reduces to the results (M1 = M i in this instance) C* --

lO00p* Mi '

1000 m* -- ----~i"

(3.5.2)

From Eqs. (3.5.1) and (3.5.2) we obtain two relations needed in our later derivations, namely,

c*(T, P) p*(T, P) c i ( T , p ) X i = p(T, P)

Y':~jn j M j Mi ~ j n j

,

m*xi -mi = ~ Mi .

(3.5.3)

We now recapitulate three different ways of specifying the chemical potential in canonical form, relative to reference chemical potentials. For q - x, c, m we use Eq. (3.4.8), and we also adopt the special case xi~ - x i - 1. This leads to the set of relations that appear to be different, but that must ultimately be shown to be equivalent, namely,

lzi(T, P, xi) -- lzi(T, P , x * ) -k- R T l n ? ' i ( T , P, xi) -t- R T l n x i ,

(3.5.4a)

lzi(r, P, ci) - # i ( r , P, c?) nt- R T l n y i ( T , P, ci) + Rrlnci(T,

P),

tz/(r, P, mi) - a/(r, P, mi~ + RT In • + R T lnmi.

(3.5.4b)

P, mi) (3.5.4c)

Analogous developments involving standard states at unit molarity or molality give rise to complications that are to be explored in Exercises 3.5.1 and 3.5.2.

STANDARDIZATION OF THERMODYNAMIC ANALYSIS FOR NONIDEAL SOLUTIONS

3.5.1

171

Equivalence of the Various Formulations; Gauge Invariance; Requirements for Reference Chemical Potentials

In accord with the introductory comments to this section we are mandated to equate the three formulations in Eq. (3.5.4). This leads to the relations

#i(T, P, xi ) q- R T l n y i ( T , P, xi)-t- R T l n x i :- #i(T, P, ci@) + R T l n yi(T, P, Ci) -Jr-R T l n c i ( T , P) -- #i(T, P, mi@) -Jr-R T l n y i ( T , P, mi) q- R T l n m i .

(3.5.4d)

Any formulation that satisfies (3.5.4d) represents an acceptable expression that guarantees the invariance of /Zi under a change of composition variables. By analogy to other field theories we shall refer to this process as maintaining the gauge invariance of the chemical potential. There obviously is considerable leeway in how to proceed; hence, we note that (3.5.4d) may sensibly be broken up into two portions: a part which relates the various reference chemical potentials ~i (T, P, qff) at fixed compositions, and a part involving relations between composition variables RT ln(yiqi), in which qi = xi, ci(T, P), or mi. Interrelations between the reference chemical potentials are thus found by dealing with the i th species in pure form, for which composition variables play no role. We specialize Eqs. (3.5.4a) and (3.5.4b) by setting x i - xi~ - 1, c i - ci~ and equating the resulting relations, taking cognizance of the discussion ahead of Eq. (3.4.10b), namely, that }4 (T, P, xff) -- 1. This yields

#i (r, P, c*) -- #* (T, P, ci@) + RT ln yi (T, P, c*) + R r ln c* (T, P) 9

~X

-- tZi (T, P, xi ) ~ lzi (T, P) =-- lzi (T, P),

(3.5.5)

where we have now introduced by definition a shorthand notation in the second line, which pertains to the chemical potential of pure i at temperature T and pressure P. The preceding equation may be solved for

#i (T, P, ci@) -- #* (T, P ) - RT ln vi (T, P, c*) - RT ln c* (T, P) =/zi_,C(T, P),

(3.5.6)

in which the last expression serves as a shorthand notation for the three terms in the middle. Eq. (3.5.6) conveys important information" it shows how//,i (T, P, C/@) may be determined from the chemical potential of pure i at temperature T and pressure P; Eq. (3.5.6) further involves the activity coefficient and concentration of pure i under the same conditions. In Section 3.7 we indicate how to determine vi(T, P, c i ); it will be shown that this quantity ordinarily does not deviate much from unity [as is already evident from Eqs. (3.4.3) and (3.4.7) by which vi(T, 1, c*) - 1]. Further, Eq. (3.5.6) for an ideal solution reduces to Eq. (2.5.12b)

172

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

in the limit of infinite dilution, showing the overall consistency of the approach. Here c* is specified by Eq. (3.5.2); therefore, -RTln(yic*) in Eq. (3.5.6) is known. Thus, lzi(T, P, ci~) is now directly linked to/z*(T, P) and is abbreviated as / z i*C(T, P). We will also show in Eq. (3.5 . 17) how to determine ai(T, P , ci~) in terms not involving c/~ The above findings are direct consequences of the invariance requirements imposed on/zi. The awkward term/Z i (T, P, r has now been eliminated. As the second step in the process of examining gauge invariance effects, we equate Eqs. (3.5.4a) and (3.5.4b) for any arbitrary concentration xi and ci, and introduce Eqs. (3.5.6) and (3.5.5). This provides a relation between activity coefficients,

yi ( T , -fi~c~[) + R T l n

y~ ( T , -fil c i )

= RTln?'i(T, P, xi) + RTlnxi.

(3.5.7)

Or equivalently, in view of (3.5.3), yi(T, PI c*)

-- yi(T, P, xi)

ci(T, p ) Xi

= yi(T, P, xi)[ p*(T' P) y ~ j n j M j ] p(T, P) Mi ~--~j nj "

(3.5.8)

In later sections we show how to determine yi(T, P, c*). However, Eq. (3.5.8) is almost never used: It simply demonstrates that yi(T, P, ci) in Eq. (3.5.4b) is directly related to yi(T, P, xi); their interdependence is simply a result of requiting gauge invariance. The interrelations (3.5.6) and (3.5.8) guarantee that Eqs. (3.5.4a) and (3.5.4b) actually yield physically identical predictions. A similar set of relations is found by equating (3.5.4c) with (3.5.4a); one need merely replace ci in Eqs. (3.5.5-8) by mi, and use the appropriate formula in Eq. (3.5.3). This yields

~i (T, P, mi0) -- lZ* (T, P) - RT In Yi (T, P, m*) - RT lnm T' ~i, m (T, P)

(3.5.9)

and m* M1 yi(T, P, mi) = yi(T, P xi) ---~ xi -- yi(T P x i ) ~ X l . yi(T, P, m~) ' mi ' ' mi

(3.5.10)

Remarks made in conjunction with Eqs. (3.5.6) and (3.5.8) apply to (3.5.9) and (3.5.10) with obvious modifications. As in (3.5.6) and (3.5.9) we rewrite

STANDARDIZATION OF THERMODYNAMIC ANALYSIS FOR NONIDEAL SOLUTIONS

173

Eq. (3.5.4) as follows [/z*(T, P) =_ lzi(T, P, x~)]"

I~i(T, P, xi) -- /z i*(T, P) 4- R T l n v i ( T , P, Xi) @ R T l n x i ,

(3.5.11 a)

I~i(T, P, ci) -- l.ti.c ( T , P ) + R T l n y i ( T , P , ci) + R T l n c i ( T , P), # i ( T , P , mi ) _ ]z i, m ( T , P ) + R T l n

(3.5.1 lb) Yi ( T , P , mi)

+ R T lnmi.

(3.5.1 lc)

These relations are a restatement of Eq. (3.4.8) based on Eqs. (3.5.6) and (3.5.9); clearly they involve the reference chemical potential. 3.5.2

Gauge Invariance for Standard Chemical Potentials

So far we have used only reference chemical potentials. Matters can be simplified by introducing instead the standard chemical potential; we therefore begin with equations based on Eq. (3.4.6). We rewrite xi~ = x i _= 1 and/Z i (T, 1, x i ) =~ /z i*(T, 1)', then I~i(T, P , x i ) -- Iz i*(T, 1) + R T l n F i ( T , P xi) + R T l n x i ,

(3.5.12a)

lzi(T, P, ci) -- lzi(T, l, ci@) Jr- R T l n I ' i ( T , P, ci) + R T l n c i ( T , P),

(3.5.12b)

lzi(r, P, mi) - lzi(r, 1,m/O) + R r l n G . ( T , P, mi) + R T lnmi.

(3.5.12c)

The procedure used earlier in this section will now be repeated: specialize (3.5.12a) and (3.5.12b) to pure materials under standard conditions by setting P - 1, x i - 1, c i - ci0, with G.(T, 1 , x * ) = 1 and G.(T, 1,c i) -ai(T, 1, c*)/c*(T, 1) -- 1 on account of Eqs. (3.4.5) and (3.4.10a). Equating the resulting relations, one finds

lzi(T, 1,c/@) --/z i~< (T, 1) - R T l n c i9 (T, 1) ~ / z i~ C (T, 1).

(3.5.13)

Equation (3.5.13) bears the important message that lzi(T, 1, ci@) is related to the chemical potential of pure i, corrected for by a quantity that is specified by Eq. (3.5.2); this is a direct consequence of gauge invafiance. Eq. (3.5.13) shows in detail how to determine standard chemical potentials in terms of molarity. Eq. (3.5.13) is also consistent with Eq. (2.5.12b) as applied to pure materials. Next, equate (3.5.12a) with (3.5.12b), taking account of (3.5.13); we obtain:

R T l n F i ( T , P, ci) + R T l n

[ci(T,P)] ~*(-T-, 1)

= R T l n ~ . ( T , P, xi) -Jr-R T l n x i ,

(3.5.14)

174

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

which, on account of (3.5.3), may be rearranged to read

~.(T, P

'

Ci) - -

Fi(T P X i ) r (T, 1) ' ' ci(T, P)

=~(T,

xi

P, xi)[ p*(T' 1 ) ~ j n j M j l p(T, P) Mi Y~j nj

"

(3.5.15)

Equation (3.5.15), though not frequently used, demonstrates the relation between activity coefficients expressed in terms of molarity and those expressed in terms of mole fractions. One other quantity of interest remains to be specified, namely ai(T, P, ci@). For this purpose we use Eq. (3.4.7) to rewrite (3.5.15) as

yi(T, P , ci)ai(T, P, ci@) - yi(T, P , xi)ai(T, P ,

x*)[c . ( T , p1)) xi ] 9 c*(T,

(3.5.16)

We now consider the special case Xi - - X i* - 1, Ci - - Ci* , noting that yi(T, P x*) -1, as can be demonstrated by the reader. We introduce (3.5.2) and solve to obtain

P,x*)Ip*p*(r, 1) (T, P)

ai(T, P, ci~ -- ai(T, Yi-(7"] -fi] c*)

Xi

*C

~ ai

(T, P).

(3.5.17)

As shown earlier, the activity of i in the reference state ai (T, P, c/~ may be related to the activity of pure i at the same T and P, as is explicitly indicated by the abbreviated notation on the right. Eq. (3.5.17) is another important consequence of gauge invariance. We show later how ai and Yi may be determined experimentally. We therefore at this point have completed the task of specifying the quantity ai ( T , P , ci@) which we had introduced in conjunction with Eq. (3.4.7). In a similar vein one obtains relations that are based on the use of Eqs. (3.5.12a) and (3.5.12c), namely,

Ixi(T, 1,m/O) -- #i, (T, 1) - R T l n m i9 -- #i,m *

~.(T, P, mi) =1-'i(T, P, x i ) ~m i x i mi

--/~(T,

( T , 1)

Mix1 P, x i ) ~ , Mi

ai(T, P,x*) ,m(T P) ai (T, P, m?) - Yi iT; -P; m*) =~ai ' '

(3.5.18) (3.5.19)

(3.5.20)

thereby providing interrelations for lzi(T, P, mi@), l~(T, P, mi), and ai(T, P, m/~ in terms of corresponding quantities for mole fractions. One should note that in all of the expressions for the chemical potential the quantity m/~ has been eliminated.

STANDARDIZATION OF THERMODYNAMIC ANALYSIS FOR NONIDEAL SOLUTIONS

175

Equations (3.5.12a-c) may now be rewritten by (i) introducing Eqs. (3.5.13), (3.5.18); (ii) reintroducing the notations ai(T, P, qi~ --a*q(T, P) with q x, c, m, as specified by Eqs. (3.5.5), (3.5.17), (3.5.20); (iii) using the relations of Eqs. (3.4.7), ~.(T, P, qi) - yi(T, P, qi)a *q (T, P). Then tzi(T , P , xi) _ lz ,x i (T, 1 ) + R T ln[yi(T, P, xi )a*X(T, P)xi] = / z i* x (T, 1) 4- R T lnai(T, P, xi),

(3.5.21a)

lzi(T, P, ci) -- #*C(T, 1) + R T l n [ y i ( T , P, ci)a*C(T, P)ci(T, P)] = lz i*c(T, 1) + R T l n a i ( T , P, ci) ,

(3.5.21b)

lzi(T, P, mi) l lZ,im (T, 1) -k- R T l n [ y i ( T , P, mi)a*m(T, P)mi] (3.5.21c)

-- lz*m(T, 1) + R T l n a i ( T , P, mi). 9q

,q

Note that a i (T, 1) ~ 1 for all q; hence, the quantities a i (T, P) in (3.5.21) will ordinarily not differ significantly from unity. 3.5.3

Alternative Formulation

Yet a different formulation may be established by combining Eqs. (3.5.13) and (3.5.21 b) to obtain , [ ,c ci(T,P)] lzi(T, P, ci) -- lzi (T, 1)-+- R T l n yi(T, P, ci)a i ( T , P ) ~ , ( T , 1) , (3.5.22a) and similarly, by combining (3.5.18) with (3.5.21c) we obtain

[

mi]

lzi(T, P, mi) -- #i*(T, 1) + R T l n yi(T , P , mi)a*m(T, P)~_, 9 mi

(3.5.22b)

Note that/Z i*x (T, 1) in Eq. (3.5.21a) is identical with/Z *(T i , 1) . 3.5.4

Discussion

In the above treatment we have specified the chemical potentials in two different ways: Eqs. (3.5.11) involve the reference chemical potential, whereas Eqs. (3.5.21) relate/Z i to standard chemical potentials. In each case, precautions were taken to ensure a unique specification for #i: the reference chemical potentials were thus written out in terms of tZi9x (T, P) ~ lzi(T, P, x i*) ~ lZi* ( T , P ) for the pure component i through Eqs. (3.5.6) and (3.5.9). If desired the activity coefficients yi(T, P, ci) and yi(T, P, mi) may be related to yi(T, P, xi) via Eqs. (3.5.8) and (3.5.10); the product vi(T, P, qi)a/q (T, P) may be ascertained

176

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

experimentally as discussed below in Section 3.7. The standard chemical potentials lzi*c(T, 1) and /~i*m(T' 1) were specified in terms of ~i9X(T, 1)__ P'i*(T, 1) through use of (3.5.13) and (3.5.18). If desired,/-}(T, P, ci),/-) (T, P, mi) may be written in terms of Fi(T, P, xi), as shown in Eqs. (3.5.15) and (3.5.19); and a i9c (T , P), a i, m (T, P) may be specified in terms of a i, x (T, P) as shown in (3.5.17) and (3.5.20). Thus, one can always correlate activities and activity coefficients expressed in terms of molarity or molality with those involving mole fractions. There finally remains the problem of guaranteeing that the chemical potentials specified so far shall actually remain gauge invariant with respect to reference or standard chemical potentials that differ from those adopted in the present section. This matter is to be handled in Exercise 3.5.5. That such invariance can always be maintained should become clear on reflection of the meaning of Eqs. (3.5.5), (3.5.9), (3.5.13), (3.5.18), in this Section, and of Eq. (3.4.14). EXERCISES

3.5.1 Invoke the invariance principle for chemical potentials by comparing Eq. (3.4.20) with Eq. (3.5.4a). Find an expression relating y/(T, P, ci) to yi(T, P, xi). H o w do your results differ from those cited in this section in the text? 3.5.2 Repeat Exercise 3.5.1, by comparing Eq. (3.4.22) with Eq. (3.5.12a). Relate lzi(T, 1, ci -- 1) to tzi(T, 1 , x * ) and Fi'(T, P, ci) to ~ ( T , P, xi). H o w do y o u r results differ from those cited in the text? 3.5.3 Repeat Exercise 3.5.1 when molality is used in place of molarity and/Z i is referred to unit molarity as the reference chemical potential. How do your results differ from those cited in the text? 3.5.4 Repeat Exercise 3.5.2 when molality is used in place of molarity and/Z i is referred to unit molality as the standard chemical potential. How do your results differ from those cited in the text? 3.5.5 Discuss the possibility of selecting different gauges in which ~ i 9c ' /Li.x ' /s .m are interrelated by expressions which differ from Eqs. (3.5.5), (3.5.9) and (3.5.18) and show how these alterations may be compensated for in equations interrelating or specifying the corresponding activity coefficients. 3.5.6 Derive interrelations involving yi(T, 1, ci), yi(T, 1,mi), and yi(T, 1,xi).

3.6

Reformulation of the Thermodynamic Description of Nonideal Solutions

The complications of the last section arose from the specification of/Z i in terms of the canonical relations (3.5.4), together with the use of the quantity/Z i ( T , 1, q ? ) as the standard chemical potential; one should recall that q/e had been so chosen that ai (T, 1, q ? ) -- 1. This treatment is perfectly self-consistent; however, a simpler alternative exists. One may return to the fundamental relation (3.4.2) and

REFORMULATION OF THE THERMODYNAMIC DESCRIPTION OF NONIDEAL SOLUTIONS 177

select for the concentration variable q+ the reference state q/* for the pure material. With this choice, and in view of the relation (3.4.3), Eq. (3.4.2) becomes

[ai(T, P, qi) ] lzi(T, P, q i ) - lzi(T, 1,q*) + RTln L q~i/~;-l~ .

(3.6.1)

Note that x * - 1 and that c* and m* are specified by Eq. (3.5.2). Since ll,i (T, 1, qi ) is the chemical potential of pure i under standard conditions this quantity should be the same, independent of the choice of concentration units. A proof that this is the case was provided in the derivation of Eqs. (2.5.12c), (2.5.14c). We therefore write

lzi(T, P, qi) -- lzi (T, 1) + RTlnni(T, P, qi),

(3.6.2)

in which we have set

hi(T, P, qi) =

ai(T,P, qi) qi (T, 1)

(3.6.3a)

= yi(T, P, qi)a *q (T, P) = re (T, P, qi)

qi(T,P) qi (T, 1)

qi(T,P)

(3.6.36)

q/*(T, 1)

P) =--yi(T, P, qi)A *(q) qi(T' qi*(T , 1)'

(3.6.3c)

and in which we have also reintroduced A *(q) =-- a i ( T , P, q*) =- a *q (T, P). An alternative derivation of Eq. (3.6.2) is achieved by introducing Eqs. (3.5.13) and (3.5.18) into Eqs. (3.5.21b) and (3.5.21c), respectively. Written out in full, Eq. (3.6.2) then read

lzi(T, P, xi) -- lZi* (T, 1) + RTln[yi(T, P, xi)a i* X (T, P)xi] --/Z i

(T, 1) @ RTln[~.(T, e , x i ) x i ] ,

(3.6.4a)

, [ ci(T,P)] tzi(T, P, ci) - lzi (T, 1 ) + RTln yi(T, P, ci)a*C(T, P)-ffi,--CT-' 1) ,

[

ci(T,P)] , ci(T, 1)

--/z i (T 1) + RTln Fi(T P c i ) - - - ~ ~

,

, ,

,

[

,

[

(3.6.4b)

mi] #i(T, P, mi) - #i (7', l) + RTln yi(T, P, mi)a*m(r, P)__-7 mi

m,]

-- ~i (T, l) + R T l n /-)(7', P, mi)mU .

(3.6.4c)

178

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

It should be obvious that when q = x, the treatment of this section and that of Section 3.5 are identical. Moreover, the above equations reduce, as they should, to the formulations (2.5.13b), (2.5.15b) for ideal solutions. The above formulation has the great advantage that the standard chemical potential is now independent of the mode of specifying the composition of solutions. The standard state in each instance refers to that of pure i at temperature T and unit pressure. The price paid for this simplification is that the interrelations between ai (T, P, qi) and qi are now slightly more complex than those involving ai(T, P, qi) and qi when q = c, m. The ai introduced in Eq. (3.6.3) will be termed the relative activity. Note that whereas ai (T, 1, c*) and ai (T, 1, m*) differ from unity, rli(T, 1, q*) -- 1 for all qi. Equations (3.6.3a-c) also show explicitly that ai(T, P, xi) = ai(T, P, ci) = ai(T, P, mi) all have the same numerical values, independent of the composition units, whereas the corresponding ai differ numerically. Thus, the use of relative activities has much to recommend it. EXERCISES

3.6.1 Determine the choice of gauge by which Eqs. (3.6.2)-(3.6.4) may be directly derived from Eqs. (3.5.3) and (3.5.9). 3.6.2 For a two-component solution, the following equation has been proposed: ln(al/A~ xl) --lnxl + (B/RT)x 2, B -- B(T, P)is a parameter. Determine (a) Yl (T, P, Xl), (b) ?'2(T, P, x2), (c) P1 (T, P, Xl). 3.6.3 From the representation in Exercise 3.6.2 determine ln[al (T, P, Cl)/A~ c)] as well as ln[al (T, P, ml)/A~m)], and thence find ?'1 (T, P, Cl), ?'1 (T, P, ml), ?'2(T, P, c2), ?'2(T, P, m2). 3.6.4 For the representation shown in Exercise 3.6.2 determine G1, H1, S1, I7'1,/~1, A1.

3.7

Characterization of Equilibrium in Nonideal Solutions

So far we have described several methods for determining the chemical potential of species in nonideal solutions. This now provides the groundwork for the study of equilibrium constants. As might be expected, the large variety of ways in which these chemical potentials may be specified is reflected in many different ways for defining equilibrium constants. As usual, care will have to be taken to ensure a proper interrelation between the different specifications.

3.7.1

General Approach

In most general terms, one starts with the overall criterion for chemical equilibrium, (Y-~1Vllld)eq = 0, developed in Section 2.9. Here again the vl indicate stoichiometry coefficients for the generalized chemical reaction Y-~lviA1 = 0, where the A1 represent all of the participating species; vl is positive or negative according as the corresponding AI represents a product or a reagent for the reaction as

CHARACTERIZATION OF EQUILIBRIUM IN NONIDEAL SOLUTIONS

179

written. The most straightforward approach involves use of Eqs. (3.4.1) or (3.4.2), in which, for the time being, q+ is simply some convenient reference value of the composition variable. Starting with Eq. (3.4.1) the equilibrium state is characterized by

I al(T, P, ql) ] ~-~v,#,(T, P,q+) + Z vIRTln ~)TI fii ~-) 1

l

-0,

(3.7.1)

eq

which leads to many possibilities for constructing equilibrium 'constants'. We confine ourselves to the following" (i) One may rewrite the preceding expression as In/((T

P) = - Z Vl#l(T, P, q~-) ' RT 1

Z

l

VlIn

al(T, P, ql) ] a/(T-~ P, ~ - ) eq'

(3.7.2a)

in which an equilibrium constant K has been defined in terms of the particular reference chemical potential, thereby being independent of the composition of the system. Clearly K depends parametrically on the temperature and pressure. Equation (3.7.2a) may be reformulated as

[al(T,P, ql) lv' K(T, P)- I-I al(V, PI q~-) eq

(3.7.2b)

l

By contrast, if Eq. (3.4.2) is chosen as the starting point one arrives at the expressions ln~

Vl#l(T, l, q~-)

= - Z

(3.7.3a)

l or

al(T, P,_ql_2_]vt l

al(T, i~ q?)

(3.7.3b)

eq

On the other hand, one is also free (ii) to return to Eq. (3.7.1) to define a quantity InK(T,

P)=-ZVl[ l = Z l

lzI(T'RTP'q~-) -lnal(T

vl ln[a/(T, P, ql)]eq'

, q/+)]

(3.7.4a)

180

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

in which the definition of the equilibrium 'constant' K now includes the activity of the various species at their reference compositions q/+. Eq. (3.7.4a) may be written as

K(T, P)- I-I[al(T, P, ql)]eVlq.

(3.7.4b)

l

When Eq. (3.4.2) is adapted to this formulation one obtains

lnEa -- -- ~

vt[ IxI(T' I'

-- lnal(T, l, q+)],

(3.7.5a)

l or

Ea(T, P ) - I-I[al(T, P, ql)]e ;.

(3.7.5b)

l

Comparison of Eqs. (3.7.4b) with (3.7.2b) or of Eq. (3.7.5b) with (3.7.3b) leads immediately to the schematic interrelation K I-[(a+)V'--K~a,

(3.7.5c)

l

where the parametric dependences on T, P, and qt+ have been suppressed. As indicated above, an enormous variety of equilibrium constants may be constructed, depending on how one specifies composition variables, what value is selected for q+, and whether one elects to refer/zi to a standard or to a reference chemical potential. This shows that while the equilibrium constant is a useful quantity for characterizing chemical equilibrium, it is not a fundamental concept in the thermodynamic sense, since it cannot be uniquely specified. To prevent proliferation of so many different quantities, we shall henceforth restrict ourselves to equilibrium parameters such as (3.7.3a) or (3.7.5a) that are related to the chemical potentials of the various species in their standard state; this is an almost universally accepted practice.

3.7.2 EquilibriumConstants Referred to the Standard State If one elects to set q+ - q* in Eq. (3.7.3a), then ai(T, 1, q*) -- q* in (3.7.3b); moreover, lzi(T, 1, q*) is the chemical potential of pure i,/x*(T, 1), regardless of the choice of qi (see Section 3.6). One then obtains the result l n / C - -(RT)-I Z 1)lbl,i(Z, l

1)- Z vl[lnal(T, P, ql)]eq' l

(3.7.6a)

CHARACTERIZATION OF EQUILIBRIUM IN NONIDEAL SOLUTIONS

181

or (3.7.6b)

l -

ql eq - -

l

q?(T, 1) eq

.

(3.7.6c)

One should note that E(T, 1) here involves the relative activities. When mole fractions are adopted as composition variables Eq. (3.7.6b) exhibits a special feature: whenever q indexes a pure condensed phase (denoted by s) the corresponding factor in the product reduces to [ys(T, P, qs)a*(T, p)]vs _ [Fs(T, p, qs)]v', which will later be shown not to differ significantly from unity. Thus, the only terms that contribute substantially to (3.7.6b) are factors relating to species actually dissolved in solutions. The above scheme has the further advantage that only a single/C(T, 1) is invoked, regardless of what composition variable is selected, and that, according to Eq. (3.7.6a), this quantity varies only with temperature.

3.7.3 Equilibrium Constants, Hybrid Procedure The second scheme, which is more generally used, involves a hybrid procedure patterned after the methodology of Section 2.11. Here one distinguishes between pure condensed phases, indexed by the symbol s, and components forming homogeneous mixtures, indexed by the symbol j. For the pure condensed phases one adopts Eq. (3.6.2) in the specification of the chemical potential; for species in solution it is conventional to introduce Eq. (3.5.21). The equilibrium condition for the reaction ~-~lvlAl = 0 is now specified by

O -- Z Vsll~s -+-Z Vj . j l s

j ,

=

(T,

eq *q

+

s

(T,

j

+ R T ~ v s l n [ O*(T' P ) ] + R T Z v j l n [ o j ( T ' P)]eq" s q*(T, 1) J

(3.7.7)

This may be rearranged to yield the following definition of an equilibrium constant:

lnKq =__-(RT)

1

- [ s~

Vs#s(T, *

1) + ~

J

vjlzj (T, 1) *q

=Zvsln[a*(T'P)]+Zvjln[aj(T,P, s q*(T, 1) J

]

qj)]e q,

(3.7.8a)

182

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

or alternatively,

Kq(T, 1 ) - I-I[cts (T, P, q.)]Vs i-i[aj(T ' p, qj)]Ve; s j = { ~ [ y s ( T , P , q * ) a * q ( T , P ) ] vs

,q

vj [

Fl[yj(r, e, qj)aj

•

(T, P)]eq / R q j

J

Vj

[eq'

(3.7.8b)

J

where once again for pure condensed phases s, (qs ~q's) - 1. Note that, as in the case of ideal solutions, the equilibrium constant involves a ratio of factors for the equilibrium concentration variables q j, raised to the appropriate power; however, in the present case this ratio is preceded by a host of factors, enclosed in curly brackets, that attend to the nonideality of the constituents engaged in the chemical reaction. Departures from ideality of the pure components are discussed below; for intermixed components one must look up in appropriate tabulations values of the various activity coefficients Fj (T, P, qj) -yj(T, P, qj)a; q (T, P). In principle, then, we have arrived at an appropriate formulation of the equilibrium constant for nonideal cases. In the above approach every choice of composition variable q carries with it a different equilibrium parameter Kq; more will be said about equilibrium parameter choices at a later stage. 3.7.4

Free Energy Changes

An alternative, equivalent point of view emerges by examination of the interrelation between A Gd and ln K~. On writing A G d - Y~l vl#l and utilizing (3.6.2) and (3.7.6a), one finds

AGa - Z vll~l(T, 1) + RT ~ vt In at(T, P, ql)

l

(3.7.9a)

l

= -RTlnK_.(T) + RT Z vl In ctl(T, P, ql) l

(3.7.9b)

= - R T Z vl ln[n/(T, P, ql)]eq

l + RT Z vt In at(T, P, ql).

1

(3.7.9c)

CHARACTERIZATION OF EQUILIBRIUM IN NONIDEAL SOLUTIONS

On introducing the definition Y]l vl#~ (T, 1) -- AG ~ the important relation

183

Eq. (3.7.6a), one obtains

AG O* ln/C - - ~ .

(3.7.10)

RT

From the definition we find that A G O* is the differential Gibbs free energy per unit advancement of the reaction Y]l vtAz - 0 when all components are in their pure state under standard (unit) pressure. That such a system may not be experimentally accessible is not of concern, since one can simply look up tabulated values of #~(T, 1) for all the pure components and thereby obtain AG ~ Using Eq. (3.7.10), Eq. (3.7.9b) becomes

AGd -- AG O*+ RT Z Vl In al(T, P, ql).

(3.7.11)

l We return to Eq. (3.7.10) to note the very fundamental interrelation between (i) the free energy change per unit advancement of the reaction, as specified by Y]l viAl -- 0, when all participating species are isolated and maintained at standard conditions, and (ii) the natural logarithm of the equilibrium constant pertaining to the reaction in question. One is equally at liberty to make a distinction between pure condensed phases and homogeneous mixtures. In that event one obtains from (3.7.8a) the relations

A G d -- Z

VslZs -k- Z s

- Z

VjlZ j j

,

Vs#s (T, 1) + Z

s

*q

vj#j (T, 1)

j

+ RT Z Vs In as (T, P, q*) + RT Z vj lnaj(T, P, qj) s

j

= - R T l n K q ( T , 1) + RT Z

Vs lnas(T, P, q*)

S

4- RT Z l)j lnaj(T, P, qj).

(3.7.12)

J If one now sets AG dq - Y]s Vslz*(T, 1) + Z j 1)j]Zj*q (T, 1), one obtains an analogue of Eq. (3.7.10), namely,

In

g q ( Z ) -- -

Ao q e-------~'

(3.7.13)

184

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

and (3.7.12) may be rewritten as

AGd - A G Oq + R T Z

Vs lnas(T, P, q*) S

+ RT E

1)j lnaj(T, p, qj).

(3.7.14)

Note the distinction between A G ~ and AG O* by referring to Eqs. (3.5.21) and (3.6.2); see also Exercise 3.7.1. 3.7.5

Activities and Equilibrium Constants for Pure Condensed Phases

By way of review we see from Eq. (3.7.8b) that the equilibrium constant involves products of factors of the following forms" (i) the quantities Fs(T, P, q * ) y s ( T , P , qs*")as*q (T, P), which relate to pure condensed phases, (ii) terms involving the activity coefficients yj(T, P, q j), which correct for gross deviations from ideal properties of species making up homogeneous solutions, (iii) terms involving aj q (T, P), which, by (3.5.17) or (3.5.20), relate to the activities and activity coefficients of,pure j at pressures other than one atmosphere, and (iv) the usual products r-I j q/~J[eq that involve concentration units and which constitute the equilibrium constant for an ideal solution. We shall now show that the products in (i) generally differ only slightly from unity, so that for all but the most accurate calculations, these contributions are customarily omitted from the right-hand side of Eqs. (3.7.6b) and (3.7.8b). This step is frequently summarized by the statement that "the activity of all pure condensed phases is unity" but such a claim is not really tenable. In fact, we now provide a procedure to determine the activity of condensed phases, for use in accurate analyses of experimental data. Toward this goal we now introduce Eqs. (3.4.7), (3.4.5), (3.5.17), and (3.5.20); in each instance we may write

?'i(T, P, qi)a *q (T, P) -- ~.(T, P, qi) -

a i ( T , P , qi)

(3.7.15)

qi

with q - x, c, m and a i:~X (T, P) -- a i:r (T, P). In the further evaluation for a pure condensed phase we write d #s IT -- Vs d PIT -- R T d In as ( T, P, q* ) [T . Then, on integrating from P -- 1 to the pressure of interest we find

In[as(T, P,q*)] - I n [ a s ( T , 1, qs*)]- ~

i f PVs(T, p ) d p .

(3.7.16)

Ordinarily, for condensed phases Vs changes so little with pressure that it may be regarded as sensibly constant. Furthermore, according to (3.4.3), as(T, 1, q*) --

CHARACTERIZATION OF EQUILIBRIUM IN NONIDEAL SOLUTIONS

qs*. For q -- x, x~9 --= 1; for q -- c, q* -- c s*(T, 1) 9 and f o r q - m , q * = m Eq. (3.7.16) becomes In Fs (T, P, q*) -- l n l a s ( T ' P ' q * ) ] ~ q*

(P - 1)V(T, 1, q*)

RT

185

s* . Then

(3.7.17)

Ordinarily, the ratio on the right-hand side is quite small, especially if the total pressure does not deviate greatly from unity. In that event,

Fs(T, P, q,) _ as(T, P, q*) = as (T, P, q*) ~ 1,

q;

(3.7.18)

i.e., the activity coefficients Fs and relative activities for pure condensed phases differ very little from unity. Also, Eqs. (3.7.18) and (3.4.3) show that Fs(T, 1, q's) - as(T, 1, qs*) - 1, i.e., the activity coefficient Fs and relative activity of all pure condensed phases is unity under standard conditions P = 1; in these circumstances the requirement as (T, 1, q*) -- q*s is automatically recovered. Eqs. (3.7.17) or (3.7.18) provide a quantitative means for evaluating the product Vs ( T, P , qs*")as*q (T, P), appearing in Eq. (3.7.8b). It should now be clear why this product is usually omitted from further consideration. The corresponding determination of the products for species in a mixture, y j ( T , P , qj*")aj*q (T, P), is more involved and will be considered in detail beginning with Section 3.11. For gaseous species the simplest procedure is to substitute fugacities in place of activities. This brings the machinery of Section 3.1 to bear on the problem. Formally, one may adhere more closely to the methodology of the present section by writing fi = Pi ( f i / Pi ) = Vg ( T, P, Pi ) Pi . The quantities Vg ( T, P, Pi ) -fi (T, P, Pi)/P are specified as shown in Section 3.1. 3.7.6

Discussion

At this point the troublesome question arises once more as to the significance of K or/C o r Kq when its numerical value and even its functional form depends on so many arbitrary choices. Here one must keep in mind that K or/C o r Kq was introduced in the process of establishing the differential Gibbs free energy change A Gd accompanying the chemical reaction }-~l viAl = 0. As is evident in (3.7.10) or in (3.7.13), the quantity AG o* - - R T l n l C , or A G ~ -- - R T l n K q has precisely the same functional form as the terms in Eq. (3.7.9c), and as the second and third term on the right of Eq. (3.7.12), except that in/C o r Kq, the Vj and qi are to be evaluated for the equilibrium state. Thus, the vagaries of choice adopted for/C o r Kq must be exactly matched in the specification of activities, activity coefficients, and q j ' s used in Eqs. (3.7.9) and (3.7.12). It follows that AGd is indeed invariant under changes in standard states, reference states, or methods of specifying the composition of the system. Since A Gd and other thermodynamic

186

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

state functions derivable therefrom (such as ASd or AHd) are the fundamental quantities of interest, the arbitrariness of/C or Kq causes no difficulty other than being a nuisance. It should be remembered that, once a choice of units and of standard state has been made, a value of/C or Kq >> 1 implies that A G O is a large negative quantity, and hence, that A Gd is also likely to be large and negative. Thus, equilibrium will be established after the pertinent reaction has proceeded nearly to completion in the direction as written. Conversely, for values of/C or Kq << 1, equilibrium sets in when the reaction is close to completion in the opposite direction. Thus, the equilibrium constant serves as an index of how far and in what direction a reaction will proceed, and this prediction does not depend on the arbitrariness discussed earlier. It should be clear that the equilibrium constants do not in themselves possess the same fundamental importance as the differential Gibbs free energies. However, the full utility of equilibrium constants will not become clear until some illustrative examples are provided below. Another important function of the equilibrium parameter involves the fact that the various concentrations or activities of species in solution at equilibrium are no longer independent. For example, in considering the equilibrated state among three dissolved species, as given by AzB -- 2A + B, one can no longer vary AzB, A, and B independently. Rather, having added AzB to a solvent a process occurs by which all three units ultimately are present at concentrations consistent with the specified equilibrium parameter. 3.7.7

Review Based on Mole Fractions

We summarize the principal findings, based on the use of Eq. (3.4.23) wherein mole fractions serve as composition variables and standard conditions are imposed: the general equilibrium requirement Z l ( 1 ) l l . Z l ) e q = 0 leads to the relation

y ~ VllZl (T, 1) + RT l

(/z)

Vl lnal(T, p, Xl)

-- O,

(3.7.19)

eq

which may be split up into components s that appear solely as pure condensed phases and components j, that constitute the solution. On rearranging Eq. (3.7.19) according to this pattern one finds (for q -- x only!)

J

= In Kx -- ~

Vs lnas(T, P, 1) + ~

s

(3.7.20a)

vj lnaj(T, P, xj) eq

j

or

vj

Kx(T, 1) -- 1 7 as(T, P, 1) vs 1-I aj(T, P, xj) s

j

. eq

(3.7.20b)

CHARACTERIZATION OF EQUILIBRIUM IN NONIDEAL SOLUTIONS

187

This represents the fundamental expression for the equilibrium constant. We also write

Vs#*(T, 1 ) + ~

AG,~* = ~ s

vjlz~(T, 1).

(3.7.21)

j

Then AG O* In Kx(T, 1) -

RT

(3.7.22)

and

AGd - AG O* + RT Z Vs lna,(T, P, 1) + E vj lnaj (T, P, xj) 9 (3.7.23) eq s j This summarizes the relations that are found to be most useful in the thermodynamic analysis of chemical equilibrium.

3.7.8

Pressure and Temperature Variations of the Equilibrium Parameter

The definitions of the equilibrium parameters for nonideal systems involve the chemical potentials of the pure constituents that undergo the chemical reaction of interest. Thus, they are either exactly the same, or differ only slightly, from those adopted for ideal systems. For this reason the methodology and the results of Section 2.11 may be taken over (with appropriate minor modifications, as necessary) and need not be repeated here. There remains the apparent contradiction of having the left-hand side of equations such as (3.7.6) or (3.7.8) specified in terms of unit pressure, while the righthand side involves an arbitrary pressure P. This matter is left as an exercise 3.7.5, that is to be resolved by the reader. EXERCISES 3.7.1 Compare and contrast Eqs. (3.7.11) and (3.7.14) and specify the standard or reference state that is associated with Eq. (3.7.14). 3.7.2 Provide estimates showing under what conditions yj(T, P, q)a; q (T, P) will deviate from unity by more than 10%. 3.7.3 Invent a semipractical scheme by which any reaction could in principle be carried out by holding all reagents and products at standard conditions. (Hint: recall van't Hoff's scheme.) 3.7.4 Write out expressions for Kx, Kc, and Kp for homogeneous gas phase reactions in terms of fugacities or fugacity coefficients, based on derivations analogous to those of Sections 2.9 and 2.11.

188

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

3.7.5 How can one reconcile the fact that the equilibrium constants on the left of equations (3.7.6) or (3.7.8) are specified at unit pressure while the functions on the fight involve any arbitrary pressure P ? Hint: consider the analogy with the concentration variable ci (T, P). What this function really specifies is the concentration variable under the existing experimental conditions, whose value happens to vary when temperature and pressure are altered. From the viewpoint of the experimentalist the T and P dependence is not germane to the experiment; all one needs to know is the concentration of i under the specified experimental conditions.

3.8

Variation of Activity, Activity Coefficients with Temperature and Presssure

The techniques employed earlier will now be used to investigate changes of activity and activity coefficients with temperature and pressure. For simplicity we abbreviate ~.(T, P, qi) by F/(q) and ai(T, P, qi) by a[ q). By straightforward combinations of Eqs. (1.10.13), (3.4.5), (3.5.1), (3.5.2), (3.5.12a), (3.5.19), with q -- x, m, we find that at constant temperature and composition

_(Olna} m))

0 In F/(m)

OP

OP

T,mj

(O,na:X')

_ T,m;

OP

OP

T,xj

T,xj

(Zi(T, P, mi) - f/i*(T, 1) RT

(3.8.1)

Note that the chemical potentials were specified relative to standard conditions. Thus, the differentiation (O#i/O P)z,xj -- Vi must be executed with care, since the partial molal volume of i in solution is taken at pressure P, so that l?i - 17'/(T, P), while for pure i, Vi* - Vi* (T, 1). When referring to concentration as to the unit of choice, we employ Eqs. (1.20.15), (3.4.5), (3.5.1), (3.5.2), (3.5.12a), (3.5.15), and the abbreviations L _= lim P --+ 1, as well as p = p (T, P), p* = p* (T, 1), to write ,..,

,..,

_ (01na:C )

OIn Fi (c))

OP

OP

T,xj

OP

T,xj

_ (Olnp

81np*)

+L

OP

T

o, ) T,xj

V(T, P, ci) - Vi*(T, 1) +L RT

_ (Olnp OP

T,x;

)T,xj

(*) OlnPi OP

T (3.8.2)

CALORIMETRIC FUNCTIONS OF STATE IN CHEMICAL PROCESSES

189

In like manner we find

(

01n~ .(m)) OT P,mj

_ ( Olna~x) )

_ (OlnFi (x)) OT

P,mj

OT

OT

P,xj

Hi(T, P, mi) - H*(T, 1) RT 2

P,xj

(3.8.3)

and

_ (O,n#)

0 In Fi (c))

OT

OT

P,xj

_

P,xj

(01np*) OT

P,xj +

OT

_(01np) p OT P,xj

_(Olna~ x) ) _

_

OT ISli(T,

P,xj

P, ci) -- H*(T, 1) RT 2 +

( 0*lnPi ) OT

( ) 0 lnp

p

OT

P,xj

(3.8.4) Here also, Hi, being derived from lZi (T, P, qi ), is the partial molal enthalpy of i in solution at pressure P, while H/*, being derived from lzi(T, 1, x*), is the enthalpy of pure i at unit pressure.

3.9 Calorimetric Functions of State in Chemical Processes 3.9.1 StandardEnthalpies of Elements Up to this point we have considered the thermodynamic characteristics of chemical processes that have been brought to a state of equilibrium among the participating chemical species. We now focus on a rather different subject, namely, the characterization of chemical processes themselves. This requires consideration of procedures for calculating the enthalpy change and related thermal characterizations for a change of one mole of a participating species in a given chemical reaction. 1 By way of preliminaries we must first consider the individual thermal properties of all the elements and compounds participating in the reaction. Since one cannot assign a unique value to the energetics of any material it is necessary to adopt conventions that lead to a proper specification of the energy changes in any chemical reaction. These matters are best illustrated by several examples provided below.

190

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

We begin with the characterization of pure elements: by convention, at any temperature T of interest the enthalpy of a pure element in its most stable state at that T and at a pressure P of one bar shall be assigned the value zero. Since the chosen pressure of 1 bar represents the selected standard state, we designate the standard molar enthalpy of pure i at temperature T by the symbol ~ .i,T 0 _ 0. However, the asterisk is ordinarily omitted, it being understood that element i is in its pure state. One must be careful in the application of this rule: for example, the stable configuration of carbon at room temperature and ambient pressure is graphite, not diamond. In the same vein, sulfur under these conditions is stable in the rhombic habit, and Sn, in the white rather than the grey crystalline state. As another example, Br2 at P = 1 bar and 300 K is a liquid, while as a participant in reactions at 500 K Br2 is in the gaseous state. 3.9.2

Standard Enthalpies of Formation of Compounds

Standard enthalpies of formation of compounds are determined by considering the chemical reaction that produces the material of interest from its constituent elements. As an example consider the formation of carbon dioxide C(s, graphite) + O2(g) -- CO2(g).

(3.9.1)

When the reaction is reported at a total pressure of 1 bar at T -- 298.15 K (the reader is asked to design a hypothetical set of conditions such that the total pressure during the reaction remains at 1 bar; see Exercise 3.9.2) calorimetric measurements yield a value of -393.5 kJ per molar advancement of the reaction Since by convention /_)0 i,T - 0 for graphite and for O2 gas at one bar the measured enthalpy is taken as the standard (molar) enthalpy of formation, A /-)~,298.15 ---- -393.5 kJ, of one mole of CO2(g) at a pressure of one bar at 9

298.15 K. 3 More generally, The standard (molar) heat of formation AlYt~,T of a compound is the measured enthalpy change in the formation of one mole of the compound at a pressure of one bar and at temperature T from its constituent elements in their most stable state. In the above example the CO2 formation was exothermic; the enthalpy of the compound is more stable than that of the elements from which it is formed. A contrary example is furnished through the reaction 2C(s, graphite) + H2(g) = C2H2(g),

(3.9.2)

for which A Hf298.15 ~0 --- +226.7 kJ/mol at room temperature, showing that C2H2 is energetically less stable than hydrogen gas and graphite.

CALORIMETRIC FUNCTIONS OF STATE IN CHEMICAL PROCESSES

191

The actual determination of heats of formation of compounds may require a rather involved methodology: (1) If gaseous elements are involved one calculates for each gas the appropriate A H in changing from an ideal gas state at temperature T and at 1 bar to the real gas under the same conditions, using the procedures cited in Section 1.13. 4 (2) The enthalpy of mixing of the pure elements at T and 1 bar are determined as specified by Eqs. (2.5.7-9). If deviations from ideality are important the procedure of Section 3.13 must be utilized. (3) The enthalpy change must be computed for altering each of the reagents from T and 1 bar to the conditions of the reaction, TR and PR. For this purpose one generally employs a relation of the form

AH

-- f TTRCp d T

+

f pPR(V

- TVol) dP,

(3.9.3a)

where the second term derives from Eq. (1.13.17). (4) The reaction is carried out, starting with the reagents under the conditions specified in (3.9.3a); the resulting enthalpy of formation of the compound is determined calorimetrically. (5) The enthalpy change is computed via Eq. (3.9.3a) for bringing the product from the final equilibrium configuration at TR and PR back to temperature T and 1 bar. (6) For a gaseous product one determines the enthalpy change involved in bringing gases from their actual state to their ideal state at T and 1 bar. (7) To list the enthalpy of formation at some temperature other than T one may use the relation

AIY-I~

- AIYt~ --

fT Tt Cp(T)

dT.

(3.9.3b)

The overall enthalpy change determined in all the above steps is clearly very different from the equilibrium quantity A Ha introduced in Section 2.9. 3.9.3

Standard Enthalpies of Reaction

Using the above procedure an extensive tabulation of A/-)~ T i values has been built up for a large collection of compounds. Once that is available the standard enthalpy change for any chemical reaction 1 Y-~i vi Ai --+ 0 of interest may be established from the general relation

A[~ROx(T) -- ~ Vi A [7-If,i(T). 0 i

(3.9.3c)

As an example, consider the reaction N2H4(1) + O2(g) -+ N2(g) + 2H20(1).

(3.9.4)

Aside from setting at zero the standard enthalpies of the elements one requires the quantities A ~ ~f , i - 50.42 kJ/mol and -285.84 kJ/mol for hydrazine and

192

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

water respectively at 298 K and 1 bar. On adding up these values algebraically (with due regard to the fact that the vi for reagents are negative), one finds that A H ~ - -622.09 kJ/mol. Here the '/mol' designation really shows that the calculation applies to one mole of advancement of the reaction as written (i.e., with ~. = 1 mol). It thus involves one mole of hydrazine or oxygen as reagents, and one mole of formation of nitrogen, while the enthalpy change per mole of water is half of the above. Again, these enthalpy changes must be clearly distinguished from the quantity A Hd. A very interesting example has been provided by P.A. Rock. 5 It deals with the isomerization reaction of n-butane to isobutane CH3CH2CH2CH3 --+ CH3CHCH3.

(3.9.5)

I CH3 Reaction (3.9.5) is accompanied by several competing reactions, so that it is difficult to determine the standard enthalpy of isomerization by direct experimental measurements. To address this problem on can determine the combustion of both isomers under standard conditions: 13 CH3CH2CH2CH3(g) + -:--O2(g) ~ 4CO2(g) + 5H20(1), z

13 CH3CHCH3 + -~-O2(g) -+ 4CO2(g) + 5H20(1),

A/-t011, (3.9.6a)

AH~

(3.9.6b)

I CH3 One then adds the measured enthalpies algebraically to obtain A H ~ -- AH~ AH~ -6.862 kJ/mol, showing that the isomer is the slightly more stable configuration. 3.9.4

Temperature Variation of Standard Enthalpy Changes

Generally, standard enthalpy changes are tabulated at T = 298.15 K. To convert these values to any other temperature we use the basic relation dI2IiIP -CipdT, to write, as in Eq. (3.9.3c), d ( A H ~ -- d(AC ~ dT, with AC"~ - Y~i l)iC~ , to obtain A/4~

A/4~

A~~

(3.9.7)

The temperature variation of the heat capacity of each participant in the chemical reaction must be specified so that the integration can be carried out. Here one must be especially careful not to extend the temperature interval of integration over such wide limits that changes in phase or in the most stable form of the elements or compounds are encountered.

CALORIMETRIC FUNCTIONS OF STATE IN CHEMICAL PROCESSES

3.9.5

193

Applications

It hardly requires much commentary to note that thermochemical information is largely used to categorize the usefulness of chemical reactions as sources of energy. Much of the energy needs of the world are met through combustion of petrochemicals or coal. The enthalpy values then provide listings to show what types of combustion processes yield the optimal energy outputs. Tabulations of this type also indicated a large enthalpy output when dimethyl hydrazine is combined with dinitrogen tetroxide. The reaction is spontaneous, a feature that led to its use in the Apollo Lunar Lander missions. Ammonium perchlorate is another energy source of considerable use: its decomposition gives rise to a lot of gaseous products, that, together with the energy released, is used as a thruster in rockets. 3.9.6

Entropy Changes in Chemical Reactions

Entropy changes accompanying a particular chemical handled by writing A S ~ ,RX -- Z

1)i ~0,,v-

reaction Zi 1)iAi --+ 0 are 3( . 9 . 8 )

i

The individual molar entropies under standard conditions of the various participating species are generally available in tabular form; these are determined as shown in Section 1.17. Alternatively, they are found via an empirical equation of state, as in Eqs. (1.13.10), (1.13.12), followed by an integration; the temperature of the reaction is a parameter. As explained in Section 1.17, if nuclear effects may be excluded, or any frozen-in disorder remains undisturbed (or any nonequilibreated condition is not altered), one may set the entropy o f all participants in the reaction, whether they be elements or compounds, equal to zero at T = O. This then establishes an absolute scale for the tabulated entropies of the reagents and products at any temperature T. The reader is asked to set up strategies for determining AST,Rx for reactions under other than standard conditions; see Exercise 3.9.7. As an example one may cite the reaction 1 Zn(s) + ~O2(g) --+ ZnO(s), Z

(3.9.9)

for which ~0298.15 (J/K mol) - 41.63, 205.03, 43.64 and ~,0P,298.15 (J/K mol) 25.40, 29.35, 40.25 for Zn, O2, and ZnO respectively. Thus, for the reaction as written, AS~ =-41.63102.52 + 4 3 . 6 4 - - 1 0 0 . 5 1 J/Kmol. Also, ignoring any change of heat capacity with temperature we calculate at 1 0 0 ~ A8~ -- A8~ + AC 298.15 0 ln(398 915/298 " 1 5 ) - - - 1 0 0 . 5 1 + ( 4 0 . 2 5 - 1 4 . 6 7 - 25.40)1n 1.134 = - 1 0 0 . 5 6 J / K m o l , a very small change in the entropy over a 100 ~ interval.

194

3.9.7

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

Standard Gibbs Free Energy Changes

The standard Gibbs free energy changes accompanying a chemical reaction are handled in a manner completely analogous to that described in the preceding subsections. By convention, at any temperature T of interest the Gibbs free energy of a pure element in its most stable state at that T and at a pressure P of one bar shall be assigned the value zero.

Sometimes the requirement of the most stable configuration is dropped, but this cannot be done if one also involves calculations of enthalpy changes. The standard Gibbs free energies of formation of compounds are based on appropriate synthetic reactions in a manner analogous to the determination of enthalpies of compounds from their elements. For more general reactions E i Vi A i ~ 0 one writes

A a ~ ( T ) -- Z

1)i Aa~,i (T),

(3.9.10)

i where the Gibbs free energies of formation of the compounds, A G~, T,i' are available from tabulations, usually compiled at 300 K; the entries are generally based on calculations via A G}, r,i - A/-)~, r,i - T A S}, r,i" Other methods include emf determinations, taken up in the next chapter. As an example, consider the reaction at 298.15 K: C2H5OH(1) -k- 302(g) --+ 2CO2(g) at- 3H20(1).

(3.9.11)

One may consult tabulations of molar Gibbs free energies to find A G~ (kJ/ mol) = - 1 7 4 . 8 9 , 0, -394.36, -237.18 for the species written out left to right in Eq. (3.9.11). Accordingly, AG098.15,RX : 3(-237.18) + 2(-394.36) (-174.89) = -1325.37 kJ/mol at 298.15 K. The free energy change at 50~ is found by consulting the entropy listings for the species in the order written in Eq. (3.9.11), left to right: S~ ( J / m o l ) - 160.7, 205.03, 213.64, 69.91. Neglecting the changes of entropy with temperature over the indicated interval we then note that AG~ = -1325.37 - (50)[3(69.91) + 2(213.64) 3(205.03)- 160.7] = -1318.41 kJ/mol, a small change relative to the room temperature value. 3.9.8

Tabulations

With assignment of zero for the standard enthalpies of pure elements one may equate A t ~ i ,, T with the standard enthalpy of the compound species i ' ~ 0i,T" This quantity at temperature T may then be taken relative to the molar enthalpy at 298 K or at 0 K, /_)0i,298 or /_)0i,0. The difference is ordinarily tabulated as

CALORIMETRIC FUNCTIONS OF STATE IN CHEMICAL PROCESSES

195

or as ( ~ 0i,T _ 171,Oo) , / T, since these ratios vary only slowly with temperature. Of course one also encounters tabulations of (/4~ 8 - / 4 ~ 0) / T but they tend to be less complete. One may then adjoin the standard molar entropies to obtain (/~o, T _ ~ oi , 2 9 8 ) / T

'

~0

t,T

_

9o

/~0

_/~0

i,T

i,298

i,298

T

~0

_

(3.9.12a)

t,T

T

and ~9 ~ t,T - I~ i,O

1:I ~ - f t ~ 1,T i,O _

T

~0

(3.9.12b)

t,T"

T

For interpolation purposes one may use the relationship, valid at P - 1 bar: 0~

- 0~

S~

--

dT.

(3.9.13)

We call attention one last time to the difference between the change in Gibbs free energy invoked here and the differential Gibbs free energy expression that was set up in Section 2.9, while discussing chemical equilibrium. 3.9.9

Integral vs. Differential Heats of Dilution

Heats of solution are not constant but generally vary with concentration of the components. For example, when HC1 is dissolved in water A H / m changes from - 1 7 . 9 to - 1 7 . 4 kcal/mol as one proceeds from unit molality to infinite dilution. To handle such cases one distinguishes between integral heats of solution, A H / m , and differential heats of solution (OAH/Om)T,p that pertain to the addition of an infinitesimal amount of solvent to a solution of molality m. The two heats may be related in the following manner: let H, H~, H I represent the enthalpy of a binary solution and of components 1 and 2 in pure form relative to any arbitrary standard value for the enthalpy. Then the enthalpy of mixing of n l moles of solvent and n2 moles of solute is given by

A H -- H - nlI4~ - n214~,

(3.9.14)

so that the differential heat of solution with respect to mole numbers reads -

On2

T,P,nl

~ --

H~

T,P,nl -

=

r162

= L 2 - L 2,

-

-

-

~ )

(3.9.15)

where the enthalpies were determined relative to those at infinite dilution, ~o.

196

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

For the solvent one finds by analogy: OAH

Onl 3.9.10

-- /-tl-

(3.9.16)

/-'tt"

Adiabatic Combustion of Methane

As an illustration of the above exposition we consider the adiabatic combustion of methane and ask for the final temperature that may be achieved in this process. The relevant reaction is written as CH4(g) + 202(g) ~ CO2(g) + 2HeO(g),

= - 1 6 5 . 2 kcal/mol. (3.9.17) Initially CH4(g) is heated in an inert atmosphere to 573 K. Air is then admitted and the reaction is allowed to go to completion. For later use one requires molar heat capacity tabulations for all the species in the above reaction. Since a direct determination of the final temperature is difficult we consider a sequence of steps for which the enthalpy changes may be readily determined, and which are then summed. (i) Cool the products from the final temperature T f to 298.15 K. (ii) Cool the reagents from the starting temperature of 573.15 K to 298.15 K. (iii) Carry out the reaction at 298.15 K; here the tabulated heats of formation may be used in conjunction with Eq_ (3.9.3c) to determine the corresponding enthalpy change, given by A H -- A H i + Anii -t- Aniii - - 0 for an adiabatic process. This may be translated into the following equation:

98

AH~

98

-- - A / 1 0 9 8 - n [ c n 4 ( g ) ]

f298

Cp[CH4(g)]dT

J473

- n [02

(g)] f

298

./473 298

-

n [N2(g)]

fd473

Cp [02 (g)] d T Ce [N2(g)] tiT.

(3.9.18)

Here one must take into consideration that all the oxygen was used up in the combustion process, whereas n[N2(g)] - 2(0.79/0.21) is the number of moles of N2 present in the air that was used in the combustion process. On inserting the relevant relations for the various molar heat capacities and inserting the known value for the molar enthalpy of step (iii) one may numerically determine the upper limit, T f , of the integrals on the left of Eq. (3.9.18). This yields a value of TZ -2265 K, which is an upper limit, since the reaction is not totally adiabatic, is not 100% complete, and because some dissociation of the products takes place at the elevated temperatures achieved in the combustion.

EQUILIBRIUM CALCULATIONS

197

REFERENCES AND COMMENTARY 3.9.1. Up to this point we have designated the chemical reaction as Z i viAi --0. This was to indicate that we treated the process as having come to a quiescent state, only small deviations from equilibrium being permitted. In this section we consider actual chemical reactions that are permitted to proceed on a macroscopic scale; the corresponding reaction will be designated as Zi 1)iAi ~ O. 3.9.2. Devise a hypothetical set of conditions under which reaction (3.9.1) can be carried out at a pressure of one bar. 3.9.3. Sometimes, for convenience, the designation T = 298.15 K is abbreviated to read 298 K; occasionally one even encounters the designation 300 K, which is taken to be room temperature. 3.9.4. This step is necessary because, by convention, the standard state of a real gas is actually the hypothetical state of the ideal gas at one bar. Allowance must be made for the difference between these two states in exact determinations. Since the enthalpy of an ideal gas is independent of pressure, the difference between its value and that of the real gas may be determined by integrating Eq. (1.13.17) between pressure 0 and pressure P. In general this value tends to be very small compared to the contribution from all other steps. 3.9.5. RA. Rock, Chemical Thermodynamics, University Science Books, Mill Valley, CA, 1983, p. 195 ft. 3.9.6. Describe procedures that can be used to change from standard enthalpy changes to standard energy changes. 3.9.7. To determine entropies of reactions under conditions different from 298.15 K and 1 bar, revert to appropriate expression for the variation of the individual entropies, Si, with T and P. Identify two methods that can be used for this purpose. Set up the requisite expressions and discuss your strategies. 3.9.8. Devise strategies for determining enthalpy changes and Gibbs free energy changes at pressures different from 1 bar.

3.10

Equilibrium Calculations

The representative calculations provided below illustrate some general features of equilibrium calculations.

3.10.1

Dissociation of Water Vapor

In this calculation we neglect the deviation of the vapor phase constituents from ideality. Let ot represent the degree of dissociation of water into its elements. Then the following scheme may be devised for the study of equilibration conditions:

198

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

Equation

2H20(g)

Relative m o l e n u m b e r s

1 -ot

=

2H2(g)

+

O2(g) c~/2

ot

Total: 1 + or/2 Mole fractions

Partial p r e s s u r e s

1 - - Ol

Ol

a/2

1 + ~/2

1 + ot/2

1 + c~/2

~

1-or

1 +~/2

Concentration

P

et

~

1 +~/2

P

c~/2 p

~

1 +~/2

1-or

ol

o~/2

V

V

V

Here P is the total pressure of the gas phase. The equilibrium 'constant' for the reaction as written is given by Ot3

Kx = 2(1 - c~)2(1 + or/2)'

P22 P02 Kp

ot3 P 1

p2 H20

Kc = 2(1

2(1 - c~)2(1 -k- c~/2) '

t3/3 - ot)2g"

(3.10.1a) (3.10.1b)

(3.10.1c)

Experimental work shows that at P = 1 bar, c~ -- 5.04 x 10 -3 and 1.21 x 10 - 2 at T -- 2000 K, 2200 K. Insertion into (3.10. lb) yields Kp -- 6.45 x 10 -8, 9.02 x 10 -7 bar at those two temperatures. Thus, the degree of dissociation is very tiny even at those elevated temperatures; the small value of Kp likewise shows that in the reaction as written the reagent heavily outweighs the products at equilibrium. Once Kp has been determined one can calculate the average enthalpy of dissociation close to 2100 K, associated with the equilibrium state, according to the relation A H ~ -- [ R T 1 T z / ( T 2 - T1)] l n [ ( K p ( T z ) / K p ( T 1 ) ] - 482 kJ/mol. The associated Gibbs free energy change at 2000 K is given by A G o - - R TIn Kp -275 kJ/mol, showing that the reaction proceeds spontaneously in the direction opposite to that written down above. One should note that all the above values change as the total pressure P is altered; the changes can be monitored through experimental observation. 3.10.2

Decomposition of Calcium Carbonate

As another excellent illustrative example consider the decomposition of calcium carbonate, CaC03 (s) - Ca02 (s) + C02 (g),

(3.10.2)

EQUILIBRIUM CALCULATIONS

199

for which we set up an equilibrium constant a (CaO, s) K -- f(CO2,

g)

a(CaCO3, s)

.

(3.10.3)

The fugacity is found from Eqs. (3.1.6,7): l n ( f / P ) - fo P ~ b d P = b P RT RT'

f = exp ~b .P -P RT

(3 10.5a) "

To calculate the fugacity of CO2 we introduce the following empirical equation: P 17' -- R T + bP,

b - 42.7 cm 3/mol,

(3.10.4)

with which, one finds f--

Pe ~

(P in bar).

(3.10.5b)

We next attend to the ratio a (CaO, s) a (CaCO3, s)

expI

]

(3.10.6a)

where Eq. (3.7.17) had been invoked, based on the assumption that the molar volumes are sensibly constant. Density measurements yield to the values iS'(CaO, s) - 16.76 cm 3/mol and I7'(CaCO3, s) - 34.16 cm 3/mol, so that a (CaO, s) a (CaCO3, s)

= exp

[y](P - 1)]

I

T

y - - 0 . 2 1 2 bar- 1

(3.10.6b)

Putting (3.10.3), (3.10.5b) and (3.10.6) together one finally obtains Kp _ (peO.52P/T) e -0.212(P-1)/T

(P in bar).

(3.10.7)

At 1300 K the equilibrium pressure is P = 3.403 bar; then the second exponential factor in Eq. (3.10.7) has the value 0.9996. This verifies by an illustrative example that the ratio of activities, such as Eq. (3.10.6a), can ordinarily be set equal to unity. The equilibrium constant at 1300 K has the value K = 3.406. For any other temperature the equilibrium constant can be found by inserting Eq. (3.10.4), at the temperature of interest. 3.10.3

Synthesis of Ammonia from the Elements

The reaction under study is given by 1 3 ~N2(g) q-- ~H2(g) - NH3(g),

(3.10.8)

200

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

which provides an example of how equilibrium states may be characterized selfconsistently. Experimental studies have shown that at 450 ~ and at 300 bar the equilibrium concentration of NH3 achieved with an initial 3 : 1 mixture of H2 : N2 stands at 35.82 mol %. Under the prevailing conditions the gaseous fugacity coefficients are specified by y(H2) = 1.08, y(N2) -- 1.14, and y(NH3) = 0.91, as determined by the methodology of Section 3.1. The equilibrium constant for reaction (3.10.8) is specified by

Kp=

f(NH3) f 1/2 (N2) f3/2 (H2) v(NH3) n(NH3) n y1/2(N2)y3/2(H2) nl/2(N2)n3/2(H2) p '

(3.10.9)

where n(X) is the mole number of species X, n is the total mole number of all species, and P is the total pressure. Initially, n0(N2)/n = 1/2, n0(H2)/n = 3/2, no(NH3)/n = 0. Let x be the fraction of NH3 prevailing at equilibrium, such that n(N2)/n -- (1 - x)/2, n(H2)/n = 3(1 - x)/2, n(NH3)/n = x, and n = 2 - x. Then at 300 bar 0.91 x 2-x bar -1. (1.14)1/2(1.08)3/2 [(1 - x)/211/213(1 - x)/2] 3/2 300 (3.10.10) Under prevailing conditions it was found that x = 0.5275, whence Kp = 6.79 x 10 -3 bar-1. Thus, under these conditions the production of ammonia from its elements is far from complete. The above also illustrates the arbitrariness in the specification of Kp. The numerical value of the equilibrium constant would have been different if another pressure unit had been used or if some other method of expressing concentrations had been employed. Nevertheless, once the equilibrium constant has been determined for a particular set of physical parameters, this same value can be used to determine x for any other set of conditions at the same temperature. The above value of Kp holds so long as P is expressed in bars. It is this feature that renders the concept of equilibrium constants very useful.

Kp --

3.10.4

Hydrolysis of Ethyl Acetate

As an example of equilibrium in solution consider the following hydrolysis reaction: H20(1) + ESAC(aq) = ACOH(aq) + ETOH(aq),

(3.10.11)

where ESAC - CH3CH2OCOCH3, ACOH _-- CH3COOH, and ETOH -CH3CH2OH stand for acetic acid ester, acetic acid, and ethanol respectively. The

DETERMINATION OF ACTIVITY COEFFICIENTS

201

corresponding equilibrium constant has the form given by Eq. (3.7.8b),

Km

--

aACOHaETOH aH2OaESAC

=

mACOHmETOHFACOHFETOH

.

(3.10.12)

mH2OmESAC /-'H20FESAC

Under standard conditions mH20 = 55.54 mol/1 and as a good first approximation FH20 -- 1. For nondissociating chemicals of the type treated here and in sufficiently dilute solutions the deviations from ideality may be ignored. One can then determine Km from a measurement of the observed molality of each component in a very dilute solution. This value of Km applies to any other solution generated under the same prescribed conditions of temperature and pressure. Once Km has been fixed in a given experiment the activities of the three solutes under other conditions are interrelated as shown in Eq. (3.10.12).

3.11

Determination of Activity Coefficients by Vapor Pressure Measurements

Up to this point the treatment of activities and activity coefficients has been formal, since we have not specified how these quantities may be actually be experimentally determined. We now address this matter.

3.11.1

Vapor Pressure Measurements Based on Raoult's Law

One commonly used method involves a measurement of the vapor pressure of a binary solution; at a fixed temperature the chemical potential of each species in the gas and in the vapor phase is the same. We employ Eqs. (3.6.4a), (3.6.2) with a--= a, and (3.1.4a) to write /t~

-

#*X(T,

1)-

RTln[yi(T, P, xi)a*X(T, P)xi] - RTln fi(T, P, Pi)

= RTln[ai(T, P, xi)/fi(T, P, Pi)].

(3.11.1)

We next cite the equilibrium condition involving pure i in the liquid and vapor phases, for which xi - 1 and Pi - Pi*, so that #~

#*X(T,

l)-

RTln[yi(T, Pi*, 1)a'X( T,

Pi*)] -

RTln fi(T, Pi*, P*)

= n r ln[ai(r, Pi*, 1 ) / f / ( r , Pi*, Pi*)]"

(3.11.2)

Equating the above relations leads to

fi(T,P, Pi) ] ai(T, P, xi) - ai(T, Pi*, 11 fiiT,, P?I Pi*) "

(3.11.3)

202

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

p?

E

p~

B C

0

Fig. 3.11.1. Diagram illustrating the determination of activity coefficients relative to the pure solvent (ACB) with vapor pressure P*, and to infinite dilution (DFE), based on Henry's law, associated with the extrapolated pressure P$i"

As discussed earlier, the activity of a pure condensed phase is ordinarily close to unity, and the ratio of fugacities may usually be replaced by the ratio of vapor pressures. In these circumstances one finds the approximate relations

P/ ai(T, P, xi) ,~ p,~

or

P/

yi(T, P, xi) ,-~ Xi P/* '

(3.11.4)

which are cited in elementary texts. The physical significance is perhaps best illustrated in Fig. 3.11.1. The quantity xi P* is represented by the arrow A---~,which represents the vapor pressure that would prevail if the solution were ideal. By contrast, Pi (arrow of length ~ ) represents the vapor pressure of the actual solution. It is clear that Raoult's Law is involved in these calculations. The present experimental approach is useful primarily when the species of interest is the solvent. In more accurate calculations one must determine the fugacities by the methods of Section 3.1 and the activities of the pure phases by the methods of Section 3.7. 3.11.2

Vapor Pressure Measurements Based on Henry's Law

An alternative procedure rests on the choice of a reference solution at infinite dilution. This method is useful if the component under study is the solute. As mentioned in Section 3.3, experimental investigations have shown that at very great dilution the vapor pressure of the solute obeys Henry's Law, Pi = KHXi, in the limit xi ~ O. The straight line OP * of Fig. 3.11.1 shows the vapor

DETERMINATION OF ACTIVITY COEFFICIENTS

203

pressure of i if this component were to obey Henry's Law over the entire composition range. Based on this scheme one selects as a standard the hypothetical Henry's Law substance i in pure form, that is produced by extrapolating Henry's Law to xi -- 1, where its vapor pressure would be P/~. The fact that actual solutions do not behave in this manner is no deterrent to the use of P/~ rather than P/* as a reference state. This choice, rather than being an absurd procedure, is a matter of convenience. For, near infinite dilution the solute molecules are essentially isolated and cannot interact. The fictitious Henry's Law material thus has properties akin to an assembly of ideal, noninteracting units, that serves as a good reference standard. In the present case Eq. (1) remains applicable, as before. However, the gas phase of the hypothetical material would have a chemical potential of the f o r m / z i - / Z ? P(T) + R T In 3~(T, Pi$, Pi~) and in the condensed state, the form (3.5.21a), with P replaced by have #OP(T)-

Pi~. Hence, in place of Eq. (3.11.2) we now

#*X(T, 1 ) - RTln[yi(T, Pir lla*X(T, P / e l ] - RTln fi(T, Pir = RTln[ai(T, P*, 1)/f(T, Pir P/el].

Piff)

(3.11.5)

When account is taken of Eq. (3.11.1) one obtains

ai(T,P, xi)--ai(T,P*,l)

[ fi(T,P, Pi)~) )~iT, P ? I "

(3.11.6)

To a good degree of approximation this relations reduces to

Pi Pi ai(T, P, xi) ~ __~ or yi(T, P, xi) ~ xiPi ~.

(3.11.7)

To use this result one must execute detailed measurements of the vapor pressures of the actual solutions at great dilution, where Henry's Law is obeyed. Extrapolation of the resulting straight line to the composition of pure solvent then establishes P/~. Now the vectors ~ and D-~ in Fig. 3.11.1 correspond to xi Pi~ and Pi respectively. A measurement of Pi at the particular composition xi then yields the activity or activity coefficient in this regime. For more precise work Eq. (3.11.6) must be used, in the manner discussed earlier. One must not adhere to the mistaken notion that the analysis leading to Eqs. (3.11.3), (3.11.4) should be used in the upper range of xi and that the analysis leading to Eqs. (3.11.6), (3.11.7) should be used in the lower range. The two approaches are based on the use of different reference states, Pi* and Pi~, and are therefore not interchangeable. One must stay with one or the other scheme to obtain internally consistent results. The choice is simply one of convenience, depending on whether one investigates the properties of i in concentrated or dilute solutions.

204

3.11.3

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

Activity Coefficients Based on Molarity or Molality

In principle, one can use Eq. (3.11.6) in Eqs. (3.5.8) and (3.5.10) so as to obtain yi(T, P, ci) and yi(T, P, mi). However, one can determine activities and activity coefficients relative to ci or m i directly from experiment. This involves the use of Eq. (3.5.21b) that specifies/Zi for the ith species in the condensed phase, and correlating this relation with Eq. (3.1.4a) for the gas phase. This yields - #*C(T, 1) -- RTln{yi(T, P, ci)a*C(T, P)ci(T, P)}

p~

- RT ln fi (T, P, P~).

(3.11.8)

Here we have adopted as a reference state the hypothetical solution in which the Henry's Law region is extrapolated to the value ci - 1, as indicated in Fig. 3.11.2. We thus set the corresponding pressure as Pi - P~. For such a hypothetical solution at a total pressure P~ P~

-

~i

9c (T, 1) -- RTln{yi(T, P', 1)a i,C(T , p,)} - R T l n fi.(T, P', P[]).

(3.11.9)

Juxtaposition of (3.11.8) and (3.11.9) yields the relation

yi(T, P, ci)a*C(T, P) = Yi (T, P', 1)a*C(T, P')

fi.(T, P, Pi) f i ( r , P', P~)ci(T, P)'

R

/ Z

// /

"

(3.11.10)

I.//

//I '

. . . . . . .

F

I

c

I ....

0

Ci=l

Ci

---

Fig. 3.11.2. Diagram illustrating the determination of the activity coefficient y i { T, P i , Ci }; see text.

DETERMINATION OF ACTIVITY COEFFICIENTS

205

which simplifies to the following very approximate relationship"

Pi

?'i(T P Ci) ~ '

'

(3.11 11)

Ci P i D "

The calculations are illustrated in Fig. 3.11.2 by vectors G]/G--I~. Finally, the activity coefficient with respect to molality may be determined in precisely the same manner as sketched above. One equates (3.5.21 c) with (3.1.4a) to obtain the analogue of (3.11.8), with c replaced by m. One then chooses as a reference solution the hypothetical Henry's Law case in which the straight line region of the P versus m i curve at low m i is extrapolated to m i = 1. Let the corresponding hypothetical vapor pressure be Pie . This leads to an equation of the form (3.11.10), (3.11.11) and to the relation

?'i(T, P, mi)a*m(T, P) -- gi(T, P', 1)a*m(T, P')

fi(T, P, Pi)

(3.11.12) f i ( T , P ' , p/O) mi ,

and to the very approximate relation

?'i(T, P, mi) ,~

Pi mi P?

.

(3.11.13)

The calculations are illustrated in Fig. 3.11.3 by the vectors ~/J--~. In passing from (3.11.10) to (3.11.11) or from (3.11.12) to (3.11.13) one is forced to set P - P' (which is not a very restrictive assumption), and to assume

I K

f

. . . . .

9

I

!J

I

0

i

mi=l mi _._....~.

Fig. 3.11.3. Diagram illustrating the determination of the activity coefficient y i ( T , P i, m i); see text.

206

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

that Yi (T, P, 1) = 1 when Ci = 1 or m i = 1. The latter assumptions are extremely questionable, because at unit molarity or molality one is generally far removed from ideality. However, one can improve the procedure by an iterative technique, employing (3.11.11) or (3.11.13) to determine ?'i(T, P, 1), and then substituting this value in (3.11.10) or (3.11.12) respectively. Using the condition P = P' and replacing the terms in braces with P i / c i P[~ o r P i / m i Pie, respectively, yields a better value of ?'i (T, P, ci) or of ?'i (T, P, m i). If desired, further iterations may be performed, by concentrating on measurements near unit molarity or molality that yield new values of ?'i (T, P, 1) which may then be used on the fight-hand side of (3.11.10) or (3.11.12) for improved determinations of yi(T, P, qi). 3.11.4

Summary; Numerical Example

After the welter of different reference states, activity coefficients, activities, methods of specifying compositions, and deviations from ideal behavior, the reader may be excused for wondering whether a unified description of physical phenomena has in fact been achieved. This may be checked by noting whether the change in chemical potential for any given process is indeed independent of all arbitrary features that have been introduced in Sections 3.4-3.10. We shall employ a simple numerical example to check on the overall consistency, employing well established data relating to the Br2-CC14 system at room temperature. The vapor pressure of Br2, corresponding to various compositions of the liquid phase, as read off from a published graph*, is shown in Table 3.11.1, and Table 3.11.2, part (a); also included is the value of the vapor pressure P/~ attained by extrapolation of the Henry's Law region to x - XBr2- 1. Shown in Table 3.11.2 are corresponding activities and activity coefficients based on Raoult's Law, when Br2 is considered as the solvent, or based on Henry's Law, when Br2 is considered as the solute. The calculations were carried out in conformity with Eqs. (3.11.4) and (3.11.7), which involve the assumption that Br2 is an ideal gas and that the deviations of a i9x (T, P) from unity may be neglected. One should observe that the activities and activity coefficients in schemes (a), (b), and (c) are widely different. As an example, consider the change in chemical potential A # when the mole fraction of Br2 is increased from Xl - 0 . 4 to x 2 - 0.8 at constant temperature. Then, according to Table 3.11.2, for the gas phase A # - R TIn Pf/Pi = Table 3.11.1 Tabulation of Br2 vapor pressure as function of Br2 mole fraction in Br2-CCI 4 solutions at room temperature*

x P (atm)

0 0

0.2 0.095

0.4 0.145

0.6 0.183

0.8 0.217

1.0 0.280

(1.0) (0.539 = P:~)

*Data from G.N. Lewis and H. Storch, as cited by G.N. Lewis and M. Randall, Thermodynamics, 2nd edition. Editors K.S. Pitzer and L. Brewer. McGraw-Hill, New York, 1961, p. 254.

DETERMINATION OF ACTIVITY COEFFICIENTS

207

Table 3.11.2 Tabulation of activities and activity coefficients for Br 2 in Br2-CCI 4 solutions at room temperature (a) Gas

(b) Solvent

(c) Solute

x

P (atm)

y

a = P/P*

y

a = P/PD

y

0 0.2 0.4 0.6 0.8 1.0

0 0.095 0.145 0.183 0.217 0.280 = P*

1 1 1 1 1 1

0 0.339 0.517 0.653 0.775 1.00

1.92* 1.69 1.29 1.09 0.97 1.00

0 0.185 0.269 0.339 0.402 0.520

1.00 0.925 0.672 0.564 0.510 0.520 w

1.0

0.539 = P$

*Calculated as 0.539/0.280. w as 0.280/0.539.

RTln(O.145/O.217)a = RTln0.668. For case (b) where Br2 is considered as the solvent, A # = R T l n ( a f / a i ) b -- R T ln(0.517/0.775) = R T ln0.667. For case (c) where Br2 is considered the solute A # = R T l n ( a f / a i ) -- R T ln(0.269/ 0.402)c = R T In 0.669. The preceding calculations show explicitly that although the numerical values of activities and activity coefficients in Table 3.11.2 differ greatly for the different cases, the A # values are identical within experimental error. Thus, only A # is of fundamental significance; activities or activity coefficients have only a relative significance. If deviations from ideality had been ignored the value A # = l n ( x z / x l ) = R T l n ( 0 . 4 / 0 . 8 ) = RTln(0.500) would have been obtained, which is appreciably off the mark. One should also note that whereas for case (b) V tends to be greater than unity, }, is less than unity for case (c).

3.11.5

Correlation between Activity Coefficients for Components in a Binary Mixture

Frequently it is more convenient or only possible to measure activities or activity coefficients for a component that differs from the one in which the experimentalist is interested. In that case it is expedient to use the Gibbs-Duhem equation for a binary mixture. For a two-component system at constant T and P, we find n] d # j = - n 2 d#2. On account of Eq. (3.6.2) this may be rewritten as d l n a l ( T , P, ql) - n2 d l n a 2 ( T , P, q2)

(3.11. 4)

t/1

where q = x, c, m. Integration then yields [ Ctl(T' P' q_~~) } _ In ~iiT,, PI q~l) -

;

d In a2(T, P, q2).

(3.11.15)

208

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

As a special case we may consider the specification of composition via mole fraction, starting with a pure phase:

{

In al(T, P, xl) i l l (T, P, 1)

} f0 _ _

x: ~x2 , dlna2(T, P, x;). 1 -- X 2

(3.11.16)

Thus, if a2 is known as a function of x2, and on setting al (T, P, 1) -- a *1X (T, P) 1 the definite integral may be evaluated by numerical methods, yielding a l ( T , P , xl). 3.12

Oxidation Boundary for Magnetite-Zinc Ferrite Solid Solutions

In this section we cite a somewhat esoteric example to show how the standard chemical potentials~which so far have been left unspecified--can actually be determined experimentally. Consider the thermodynamic properties of a Fe304ZnFe204-Fe304 solid solution at the oxidation boundary separating hematite (Fe203) from magnetite-based zinc ferrite ZnFe204-Fe304 solid solutions. 1 We proceed with the following steps. 3.12.1

The Oxidation Process

We first write down the chemical reaction that schematizes the oxidation process, namely: (Fe304)l-x. (ZnFe204)x (s) -~-

1--x

O2(g)

3(1 - x)Fe203(s) + xZnFe204(s). 2

(3.12.1)

The solid solution (ss) on the left has been represented as a mixture consisting of 1 - x moles of magnetite and x moles of zinc ferrite per formula unit of ss. It is assumed that the zinc ferrite moiety remains passive while any changes relating to the oxygen/cation composition occur exclusively in the magnetite component. The free energy change accompanying reaction (3.12.1) at constant T and P is specified by

1-x AG(1)--/x[(Fe304)l-x. (ZnFe204)x (s)] - ~/x[O2(g)] + ~ / z 3 -( 1x)

[Fe203(s)] + x/z[ZnFe204(s)].

(3.12.2)

For solids we adopt the expression Eq. (3.5.21a), lzi(T, 1 ) - /z*(T, 1)-tR T l n a i ( T , P,x), and for gases we set lzg(T, 1) - / z ~ + R T l n P , thereby

OXIDATION BOUNDARY FOR MAGNETITE-ZINC FERRITE SOLID SOLUTIONS

209

ignoring deviations from ideality in the gas phase. For simplicity we further assume that we may set /z[(Fe304)l-x. (ZnFe204)x (s)] = x#[ZnFe204(s)] -t- (1 - x)#[Fe304(sss)],

(3.12.3)

where/~[Fe304(sss)] stands for the chemical potential of the Fe304 component that forms the spinel solid solution (sss) with ZnFe204, which must be carefully distinguished from/z*[Fe304(s)], that represents the chemical potential of pure magnetite. On applying (3.12.3) to (3.12.2) we find AG(1) -- - ( 1 - x)#[Fe304(sss)] - - T /-z lx

3(1 - x ) 1-x

-

~

R

T

[Fe203(s)]

1 -- x/zOP[O2(g)] 4

-- --(1 -- x)#* [Fe304(s)] +

[O2(g)] -t- 3(1 ~ / z 2-- x)

#* [Fe203 (s)] - (1

-

In ct[Fe304 (sss)]

x)RT

In P[O2(g)].

(3.12.4)

At equilibrium A G(1) = 0. Here P[O2(g)] stands for the oxygen pressure prevailing during the oxidation. 3.12.2

Oxidation Step under Standard Conditions

Consider the following oxidation step that converts magnetite to hematite, carried out under the conditions prevailing at the oxidation boundary: 1

2Fe304(s) + =O2(g) -- 3 Fe203(s), z for which the corresponding free energy change is AG(5)--2/z*[Fe304(s)]-

1. OP[O2(g)]

-

(3.12.5)

-RTlnP[O2(M-H)]

+ 3#* [Fe203(s)] - 0,

(3.12.6)

where P[O2(M-H)] is the oxygen pressure prevailing at the magnetite-hematite boundary. In the literature the following empirical formula has been proposed: log P[Oz(M-H)] = - 2 4 , 6 3 4 / T + 13.96 bar. When this expression is inserted in (3.12.6)one finds -/z*[Fe304(s)]- ~1/zOP[O2(g)] -t- 3/z*[Fe203(s)] 2.303RT[ 24,634 ] 4 T + 13.96 .

(3.12.7)

210

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

This solves the task at hand, showing how the appropriate combination of chemical potentials under standard conditions has been determined empirically at any temperature T.

3.12.3

Oxygen Fugacity at 1500 K

We can go a step further to specify the equilibrium vapor pressure at the twophase boundary: insert (3.12.7) into (3.12.4) with AG(1) = 0, so as to solve for log P [O2(M-H)] - - 4 log ct[Fe304(sss)] +

-

T

at- 13.96 .

(3.12.8)

In first approximation a = 1, so that the first term on the fight drops out. To improve on this step one may use Eq. (3.7.16) to determine the logarithmic contribution, but in view of the approximations introduced earlier it is doubtful whether such a step is justified. As a typical value one finds that at T = 1500 K, P(O2) = 3.45 x 10 -3 bar. REFERENCE

3.12.1 E Wang, Q.W. Choi, J.M. Honig, Z. anorg, allg. Chem. 550 (1987) 91. 3.13

Activity of Solvent and Solute from Lowering of the Freezing Point of the Solution

3.13.1

Activity of the Solvent

The activity of the solvent may be directly written down in analogy to Eq. (2.8.16). One should recall, however, the various approximations that had been introduced in deriving this expression. These generally hold because the fractional changes in the lowering of the freezing point are small. Here we replace x2 ~ - ln(1 - x2) - lnxl by - lnal x) to write ~

-lnal

-- RT~ +

~

RTf

~l

~s

2-R

2

-- - ln

-lnxl,

(3.13.1)

~

in which O =- To - Tf represents the freezing point depression, L1 is the molar heat of fusion of the pure solvent, and all other symbols retain their standard significance. If desired, one may everywhere replace In F(x) by In y(x) + lna~X. One notes that when O has been experimentally determined the activity coefficient

F(1x) can be found.

ACTIVITY OF SOLVENT AND SOLUTE

3.13.2

211

Determination of Activity Coefficients at Temperatures Other than the Freezing Point

The difficulty with Eq. (3.13.1) is that it yields F1(x) only at the freezing point TU of the solution. One generally wishes to know the activity coefficient at some

F(X)(T~) in terms t7I~)/RT2. Then,

standard temperature T~. The specification of done by setting (0 lnF1/OT)l,p In

-

-

-- (/-)1

--

FI (Tf (Ts.______~)= _ fT~ '~ FI1RT 2 - I1~ dT

of

(constant P).

F(lX)(Tu)

is

(3.13.2)

Here /-tl -- /-t~ is the enthalpy difference in changing the solution from a pure state, x - 1, to a final composition x corresponding to temperature Ts. This difference must be known as a function of temperature T before the integration can be carried out. For this purpose we next invoke Kirchhoff's law"

(/-)1 -- /-I~)T -- (/-)l -- I2I~)Tf Jr- ( C l p -- C~p)(r - Tf),

(3.13.3)

with which we obtain In

F1 ( T s ) rl(rj)

=

_ (H1 -

Tf)

H~)T r (T -

RT,

+

Clp -

R

d~p [ Ts - rf _ ln(Ts/ Tf )], Ts

(3.13.4)

which solves the problem at hand. Note the many parameters that must be specified to use this relation. 3.13.3

Activity of the Solute

If the activity of the solvent is known that of the solute may be determined by using the relation

d ln a~m) - d ln a~X) =

__ Xl__

X2

(x)

dlna 1 .

(3.13.5)

To facilitate the calculations, normally carried out in terms of molality we reintroduce Eq. (3.5.1) to write

Xl

x2

=

1000

mM1

=

bl

m

,

(3.13.6)

where m is the molality of the solute, and M1 is the gram molecular weight of the solvent. Also, we rewrite Eq. (3.13.1) in the form - lnal x) -- - lnal m) -- b269 -+- b369 2,

(3.13.7)

212

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

in which b2 and b3 are the multipliers of 69 and of 692 in Eq. (3.13.1). On differentiation of the above one finds -dlna

1( m )

_

(b2 + 2b369)dO

(3.13.8)

so that Eq. (3.13.5) may be rewritten as dO

dO

. = . d l n a 2( m ) _- .bl m (b2 +. 2b369)dO ~m F c O m ,

(3.13.9)

with ~-1 _ bl b2 and c - 2bib3. In further manipulations it is convenient to introduce a quantity j -- 1

69

(3.13.10)

~,m

so that

dj

dO m

~m dO

6) + ---zdm)~m z

dO

69 + ~dlnm )~m )~m

(3.13.11a)

+ (1 - j ) d l n m ,

)~m or

dO

~m

=(1 - j)dlnm

(3.13.1 lb)

- dj.

Now introduce (3.13.1 lb) into (3.13.9) to find d ln F2(m) - d l n a ~ m) - d l n m -

dm dO - j ~ - dj + cO~. m

(3.13.12)

m

The independent variable involved here is the molality. Hence, one integrates from very dilute solutions, m = 0 to a final value m. For very dilute solutions Eq. (3.13.7) reads - l n x l = b 2 0 = - l n ( 1 - x2) ,~ x2 ~ x 2 / x l ,~ m / b l . It then follows from (3.13.10) that j - 0 for m - 0 . The integration over m of Eq. (3.13.12) thus yields the final desired result (In F ( m ) vanishes at infinite dilution): In

F2(m) ( T f ) -

0 ~.m

1-

fomj-m- d m

+ c

fom~~m

dO.

(3.13.13)

The integrations can be executed numerically after the freezing point depression (9 has been determined empirically as a function of the molality m and vice versa. This then yields the activity coefficient of the solute at the freezing point of the solution.

ACTIVITY OF SOLVENT AND SOLUTE

3.13.4

213

Activity Coefficient of Solute at Arbitrary Temperatures

To find the activity coefficient freezing point we set

/-,(m) at

y~ln

a standard temperature Ts other than the

(3.13.14)

alm) (Tf ) '

whence

dlna~m)(Ts) _ We now set x l/X2

--

x__l.1dlnalm)(Ts) _ _x__[ d l n a l m ) ( T f )

x[ dy.

X2

X2

x2

(3.13.15)

b l / m and determine the quantity

d ln F2(m)(Ts) - d ln[a~m) (Ts)] - d lnm bl [d lnal m) (Tf)] -- bl dy m

This expression correlates/-,(m)

at

d lnm

(3.13.16)

m

the standard temperature Ts with the activity of

the solvent al m) at the freezing point of the solution. We can set ( b l / m ) d lnal m) =

d ln a~m) to obtain bl d ln F2(m) (Ts) - d ln[a~m) (Tf )] - d lnm - ~ d y m

-- d ln[/-'2(m) (Tf)] -- bl dy.

(3.13.17)

m

Next, integrate from m - 0 to m. At the lower limit/-,(m) vanishes. We obtain

__

1, so that its logarithm

ln[1-'(m)(Ts)] - ln[F(m)(Tf )] - bl f0 m --1 dy,

(3.13.18)

which is the final desired relation. The variation of m with y must be determined empirically; this generally is a laborious procedure, but in principle is straightforward.

3.13.5

Elevation of the Boiling Point

Procedures analogous to the above apply to the determination of activities from the boiling point elevation. The requisite changes in the derivations should be obvious and will not be detailed here.

214

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

3.14 Mixing in Nonideal Solutions In this section we set up procedures that can be use later for constructing phase diagrams; the results are also of intrinsic interest. We study the formation of a nonideal solution from its pure constituents; these results should be contrasted with those of Section 2.5. All thermodynamic properties of interest will be derived from the Gibbs free energy of mixing, specified by n

Aam

--

Zni[txi(T,

P, xi) - tx*X(T, P)],

(3.14.1)

i=1

in which /d,i is the chemical potential of constituent i in the nonideal solution, whose composition is specified by mole numbers n i and mole fraction xi, and /d, i9x is the corresponding chemical potential of each constituent in its pure form at the same temperature and pressure. Readers are invited to carry out corresponding analyses based on molarity or molality.

3.14.1 ThermodynamicFunctions of Mixing We begin with Eqs. (3.6.3a) and (3.6.4a). As stated in Section 3.4, for pure materials G. (T, 1, 1) = ai (T, 1, 1) = 1, G (T, P, 1) = ai (T, P, 1), so that for pure components/zi9x (T, P) _ /d, i.x (T, 1) + R T In G (T , P , 1). On inserting this relation and Eq. (3.6.4a) into (3.14.1) we find

AGm--RTZniln{ i

G(T'P'xi) } ~'(T'-P"liXi "

(3.14.2a)

To simplify the notation we in the remainder of this section let/-) stand for the ratio G'(T, P, x i ) / G ( T , P, 1); in any case, the denominator normally is very close to unity, so that in all but the most meticulous calculations this particular term may be dropped. Thus, we write

AGm -- R T Z

ni ln(G.xi).

(3.14.2b)

From (3.14.2) we readily obtain

OAam) ASm -- -

3T

P,xi

= - R 5-',_,ni ln(xiI

- R r 5-',_.,ni

i

i

OIn/-) ) OT

P,xi (3.14.3)

AHm -- - T 2 [

(o(AGm/T)] OT

P,xi

=-RT2

y~ni

i

OlnG. ) 3T

(3.14.4)

P,xi

MIXING IN NONIDEAL SOLUTIONS

A Vm --

O(AGm) 0P

215

-- R T T,xi

ni

OP

i

(3.14.5)

'

T,xi

(3.14.6)

A E m - A H m - P A Vm,

(ACP)m---eT~ni[2( i

olnFi)OT P,xi

0T 2

(3.14.7) P,xi

A corresponding equation is obtained for (ACV)m. The various partial derivatives called for in Eqs. (3.14.3-7) may be determined via Eqs. (3.8.1), (3.8.3). The above expressions all correctly reduce to the results of Section 2.5 for ideal solutions.

3.14.2 Mixing in Binary Solutions; the Margules Equation We examine Eq. (3.14.2) in greater detail for a binary solution. Define A G m A G m / ( n l + n2), so as t o achieve a symmetric formulation, relative to a standard state at unit pressure: AGm RT

- - Xl

lnxl +

X2

lnx2 + xl In/'1 +

X2

In F2.

(3.14.8)

We now need explicit relations for/-'1 and F2. The simplest useful formulation is an expansion of In F1 and of In F2 in a power series in x l and x2 as shown below"

In/-'1 - lnal - lnxl In F2 -- In a2 -

--

In x2 --

D l x 2 --t- B l x 2 + . . . ,

(3.14.9a)

D 2 x l -t-- B2x 2 at - . . . .

(3.14.9b)

Note that In F1 is expanded as an ascending power series in x2 and In F2, in an ascending series in x l. This ensures that as x2 --+ 0, F1 ~ 1 and as x l --+ 0, F2 ~ 1, as required. The series (3.14.9) as written will be useful if, for successive members Clx23 + Flx~ + . . . , the inequality 1C1 I, IF1 I , - . . ~ ]B11 applies. The coefficients on the fight of (3.14.9) are not arbitrarily adjustable. This may be shown by solving (3.14.9) for In ai = In xi + Di (1 - xi) + Bi (1 - xi)2 with i = 1, 2, and then constructing the expression Xi d l n a i -- xi d l n x i - (Di + 2Bi)xi dxi 4- 2 B i x 2 dxi.

(3.14.10)

Next, from (3.14.10) form the sum xl d l n a l + x z d l n a 2 , which at constant T and P must be made to vanish, so as to satisfy the Gibbs-Duhem relation. Also, xl d In x 1 + x2 d In x2 - d x l + dx2. Thus, 0 - d x l + dx2 - (D1 + 2B1)xl d x l - (D2 + 2B2)x2 dx2 + 2 B l x 2 d x l + 2B2x 2 dx2.

(3.14.11a)

216

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

Now set x 2 - - 1 - - x l , dx2 -- - d x l , cancel out the common multiplier dxl, and rearrange the resulting term in ascending powers of x l. This yields D2 - [D1 -I- D2 + 2(B1 - B2)]Xl 4- 2(B1 - B2)x 2 --O.

(3.14.11b)

The coefficients of x ~ x~, x 2 must vanish separately if (3.14.11)is to vanish for arbitrary Xl in the interval 0 ~< Xl ~< 1. This requirement is met by setting D1 -- D20 and B 2 - B 1 = B , which leads via Eq. (3.14.9) directly to the Margules (1895) equations" In F1 -- Bx2;

In F2 - Bx 2,

(3.14.12a)

F2 -- exp(Bx2).

(3.14.12b)

or

F1 -- exp(Bx2); Eq. (3.14.8) now reads

A G m / R T -- Xl lnxl +

X2

lnx2 -k- Bxlx2.

(3.14.13)

This is the fundamental equation that forms the basis of much of our subsequent discussion.

3.14.3

Pictorial Representations of the Margules Relations

We note that for ideal solutions, B -- 0, for which the variation of A G m / R T with x2 is represented by curve 1 in the accompanying diagram, Figure 3.14.1. Since Xl, x2 < 1, AGm is necessarily negative: thus, the mixing process for components to form a homogeneous solution occurs spontaneously. For B < 0, the term B x l x 2 renders AGm e v e n more negative than for the ideal case; this situation is represented by curve 5 in the diagram. However, a very interesting situation arises for B > 0; for, when B becomes sufficiently large, the simple monotonic variation A Gm with x2 gives way to a more complex behavior in which there is first a local minimum and ultimately, an absolute maximum in A Gm at x - 1/2, followed by another minimum. This situation is depicted by curve 4 on the diagram. Curve 3 is discussed later.

3.14.4

Phase Separation

We now touch on a point that is extensively discussed in Section 3.15. Reference is made to curve 4 of Fig. 3.14.1, which is redrawn in Fig. 3.14.2, where we set x - x 2 . Note the symmetrically displayed minima in A Gm that occur at mole fractions x ~ and x ' . Clearly, an attempt to make up a homogeneous solution of composition x in the range x ~ < x < x" at fixed T and P will result in a more positive value of A Gm than formation of a heterogeneous mixture of composition x ~

MIXING IN NONIDEAL SOLUTIONS

217

+0.1 B = 2.88 I /

/|

\

/

-0.1

|

| B=2

-0.2

I- -0.3 n,"

E

K.9

B=1.15

< -0.4

-0.5

|

O

-0.6

-0.7 B = -0.33 -0.8

0

0.2

0.4

0.6

0.8

1.0

X2

Fig. 3.14.1. The Gibbs free energy of mixing of two components in a nonideal solution, as specified by Eq. (3.14.13) for various values of B.

,.,.,

A Gm RT

0

I I I I I I i

I I I I

i i

I !

l

x'

x'"

x"

I I

1

Fig. 3.14.2. Illustration showing the reduction in the reduced Gibbs free energy of mixing when a mixture of mole fraction x m splits into two phases of composition x p and x " in mutual equilibrium.

218

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

and x" (present in appropriate quantities to preserve the overall composition x of the hypothetical homogeneous solution from which these two phases are derived). Since A Gm is reduced through separation of the solution into two distinct phases, which renders A Gm a s small as possible, such a process will occur spontaneously. Thus, under the specified conditions for x ~ ~< x ~< x" the solution splits into two components, x ~ and x ' , whose compositions correspond to the minima of the curve; the two distinct phases x - x ~ and x - x" are in mutual equilibrium. The specific compositions x ~ and x" of the heterogeneous phases can be determined through the requirement that x be a minimum at these two points" (OACJm/OX) -- O. Because of the symmetry inherent in Eq. (3.14.13), x' - 1 - x " . On setting x2 - x - 1 - X l in Eq. (3.14.13) and then setting ( O A G m / O X ) T / R T -0, one arrives at the result

xt t ] In 1 - x "

- B ( 2 x " - 1),

(3.14.14)

which is a transcendental equation that must be solved numerically for x = x" once B is specified. As B is increased x ~ and x" move symmetrically away from the value 1/2 toward the values 0 and 1 respectively; for, as B --+ c~ in Eq. (3.14.14), x" must approach unity. Also, as is seen from Fig. 3.14.1, there exists a critical B value, Be, for which x ~= x" = 1/2. Thus, when B < Be no phase separation is encountered. The critical value, Be (which separates the U-shaped curves from those that have minima away from x = 1/2) is determined from two requirements that must be imposed on Eq. (3.14.13): O2AG

m

Ox 2

=0--

1

1 -Xc

~

1 Xc

2Bc

(3.14.15a)

and

03AGm Ox 3

=0-

1 (1 - - Xc) 2

1

X2'

(3.14.15b)

whose solutions lead to the values 1 X c = 2,

Bc - 2.

(3.14.15c)

The corresponding variation of A G m / R T with x2 is shown as curve 3 in Fig. 3.14.1. On surveying the above results it is obvious that the very simple model introduced here provides a semiquantitative framework for understanding the origin of phase separation phenomena in real binary solutions.

MIXING IN NONIDEAL SOLUTIONS

3.14.5

219

Fugacity of Gas Phases

We return briefly to Eq. (3.14.9) which b e c o m e s (with D - - 0 , B1 -- B2 --: B) l n a l -- l n x l + Bx 2.

(3.14.16)

Plots of al versus Xl are shown in Fig. 3.14.3 for B = 0, 1.15, 2, 2.88. One notes that for B > Be -- 2 there exist two values of x l, n a m e l y x 1 and x "1 that are associated with the same activity. Since #1 - #1 § R T l n a l , the equality a 1(X~l) -- a 1(X'l') implies an equality of chemical potentials" # 1(X~l) - # 1(x'(), that is again an indication of two discrete phases of composition x '1 and x "1 in mutual equilibrium. Note that Fig. 3.14.3 is also representable as a vapor pressure diagram of P1/PI* versus Xl. This is so because when equilibrium prevails #~ + R TIn a l --

#oP + R T l n P 1 ; on setting al -- FlXl and rearranging one finds XlF1/P1 al/P1 - e x p [ ( # ~ - # I ) / R T ] =-- C. For pure liquid 1, Xl - 1, F1 ~ 1, al -- 1, P1 -- PI*, thereby rendering C - 1 / P ~ , so that finally P1/PI* -- a l, which proves the assertion. Thus, Fig. 3.14.3 is equivalent to the standard diagrams that show departures from Raoult's Law. The interpretation of curve 4 on the diagram now

1 92

I

iI

\

\

\|

! 10 I

0.8

\\

...............

\,,~,,

Xl~

/ /

0.6

0.4

0.2

0

Ir'/'/i ' 0 x~ 0.2

I 0.4

Xl

I 0.6

r

i 0.8

1 x~' 1.0

Fig. 3.14.3. Activity of component 1 versus mole fraction X l in a binary mixture, as determined by Eq. (3.14.16).

220

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

runs as follows: as the solution is rendered increasingly rich in component 1, the partial pressure of component 1 in the vapor phase rises much more rapidly than would be the case for an ideal solution. In the range x '1 ~< Xl ~< x "1 , P1 / P~ remains fixed, corresponding to the fact that there now exist three phases in mutual equilibrium, i.e., two liquids and the vapor. As the solution is made increasingly rich in component 1, the condensed phase of compositionx "1 grows at the expense of ' Ultimately, for compositions Xl > x II1 one reverts the phase at composition x 1. back to a single liquid phase, and P1 again varies with Xl until the value P1 - P{ is reached at Xl = 1. Note further that as x l --+ 1, all curves in Fig. 3.14.3 merge with the one for which B - 0; this agrees with the experimental fact that Raoult's Law always holds in this range. Similarly, as Xl --+ 0 one obtains a straight line region consistent with Henry's Law; here as Xl --+ 0 the slope varies with each solution. For B < 0 negative deviations from Raoult's Law are encountered, but for B > 0 one finds positive deviations with slopes that become steeper, the greater the positive departure from ideality. It is remarkable how such a simple model as the above can rationalize so many experimental observations.

3.14.6

Entropy of Solution

Finally, we note an important point: as long as B is taken to be independent of T the term - R ( x l lnxl + x2 lnx2) in Eq. (3.14.13) represents the entropy of mixing, which, in this approximation is the same as for an ideal solution. The term BXlX2 is then to be identified with A [ t m - - A G m Jr- T A S m . We may then rewrite Eq. (3.14.13) as AG

m --

R T { x l lnxl + X2 lnx2} +

tOXlX2,

(3.14.17)

in which we have set B =_ w / R T and AHm -- R T B x l x 2 -- tOXlX2. On introducing the definitions for mole fraction we find [AG -- n A G -- (nl + n 2 ) A G ] A G m = R T ( n l lnxl -+- n2 lnx2) -4- w ( n l n 2 / n ) .

(3.14.18)

Clearly, in this oversimplified model all of the deviations from ideality are to be associated with the presence of a nonzero enthalpy of mixing term, namely w(nln2/n). Several ad hoc methods have been proposed to simulate a contribution of the entropy to nonideality while retaining the present model. In the simplest representation one allows the coefficient B in Eq. (3.14.13) to be a function of T, B ( T ) . It is then a simple matter to check that (w ~ represents the temperature derivative) A ~ m _ _~O A G m _-- _ R[Xl lnxl -f- x2 lnx2] - WtXlX2 OT

(3.14.19)

MIXING IN NONIDEAL SOLUTIONS

221

is the molar entropy of mixing, which is still symmetric in the two mole fractions but does add a correction term to the ideal molar entropy of mixing. The molar enthalpy of mixing is found from the customary relation O ( A G m / T ) / O T -- A t T I m / T 2, yielding (3.14.20)

AISIm -- ( w - w ' T ) x l x 2 ,

which is now a symmetric, T-dependent quantity. Usually w ~ is of the order of 10-2w, but, when multiplied by generally prevalent values of T, is of order w and hence not negligible in Eq. (3.14.20).

3.14.7 Asymmetric Mixing Functions So far the mixing terms have been strictly symmetric on interchanging the designations of components 1 and 2. This is scarcely the case in real life. A degree of asymmetry is introduced by requiting that the free energy of mixing have the form, termed the F l o r y - H u g g i n s model, G - G* -

RT[nl

ln991 4- n2 In go2] 4- A(T)(nl ~0 4- n2V~

(3.14.21a)

with (/9i ~

n i Vi ~

(3.14.21b)

nlVO + n 2 V O

being the volume fractions of the two components, wherein the i?i~ are the molar volumes of component i in pure form. We shall neglect the temperature and pressure dependence of these quantities. It is important first to determine the chemical potentials associated with the Flory-Huggins relation. By standard methods it is may be shown that #1 - #~ = lnq91 + ~o _ ~o q92 + ,4(T) ~~176 RT

(,1o

RT

(3.14.22) '

with a similar expression for the second component, in which the subscripts 1 and 2 are interchanged. The reader should check that Eq. (3.14.22) and its analog correctly reproduce the Gibbs free energy as A G m - n l l z l + n2/z2, which is not obvious from the form of Eq. (3.14.22). Moreover, it must be checked whether the Gibbs-Duhem relation nl d # l 4- n2 d/z2 = 0 holds. To show that this is the case is a tedious exercise in standard calculus (see Exercise 3.14.5). From Eq. (3.14.21 a) one obtains the following expressions: the entropy of mixing is specified by S - S* - - R ( n l

dA(T)

lngol + n2 In go2) - (nl ~0 + n 2 ~ O ) q g 1 9 9 2 ~ dT

(3.14.23a)

222

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

and the enthalpy of mixing, as derived from Eq. (1.20.15a), is given by

H - H * - (nl"~r~+ n2 f,r~

T d~(T) d-----~-] "

(3.14.23b)

It is easy to see that Eqs. (3.14.21) and (3.14.23) may be converted to molar form by replacing all ni with xi. The resulting expressions, considered as a function of Xl and x2 are clearly asymmetric in the two mole fractions. In actual calculations it is expedient to introduce the ratio rl - I?~ ~ and to set I?~ (T) -- A(T). On switching to the molar expression Eq. (3.14.21a) reduces to the asymmetric form A G m - R T [ x l l n x l -+- x 2 l n x 2 at- Xl lnrl - l n ( r l X l -4- x2)]

+ A(T)

XlX2 rlXl + x2

,

(3.14.24)

from which one obtains in the customary manner

tgm

-- - R [ x l lnxl Jr x2 lnx2 + Xl lnrl - l n ( r l X l + x2)]

- A ~(T)

(3.14.25)

XlX2 , rlXl + x2

and AHm - - [ A ( T ) -

TA'(T)]

XlX2

(3.14.26)

rlXl + x2

The asymmetries now arise in both the entropic and enthalpic contributions. The above correctly reduce to Eqs. (3.14.13), (3.14.19), (3.14.20) when rl = 1. A set of representative calculations for the reduced Gibbs free energy of mixing with rl - 0.1 is shown 6 in Fig. 3.14.4 for several A/RT values. One should note the changes in the degree of asymmetry with variations in the choice of parametric values. The tick marks indicate the location of the minima in these curves.

3.14.8 Summary It is quite remarkable how many firm deductions have been based on a single hypothesis. Starting with the Margules formulation (3.14.12) for Fi, thermodynamics leads directly to the specification of A G m , a s shown in (3.14.13). All other mixing functions are then found from (3.14.3-7). When phase separation does occur the composition of the two phases in equilibrium is specified by Eq. (3.14.14). The critical value of B required for incipient phase separation, and the critical composition of the mixture are specified by Eq. (3.14.15c). Finally, one may construct diagrams such as shown in Fig. 3.13.3 by which deviations from Raoult's Law are predicted. Effects leading to asymmetric formulations may be introduced via the Flory-Huggins formulation. The foregoing analysis is a beautiful illustration of the power of thermodynamic methodology.

MIXING IN NONIDEAL SOLUTIONS

223

AGm/RT 5.0

45 4.0~3.5

q=0.1

l

J

~

-

10

]

1

-

3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5

, .

0.0

0.2

0.4

I 0.6

=

I. 0.8

, : 1.0

Mole fraction x l

Fig. 3.14.4. Variation of the molar Gibbs free energy of mixing with composition of a binary mixture, as determined according to the Flory-Huggins model, Eq. (3.14.24), with r 1 = 0.1 and for various values of A / R T. Tick marks indicate location of minima.

EXERCISES 3.14.1. Try to 'improve upon' the Margules model by including a cubic term in the expansion of Eqs. (3.14.9a,b), and note the resulting complications. 3.14.2. Derive expressions for ASm/RT, At~m/RT, AEm/RT assuming that w is a constant. Sketch the resulting curves for B -- - 1 , 0, 1.5, 3 at T -- 300 K and comment on the nature of your results. 3.14.3. For certain types of high polymer solutions the chemical potentials of the solvent (1) and solute (2) are given by the approximate relations

G2 - G~ (t91 : ~

P) + RT[ln~p2 - (r - 1)ln(1 - ~P2)] + rw(1 - ~p2)2,

N1 , N1 + rN2

~ 2 :~

rN2 9 N1 + rN2

Here N1 and N2 are the number of solvent and solute molecules respectively. Each polymer is assumed to consist of r monomers, and the volumes of each monomer and each solvent molecule are assumed to be roughly equal. (a) Relate ~Pl and ~P2 to Xl and x2. (b) Determine the entropy of mixing and compare the result with that for an ideal solution. (c) Determine the enthalpy of mixing and compare it to the Margules formulation.

224

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

3.14.4. It is found empirically that the vapor pressure of component 2 nonideal solution may be specified by P2 - (2x2 - x 2) P ~ . (a) Determine its activity coefficient relative to the standard state. (b) Find an analytic expression for P1 in terms of x l. (c) Plot out the partial pressures as a function of composition and note their shape. 3.14.5. Prove that Eq. (3.14.22) and its counterpart for the second component do together satisfy the relation G = nl/~l + n2/~2 and the Gibbs-Duhem relation. Note in this connection that ~01 and ~02both involve n l and n2. The derivations are straightforward but tedious. 3.14.6. The calculations were performed by Mr. Joseph Roswarski at Purdue University.

3.15 3.15.1

Phase Stability: General Consequences of Deviations from Ideality Qualitative Free Energy Curves

We had earlier encountered specific examples where deviations from ideal behavior of binary solutions led to the phenomenon of phase separation. We now introduce several generalizations on the basis of qualitative sketches introduced below. In ideal solutions the mixing process is rendered spontaneous through the positive entropy of mixing A S m / R - - { x l lnxl + x2 lnx2} > 0; here A H m -- O. The molar free energy for ideal solutions reads ~d - R T { x l lnxl + x2 lnx2}

=-TA3o+5*.

-I- (X1/Z~ -I- X2IZ~)

(3.15.1a) (3.15.1b)

Qualitatively, Eq. (3.15.1) may be represented as shown in Fig. 3.15.1(b), where energy is plotted schematically versus x2 -= x. The figure shows the quantity - T A S o versus x as the bottom curve; the sloping baseline (top curve) for G* is obtained from the sum xl/z 1 + X2/Z2, in which the #~ are held fixed. The resultant graph, shown in the middle, G - - T A S 0 + G*, is a skewed U-shaped curve. For nonideal solutions two corrections are inserted" (i) An excess entropy term Se must be added to AS0 to account for deviations from ideality, as already alluded to in Section 3.14. The result may be simulated by the expression AS - Xl lnxl + x2 ln(rx2), where r is a suitable parameter. One now obtains a skewed - T AS curve of the type shown in Fig. 3.15.1, parts (a) and (c). (ii) The enthalpy of mixing no longer vanishes but depends on x2 = x, which is simulated by the general expression A H - w x ( 1 - x)/[1 + ( r - 1)x] of Eq. (3.14.26), which reduces to the last term of Eq. (3.14.17) when r - 1. On introducing these corrections one obtains a new curve, sketched in part (a) where w < 0, and in part (c) where w > 0. The resultants obtained on adding up the sloping baselines, the entropy, and the enthalpy contributions are indicated by curve G. In part (a) G is simply more negative and more skewed than for part (b), but in part (c) a nonmonotonic variation of G with x is obtained, because over certain ranges

PHASE STABILITY: GENERAL CONSEQUENCES OF DEVIATIONS FROM IDEALITY

225

AH

T

T

>,,, 0 t-" LLI

O t-tlJ

-TAS

-TAS0

0

X2

1

0

X2

(a)

1

0

(b)

X2

(c)

Fig. 3.15.1. Schematic representations of several (free) energies of binary solutions under various assumed conditions: (a) AH < 0, (b) A l l / = 0, (c) A / / > 0. Note the nonmonotonic change of G in the latter case.

of x large positive A H values outweigh the negative contributions associated with - T AS. As already indicated in the last section, and as will again be shown shortly, such a situation signals the onset of phase separation. 3.15.2

The Lever Rule

In this connection it is of some interest to establish how much of each phase must be present to form a heterogeneous mixture of average mole fraction x. The reader m a y consult Table 3.16.1, in deriving Eqs. (3.15.2). Consider n = nA -4- nB moles of A and B with an overall mole fraction 1 - x n A / n . Let a two-phase mixture be formed such that the mole fraction of A in the first phase and in the second phase is given by 1 - x ' - n~A/(n~A + n~B) and by 1 - x " - n A~t/(n~ + n ~ ) , respectively. Let the fraction of the total mole numbers in the ~ phase be 1 - f - (n 'A -k- n ~ ) / n . Then one finds that nA - n(1 - x), , - n(1 - f ) ( 1 - x'), and n A " - n f (1 - x " ) . Conservation of mole numbers nA " On substituting and solving for f one obtains requires that n A n A -+- n A" -

-

X ~X

f =

x

,,

-x

X

f!

1 - f -- x " which results are known as the L e v e r Rule.

!

,,

(3.15.2a)

my

x"

(3.15.2b)

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

226

Table 3.16.1 Compositional relations for binary mixtures Mole fractions, total

nA

- - (1 - - n ) x

nB=

nx

Phase I

Phase"

Mole number

f f n A, n B

u u n A, n B

Mole fractions

xl _ A -

I nA -- 1 -- x I i ~ nA § nB

-

-

u _ XA

-

-

-

u nA -- 1 -- x u u u nA § nB -

-

,

I

Total mole numbers per phase Mole fraction for each phase Overall composition of system Conservation of mole numbers

nB I I I = x x B -- nA § n B

II _ XB --

nf

nlt

_

I

B n!

1 _

--

f

nf §

f

nA

Final interrelations

n A =n(1

=

xl)n ' +

-

f)n

§

(1 -

x 1 ' ) n l'

(1 - x ' ) f n

--x)

:

n

--x § nB

:

nB -n

x?l :

xtn t

§

x t t n tt

- - x I (1 - - f ) n § x 1I f n nB =nx

n ' = n ( 1 -- f )

n~

n II

n If =

nB = nB § n

(1 -

' (1-x

nI §

xB = ~ nA

n

- - (1 - x ' ) ( 1

' =n nA

B

nB

--

tt = n tA § n A

x)n

=

nA = 1 --x

(1 -

It

n fl

-n-

nA §

nA

I/

--n A §

nI n II =

x A --

Interrelations for mole fractions

3.15.3

t

--n A §

It

n

u u = x nA § nB

n" = nf ')

nix t

" = n " (1 - x ' ) nA nBtt :

nttxtt

Homogeneous vs. Heterogeneous Solutions

For a given set of conditions, can one predict whether a given alloy is homogeneous or not? Consider a solution of mole fraction x0 for a system for which Fig. 3.15.2 is relevant. If the alloy were simply a mechanical mixture of the pure components A and B, then according to the Lever Rule the free energy of the sy_stem would be given by the intersection (~* of the straight line joining O-*A to O~ with the vertical line at x - x0. If instead the phase mixture involved two solutions of compositions x a and x a" , then the same Lever Rule yields the free energy of the alloy designated as Ga. S i n c e Ga < G*, this new state is more stable than the original one. Analogous remarks apply to another heterogeneous alloy whose two compositions are represented by x b, and x b"", the corresponding free energy, Gb, is still lower. Continuing this process it is found that at the composition x where the two phases merge into a single homogeneous solution of composition x0, the free energy attains its lowest possible value, G0. This is the stable configuration. For the type of free energy displayed in Fig. 3.15.2, 02G/Ox 2 > 0 for all x; hence, a straight line joining any two points on the curve always lies

PHASE STABILITY: GENERAL CONSEQUENCES OF DEVIATIONS FROM IDEALITY

227

xo

g

g;

U3 I11 I1) . . . . . .

U_

w!

r .13 .13

(.9 _

Go

0

XEX 2

xo

1

Fig. 3.15.2. Schematic representations of the change in Gibbs free energy with composition for a binary homogeneous liquid solution.

above the points on the curve between the intersection points. When the free energy versus composition curve is U-shaped the homogeneous solution always has the lowest free energy. By contrast, consider the free energy curve as a function of composition as sketched in Fig. 3.15.3, which reproduces curve G in Fig. 3.15.1 (c). If an alloy of composition x0 between the two minima were to exist as a homogeneous solution, its free energy would be given by the point Go which falls on the curve. On the other hand, if the alloy were a heterogeneous mixture of composition x a and x a" for component 2, the free energy of the system would be lowered to G a . By choosing x ~ and x" to be more widely separated in composition one progressively lowers the free energy until a minimal value G is reached when the two phases are of composition x ~ and x ' ; here the straight line connecting these particular mole fraction values forms a tangent to the two curves near the local minima. Any attempt to spread the composition of the two phases further will lead to a " with a corresponding value rise in free energy, as is illustrated for x b and x b, for Gb. The stable state of the system under study may thus result in a heterogeneous mixture of composition x ~ and x ' . In fact, in the range x t < x < x ' , the alloy consists of two phases of composition x ~ and x" in the proportions (x - x t) / ( x " x') and ( x " - x ) / ( x " - x ~) given by the Lever Rule. Note that the proportions change linearly with composition x. However, in the ranges x < x' or x > x ' , the homogeneous solution is stable, as is seen by inspection of Fig. 3.15.3.

228

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

x'~ Gb

-rf

~

I

I I !

0

xl

I I I

Xo

x" x

Fig. 3.15.3. Schematic representations of the change in Gibbs free energy with composition for a binary heterogeneous liquid solution. A single phase is stable for 0 ~< x ~< x p and x " ~< x <~ 1; a biphasic mixture is encountered in the intermediate composition range.

The tangency effect described above has a physical interpretation: it specifies the slope of the free energy curve G vs. x at the points of tangency x ~ and x~; in other words, it is the chemical potential of the solution at the two compositions x ~ and x ~. The two chemical potentials are obviously identical, which is the requirement that must prevail at equilibrium. 3.15.4

Distribution of a Solution Among Two Phases

The changeover of the A-B alloy system from an A-rich to a B-rich homogeneous phase with composition at fixed temperature is easily visualized with the aid of Fig. 3.15.4, in which we represent the fraction f of the total mole numbers in phase ~ and phase ~. As pure B is added to pure A, the B atoms form a homogeneous solution in A up to the composition x ~at which the solution is saturated with B. Any further addition of B results in the formation of a second phase for which x - x ~, in which units of type A are considered dissolved in B. With further addition of B the proportion of the second phase at fixed composition x" increases at the expense of the first until the phase boundary at x - x" is reached. For x > x ~ only the homogeneous solution of B containing A is stable. Thus, in the heterogeneous region the compositions of the two phases of the mechanical mixture remain constant, but the relative amounts of material in each phase changes with alterations in x. 3.15.5

Generalization to Several Components

The presentation may readily be generalized to a system in which more than two phases appear. A typical example is shown in Fig. 3.15.5; here the homogeneous and heterogeneous composition ranges are delineated by use of an imaginary

PHASE STABILITY: GENERAL CONSEQUENCES OF DEVIATIONS FROM IDEALITY

229

Comp. B

Comp. A

.........

1-f

!

0

X'

1

X" X --=X B

Fig. 3.15.4. Variation of the makeup of a binary system with relative composition (e.g., mole fraction), shown as a plot of 1 - f and of f vs. x.

/ A

!

! I I I

I I I

!

cz+,8

, ~', l I I I I

'

~+y I I I I I

x

,y

,y+~,

i I I I I

I I I I

I

Fig. 3.15.5. Gibbs free energy as a function of mole fraction x when several homogeneous phases of different composition are formed. Stable single phases are indicated by cross hatching on the x scale.

string that is tightly wound around the curves between points A' and B' in the diagram to exhibit all possible c o m m o n tangent constructions. It is customary to designate phases consecutively by Greek lowercase letters in alphabetical order. The various phases and their composition ranges are indicated on the diagram. In general, if A H is small, as is likely to be the case for homogeneous solutions, the free energy, plotted as a function of composition, forms a broad, shallow U-shaped curve, and the ranges of composition over which the single phases are

230

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

Y

I

1 o~+/~

0

L

q

Fig. 3.15.6. Gibbs free energy as a function of mole fraction x when several homogeneous phases of different composition are formed, one of which is a compound of composition A l _ x l Bx 1" Stable single phases are indicated by cross hatching on the x scale.

stable are large. However, when a compound is formed, deviations from the appropriate stoichiometric ratio remain very small. This is reflected in the extremely sharp rise of the free energy as the composition is changed even very slightly from the appropriate stoichiometric ratio. A situation of this type is depicted in Fig. 3.15.6; it is seen that the composition x = Xl is stable over only a narrow range. The composition of the compound does not necessarily coincide with an ideal stoichiometric value; for example, the compound CuA12 does not exist, but a compound on the Al-rich side of this value is stable.

3.16

Discussion of Several Types of Phase Diagrams

Some general features of standard phase diagrams will now be correlated with the free energy curves depicted in Section 3.15. We immediately specialize to systems where the liquid phase is homogeneous throughout its composition range; the corresponding free energy curve is then U-shaped, as depicted in Fig. 3.15.2. In our first example, Fig. 3.16.1, the solid state also exists only as a homogeneous solution. Quite generally, as the temperature is lowered the free energy curve of the solid moves past that of the liquid [Why? See Exercise 3.16.6] and the shape of each curve (i.e., the skewness of the U shapes) will also be altered. Thus, with diminishing temperature the two free energy curves will ultimately

DISCUSSION OF SEVERAL TYPES OF PHASE DIAGRAMS (a)

(b)

T1

231 (C)

TA

.......~

XB ~

B

T2

q

9

A

-

A

XB

u

i

d

,I

,,--

ix~ XB ixB

.....

B

i

(d)

I

(e) TB

.... i_

Tu :;

T3

TA

ml

. . . . ,s. . . . ,,-'[ . . . . . . . . •

TA ....

TI

Solid 9 Solid

! !

A

XB

B

A

XB ~

B

TB

x 7121-i E

T3

t t

A

x0 XB

~. . . .

B

F i g . 3 . 1 6 . 1 . Correlation of a standard phase diagram with the temperature change of free energy curves of simple liquid and solid phases.

intersect, so that a common tangent construction is called for that indicates the presence of biphasic mixtures. In Fig. 3.16.1 (a) the temperature T1 is sufficiently high that the entire free energy curve for the liquid, Ge(x), lies below that for the solid, Gs(x); at any composition x, the system is stable in the liquid state. As the temperature is lowered to a value TA (part (b)), the two free energy curves touch at the composition x - 0; solid and liquid now coexist for the pure phase, so that TA represents the melting point of pure A. However, for x > 0 the free energy of the liquid remains below that of the solid; beyond x -- 0 the solution remains in the liquid state. As T is reduced further to the value T2 (part (c)), the Ge(x) and Gs(x) l It curves intersect, such that in the composition regions 0 ~< x < x B or x B < x ~< 1 a single homogeneous solid or homogeneous liquid phase respectively is stable; in the intermediate range the common tangent construction shows that the system divides into a phase mixture involving a solid of composition x B in equilibrium with a liquid of composition x B. " With a further decrease in temperature to the value TB (part (d)), the curves Ge and Gs touch at XB -- 1; this is the melting point of pure B. For T < TB (part (e)), G~ < Ge for all x; the homogeneous solid phase is now the stable one over the entire composition range. The preceding information may be assembled into an equilibrium phase diagram shown in part (f), which is typical for this type of system. The diagram shows the T-x regions in which the homogeneous solid or liquid is stable; the two corresponding boundary lines are known as the solidus and liquidus: the T-x

232

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

region between these represents an unstable or forbidden range called a miscibility gap. A tie-line connects the solid of composition x B with the liquid of differ" with which it is in equilibrium. The lens-shaped 'forbidden' ent composition x B region thus traces out the range of the tangent construction at each relevant temperature. The vertical line indicates what happens as the temperature of the system of fixed composition x0 is lowered. Initially the liquid phase is stable down to temperature Tu. On further slow cooling the system separates into two phases of composition x 0' and x 0" in amounts such that the average x0 composition is maintained. On further cooling the x ~ values shift to the right along the lower curve, and the x" values similarly shift to the right along the upper curve; their relative amounts change as well, according to the lever rule, so as always to maintain the same average composition x0. Ultimately at temperature 7) a final two phase mixture of composition x0-"~ and x0-"" is attained. As one cools beyond this point a solid solution of uniform composition x0 is maintained that remains stable down to the point where another phase separation, indicated by the dome-shaped dotted curve, may be encountered. The Ag-Au, Cu-Ni, Au-Pt, and U-Zr alloys are representative examples of such systems. Variants of this scheme are encountered for alloys where Gs(x) has a significantly smaller curvature than Gl(x). As shown in Fig. 3.16.2, with diminishing temperature G s (x) moves past G l(x); in this process there are at first one and then two intersections between the two curves. These call for the usual common tangent construction. With diminishing T these regions move inward until they meet at a common point. The relationship between the free energy curves and the phase diagram on the right should be clear; again, the regions between areas a and b in part (b), or between c and d, or between e and f in part (c) represent miscibility gaps. These lens-shaped regions represent 'forbidden' zones in the sense described above--no homogeneous single phase is encountered under the conditions covered in those particular T - x ranges. Fig. 3.16.2(0 may be viewed as two back-to-back phase diagrams of the type shown in Fig. 3.16.1 (f), and can thus be interpreted in a similar manner. The inverse situation is depicted in Fig. 3.16.3; it arises when the curvature of Gs(x) significantly exceeds that of Gl(x). In the above two cases the solidus and liquidus curves meet tangentially to an isothermal line at a congruent point; the solution freezes at this temperature without any change in composition. Au-Ni alloys exhibit the behavior depicted in Fig. 3.16.2. The melting point of the intermediate composition is below that of either pure material. Inasmuch as A He > 0 for this case, the solid solution for T < Tc is less stable than a mixture of phases; this is indicated by the dotted curve at the bottom of the diagram. A second case frequently encountered involves a solid for which the components are only partially miscible and in which no intermediate phases are formed. The liquid free energy curve exhibits a maximum at an intermediate composition, and the solid free energy curve is U-shaped. The relevant free energy curves

DISCUSSION OF SEVERAL TYPES OF PHASE DIAGRAMS (a)

[•)

TA /

(c)

(d)

Aa

B

~

T2

ml

sTB

'sl _ XB

233

b

1 XB ---~-

(e)

I I I

s II+s, ._ I l l+sls A XB

B

T3

! !

!

TA ~,,~ b

I

~a ..................... ~

= ~

T2

e

o_ E 1-9

I I I L

A XB

_-.--

y

B

A

XB

B

:

TB l T1T

_

A

~ ",~ - " " ./ + B ' , . XB

T3 -Tc ~

B

Fig. 3.16.2. Correlation of a phase diagram developing a minimum with the temperature change of free energy curves of liquid and solid phases.

[.

.Solid .

.

(b)

ma

Tb Tc ~Td E \/'Solid

/~ 0 A

(d)

I .

.

.

.

,

.

l

0 A

50

100 B

Mole Percent B

1 B •

~

Fig. 3.16.3. Correlation of a phase diagram developing a maximum with the temperature change of free energy curves of liquid and solid phases.

234

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

(a)

T1

(b)

I

T2

(c)

T3

I

i

~ I I+ Otl A

XB~

(d)

B

T4

jJ

Ax'

x"

,(e)

XB

B

T5

J

I

A

XB"

T1

TA

T2 TB

I Ol1

I I

t I

Oil I I+Otl l!1+r A x' X B ' ~ X '''

I 1

ll~ 2 X.... B

i

x'

T3 T4 T5

......

!

i A

B

I XB " - - "

x ....

B

A

XB

:

B

Fig. 3.16.4. Derivation of a phase diagram from the temperature change of free energy curves of a liquid and solid that are only partially miscible.

are shown in Fig. 3.16.4. At temperature T -- T1, Gt(x) < Gs(x) for all x; the homogeneous liquid phase is stable over the entire composition range. At a particular value T -- T2, somewhat below the melting point of pure A, the free energy curves for solid and liquid intersect, as shown in part (b). By the common tangent construction, we see that for 0 ~< x ~< x ~ the homogeneous solid alloy is stable; for x ~ ~< x ~< x" solid of composition x ~ is in equilibrium with liquid of composition x ' ; for x > x" the homogeneous liquid phase is stable. Part (c) is typical of a temperature T3 at which Gs and G~ intersect both at the A-rich and at the B-rich ends of the diagram. Here, homogeneous solid alloys exist for the composition ranges 0 ~< x ~< x ~ and x"" <~ x ~< 1; solid of composition x ~ is in equilibrium with liquid of composition x", and liquid of composition x m is in equilibrium with solid of composition x " ' ; in the range x" ~< x ~< x m the homogeneous liquid phase is stable. Further cooling narrows the homogeneous x " - x m range until it vanishes at temperature T4 where x" - x m. This point, known as the eutectic temperature, is the lowest temperature at which an alloy still remains liquid. This liquid of composition x m is in equilibrium with two solid phases of composition x ~ and x m~. The eutectic solid is a heterogeneous mixture of two solid phases of composition x ~, x " . For T - T5 < T4, only the solid is stable, but in the composition range x ~ < x < x"" of part (e) one encounters a heterogeneous mixture of two phases of composition x ~ and x"" in mutual equilibrium. The corresponding phase diagram is shown in part (f). Ag-Cu, Pb-Sn, Pb-Sb, A1-Si, and Cr-Ni are examples of systems exhibiting those properties which arise when the end mem-

DISCUSSION OF SEVERAL TYPES OF PHASE DIAGRAMS (a)

'

-.l I+"! A

TI

b)

/

T2

235

T3

/

I I I

I

/ XB -"--'"

I I I

~ 11+~ I'l+p

. . . . . . .

9

B

A

XB--~"

A

B

XB- - ~

I I I

! ~/~ B

TI T2 T3 E T4 ~ A

XB ~

B

A

XB ~

B

Fig. 3.16.5. Derivation of a phase diagram from the temperature change of free energy curves of a liquid and two solid phases.

bers crystallize in the same structure. The e region in Fig. 3.16.4(f) covers the T - x range of the homogeneous liquid phase; ~1 and Or2 cover conditions under which homogeneous solid A-rich or B-rich alloy phases are stable. The remaining parts of the diagram involve 'forbidden', two-phase zones. One also encounters such a system for two materials A and B which crystallize in different habits; the construction of the corresponding phase diagram is indicated in Fig. 3.16.5. Conventionally, the solid solution occupying the left portion of a phase diagram is labeled as the or-phase. Subsequent homogeneous phases occurring to the right are labeled by successive lowercase letters of the Greek alphabet,/3, V, ~, etc. The reader should be able to construct and interpret the phase diagram in Fig. 3.16.6 on a similar basis. This situation arises mainly when the melting points of pure A and B differ considerably and the solid phase is inhomogeneous. Diagrams of this type are classified as belonging to the peritectic type; the temperature Tp is the peritectic temperature; above it no solid Or2 phase is formed for any alloy composition; below it there is a wide composition range over which the homogeneous solid Or2 alloy system is stable. The ~ + Or l , ~ + Or2, and Ot 1 -+-Or2 regions of the phase diagram show T - x ranges over which biphasic mixtures are encountered. The remaining regions indicate where monophasic solid or liquid solutions are to be found. We next turn to more complex cases involving formation of intermediate compounds. In Fig. 3.16.7, the relative positions of the free energy curves for or,/3,

236

3. C H A R A C T E R I Z A T I O N

(a)

Tp

G

A

f.

.

!

$

I

~ ..,

T2

I

$

~1~1+o~1i

SOLUTIONS

(c)

(b)

I i

OF NONIDEAL

I

i + i

I

.

XB ~

.

1

B

A

f

XB - - - ~

1 0~1+0!2 I 012 1012'll . .!

B

A

~

B

(d)

TA .(e)

T3

.

.

.

.

I .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

T1

.

I mp

T2

s

Otl I

A

O~l+ff2 XB -~

TB T3

1I if2 B

A

XB

B

Fig. 3.16.6. Derivation of a phase diagram from the temperature change of free energy curves of a miscible liquid and two partially miscible solid phases. The melting points of A and B are widely different.

and F, and of liquid phase are such that an initial intersection of the Ge, curve occurs with the G s curve for phase/~, as shown in part (b). On further cooling, more intersections arise as shown in parts (c) and (d); this leads to the subsequent appearance of the a and y alloys. At still lower temperatures, two eutectics are formed, first one between/~ and ot and then, a second between/3 and F- Finally, for T = Ts, one obtains three homogeneous alloys phases in three composition ranges near x = 0 (ol), x ~ 1/3 (fl), and x = 1 (y), as well as two-phase regions in the intermediate composition ranges. The reader should carefully note how the phase diagram of Fig. 3.16.7f is assembled from an examination of parts (b)-(e). Exercise 3.16.5 calls for the generation of free energy curves whose gradual intersection leads to a peritectic diagram of the type illustrated in Fig. 3.16.8. Phase diagrams of this type are commonly encountered in the literature; they tend to arise when the melting points of the pure components are very different. One should carefully note the very distinct domains in which monophasic and biphasic regions are present. A general feature of the phase diagrams explored in Figs. 3.16.7 and 3.16.8 is the existence of a maximum, corresponding to the composition of the intermediate /3 phase. This reflects the composition at which Ge, and G s first become tangent to each other. However, this point generally does not coincide exactly with the minimum in G; the composition of the intermediate phase then does not agree precisely with the ideal stoichiometric formula for the phase.

~ ~

t~

c

9

or)

o

o

~...i

~"

Z

~>

;:l> .

.

.

.

> ................

Temperature Mi

N

~

~e

r--

1

N

or)

(,#) m ~

~

~.~

~f

"~

~

Temperature

>

>

~

. . . . . . .

>

........ or)

~<

a

" h"

I

r ~"

+

--I

--I

--I ---I~

--t

1

"

l

238

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

(a)

T1

(b)

(c)

T2

T3 7

!

A

XB--'-~

B

A

xB~

B

A

I-"3

_

.

!

,

XB~

.... T4

B

T1

T5 I

T2

?,

""

T3 T4

Ik,.\ ~

1 I" A

', ii

T5

,,., ,,[

I"+~I

. Y! I+, . . I. I.l~+~l XB ~

B

A

................ XB ~

B

A

XB ~

Fig. 3.16.9. Derivation of a complex phase diagram from the temperature change of free energy curves of a miscible liquid and three solid phases.

Finally, we refer the reader to Fig. 3.16.9 for a study on how the diagram of part (f) arises. More complex types of phase diagrams will not be explored here, but the general methodology should be clear from the preceding examples. EXERCISES 3.16.1. Examine Fig. 3.16.10(a) and answer the following questions: (a) Identify the areas labeled 1-6. (b) Describe and interpret the events occurring at temperatures TA, TB, Tc. (c) Describe the events taking place when alloys of compositions L, M, N are cooled from T > TA to T < TD. 3.16.2. Label all areas of the peritectic diagram of Fig. 3.16.10(b). State what happens as liquids of composition a, b, c, d, e are cooled from temperature T1 to temperature T2. 3.16.3. On phase diagram 3.16.10(c) identify the regions 1-5. Can such a phase diagram arise if the solid free energy curve is U-shaped and the liquid free energy curve displays a maximum at intermediate compositions? 3.16.4. Identify the regions 1-11 of the phase diagram, Fig. 3.16.10(d). What feature in the free energy curves gives rise to the very narrow line running vertically at composition x = 1/2? 3.16.5. Construct a set of free energy curves that lead to the formation of the compound peritectic phase diagram depicted in Fig. 3.16.8. 3.16.6. On the basis of Eqs. (3.14.2b) and (3.14.3) provide a rationale why the free energy of a solid phase tends to shift more rapidly with temperature than that of a liquid phase.

B

VARIATION OF MUTUAL SOLUBILITY WITH TEMPERATURE a

a b

b

LMN

d

e

,,

,,

,,-

i!

tI

T1 ,..... 900

TB T(K)

Tc

1

I

11 l \ 600 T2 500

TD

'

I

~

I l I

! I

I

"~

l

~l

I

-

800 700

XB

c

-I

1000

I

0

239

|

I

I I I

g

I

I

i

\ ~ , 6",{

0.2 0.4

0

0.6

0.8

1

XB

T1 T2

1

T3

I T5 T T6

T4 T7 . . . .

A 0

B XB

1

T9

A

1/4 1/3

1/2

4/5

B

XB

Fig. 3.16.10. P h a s e d i a g r a m s to be rationalized in t e r m s of the q u e s t i o n s p o s e d in Exercise 3.16.4.

3.17

Variation of Mutual Solubility with Temperature; Second Order Transitions

As a rule of thumb, a rise in temperature broadens the homogeneity range of a solution that is heterogeneous at low temperatures. This is so because at higher T the negative contribution to the free energy arising from - T A Sm will begin to compete with and ultimately outweigh the positive contribution arising from A Hm which was primarily responsible for the initial phase separation. The net effect is to bring the local minima, shown in Fig. 3.17.1, together until they merge for x = 1/2 at a critical temperature, above which the solution is homogeneous throughout the entire composition range. We wish to quantify the temperature dependence of the homogeneity range of a binary solution on the basis of a very simple model.

3.17.1

The Bragg-Williams Approximation

These matters may be attended to quantitatively in the so-called Bragg-Williams approximation. On temporarily ignoring the variation of G* with x in

240

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

Eq. (3.14.13) one obtains the relation (B =_w/RT)

AC_r m

Cr m -~__1.*

RT

RT

/13

: (1 - x) ln(1 - x) + x lnx + ~--~x(1 - x).

(3.17.1)

Plots of (AGm/RT) versus x for variety of values for the dimensionless parameter RT/w are exhibited in Fig. 3.17.1. We note that either at very low temperatures or for very large positive w the last term outweighs the entropy term, except near the end points where x --~ 0 or x ~ 1. Correspondingly, the major portion of the top curve in Fig. 3.17.1 is positive, but there are two minima symmetrically disposed along the negative portion of the curve. For RT/w > 0.37 the curve lies entirely in the negative domain, and for RT/w > 1/2, it no longer exhibits local minima away from x - 1/2. In the latter case 02 A Gm/O2x > 0 for all x, so that the alloy remains homogeneous over the entire composition range below the critical value of (RT/w)c = 1/2. When account is taken of the variation of G* with x, the curves in Fig. 3.17.1 become skewed with respect to x, as was illustrated in Fig. 3.15.1.

+~

'

'

'

' RT/VV '

+0.6,

'

'

'

'

,

,

0.20

+0.5

+0.4 +0.3 +0.2

n," 1(..9

0.30

+0.1

0.33

0

1(.9 v

0.40

-0.1

0.45 -0.2

0.50

-0.3

0.66 -0.4

[ -0.50

I

I

I

I

0.1

0.2

0.3

0.4

1 oo I 0.5 X

I

I

I

I

I

0.6

0.7

0.8

0.9

1.0

-~

Fig. 3.17.1. Variation of reduced molar Gibbs free energy of mixing with composition for a mixture with to > 0. Different curves correspond to different R T/to values. Dashed curve shows the locus of points for local minima.

VARIATION OF MUTUAL SOLUBILITY WITH TEMPERATURE

241

0.5

0.4

0.3 RT 0.2

0.1

0

0.2

0.4

0.6

0.8

1.0

X

Fig. 3.17.2. Separation of phases for a binary mixture. Region inside curve represents domain of heterogeneous phases.

The locus of the minima is shown as a dotted curve in Fig. 3.17.1. Any horizontal line within its boundaries represents the composition range over which the mixture is heterogeneous for a fixed value of RT/w. The plot of RT/w versus x obtained from the dotted curve is shown in Fig. 3.17.2. For constant w this figure traces the variation of the immiscibility range with temperature. For compositions within the dome-shaped curve the binary solution is unstable and breaks into two phases whose compositions, x ~ and x", are located by the intersections of the horizontal line, at the RT/w value of interest, with the curve. As T is lowered the two phases approach the pure limiting compositions x - - 0 and x = 1, which would be attained, if the system could be equilibrated, at T -- 0. 3.17.2

Variation of the Gibbs Free Energy with Temperature in the Bragg-Williams Approximation

For a quantitative assessment of the variations of G with temperature at fixed overall composition, one must deal with Eq. (3.17.1), which is satisfactory as it stands for the T range RT/w > 1/2. However, at temperatures RT/w < 1/2 the splitting of the solution into two phases of composition x ~ and x" must be taken into account. The relative amounts are specified by the lever rule, Eq. (3.15.2). This phase separation introduces complications because x ~ or x" are now no longer arbitrary variables; instead, they are fixed by the solution temperature. In the subsequent analysis it is important to keep track of quite a few quantities that are listed in Table 3.16.1 for easy reference. For a two-phase heterogeneous mixture we write G - G' + G",

(3.17.2)

242

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

in which G ~ and G" are the Gibbs free energies for the two phases of composition x B~and x B." According to Eq. (3.14.18), !

G ~-

'~*

'~

+

+ RV{

~ln A

~+

~ln

+

!

nAn B

(3.17.3)

with a corresponding expression for G". Referring to lines 2 and 8 of Table 3.16.1, and introducing the definition of n'G ~ - G' on the left of Eq. (3.17.3), we find

n 'G' - - n A' G-*A + n B'G-*B + R T n' { ( 1 - x ') l n ( 1 - - x ' ) + x

lnx'}

+ w(T)n'x'(1 - x'), n'(~'--

(3.17.4a)

" "~*A + nBG " ~ ~ -Jr-R T n " {(1 - x") In(1 - x") + x " In x" } nAt.r

+ w ( T ) n " x " ( 1 - x").

(3.17.4b)

Here we have assumed that w ( T ) is the same function for both phases. This is probably a rather poor approximation; it has been introduced to keep the subsequent operations manageable. (Brave readers are invited to explore the complications that arise when different w values are assigned to the two phases.) Accordingly, the total free energy reads G -- n~G ~+ n ' G " -- nG, so that by lines 6 and 8 of Table 3.16.1, G---

n

--- XA ( ~

+~

n

-+- XB ( ~

+ RT(1 - f ) { (1 - x') ln(1 - x') + x' lnx'} + R T f { ( 1 - x")ln(1 - x") + x" lnx"} + w(T){(1 - f ) x ' ( 1 - x ' ) +

f x " ( 1 - x " ) }.

(3.17.5)

Since the minima in Fig. 3.17.1 are symmetrically displaced with respect to x - 0 and x - 1, we now set x' - 1 - x ~' and 1 - x ~ - x ' . Again such a rather drastic assumption has been introduced to keep subsequent~ mathematical~ manipulations simple. Eq. (3.17.5) then reduces to (G* = xAG*A + xBG B) G G* R T = R--T + { ( 1 - x ") l n ( 1 - x ' ) + x " l n x " } + wR( TT ) x ' ( 1 - x ' ) .

(3 17.6)

The preceding equation, somewhat fortuitously, has precisely the same form as Eq. (3.17.1); however, it differs from the earlier version in an important aspect: x ' ( T ) is not arbitrarily adjustable but rather is the solution of Eq. (3.14.14) when R T / w < 1/2. Thus, Eq. (3.17.6) depends solely on T because the composition x does not occur as an independent variable. By contrast, for homogeneous

VARIATION OF MUTUAL SOLUBILITY WITH TEMPERATURE

243

solutions where the variation of G with T is given by Eq. (3.17.1), x is a truly independent variable, specified by the overall composition of the solution that is controlled by the experimenter. Equation (3.17.6) can now be used to show how G / R T varies with temperature by numerical solution of the transcendental equation (3.14.14). This variation is not of particularly great interest. Rather more to the point is a study of the enthalpy changes with temperature. Proceeding by standard methodology, one obtains the enthalpy as H - -TZ[O(G/T)/OT]. Here one must be careful to recognize that for RT/w < 1/2, x" - x" (T) is an implicit function of temperature. Accordingly, the differentiation process yields

171 171* { [ x" ] w(T) (l _ 2x,,) } dx" RT 2 =-RT----~+ In l - x " + RT dT +x'(1-x')[(

1 dw(T))

RT

dT

w(T)]

RT 2 .

(3.17.7a)

On account of Eq. (3.14.14) the central term in braces drops out; Eq. (3.17.7a) may then be rearranged to read

/Q- XA/QI nt- XB/-tI~-~- [to(T)- T dw(T)] ~ dT

x"(1 -

x") .

(3.17.7b)

In the literature on the subject it is conventional to introduce a quantity termed the degree of order, or order parameter, Sp, defined by

Sp ~

2X" -- 1

or

x" -- ~(1 1 + Sp)

( - 1 ~< Sp <~ 1),

(3.17.8)

with which the equilibrium condition (3.14.14) assumes the simpler form 1 +Sp] w In 1 - Sp -- --R--~Sp.

(3.17.9)

Equations (3.17.6) and (3.17.7) now read

{1

1

AGm -- RT ~(1 + Sp) ln(1 + Sp) + ~(1 - Sp) ln(1 - Sp) - ln2

/

tO

+ 4RT (1 - s 2)

(3.17.10)

and

AlClm--(w(T)-

dT

1

~(1-Sp

2)

.

(3.17.11)

244

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

Equation (3.17.9) is frequently written out in the equivalent form

tanh( wsp ) 2RT --Sp,

(3.17.12)

and one conventionally defines a characteristic temperature Tx by Tx = w/2R, on the assumption that w is constant. Equation (3.17.12) then reads

3.17.3

Tx

tanh- ~Sp

T

Sp

(3.17.13)

Variation of Enthalpy with Temperature

It should be noted that Eq. (3.17.12) is a function of temperature alone; we next ascertain precisely how A Hm varies with T. For this purpose we must first determine s p ( T / T x ) by numerical solution of Eq. (3.17.13); the resulting universal functional relationship is shown in Fig. 3.17.3. One notes that the value of Sp changes very little in the range 0 ~< T~ Tx <~0.3; x" ~ 1 in this range. As T/T~ is further increased there is a gradually accelerating dropoff in Sp, until for T~ Tx --+ 1, Sp drops abruptly to zero, corresponding to x " - 1/2. Using the sp(T) relation shown in Fig. 3.17.3 one can determine AHm via Eq. (3.17.11). The result is shown in Fig. 3.17.4 as a plot of AIY-Im/(1/4)w versus T~ T~, on neglect of the term dw(T)/dT. One notes that at first AHm is close to zero, but as T/Tx increases beyond 0.3 the enthalpy of mixing increases rapidly to its final value of (1/4)w at T -- Tx. The heat capacity at constant pressure is now found through the relation (ACm)P -- (OAHm/OT)p. In Fig. 3.17.5, where we have set w - 2R, we show 1.0

0.8

0.6 0.4

0.2

0 0

0.2

0.4

0.6 T~x

0.8

1.0

Fig. 3.17.3. Variation of order parameter Sp with T/T~. as given by Eq. (3.17.13); T~, - w/2R.

VARIATION OF MUTUAL SOLUBILITY WITH TEMPERATURE 1.0

245

]

_m|

_

0.8 0.6 H1-Hm (1/4)W 0.4

0.2

I

0 . 0

t~-t--"~ ! ! 1_ 1 t__l l 0.2 0.4 0.6 0.8 1.0 T/Tx

1.2

Fig. 3.17.4. Change of reduced enthalpy of mixing with reduced temperature, as given by Eq. (3.17.11) with d w / d T = O.

2.0

. . . . . . .i. . .

1

I

0.4

0.6

I

I

I

1.0

I 1.2

1.5

A(~m)p 1.o 0.5 0

|

0

0.2

,

0.8 T/Tx

1.4

Fig. 3.17.5. Plot of reduced heat capacity vs. reduced temperature as obtained from Fig. 3.17.4, with w ----2R.

the resulting plot of (ACm)p/R versus T~ Tx; the graph is proportional to the slope of the curve in Fig. 3.17.4. Inspection of Fig. 3.17.5 shows that as the temperature rises there is a continuing increase in heat capacity, with a sharp drop off back to zero at T - Tx. This figure has approximately the shape of the Greek capital letter A and hence is frequently called a lambda anomaly: Tx is known as the lambda point. What Fig. 3.17.5 once more illustrates is that the transition from the biphasic to the monophasic regime occurs over a considerable range of temperature. This gradual transformation is known as a second order transition that contrasts sharply with first order phase transformations discussed in Chapter 2. As the temperature is raised, continually greater amounts of energy are required to achieve a further rise in T while the system moves close to the region of homogeneous stability, the onset of which abruptly halts the heat capacity anomaly.

246

3. CHARACTERIZATION OF NONIDEAL SOLUTIONS

3.17.4 Physical Interpretation We briefly consider the physical interpretation of the foregoing analysis in which the order parameter plays a central role. At T --O, Sp - 1, so that x " - 1 and x ~- 0. As indicated by Eq. (3.17.6), this signals the separation of the AB mixture into pure A and pure B ~ t h e completely ordered state of the system. With rising T, Sp drops slightly; x ~and x" move away from 0 and 1, respectively: some B has dissolved into A and vice versa. The system has now become slightly disordered. With an additional increase in T, Sp becomes still smaller; x ~ and x" move further away from their end points, so that the two phases are now characterized by a greater degree of mixing and disorder. Finally, at Tz, S p - - 0 and x ~-- x ~ -- 1/2. The two phases have merged into one and the mixing of the two components is complete. Thus, the decline in Sp from 1 to 0 is a measure of the increasing intermixing of components A and B in the two phases. Correspondingly, At)m, AHm, A Ct,m, indicate the changes in Gibbs free energy, enthalpy, and heat capacity at constant pressure that are associated with each stage of the gradual transition from complete phase separation at T -- 0 to complete mixing at T -- T~. Although these results were derived in the context of the mixing process for binary solutions, the formulation just provided is of much more general applicability. As an example, consider a ferromagnetic domain in a specimen held at T - 0. All spins are aligned in one specific direction, with spins 'up'. States with spin 'down' do not exist. With rising temperature some spin reversals are encountered; the system has become somewhat disordered. There is a progressive shift of this type with increasing T until, at the Curie temperature, there are an equal number of spins in either alignment; the material has entered the paramagnetic state; on average there are as many 'spin-up' as 'spin down' alignments. Another illustration is furnished by the adsorption of N atoms of Hg on 2N surface sites of an inert solid. At T - - 0 the N Hg atoms are congregated on N adjacent sites as a cluster; the remaining N sites form an empty cluster. With a progressive rise in temperature Hg atoms start to move onto previously empty sites, thereby diminishing the degree of ordering. Beyond a certain characteristic temperature the Hg atoms tend to become randomly distributed, corresponding to complete disordering of the distribution of occupied sites. These examples suffice to show that a variety of physical situations can be treated by the approach of the present section, namely cases where one encounters an individual state of a system in one of two possible configurations: A or B, full or empty, up or down, plus or minus, and so on. These results again follow in a straightforward manner as an inescapable consequence from very modest beginnings the Margules formulation for activity coefficients in binary solutions. However, as with any elementary approach involving the use of many simplifying assumptions (mentioned at various stages in the derivation), one cannot expect quantitative agreement with experiment. Furthermore, the present approach, referred to as a mean field theory, neglects fluctuation phenomena that become

VARIATION OF MUTUAL SOLUBILITY WITH TEMPERATURE

247

prominent in second order transitions. However, the semiquantitative correctness of the theory is an indication that the overall approach taken here is basically sound. 3.17.5

Extension of the Model

At first glance it may appear as though the assumption regarding the equality of w(T) for the two phases ' and " in Eqs. (3.17.4a) and (3.17.4b) were excessively restrictive. It turns out 4, however, that when w is replaced by Wl and by w2 in (3.17.4a) and (3.17.4b) respectively one again recovers Eq. (3.17.6), except that in place of w(T) one must now substitute Wl (T) + f ( T ) { w z ( T ) - wl (T)} = to(T). Thus, the original T-dependent parametric function w(T) is replaced by the more elaborate t0(T); however, the basic functional dependence of G / R T on x" remains unaffected. It is for this reason that the simple model developed in Eqs. (3.17.7a,b) ff remains applicable when the restrictive assumption is relaxed as shown above. EXERCISES 3.17.1. Correlate Fig. 3.17.3 with Fig. 3.17.2. 3.17.2. Why is the enthalpy of mixing almost zero at low temperatures, as in Fig. 3.17.4, and why does it rise sharply as T approaches Tx? 3.17.3. Determine the entropy of mixing as a function of temperature, and sketch the resulting curve. Keep track of all approximations you make. Comment on your results. 3.17.4. I am indebted to Mr. Joseph Roswarski of Purdue University for this derivation.

249

Chapter 4 Thermodynamic Properties of Electrolytes

4.0

Introductory Comments

In this chapter, the thermodynamic properties of ionic solutions will be investigated. Since the interaction forces between charged species are quite strong and of long range, it is important at the outset to take account of deviations from ideality. Hence, considerable use will be made of the machinery set up in the first half of Chapter 3, in which the concept of activity and activity coefficients plays an important part. Heavy reliance is also placed on the Debye-Hfickel Theory: the final results are quoted without proof, but a derivation of the results is furnished in Chapter 9. The latter portion of the present chapter deals with the properties and characteristics of galvanic cells. Here, again, the emphasis is on fundamentals; for the myriad applications, special uses, or refined specializations the reader is referred to monographs and to review papers in the field.

4.1

Activities of Strong Electrolytes

In 1884 Svante Arrhenius advanced the then very revolutionary hypothesis that salts dissolved in aqueous solutions tend to ionize partially or completely. Such a process may be represented by an 'equation' of the form Mv+A~_(aq)= v+M z+ + v _ A z - where M and A represent the cationic and anionic constituent of the compound, z+ and z- are their appropriate ionic charges and v+ and v_, the stoichiometry numbers. The ionization process symbolized above requires that My+ Av_ itself be considered either as a dissolved species that may be equilibrated with the undissolved compound in a saturated solution or that exist as such. This formulation also hides a multitude of difficulties: in many cases the ionization process is much more complex than indicated above. For example, in the ionization of AgI one encounters in aqueous solution the species AgI, Ag +, I-, AgI~-, AgzI +, and several more exotic combinations. In what follows we limit ourselves to cases where there is a great preponderance of the elementary species over the more complex aggregates.

250

4. THERMODYNAMIC PROPERTIES OF ELECTROLYTES

4.1.1

Chemical Potential of Ionized Species; Mean Activities

One may set up the differential free energy change corresponding to the above ionization reaction by A G d -- 2 _ ,

V i l Z i - - 1) + l Z + -]- 1 ) - l Z -

-- lz i '

(4.1.1)

i

where/z i is the chemical potential of the undissolved salt or undissociated compound of species i, My+ Av_. At chemical equilibrium A Gd vanishes, so that (4.1.2)

/Z i = 1)W/Z_t_ -+- V _ / Z _ .

It is now appropriate to introduce the conventional equation /Z i - - / z * ( T , 1) + lnfii(T, P, qi) set up in Eq. (3.6.2), with q i - - x i , c i , m i . On setting i = + , Eq. (4.1.2) reads

l z - ~* + R T l n a -

(v+tz~_ + v-ix*_) +

RT(v+lnfi+

+

v_ lna_),

(4.1.3a)

where the subscript i and the parametric variations have been temporarily dropped to avoid proliferation of symbols. If Eq. (4.1.3a) is to be rewritten in conventional form, as shown above, the two sides of Eq. (4.1.3a) must match, so that one must set :r

#* - v + / x + + v_/z*,

(4.1.3b)

as the standard chemical potential for dissolved ionized species. For consistency one also adopts a mean relative activity by the relation fi _ fi~ _ (fi+) v+ (fi_) v_,

(4.1.3c)

a_ ]--IX * + v R T l n f i + , IX -- tx* + R T l n [ a+ -v+-v-

(4.1.4)

v - v+ + v_.

(4.1.5)

which allows one to write

while setting

The relation (4.1.3c) is further motivated by the following example: Consider the dissociation of CaC12 - Ca 2+ + 2C1-. The concentrations are given by c+ - c and c_ -- 2c. The geometric mean of these concentrations leads to the relation c 3 - c+c 2 - 4c 3. A similar relationship must obviously hold for the activities. Equations (4.1.3-5) imply more than meets the eye. First, one must always satisfy the Law of Electroneutrality, according to which (with z - < 0), v+z+ + v_z_

-

v+z+ -

v-lz-I : 0.

(4.1.6a)

ACTIVITIES OF STRONG ELECTROLYTES

251

Second, there is no possibility of separately determining either fi+ or fi_. For, by definition,/z+ -- (OG/On+)T,P,n_ ; that is, the determination of/z+ requires addition of only positive ions to the solution, while holding the concentration of anions fixed. However, this step, which violates the Law of Electroneutrality, cannot be carried out operationally. Consequently, one cannot deal with individual ionic activities; rather, as Eq. (4.1.3c) shows, the ionic activities occur in such a manner that only their geometric mean or appropriate logarithmic sum is involved. Third, since thermodynamic descriptions must be confined to measurable properties we may regard the quantity, #* + R T I n fi on the left of Eq. (4.1.3a) as an 'effective chemical potential' that is to be used to represent the behavior of the electrolytes. On the other hand, one does not wish to ignore the ionic nature of the solution; hence it is customary to write (4.1.6b)

# - - # * + RT lnfi~, which is in consonance with (4.1.4). 4.1.2

Mean Concentrations and Activity Coefficients

Fourth, we must refer the activity fi or fi~ to a measurable quantity in an ideal solution. For this purpose we set up relations analogous to (4.1.3c), namely, vi

] vi +

Xi -- ( X i ) + -- ( X i , +

"

m i -- (mi)V~ -- ( m i ) +v i + ( m i ) V i _-

Ci

pi

--

(Ci)•

"

--

(4.1.7a)

(Xi) v ' - , ,

vi -

(Ci)~_+(Ci)_

9

(4.1.7b) (4.1.7c)

We next adapt the arguments leading to Eqs. (3.5.1) to the present situation by defining the mole fraction for the i th positively charged species in solution by xi+ - n i + / ~ ( s ) Vsns - mi+/~--~(s) V s m s . Here a new notation has been adopted: The index s runs over all distinct c h e m i c a l compounds added to the aqueous phase, not over the ionic species i present in the solution; the dissociation process of these compounds in water is attended to by insertion of the sum Vs -- Vs+ + Vs-. Here Vs+ or Vs- are the number of cations or anions derived from the complete dissociation of the sth species My+ Av_ into the Vs+ positive and Vsnegative ions, M z+ and A z- . Thus, each mole of the compound Mv+Av_ yields (v+ + v_) = v moles of ions in solution. We assume that the solvent (s = 1) remains unionized, so that Vl = 1. Complete solute ionization is assumed to take place for all other species; the case of incomplete ionization is handled later. For all dissolved nonionic species Vs = 1; moreover, for the solvent, m l -- 1000/M1; see Eq. (3.5.2). Thus, Xi+ - - m i + [ l O O O / M 1

-Jr- v 2 m 2 -k- v 3 m 3 + . . . ] - 1 .

(4.1.8a)

252

4. THERMODYNAMIC PROPERTIES OF ELECTROLYTES

Furthermore, ml

v2m2M1

1+

--

X1 -- E s ps m s

v3m3M1

+

1000

+...

]-1

.

(4.1.8b)

1000

Accordingly, when one utilizes (4.1.7) and (4.1.8), one obtains

mi+ Xi+

miXi-

mi• Xi+

1000

1 +

v2m2M1

M1

v3m3M1

+

1000

1000

+ ....

1000 Mlxi" (4.1.9)

Next, we determine c i + / m i + by noting that mi+ - - lO00ni+/Mlnl, and ci+ : ni+ 1000/V, where V is specified in cm 3. Thus, ci+/mi+ =-- (ci+/ni+)(ni+/mi+) -- M1 n 1/ V. When eliminating V in terms of the mass of solution we sum over all component species s from which the final mixture was made up, while ignoring the fact that many of the components ionize in going into solution. Thus,

Ci+ mi+

Mlnlp ~ s Msns

Mlmlp Y]~s Msms

(4.1.10a)

and since M1 n 1 -- 1000,

ci+ mi+

cimi-

ci+ mi+

p 1+

M2m2 lOOO

M3m3

(4.1.10b)

-+- lOOO + " "

Multiplication of (4.1.10b) with (4.1.9), followed by (4.1.8b), yields

Ci+ Xi+

CiXi-

Ci+ Xi+

1000p

(4.1.11)

mzM2

MlXl [1 + 1000 + 13M3000-at-" "]

In the limit of very dilute solutions for which xl ~ obtains mi:s

1000

xi+ xl~l (Ci_....i_i)

_ 1000pl

- -

(4.1.12a)

ml

xi+ xl~l

(4.1.12b)

M1 --

mi+

1 and Ms ms / 1000 << 1, one

(4.1.12c)

Pl.

Xl-+l

We now introduce mean activity coefficients Pi

vi+

Pi-

(Yi)• ~ (Yi)+ (Yi)- ,

in strict analogy with Eq. (5a); here V+ -- v+(T, P, q+), q - x, c, m.

(4.1.13)

ACTIVITIES OF STRONG ELECTROLYTES

253

It should then be clear that the relation between activity and activity coefficient which supplants Eq. (3.6.3) now reads (omitting subscripts i for simplicity) q+ ' q, - r+ ( T' P ' q +) -q-2 v+(T, P, q+)a *q (T, P) q+

fi+(T, P, q+) --

(4.1 14)

,q

in which q - x , c, m. Here a+ (T, P) refers to the species under consideration in pure (undissociated and undissolved) form, and q* is the concentration of the pure (undissociated and undissolved) species. As usual, x* - 1; also, q*(T, P) is ordinarily close to unity. To interrelate the various v+(T, P, q) we adopt Eqs. (3.5.8), (3.5.10) in the form

Vii(T, P, ci+) -- Vii(T, P, xi• -- I Pi, (T, P) y-]snsMs 1 1/v [vi(T, P, c*)] 1/v, p(T, P) Mi ~Qs ns @.1.15)

yi+(T, P ,

mi-t- ) -

yi+(T,

P, xi•

-

{xlM1/Mi}l/v[yi(T, P,m*)] l/v, (4.1.16)

where the summation over s again serves as an explicit reminder that one sums over all chemical species added to the solution, not over the individual ionic species. Lastly, the q i are interrelated as shown in Section 3.5, after replacing q by q~:. 4.1.3

Henry's Law as Applied to Ionic Solutions

In characterizing ionic solutions one generally deals with the limiting case of great dilution, so that it is apposite to introduce Henry's Law for this purpose. The discussion of Section 3.11 must be adapted to present circumstances, as indicated by the following general example: if we elect molality as the relevant concentration variable and consider the ionization process Mv+Av_ -- v+M z+ + v - A z- we rewrite Eq. (3.11.7) in the general form

y•

_

[

, j (v+)V+(v_)V-mvP;

- 1/v

(4.1.17)

Here it is a plot of P vs. [(v+) v+ (v_)V-m v pt.] that is linear and that can be used to read off the desired mean molal activity coefficient. As earlier, P ~ refers to the extrapolated pressure reached in Henry's Law plot for the component in pure form. 4.1.4

Chemical Potentials and Standard States for Completely Ionized Solutions

To specify the chemical potential for component i that has completely ionized in solution according to the scheme M~+Av_ - v+M z+ + v_A z- we adopt

254

4. THERMODYNAMIC PROPERTIES OF ELECTROLYTES

Eq. (3.5.22b) and (3.5.20); for the concentration units usually employed for ionic solutions are molalities. The reader should carefully convince him/herself that the relevant expressions are given by 1 (this matter is not self-evident!) l z i ( T , P , m i + ) --

lz~(T, 1) -t- v i R T l n a i + ( T ,

=/x*(T, 1) + v i R T l n

[

yi•

P, mi+)

mini

P, mi+)a*m(T, P ) m ,

.

mi

(4.1.18) This relation forms the basis for subsequent developments. Consider a saturated solution of a solid (say, NaC1) dissolved in the solvent (such as water), for which Yi+ is replaced by y/s+, and m i+, by mi+.s This solution is in equilibrium with pure solid (NaC1) whose chemical potential is given by ll, i ( Z ,

P,m*) - lz*(T, 1 ) + RTln[Yi*(T, P,m*)a*m(T, P)],

(4.1.19a)

and which is in equilibrium with the saturated solution. For all but the most accurate determinations we may neglect the logarithmic term in Eq. (4.1.19a). We then find that

[

s]

lZi (T, P, mS+) - tzi (T, P, m*) -- 1)i R T In ysi+(T, p ' m si+)a i,m (T, P) mi+.m, . m i

(4.1.19b) The second term on the left-hand side represents the molar free energy change of formation of the solid from its elements; hence, the left is equivalent to the free energy of formation of the solid in the saturated solution; we therefore write, in conformity with Section 3.9, A(~0,s f,T

[

--

s]

, 1)iR T In ysi + (T, 1' m i +s ) ~mi+ m*

(4.1.19c)

where in accord with Eqs. (3.4.7), (3.4.10a), (3.4.1 lb), (3.4.12) and (3.5.20) [see also Example 3.4.1] we had set a i9m(T, 1) -- 1. From (4.1 . 19b) one can obtain A

-O,s

~O,s

Hf, T and SL r by standard techniques of differentiation with respect to T.

4.1.5

Activities of Volatile Components

As a prototype we consider a solution generated from dissolved HCl(aq) in equilibrium with HCI(g) in the gas phase. When taking account of the fact that these species are in chemical equilibrium and that the ionic species H + and C1- predominate in solution we find that OP /zHcI(T ) + R T In fHC1 ,

= I~i(T, P, mHCl) + R T l n

m-t-

y+(T, P, m+)a *re(T, P)--~-2

9 (4.1.20)

ACTIVITIES OF STRONG ELECTROLYTES

255

From the above we collect all variable terms on the left and the remainder, on the right, to obtain in an obvious notation R T l n [ y+(T, Pm , +)m+H f c1

] -- R T l n K .

(4.1.21)

Since K is fixed for a given T and P (2) one can in principle determine K by working with solutions at sufficiently low molalities so that the activity coefficient is sensibly unity. Having found K one can then operate at molalities for which V+ (T, P, m i ) is to be found. 4.1.6

Standard States for Electrolyte Solutions

As already mentioned, it is not possible to measure the thermodynamic properties of single ions. However, it would be highly desirable to set up such a compilation, so that an experimentalist does not have to measure the literally thousands of cationic-anionic combinations whose properties are of interest. Since the thermodynamic characteristics if such pairings are algebraically additive one may set up the following convention:

At any temperature the Gibbs free energy for the standard state of H + in aqueous form is taken to be zero. That is, we set A G~, v (H +, aq) -- 0. Then the thermodynamic properties of anions can be found by measuring the chemical potentials of ionic solutions containing H + in combination with different anions, and then using Eq. (4.1.3b). The anionic chemical potentials so determined can be employed as secondary standards in solutions containing different cations, and this matching process is continued as needed. Extensive tabulations constructed in this manner are available. However, this convention becomes null and void in processes where H + ions are transported across the phase boundaries of the aqueous solutions. Since the entropy of H + ions is found via the temperature derivative of the Gibbs free energy of its formation and the latter is independent of T by the above convention, one obtains AS~,~(H +, aq) - S~,7~(H+, aq) - 0. It now follows that A G~, r (H+, aq) - O. EXERCISE AND COMMENT 4.1.1. Using Eqs. (3.5.22b), (3.5.20) demonstrate that Eq. (4.1.17) correctly applies to ionic solutions. 4.1.2. Strictly speaking, one should add an inert gas in variable amounts sufficient to keep the total pressure P at a constant value during all of the measurements. Alternatively, one may employ a piston.

256

4. THERMODYNAMIC PROPERTIES OF ELECTROLYTES

4.2 Theoretical Determination of Activities in Electrolyte Solutions; The Debye-Hiickel Equation Having defined various activities and activity coefficients in solutions made up from strong electrolytes, we now turn to the determination of y+. For this purpose we briefly discuss some aspects of the Debye-Hiickel Theory.

4.2.1 Ionic Strength We first call attention to the concept of ionic strength introduced by Lewis and Randall in 1921. This quantity is defined as S ~ -

cj

2

2 Zj,

(4.2.1)

J

in which cj ( - c+ or c_ for every ion j ) is the molarity of the jth ionic species in solution and z j is the formal charge on the corresponding ion. The summation is to be carried out over all ionic species present in solution, not just over the species of interest. What renders this concept useful is the experimental fact that in dilute solutions the activity coefficient of any strong electrolyte is the same in all solutions of the same ionic strength, regardless of the chemical nature of the dissolved ions.

4.2.2 Debye-Hiickel Equation In discussing the theory of Debye and Htickel (1923) we shall defer a rather lengthy derivation of the final results to the end of Chapter 9, so as not to interrupt the flow of the underlying concepts. One may even adopt the view that the results listed below represent excellent limiting laws that are known to represent a large body of experimental data. However, this obscures the fact that an elementary exposition of the theory is presented in Section 9.5 without the use of statistical mechanics and electrodynamics, though a more detailed derivation is needed for a proper understanding of the model. We now proceed without proof. The Debye-Hiickel limiting law specifies the following relations: for the individual ionic species, the activity coefficient is specified by

z2+C43 In ?,+ --

1 + C2~/S

In y_ =

1 + C2~/S

(4.2.2a)

z2_Ce (4.2.2b)

THEORETICAL DETERMINATION OF ACTIVITIES IN ELECTROLYTE SOLUTIONS

257

whereas the mean molar activity coefficient, which is the experimentally more meaningful quantity, is given by In ?,+ -- -

z+ lz- If e J-s , 1 + c2,/

(4.2.3a)

in which, according to the methods of statistical mechanics, -

e3N2 ( 27c ) 1/2

Ce = (eRT)3/2

1000

'

(4.2.3b)

87re2a2N ) 1/2 C2 =

(4.2.3c)

1000e R T

In the above - e is the electronic charge, a the average ionic separation, N Avogadro's number, e the dielectric constant, and R the gas constant. Generally, C 2 ~ is small relative to unity; one then deals with the extreme limiting law: In y+ - - z + l z - I C e j ' S .

(4.2.3d)

On switching from natural to common logarithms and inserting numerical values for Ce and C2 one obtains the following result, valid for aqueous solutions at room temperature: log y• - -

0.5092z+ Iz_ IV'-S

(T - 298.15 K),

(4.2.4)

where a in Eq. (4.2.3c) has been set at 0.31 nm. As already noted in conjunction with Eq. (4.2.3d), the denominator is ordinarily replaced by unity; under conditions where the theory is applicable ~ << 1. If the constants Ce and C2 are to be computed at other temperatures one must take into account not only the factor in Eq. (4.2.3b) but also the variation of e with T, which is substantial for water. One should note the following: (i) Eq. (4.2.4) applies to any aqueous electrolyte solution at room temperature, but (ii) with limits of applicability generally restricted to solutions of molarity 10 -2 or less. 1 (iii) The activity coefficients for different solutions of the same ionic strength and for the same ionic valencies are the same. (iv) A plot of log y• versus ~/S for extremely dilute solutions should yield a straight line of limiting slope - 0 . 5 0 9 2 z + ]z-I in aqueous solution at room temperature. Extensive testing over a long period of time has confirmed the correctness of this prediction. (v) It has been established by use of different solvents that in the limiting case of dilute solutions one finds that - log y+ ~ e -3/2. From the context of the current discussion, it should be evident that the mean activity coefficient cited above is related to molarity. On the other hand, as is to be proved in Exercise 4.2.2, the definition of S remains virtually unaltered

T-3/2

258

4. THERMODYNAMIC PROPERTIES OF ELECTROLYTES

by switching from molarity to molality in aqueous solutions at ordinary conditions of temperature and pressure. Thus, the quantity specified by Eq. (4.2.3a) may be considered to represent either y+(T, P, c) = y(,c) or y+(T, P, m) _= y,{m). However, for very precise work, or when nonaqueous solvents are employed, or whenever T and P deviate greatly from standard conditions, the two preceding quantities cannot be used interchangeably; Eq. (4.2.3a) specifies V,(c). As has already been stressed, the Debye-Htickel relation, even, in the form (4.2.3a), is of only limited applicability. There have been many attempts to extend the range over which it remains useful; one of the most widely used versions reads In ?'+ - -

z+lz-IC~~+ 1 -+- C2 ~rS

2v+v_ ~C3m+,

v - v+ + v_,

(4.2.5)

V

in which C3 is a purely empirical constant whose value must be determined by experiment. One method for accomplishing this is explained in Section 4.10. Equation (4.2.5) is sometimes termed the extended Debye-Hiickel equation. COMMENT AND EXERCISES

4.2.1. It is sometimes asserted that the Debye-Htickel Limiting Law applies only to slightly contaminated distilled water. 4.2.2. Show that there is very little difference in the results derived in the present Section for aqueous solutions when molarity is replaced by molality. Why is this not the case for non-aqueous solvents? 4.2.3. Suggest a method for rewriting the Debye-Htickel equation so that one may easily determine the value of a for the effective ionic diameter, which may also be viewed as the average distance of closest approach between ions. 4.3

Experimental Determination of Activities and Activity Coefficients of Strong Electrolytes

Since the determination by theoretical methods of activities or activity coefficients for strong electrolytes is limited to very dilute solutions, experimental methods must be invoked to find 9/+ for m > 10 -2 molal. We shall briefly describe some of the methods in use; here the discussion is closely patterned after Section 3.13. The use of emf methods for the same purpose is described in Section 4.10, after proper background material has been developed. Once again, the choice of P - 1 bar serves as the standard pressure; in this case the activity coefficient/~ introduced in Section 3.4 is to be used. Otherwise the quantity V+ comes into use.

4.3.1

Vapor Pressure Measurements

Vapor pressure measurements may be used to determine the activity of the solvent. Equation (3.11.4) may be taken over without change:

EXPERIMENTAL DETERMINATION OF ACTIVITIES AND ACTIVITY COEFFICIENTS

-~F1 'x) ~

P1

,

P~xl

259

(4.3.1)

where PI* is the vapor pressure of pure solvent and P1 is the vapor pressure of the solvent in the presence of the electrolyte. 4.3.2

Use of Gibbs-Duhem Equation

The Gibbs-Duhem relation may be used in conjunction with the foregoing to determine the molar activity of the ionic species in a binary solution. This is necessary because a direct determination is difficult. We proceed as follows" since

dlna~ x) - dlna~ m), we use the form (T and P constant; ml - 1000/M1) 1000 ~dlna M1

wherein a~m) -- (a~m))~:, m 2

(,n)

1

+mzdln

a~m)

-0,

v2 - - (m2)+, (v2)+ + (v2)- ------v2.

(4.3.2)

Integration coupled

with the use of (3.11.4), yields

/m

mu

l

din

a~ m)

= -

/

1000 ~dlnal

(m)~ "~ -

Mlm2

s

1ooo

~dln(P1/P~). Mlm2

(4.3.3)

Thus, plots of lO00/Mlm2 versus ln(P1/P~) yield a value of the integral on the right for molalities between the lower and upper limits m l ~< m2 ~< m , ; the left-hand side is given by ln[a2(T, 1, mu)/a2(T, 1, ml)]. The problem with this approach is that changes in the vapor pressure cannot be measured with sufficient accuracy in very dilute solution. One must therefore employ this procedure in conjunction with the Debye-Htickel limiting law, which holds for m2 < m t ~ 10 -2. 4.3.3

Freezing Point Depression Measurements

Freezing point depression measurements furnish another convenient approach to determine the activity of the solvent. Eqs. (3.13.1,4) may be taken over without change because we have not had occasion to refer to the ionic dissociation process. For the determination of the activity of the ionic solute in a binary solution we modify Eq. (3.13.9) by writing ( m 2 = m; v = v+ + v_ for the compound Mv+Av_)

dlna~m ) - -1d l n a ~m) = d O q- -c~Od O v vl.m vm

'

(4.3.4)

and in place of (3.13.10) we introduce j--1

6) v)~m

(4.3.5)

260

4. THERMODYNAMIC PROPERTIES OF ELECTROLYTES

It should be verified that Eq. (3.11.13b) is replaced by 69 yam

= (1 - j ) d l n m - d j .

(4.3.6)

Next, we set a 1/v - - a+ , m+ -- v+m, m _ -- v _ m , so that m+ - m ( v vt_+V _v - ) 1/v . It follows from (4.1.14) with P - 1 bar, and with a+ replaced by a 1/u, that

[a(m)

d ln F('m) - d ln m(v++vV--)l/v

I-

a(m)) - d i n - ~(a) . (m)\

d ln(~\ m+

(4.3.7)

Thus, in place of (3.13.12) we find from (4.3.4-7), cO - d j 4- - - - d O ,

dlnF(, m) -- d l n a ~ m) - d l n m - - j d l n m

(4.3.8)

1) m

whose integration yields the analogue of (3.13.13), namely, d In F ( m )

(Tf)

--

-j

-

f0 m-j- d m

m

+ -c

O- d O . v L m -m

(4.3.9)

Here it is advisable to replace the central term by its equivalent formulation 2 f o ( J / m l / 2 ) d m l / 2 , since for strong electrolytes j / m 1/2 remains finite as m --+ 0. Finally, it is to be checked that at any temperature other than T f one obtains the analogue of Eq. (3.13.18), namely, In r~m)(T)--lnF(,m)(Tf)

blv fo m --mld m ,

(4.3.10)

in which bl = IO00/M1. 4.3.4

Solubility Measurements

Frequently, solubility measurements may be used in mixed electrolytes to obtain mean molar activity coefficients. This method hinges on the use of an electrolyte solution which is saturated with respect to any particular salt, so that the equilibrium My+ Av_ (s) -- v+M z+ + v _ A z- prevails. This situation may be characterized by (among others) use of the equilibrium constant Km specified by Eq. (3.7.8b). It is conventional either to ignore the product t e r m [ai*sm ( T , p)]vi as being equal to (at unit pressure) or close to unity, or to absorb this constant factor into the equilibrium constant as well. This then gives rise to the expression ,, K m --

a++ a v~ .

aMy.Au_

(4.3.1 la)

EQUILIBRIUM PROPERTIES OF WEAK ELECTROLYTES

261

Since the activity of pure My+ Av_ is a constant, it, too, may be absorbed into the equilibrium constant; this step finally yields

K s - - a + +a_v_ -- a+v _ (m+ Y+) v

(4.3.11b)

Inversion of this relation leads to (m)

V•

K ls/v

-- ~ .

m•

(4.3.12)

The procedure now consists in adding other strong electrolytes to the solution. Since Ks remains unaffected by this step while y(m) of the electrolyte of interest necessarily changes, m+ will change in the opposite direction. One thus measures m+ from the observed solubilities of the salt Mv+Av_ in the presence of other salts added in varying amounts. The results may then be extrapolated to infinite dilution on a plot of m+ versus ~/-S, where S is the ionic strength defined in Eq. (4.2.1). This permits an extrapolation to zero molarity w h e r e y(m) _ 1 The + mean molarity obtained from this extrapolation thus yields Kls Iv. Measuring m+ for any other value of S then provides the desired V(,m) . Two additional observations should be made. First, the methods used here treat each of the ionic types as a separate species that influences the thermodynamic properties of solutions very strongly by virtue of its associated charge. Second, it is instructive to examine the dependence of the mean molal activity coefficient for several different electrolytes as a function of the molality. Representative examples are shown in Fig. 4.3.1. One sees at first a very steep drop in y+ as m is increased, and then either a gradual or a very sharp rise in V+ as m is increased beyond 0.5. The greater the value z + [ z - [ t h e sharper is the initial dropoff. [Explain why!]

4.4 4.4.1

Equilibrium Properties of Weak Electrolytes General Discussion

Weak electrolytes are characterized by the equilibration of undissociated My+ Av_ with its ions in solution according to the schematic reaction Mv+Av_ - v+M z+ + v - A z- . There are many different ways of characterizing equilibrium conditions. Here we shall adopt Eq. (3.7.8) for the specification of the equilibrium constant; when My+ Av_ represents a pure phase we find: a

~_+a _1J_

Kq -- - - .

aMv+Av_

(4.4.1)

262

4. THERMODYNAMIC PROPERTIES OF ELECTROLYTES 1.00 0.90 0.80

"~

.~

-

.,.., 0.70 t,.Chloride

.m o

E 0.60

8 .-_>' 0.50 0.40 0.30 0.20 0.10

0

0.50

1.0

I 2.50

1.50 2.0 Molality

3.0

Fig. 4.3.1. Variation of m e a n molal activity coefficients as a function of molality for several salts dissolved in aqueous solutions.

The reader should refer back to Section 3.7 for a discussion of the standard states which have been adopted in the specification of Kq. In the event that undissociated Mu+ Au_ is present in a pure condensed state, its relative activity is equal to unity under standard conditions. So long as the pressure is not enormously different from standard conditions, a does not deviate significantly from unity. In either case, this factor may then be dropped, so that the equilibrium constant now reads (4.4.2)

Kq -- (a~+ a > ) -- (a•

Equation (4.4.2) may be rewritten in terms of activity coefficients as indicated by Eq. (4.1 9 14), omitting q*J r while also converting from a to ft. With q - x ~ c m one finds

K q " - {q+y+(T, P, q+)a*, q (T,

P)}eq'

(4.4.3)

where [a*,q (T, p)]v = [a*q (T, P)]V+[a*,q (T, p)]v_ = a, q (T, P) and a *q is determined as in Section 3.7. In the above, Kq is termed the solubility product con-

stant: under standard conditions, where a*, q - 1, !)

Ks -- {q+y+(T, 1, q)}eq"

(4.4.4)

EQUILIBRIUM PROPERTIES OF WEAK ELECTROLYTES

263

If, on the other hand, Mv+Av_ represents a dissolved but unionized species one then deals with the partial dissociation of a weak electrolyte. The discussion that follows will be restricted to solutions subjected to a pressure of one bar, in which case a *q (T, 1) = 1 for all species. The dissociation of a weak electrolyte is then characterized by an equilibrium constant in the form

g qtt' -- I - I ( q i + Yi + )eViq+ 1 7 ( q J - YJ - )eJq- -- 1 7 i i j (qoVo)e~

1, q+)eq . . . . . i26 , (qoYO)eq

q+yi+(T,

(4.4.5)

in which the subscript zero refers to the undissociated species; since the latter is electrically neutral, deviations from ideality are often neglected for these components, by setting Y 0 - 1.

4.4.2

Examples

In what follows, general principles are illustrated by specific examples. (a) The case of water is well known: here one deals with the equilibrium HzO(g) -- H + + O H - , which leads to the equilibrium constant K w -- aH+aOH- 9

(4.4.6)

aH20

It is customary to take aH20 -- 1, as discussed elsewhere, this step is strictly correct only if P - 1 bar and if no other dissolved species are present. K w has been measured carefully as function of temperature over a considerable temperature interval; for each temperature K w may be determined from conductivity or from emf measurements, the latter technique being described in Section 4.1 1. The heat of ionization per unit advancement of the ionization reaction may be determined according to Eq. (2.9.14) in conjunction with van't Hoff's Law. This requires a knowledge of how y+ changes with T. Details, based on Section 3.8, are to be handled in Exercise 4.4.1, which the reader is advised to work out in detail. (b) Another elementary case of interest involves the ionization of acetic acid (HA), which is representative of a whole class of materials that dissociate only weakly. Here one deals with the equilibrium H A - H + + A - which is characterized by the equilibrium constant KA --

all+ aAaliA

=

(VI-I+~YA)(CH+ CA-)

(4.4.7)

YHA CHA,

where A - represents the acetate ion. Strictly speaking, one should not neglect the water dissociation equilibrium which provides the common ion H +, but this contribution is usually negligible compared to the H + ion concentration generated from dissociation of HA.

264

4. THERMODYNAMIC PROPERTIES OF ELECTROLYTES

Since HA is neutral no significant error is made in setting }fftA -- 1. If we write V+ -- (VII+?'A-)1/2 we obtain from (4.4.7) CH+C A-

log ~

= log

KA

--

2 log y+,

(4.4.8)

CHA

and on using the Debye-Htickel equation (4.2.3a), we obtain the result CH+C A -

log ~

= log KA +

2(2.303)Ce yeS

CHA

.

(4.4.9)

1 @ C2%/8

Next, introduce the degree of dissociation, c~, whereby CH+ -- CA- -- COt -- CHA -(1 - or)c, c being the starting concentration. We now find COt2

log 1 - ot

= log K A @

2 (2.303) Ce ~C--d , 1 + C2~C--d

(4.4.10)

in both of which C 2 ' ~ 1 at room temperature. Equation (4.4.9) involves the ionic strength and is applicable if the solution contains other strong electrolytes with no ions in common with H + or A - . Equations (4.4.9) and (4.4.10) show the extent to which the quantity (CH+C A - / C H A ) differs from KA. (c) We turn next to hydrolysis reactions, typified by the interaction with water of the salt BA formed from a strong base BOH and a weak acid HA: A - + H 2 0 --+ HA + O H - . Here it is implied that a compensating cation such as Na + is present, so as to maintain electroneutrality. Correspondingly, if one ignores the common ion effects arising from the ionization of water one obtains 2

K H - aOH-aliA _-- a• a A - aH20

If again we set

aH20

--

1, and aliA KH --

--

(4.4.11)

a2NaAaH20 "

CHA,

( Y+ NaOH ) 2 (C +NaOH ) 2 CHA (Y+NaA)2 (C+NaA) 2

.

(4.4.12)

If BA is the salt of a weak base and strong acid the relevant hydrolysis reaction reads B + + H 2 0 = BOH + H +, which in the presence of a compensating concentration of C1- ions (assuming that BC1 is not sparingly soluble and that the common ion effects arising from the dissociation of water can be ignored) leads to the expression KH --

aB+ aH20 aBOHaH+

=

a2HClaBOH

.

(4.4.13)

a2BClaH20

If BA is the salt of a weak base BOH and a weak acid HA, the relevant reaction is written as B + + A - + H 2 0 - BOH + HA. Here it has been assumed that

EQUILIBRIUM PROPERTIES OF WEAK ELECTROLYTES

265

BA dissociates completely. Then the equilibrium constant assumes the following form: aBOHaHA

KH =

.

(4.4.14)

(a+BA)RaH20

(d) Next, we turn briefly to the case in which a pure solid A(s) is in equilibrium with undissociated A in solution, which in turn is in equilibrium with A+ and A_ according to the schematic equation A(s) -- A = v+A+ + v_A_. The equilibrium situation is characterized by /s

--/s

= V+/s

(4.4.15)

+ V_/s

In the event there is no undissociated A, we obtain the relation a++ a v-_ -- K,

(4.4.16)

where K is termed the activity product, which may be compared with three solubility products: Lx -- x++ x _v _ ,

Lc

--

C ; + C _v_ ,

t m

_ m++ m _v_

.

(4.4.17)

Thus, in each case one obtains an interrelation of the type (4.4.18)

K 1/v -- L~/Vg•

4.4.3

Thermochemistry of Solutions

Properties of ionic solutions are described in a unified manner by introducing a convention The enthalpy o f a pure liquid solvent at temperature T and at pressure P (ordinarily, 1 bar) is set equal to zero, unless the solvent participates in a chemical reaction with the solute, in which case the usual chemical conventions are applied to specify the enthalpy o f interaction.

The convention enables us to ignore the enthalpies of formation of inert solvents, which, in any event, cancel out in establishing the energetics of reactions executed in solutions. As an example consider the dissolution of n2 moles of solute (X) in n 1 moles of water, whose 'molality' as pure solvent is m -- 55.5. We write out the 'reaction' as n 2 X -+- n l H 2 0

= Xsol(m =

55.5n2/nl).

(4.4.19)

Experimentally it is reported that for n2 = 1 mol HC1 gas dissolved in n l = 10 mol of H20 at 298.15 K under a pressure of 1 bar AHs~ 5.55)-

266

4. THERMODYNAMIC PROPERTIES OF ELECTROLYTES

AH

Fig. 4.4.1. Schematic diagram for heats of solution when n moles of solvent are added to one mole of solute.

-69.3 kJ/molHC1, which quantity is known as the integral heat (enthalpy) of solution. When added to the standard enthalpy of formation, AH~ = - 9 2 . 1 kJ/mol HC1, one obtains A H f ( m -- 5.55) -- -161.4 kJ/mol HC1. On adding another 10 mol H20 the resulting integral heat of dilution is found to be A H d i l - - 2 . 5 kJ/molHC1, so that the total integral enthalpy of solution is given by AH~ -- 2.78) -- -761.8 kJ/molHC1. This represents an example of the diminishing rate of increase of the heats of dilution with diminishing solute concentration. In general a curve such as shown in Fig. 4.4.1 is encountered; one notes that as infinite dilution is approached the integral enthalpy of solution approaches a constant value. These facts call for the definition of a differential enthalpy of solution, given by A/-ti -- ( 0 AHs~ )

Oni

(4.4.20)

T,P,njr

which is simply the slope of the curve in Fig. 4.4.1. When two infinitely dilute solutions containing ions are mixed without reacting there is essentially no enthalpy of mixing for this process. However, if an interaction does occur, such that a precipitate or more solvent, of some other compound is formed, a chemical reaction has occurred that is characterized by an enthalpy change for that process. In sufficiently dilute solution these enthalpy changes depend only on the ions that are involved in the process and not on the partner ions that remain behind. It then becomes possible to adopt another simplifying convention:

The enthalpy of formation of H + ions in an infinitely dilute solution of water, A H ~ , is set at zero at all temperatures and pressures. This convention enables us to assign A H ~ values to other ions: for example, by measuring the enthalpy for forming an ilifinitely dilute solution of HC1 (the limiting value of the curve in Fig. 4.4.1) as AHso ~ -- -75.1 kJ/mol HC1, and adding

GALVANIC CELLS

267

this to the standard enthalpy of formation of HC1, AH~ (HC1, g) - - 9 2 . 1 kJ/mol, the enthalpy of formation of C1- in water at infinite dilution is then given by A H ~ ( C 1 - ) -- - 1 6 7 . 4 kJ/mol. A whole set of values for various anionic species can be similarly constructed, and these, in turn, may be used as secondary standards to be paired off with different cations for A H ~ measurements. Once these quantities are known one can readily construct enthalpies of interaction such as the precipitation of AgC1 from H + and C1-" AH~(AgC1) -A H f ~(Ag +) + A H f ~(C1-). As another example, one may determine the enthalpy of formation of the compound NaC1 at infinite dilution from A H ~ (NaC1) A H ~ (Na +) + A H ~ (C1-); this value may be checked against the determination obtained from A H f ~ AH~(NaC1, s ) + AHso~(NaC1). However, such calculations are of rather limited usefulness since processes of interest normally involve ions at finite concentrations whose properties change in the course of the interaction. These determinations are also affected by the presence of nonparticipating ions in solution, so that the machinery developed earlier and set forth later is needed to handle such a situation. 4.4.4

Entropies and Free Energies of Ionic Species

Very similar conventions may be introduced to characterize the entropies and Gibbs free energies of ionic species. Here one adopts the convention that at T -- 298.15 K and at one bar ~0 (H +) = 0 and A ( ~ (H +) = 0, f o r w a t e r as solvent. Tabulations of molar entropies and free energies may then be constructed as outlined earlier. These are also of the same rather limited applicability as the comparable enthalpies for ions. As an example, one may determine the equilibrium constant for a given ionic reaction at infinite dilution through the relation, A G~ - - R T In Kion, for which the left-hand side is first established be looking up the relevant data from the tabulations.

EXERCISE 4.4.1. Consult Section 3.8 so as to work out the details on the heat of ionization. 4.5 4.5.1

Galvanic Cells General Description

In describing the operation of galvanic cells we introduce a specific example rather than invoking the cumbersome machinery needed for a generalized approach. The example chosen for this purpose will then be generalized, and a thermodynamic analysis of the resulting processes will be furnished. Consider the operation of the Daniell Cell depicted in Fig. 4.5.1, which serves as a prototype. The two compartments are filled respectively with a saturated

268

4. THERMODYNAMIC PROPERTIES OF ELECTROLYTES

Fig. 4.5.1. Schematic diagram of the Daniell cell. The rectangular box labeled P represents a potentiometer. Anode and cathode are shown for the spontaneous operation of the cell when the potentiometer emf is slightly less than that of the cell. Zn at the anode enters the solution as Zn ++ ions and Cu ++ in the cathode compartment deposits as Cu.

ZnSO4 solution in contact with a Zn strip (left), and a saturated CuSO4 solution in contact with a Cu strip (fight). The two metallic electrodes are joined electrically to a variable potentiometer. A salt bridge that permits transfer of ionic species connects the two compartments. Electrochemical processes now ensue: the metallic electrodes interact with the ionic components according to the net reactions" Zn - Zn 2+ + 2e-, and Cu - Cu 2+ + 2e-. One finds that the tendency of Zn to go into solution, thereby furnishing electrons to the metallic wire, is stronger than the opposing tendency of Cu to do the same. Accordingly, in the spontaneous operation of the cell the second reaction is reversed as Cu 2+ + 2e- - C u . The combination of these two processes allows electrons to be transferred from left to fight through the external circuit shown in Fig. 4.1.1. This natural process can be arrested by setting up an opposing voltage in the potentiometer, thereby establishing equilibrium conditions. 1 The electron flow may actually be reversed by increasing the opposing potentiometer voltage, thus forcing the electrons to flow from fight to left, with corresponding reversals in the indicated electrochemical reactions. By minor adjustment of potentiometer settings one can thus allow the cell to operate reversibly in either direction; thereby, electrochemical processes in the cell become amenable to thermodynamic analysis. 2

OPERATION OF GALVANIC CELLS

269

Conventionally, a reaction such as Z n - Zn 2+ + 2e- that furnishes electrons to the circuit (increasing the valence of Zn) is said to be an oxidation step, and the electrode where this occurs is called the anode. A reaction such as Cu 2+ + 2e- - C u that removes electron from the external circuit (decreasing the valence of Cu) is termed a reduction step which occurs at the cathode. In the spontaneous operation of the cell the Zn 2+ ions furnished during oxidation of the metallic strip accumulate in the saturated solution on the left and combine with SO ] - to precipitate out ZnSO4. The Cu 2+ ions concomitantly removed from solution on the fight deposit on the metallic Cu strip and cause solid CuSO4 to dissolve in a compensating process so as to replace the deposited Cu 2+ ions. The resulting anionic imbalance in both compartments is compensated for by appropriate transfer of cations and anions into or out of the salt bridge, so as to maintain electroneutrality in both aqueous compartments. The net results of these processes is represented by the reaction Zn + CuSO4 - Cu -t- ZnSO4; however, the actual processes are clearly vastly more complex. The spontaneous current flow is thus a consequence of the chemical instability of Zn metal relative to a saturated solution containing Zn 2+ ions: chemical potential energy has been transformed into electrical energy flow. The above example illustrates the general characteristics of a galvanic cell: It usually consists of two ionic (or, frequently, solid) solutions in separate but interconnected compartments, in physical contact with electrodes connected to a potentiometer; frequently, the electrodes are surrounded by reactive gases. Any departures from the quiescent conditions of open-circuit conditions are accompanied by oxidation-reduction processes that keep in step with the flow of electrons through the external circuit. REMARKS

4.5.1. Strictly speaking, these are steady state, quiescent conditions, because the process is not allowed to run to completion without external constraints, but will be ignored. 4.5.2. Obviously, in everyday applications galvanic cells are operated in irreversible fashion during discharge and charge; thus, the voltages and operating conditions are not subject to the analysis provided in this chapter. Rather, the near steady state conditions considered here must be used in a thermodynamic analysis for characterizing the processes detailed below.

4.6

Operation of Galvanic Cells

We next describe the operation of galvanic cells in mathematical terms, again taking the Daniell cell as our representative example. Consider Fig. 4.6.1; for electrons to flow through the external circuit left to right the electric field F_.points in the direction of the conventional positive current flow, i.e., to the left, whereas the electrostatic potential gradient V4~ = - g points to the fight. Under spontaneous

270

4. THERMODYNAMIC PROPERTIES OF ELECTROLYTES

Fig. 4.6.1. Schematic diagram showing the direction of the emf, electrochemical potential gradient, electrostatic potential gradient, and electric field during spontaneous operation of a cell operating in conformity with Conventions 1 and 2, described below.

operating conditions involving the transfer of one Faraday (F; one farad = one mole of electrons - 96495 coulombs) in an infinite copy of the cell, the electron density, hence In ae, rises ever so slightly from right to left; under true steady state conditions no gradient in electron density is encountered in the metallic wires. Hence, the electrochemical potential gradient pertaining to electrons, given by V ( = V # - VFcb = R T d l n a e - VFcb, increases from right to left; ~'(1) > ~'(r). It is this difference in the electrochemical potentials that drives the spontaneous operation of the cell. A minimalist description of the above spontaneous operational process is given by Zn(s) + Cu 2+ (a~) + SO 2- (a~) + 2 e - ( r ) = Cu(s) + ZnZ+(a/) + SO2- (at) + 2e- (/).

(4.6.1)

Contrary to prevalent practice, we have included the electronic species in the overall reaction since under conditions where no current flow is allowed, the electrochemical potentials at the two electrodes differ, so that the electronic effects do not cancel. The solid species are indicated by (s), and the ionic activities in the two compartments, by a. A detailed rationale for writing Eq. (4.6.1) in the indicated form is provided as a footnote. 1

OPERATION OF GALVANIC CELLS

271

We next invoke the generalized equilibrium condition, ~--~i 1)i~i = 0 = Z i Vi(ll~i -+- Zi F ~ ) , where Zi is the charge on species i; for uncharged species ~'i - - //~i. Again, the stoichiometric coefficients for reagents in the chemical reactions as written are negative. The equilibrium condition as applied to the Daniell cell operation reads 2"

/Z~n(s) -+-//~uSO4(sol) @ R T l n a 2 ( r ) "--

~u(s) @//~nSO4(sol)

+ 2F~cuZ+(r) - 2F~bso2- (r) + 2 F r

@ R T l n a 2 ( l ) + 2F4)zn2+ (/) - 2F~bso2-(/) + 2F~'(/). (4.6.2)

At equilibrium the positive and negative ions (after multiplication with - 1) in solution in the left compartment are at the same electrostatic potential that preserves electroneutrality; similar reasoning applies to the ions in the right compartment. Hence, the terms involving 05 cancel out. The terms #Zn(s) + CuSOa(sol)- #Cu(s) /Z*ZnSO4(sol) may be grouped into an equilibrium parameter, namely + R T In Ka as indicated in Section 2.10 (Why the ' + ' sign?). We can then solve Eq. (2) for a (I)

2Fr (1) - 2F~ (r) =_ 2Fg - RT In K a - R T In a2 (r)

]

'

(4.6.3a)

where we have introduced the electromotive force (emf) ~s as 3 g -- ~"(l) - ~"(r).

(4.6.3b)

As already mentioned, it is this difference in electrochemical potentials, associated with the electronic species in the two compartments, that provides the basic 'driving force' for the operation of electrochemical cells. A trivial rearrangement of the above relation leads to the so-called Nernst equa-

tion g; _ RT In Ka 2F

2F

In

aZ(r)

,

(4.6.4a)

or

~ - 1 ~ ~ - ~RTln{ } F a+(r) a+(1) '

(4.6.4b)

in which we have introduced a standard emf by defining

_ __Rv

I~~ =

2F

Ac0

ln Ka = - ~ .

2F

(4.6.4c)

Several remarks are in order: (i) g is in a sense proportional to an 'open circuit voltage' that is developed by the cell, as shown in (4.6.3a). It is multiplied

272

4. THERMODYNAMIC PROPERTIES OF ELECTROLYTES

by 2 F to match the Gibbs free energy for the electrons participating in the specific reaction (1) at constant T and P. (ii) Since ~"(l) > ~"(r) we note that g > 0 corresponds to the spontaneous operation of the cell for the reaction as written. (iii) Also, since we always refer to electron transfer through the external circuit, and because of the manner in which we introduced the electrochemical potential for electrons, F signifies the magnitude of the charge on a mole of electrons. (iv) The emf cited in Eq. (4.6.4) is specified solely by the net chemical reaction abstracted from Eq. (4.6.1); electronic contributions are subsumed in the construction of ~. Therefore, A Gd just involves the electrochemical potentials of the chemical species participating in the operation of the galvanic cell. Then

AGd ~' -

2F '

(4.6.5)

again showing that a positive emf corresponds to the spontaneous cell operation. (v) The electron participation has not 'cancelled out' from Eq. (4.6.1); rather, their effect is subsumed in the definition for ~. (vi) We have used the Daniell Cell as an example; however, this discussion is capable of immediate generalization, as we show next. REMARKS

4.6.1. We must be careful about signs here; the general expression ~ i 1)iAi -- 0 requires that we set vi > 0 (<0) for products (reagents). For this reason we have placed the ionic species and the electrons in the fight compartment, as well as Zn (all of which are used up and hence are labeled 'reagents') on the left-hand side of the chemical equation. The ionic and electronic contents of the left compartment as well as Cu, involving positive vi (being generated, hence, labeled 'products') were placed on the fight side of the chemical equation. 4.6.2. We have also multiplied the quantity ~" for a single electron by F in order to put everything on a per mole basis. CuSO4 (soln) and Z n S O 4 (soln) refer to undissociated species, dissolved in the saturated solution, that are in equilibrium with respect both to their ions and to the solid species in the separate phase. 4.6.3. The term electromotiveforce is misleading and undesirable. However, it is so firmly entrenched that we will continue to use it here. Note that the definition involves the electrochemical potential of the electrons on the product side of the equation as written, minus that for the electrons on the reagent side for the reaction as written.

4.7

Galvanic Cells; Operational Analysis

The general operation of galvanic cells is symbolized by the overall 'reaction' ~ i viAi = 0, carried out only to an infinitesimal extent, which does not significantly alter the concentrations of chemical species or the emf. Advancement of

GALVANIC CELLS; OPERATIONAL ANALYSIS

273

the chemical reaction by 6)~ units involves the transfer of n F6)~ moles of electrons through the external circuit, where n is the number of equivalents, and F is the Faraday (96,495 C). The work done by the surroundings in opposing the electron transfer is n F6Mfi, the negative of the work performed by the cell. As shown in Section 1.12, for reversible processes at constant T and P, reversibly executed work other than mechanical tracks the Gibbs free energy change: 6Wn = -6G; therefore the spontaneous virtual advancement of the process by 3)~ units is specified by n F 6 ) ~ --(~G/6)~)T, p6)~, whence, as in Eq. (4.6.5), and for the reaction E i l)i Ai = O, as written, =

(4.7.1) T,P

J

The chemical potential introduced above will now be written out in terms of the hybrid system adopted in Eq. (3.7.7) and subsequent equations. (i) The standard state for pure solids or liquids participating in the electrochemical processes is that of the isolated solid at temperature T under a pressure of one bar. (ii) For materials in homogeneous solid or liquid solution the standard state is chosen for each constituent at unit activity at one bar at the prevailing temperature. (iii) For gases participating in the chemical reaction the standard state is that of the ideal gas. Then, according to Eqs. (3.7.7-8) and (4.7.1) we write (when using mole fractions, fi(T, P, Xi) ~ a(T, P, Xi) )

-

-

RTlnKx nF

RT(~ nF

Vs In as* (T, P) + ~ vj lnaj(T, P, x j ) ) . s j

(4.7.2)

Here s and j denote chemical species in the pure phase and in solution, respectively. In experiments at one bar the summation over s drops out. Conventionally, one introduces a standard emf by the expression

Aoo

RT In Kx = nF

-~0 -- - -

nF

,

(4.7.3)

so that the operation of the cell is characterized through the open circuit voltage

-

nF

Vs In as*(T, P) + Z vj lnaj(T, P, xj) .

(4.7.4)

J

Equation (4.7.4) is the generalized version of the Nernst Equation. An important aspect of this formulation is that if a cell can be set up to reproduce ionic, liquid, or gaseous reactions of interest, then a measurement of ~0 under standard conditions directly evaluates the equilibrium parameter Kx.

274

4.7.1

4. THERMODYNAMIC PROPERTIES OF ELECTROLYTES

Unified Description of Cell Operations

It is desirable to introduce a systematic nomenclature and a set of conventions that permit a unified description of galvanic cell operations. Consider as a representative example a cell at temperature T that consists of (i) a Pt electrode that is surrounded by gaseous hydrogen at pressure P and that is dipped into an HC1 solution of molality ml. The latter is connected via a salt bridge to a second compartment containing a saturated HC1 solution in equilibrium with AgCl(s), at molality mr, into which a silver electrode is immersed. The cell is at uniform temperature T. This cell is represented by the scheme Pt, H2 (P)]HCl(ml, sat H2)]lHCl(mr, sat AgC1), AgCl(s)lAg,

(4.7.5a)

where the vertical bars separate distinct phases, and the double bar indicates that the two solutions are separated by a salt bridge that generates an internal emf of its own. The specification of this potential difference is considered below; ordinarily, the emf of the salt bridge is small compared to all other contributions. The Pt electrode is inert but provides a needed interface for electron transfer to the external circuit. We now introduce several conventions. Convention 1. For the cell as written the oxidation process occurs on the left and reduction, on the right. Electrons move through the external circuit from left to right; conventional current proceeds in the opposite direction. The redox reactions are schematized through the following half reactions: at the anode (left): 89 H + ( m l ) + e-(l). The extra H + ions produced in this process are compensated for by the release of negative ions from the salt bridge or else the cations are absorbed into it; formally, this corresponds to transfer of C1- ions from fight to left across the salt bridge. At the cathode (fight) the following net process takes place: e - ( r ) + AgCl(s) = Ag(s) + Cl-(mr). The excess C1ions produced thereby are compensated for by the release of positive ions from the salt bridge or the anions are absorbed into it; formally, this corresponds to the transfer of H + ions across the salt bridge. As explained in the previous Section, the electron constituents in the half reactions should be ignored; they serve here only to balance out charges. The overall chemical reaction is thus given by 89 + AgCl(s) - Ag(s) + H+(m~ -) + C l - ( m r ) , but in the final balanced equation the compensating cationic and anionic species in each compartment must be included, as illustrated below. Convention 2. The cell emf for the cell as written is specified by the algebraic sum

of two half-cell emfs: r =r

-r

= (~l - ~r)/F,

(4.7.5b)

GALVANIC CELLS; OPERATIONAL ANALYSIS

275

Table 4.7.1 Short table of standard EMF values (in volts) BaIBa 2+

Ba = Ba 2+ + 2 e -

+2.906

MglMg 2+

Mg -- Mg 2+ + 2 e -

+2.363

AliA13+

A1 = A13+ + 3 e -

+ 1.662

ZnlZn 2+

Zn = Zn 2+ + 2 e -

+0.7628

FelFe 2+

Fe = Fe 2+ + 2 e -

+0.4402

CdICd 2+

Cd = Cd 2+ + 2 e -

+0.4029

PtITi 2+, Ti 3+

Ti 2+ = Ti 3+ + e -

+0.369

PblPbSO4ISO 2 CulCuIlIPtIH21H + AglAgBrlBrPtlCu +, Cu 2+

Pb + SO 2 - - PbSO4 + 2 e Cu + I - = CuI + e H 2 -- 2H + + 2 e Ag + B r - = AgBr + e Cu + = Cu 2+ + e -

+0.3588 t0.1852 0.0000 -0.0713 -0.153

AgIAgCIIC1PtIHgIHg2C121C1-

Ag + C I - = AgC1 + e 2C1- + 2Hg = Hg2C12 + 2 e -

-0.2225 -0.2676

Pt I2 I PtlFe 2+, Fe 3+

3 I - = 13 + 2 e Fe 2+ = Fe 3+ + e -

-0.536 -0.771

AglAg + PtlT1 +, T13+ PtIC12IC1PtlMn 2+, MnO 4 -

Ag = Ag + + e T1+ = T13+ + 2 e 2C1- = C12 + 2 e Mn 2+ + 4 H 2 0 = MnO 4 + 8H + + 5 e -

-0.7991 -1.25 - 1.3595 - 1.51

P t I S O ~ - , SO 2 -

SO 2 - + 2 O H - -- SO 2 - + H 2 0 + 2 e -

+0.93

PtIH2IOHPtIO2IOH-

H2 + 2 O H - = 2 H 2 0 + 2 e 4 O H - = 02 + 2 H 2 0 + 4 e -

+0.8281 -0.401

PtlMnO21MnO 4

MnO2 + 4 O H - = MnO 4 + 2 H 2 0 + 3 e -

-0.588

where I and r refer to the left- and right-hand compartment for the cell as written; this reflects the splitting of the cell operation into the oxidative (l) and reductive (r) steps. Convention 3. The determination of g is carried out systematically by first dealing

with the standard emfs according to

r _r _r 0,

(4.7.6)

where the overall standard emf has been written as an algebraic sum of the standard half cell emfs for the left and right electrodes of the cell operation. Extensive tabulations of the g0 values are available; an abbreviated list is provided in Table 4.7.1 for illustrative purposes. All half reactions are written out as oxidation processes; hence, when using Eq. (6), the half reaction for the process (r) must be turned around, thereby changing the sign of the emf value. Equation (4.7.6) is not unique; one could equally well have written it in the form r _ r _ q _ [r _ q], where q is any arbitrary constant. These indeterminacies are dealt with through two new conventions:

276

4. THERMODYNAMIC PROPERTIES OF ELECTROLYTES

Convention4. The emf is positive for the operating cell when electrons flow spontaneously from left to right through the external circuit.

Convention5. The standard electrode emf for the half cell operation:

1H2 (P =

1 bar) - H + (a+ - 1) + e - is set to ~ol (H +) _ 0. Emf tables can thus be constructed by coupling the half cell of interest, that are operated under standard conditions, against the standard hydrogen electrode in the reducing mode, H + (a+ - 1) + e - -- ~H2 (P - 1 bar), and determining the resulting overall emf go ' which is also the gol value for the half cell of interest. Once such determinations have been made for a limited set of half-cells, these, in turn, may be used as secondary reference standards for other half cells whose g0l values are to be established. The above procedures are best illustrated by a specific example. Consider the cell Pt, C 1 2 ( P - 1 b a r ) l H C l ( a + - 1)lAgCl(s), Ag(s),

(4.7.7)

whose operation is governed by the half reactions 2 C l - ( a _ = 1) = C12(P = 1) + 2e-,

~ol _ _ 1 3595 V,

2AgCl(s) + 2e- = 2Ag(s) + 2 C l - ( a _ = 1)

_r

2AgCl(s) = C12(P -- 1) + 2Ag(s)

go-

+0.2225 V, - 1 . 1 3 7 0 V.

Note the sign reversal in listing ~0 relative to the value specified in Table 4.7.1. The contributions from the C1- ions cancel out because these are at the same activity on both sides of the two half reactions. The electronic contributions are ignored for reasons specified earlier, but they serve the purpose of balancing charges and establishing that in this particular example n -- 2 for the reaction as written. The emf in volts is given by

) 2 Fazg

~ -- - 1 1370. - --RTln(

2

--

RTlnacl2'2F

(4.7.8)

aAgc1

a2g/a2AgC1

where the ratio ordinarily does not differ significantly from unity, and where the activity for 0 2 gas may ordinarily be replaced by the prevailing 0 2 pressure, P, over the anode. Several remarks are in order: (i) The reader should convince himself that precisely the same emf expression would have been obtained if the cell operation had been written out as AgCl(s) - 1Cl2(g) + Ag(s), with n - 1. Thus, the emf is independent of the manner in which the net reaction is balanced--a physically sensible result. (ii) The standard differential Gibbs free energy change for the net reaction 2AgCl(s) - C12(P - 1) + 2Ag(s), n - 2, is specified by AG O - - 2 F / f ~ =

GALVANIC CELLS; OPERATIONAL ANALYSIS

277

-2(96485) • (-1.1370) J/mol = 219,420 J/mol, a very sizeable quantity. Since AG o is positive we infer that the reaction under standard conditions (SC) occurs spontaneously in the opposite direction. Had the reaction been written as AgCl(s) - 89 + Ag(s) we would have obtained AG o -- 109,710 J/mol. (iii) For the cell operation in reverse, as Ag(s), AgCl(s) IHCl(a+ = 1)lCl2(a = 1), Pt one obtains g0 _ + 1.1370 V as well as AG o - - 2 1 9 , 4 2 0 J/mol. The reaction just written (with g0 > 0 and A G o < 0) proceeds spontaneously. (iv) Silver tends to react spontaneously with C12 gas to form AgC1, but under standard conditions this can be reversed by imposing an external counteracting emf slightly in excess of 1.1370 V. 4.7.2

Types of Electrodes

We briefly describe several different types of electrodes that are in common use: (a) The 'gas' electrode. In this arrangement a reactive gas is maintained over an inert electrode at a fixed pressure. As an example we have cited the hydrogen electrode 89 -- H + (ml) + e - , where Pt serves as a metallic interface. (b) The oxidation-reduction electrode. Here an inert metal wire dips into a solution containing ions in two distinct valence states, as exemplified by PtiFeZ+(x2), Fe3+(x3), corresponding to the half reaction F e Z + ( x 2 ) Fe 3+ (x3) + e - . (c) The quinhydrone electrode. This is a special case of the above situation: Here an equimolar mixture of hydroquinone (HzQ) and quinone (Q), obtained from quinhydrone (HzQ-Q), is in contact with an inert metallic electrode. The cell is represented by PtIHzQ, Q, H + (x+), corresponding to the half reaction HzQ = Q + 2H + + 2e-. (d) Metal-metal ion electrode. Here a reactive metallic electrode dips into a solution containing the same material in the ionic state. As an example we cite a silver electrode dipping into a silver nitrate solution. The cell is represented by Ag(s) lAg+ (m+), with the half reaction Ag(s) = Ag+(m+) + e - . (e) Amalgam electrodes. Here a reactive metal is dissolved in a pool of mercury into which an inert wire is dipped. Such a setup is used when the metallic element in question reacts directly with water; mercury does not normally participate in galvanic processes. As an example one may cite the setup PtiHg-Na(x)iNa+(m+), corresponding to the half reaction Na(x) = Na + (m+) + e - , where x is the mole fraction of metallic Na dissolved in Hg and m+ is the molality of sodium ions in solution. Metallic sodium obviously cannot be maintained in direct contact with water. (f) Metal-insoluble salt electrode. In this setup a metal is in contact with an insoluble salt of the metal, which, in turn, is equilibrated with the appropriate anion in solution. An example is furnished by the half cell Ag(s)iAgCl(s)iCl-(m_), with the corresponding half reactions: A g ( s ) =

278

4. THERMODYNAMIC PROPERTIES OF ELECTROLYTES

Ag + + e - and Ag + + C l - ( m _ ) = AgCl(s), for a net reaction: A g ( s ) + C1-(m_) = AgCl(s) + e-. (g) Calomel electrode. A variant of the above is the calomel electrode where a Pt wire is dipped into mercury in contact with HgzC12 paste, itself in contact with a KC1 solution. The corresponding half reactions read: Hz(g) = 2H+(a+) + 2e- and Hg2Clz(s) + 2e- = 2Hg(g) + 2 C l - ( a _ ) , adding up to a net reaction: Hz(g) + HgzClz(s) = 2Hg(g) + 2H+(a+) + 2 C l - ( a _ ) . This electrode is frequently used as a reference standard because it is more conveniently applied than the standard hydrogen electrode. (h) Glass electrode. As the name implies, it consists of a glass membrane coated with hydrated silica, which separates two solutions: the inner portion comprises Ag(s) IAgCl(s) IH+ + C l - ( a + ) l , Na + + C1- (a+)2, which is connected via a glass membrane to a compartment containing an aqueous solution with H + ions whose pH is to be found, which, in turn is connected via a salt bridge to a calomel electrode. The resulting reactions are similar to those cited in (g), except that allowance must also be made for the transfer of Na + ions between the solution and the glass bridge, thus producing a junction potential akin to that discussed in Section 4.8 below. The details go beyond the purview of our discussion. (i) Ion-specific electrodes. There exist a variety of conventional electrodes that are combined with a selection of special membranes and/or with choices of chelating compounds or of complexation agents so as to selectively control the transfer of specific ions. For example, organic complexes of Ca 2+ are used to set up cells sensitive to Ca 2+ concentrations. A special subcategory are solids maintained at high temperature that permit the transfer of specific ions at acceptably large rates. For example, at sufficiently high temperatures LaF3 crystals allow transfer through the crystal of F - ions; similarly, YzO3doped ZrO2 crystals at sufficiently elevated temperatures become conductors of O 2- ions.

4.8

Liquid Junction Potentials

A possible source of difficulty in electrochemical experiments is the presence of an emf when two dissimilar solutions in the cathode and anode regions are juxtaposed or connected by a salt bridge. Different ions in various solutions diffuse at different rates; the faster cations moving ahead of the slower ones set up an internal electric field that tends to retard their movement and to accelerate the slower cations. An analogous situation prevails for the anions. After a steady state sets in both types tend to move across the junction in the presence of an internal field; this is the origin of the junction emf. Charges are carried by different ions in proportion to their transference numbers. Consider an infinite copy of the junction in which a fraction t + of the positive charge is carried by cations with valence z + and a fraction t f of the negative

EMF DEPENDENCE ON ACTIVITIES

279

-Iz -I.

charge is transported by anions of valence z j- -

At each point in the junc-

tion under steady state, spontaneous conditions, t+/z + equivalents of species i move across from right to left and - t f / l z j I equivalents of species j are transported from left to right. The resulting Gibbs free energy change is given by t+ d/z+

d~ -

_

(4.8.1)

7z

,z-; , d " ;

for each small layer perpendicular to the junction direction that stretches from point A to point B. The total change in Gibbs free energy across the entire junction J then reads

A G j -- - F I ~ j -- RT Z.

s zT'i+dlna + -

RT Z.

t

s Izj-I t7 d ln a-f

(4.8.2)

j

in an obvious notation for the activities a. The integral runs over the entire domain A-B of the junction. The transference numbers may generally be replaced by average values t, so that @-(B) t-}- l n ~ . a +(A--------)-F --if- Z . Iz}-[ a~- (A)

RT ~i ~ i + In a+(B) ~ J --

F

. 7z

RT

(4.8.3)

J

The individual ionic activities must be estimated by use of the Debye-Htickel theory. However, Eq. (4.8.3) shows that the cationic and anionic contributions tend to cancel out. Hence, except for a truly unusual situation in which a particular ratio a(B)/a(A) exceeds a factor of ten, and/or where t/Izl for a particular species is very large compared to all others, the emf remains well below the value of R T / F = 0.0592 V at room temperature. Thus, junction potentials tend to be small compared to most emfs developed by cells. The effect may be further reduced by use of salt bridges that contain cations and anions of comparable mobility, so as to compensate for the tendency to develop internal emfs. The effect is also attenuated by employing parchment, agar-agar gels, or collodion to impede unbalancing ionic motion across the junction. In any event, junction potentials of the type described here tend to be small. 4.9 4.9.1

EMF Dependence on Activities General Description

We first set up the general expression for the dependence of the galvanic cell emfs on the activity of the components participating in the cell reaction. Starting with

280

4. THERMODYNAMIC PROPERTIES OF ELECTROLYTES

the Nemst equation we find

~_~,0 _ __RTln[OaVsS(T,P)Ha~J(T,P, mj)], nF J

(4.9.1)

where for the logarithmic argument Q at 300 K, (RT/nF)In Q = 0.05915 log Q, which includes the conversion from natural to common logarithms. The stoichiometry coefficients Vs and vj are negative for the reagents in the cell reaction as written, and the activities aj are generally referred to the molalities m j of the various components in the solid or liquid solution. The activities aj are often determined by use of the Debye-Htickel limiting law log V+ (m) -- - 0 . 5 0 9 2 z +

Iz-Iv/am

1

Sm =- -~Z zjmj. 2

(300 K),

(4.9.2)

J Thus, from a tabulation of standard emfs, from the specification of the as for pure phases as in Section 3.7, and from use of the Debye-Htickel limiting law one can calculate the emf of the particular cell of interest. Usually, the procedure is used in reverse, whereby a measurement of g yields the activity or activity coefficient for the solutes of interest.

4.9.2

Examples of Operating Cells

We now furnish several illustrations. Consider the cell Pt, H2(P)IHCI(m)I AgCl(s), Ag(s) for which the half reactions are given by 89 - H+(m) + e - and e - + A g C l ( s ) = C l - ( m ) + Ag(s), corresponding to the cell reaction 1 H z ( P ) + AgCl(s) - Ag(s) + H + (m) + C1-(m). In this case ~f _ ~0 _ R___TTIn F

aH+aCl-aAg

(4.9.3)

1/2 ' aH2 aAgC1

which simplifies considerably for measurements carried out at one bar, where 1/2 aAg/aH2_ aAgC1 -- 1; also, we set aH+aC1- -- a 2, so that

2RT

_g0 _ ~

F

lna+(m)

(P-

1 bar).

(4.9.4)

To determine the activity coefficient, without prior knowledge of ~0, rewrite the above as

2RT

-l- ~

F

l n m + --

go

2RT

- ~

F

In F+(m).

(4.9.5)

EMF DEPENDENCE ON ACTIVITIES

281

Now introduce the extended Debye-Htickel equation, Eq. (4.2.5); on setting z+ = z - -- v+ - v_ - 1 and v/-~m - ~ for the case at hand we obtain (300 K) [logm+-

s

0 "15+9~072 ~ 7 ] -

4 g ~ 0.11833 Cm+.

(4.9.6)

Measurements o f g are then taken for several HC1 solutions at different molalities; then a plot of/2 vs. m+ should yield a straight line in the range where the extended Debye-Htickel equation holds. Extrapolation of this line to m+ -- 0 yields 4g~ and the slope is proportional to C. One then returns to Eq. (5) to find the F'+ (m) values for each set of measured 4g and m+ values. Very accurate measurements of the activity coefficient thus become available. As a second example we briefly consider the operation of an electrolytic fuel cell according to the scheme: Pt, Hz(PH)INaOH(m)IOz(Po), Pt with half reactions 4 O H - (m) + 2Hz(PH) --4H20(~) + 4 e - and O2(Po) + 2H20(~) + 4 e - 4 O H - ( m ) , yielding a net reaction 2Hz(PH) + O2(Po) -- H20(s Correspondingly, 4g _ g0 _ ~ RT In 4F

[ a2

H20(1)

a22(pH)ao2(Po)

~ g 0 + R4___FTIn P22 P02

1

RT 2 F In aH20(1).

(4.9.7)

Ordinarily, the contribution from the last term is negligible. We also find A Gd -- 4 F g which, at a cell voltage of4g - 1.23 V, corresponds roughly to 475 kJ/mol. In principle, the fuel cell represents a good source for energy delivery, but there are problems in rendering such a cell truly practical. As a final example we consider the lead storage cell that involves the following configuration" PblPbSO4 (s)[H2SO4 (aq)IPbSO4 (s)IPbO2 (s)[Pb. The corresponding half reactions are given by Pb + HSO 4 - PbSO4 (s) + H + + 2e-, PbO2(s) + 3H + + HSO 4 + 2e- - PbSO4(s) + 2H20,

(4.9.8a) (4.9.8b)

for a net reaction Pb + PbO2(s) + 2H + + 2HSO 4 - 2PbSO4 (s) + 2H20.

(4.9.8c)

Omitting the contributions of the solid constituents the emf generated by this cell reads

a2 g _ ~o _ ~RT In H20 , 2F a4 +H2SO4(aq)

(4.9.9)

282

4. THERMODYNAMIC PROPERTIES OF ELECTROLYTES

which under normal conditions generates an emf of roughly 2 V. The cell is commonly used in transportation operations because it is readily reversible, can undergo many charge and discharge cycles, and provides large current densities under nonequilibrium conditions. QUERIES 4.9.1. Do the two galvanic cells Cu(s)lCu2+llfu+lCu(s) and Pb(s)lCu2+llfu+lfu(s) have the same standard emf values? Explain. 4.9.2. Is there a difference in operation and in emf for the two cells: Pt, H2(P)I HCI(xl) IIHCI(x2)IAgCI(s), Ag(s) and PtIH2(P)IHCI(xl)IAgCI(s), Ag(s)? Justify your answer.

4.10

Types of Operating Cells

We list several types of galvanic cells in common use. (a) Chemical cells. Several examples of this type have been cited in the previous sections. As an additional case we consider the cell Zn(s)IZnC12(m)IAgCl(s), Ag(s), with a net chemical reaction Zn(s) + 2AgCl(s) - 2Ag(s) + Zn 2+ (m) + 2C1-(2m). By the procedures described earlier we find

2 r _ r

. R T . In . aAg(s) . 2F 2 aAgCl(s)aZn(s)

2 _ go . R T . In . aAgts)'-. 2F a2gel(s)aZn(s)

R T ln[azn2+a21_ ] 2F

3 R T In a+ . 2F

(4.10.1 )

If one specifies the activities of the solids as described in Section 3.7 (ordinarily very close to unity) then a measurement of the cell emf yields the mean molal activity coefficient of ZnC12 in solution. (b) Electrode concentration cell. Here a mirror image cell is set up in which the electrodes differ solely in the concentration of surroundings. As an illustration we cite the cell Zn, H g ( m l ) l Z n S O a ( m ) l H g ( m r ) , Z n in which a zinc amalgam paste serves as the reactive electrode. The half reactions are given by Zn(m/) - x Z n 2 + ( m l ) + 2 x e - and xZn2+(mr) + 2 x e - - Z n ( m r ) , with the net reaction Zn(m/) -- Zn(mr); i.e., Zn has been reversibly transferred from the amalgam on the left to the amalgam on the fight. Obviously, the standard state corresponds to a t - - a r , so that g0 _ 0. Then, RT = --ln(al/ar), nF

n = 2x,

where x is the molecular aggregation of Zn in the amalgam.

(4.10.2)

TYPES OF OPERATING CELLS

283

As a second example consider the reversible transfer of gas from the left to the right compartment of the particular cell Pt, H z ( P t ) [ H C I ( m ) ] H z ( P r ) , Pt. The half reactions are given by H2 (P1) -- 2H + (m+) + 2 e - and 2H + (m+) + 2 e - -- HZ(Pr), with the net transfer H2 (Pl) - H2 (Pr). The corresponding emf is given by RT g -- ~ ln(al/ar).

2F

(4.10.3)

Clearly, g > 0, A Gd < 0 or g < 0, A Gd > 0 according as Pl > Pr or Pl < Pr respectively; in accord with intuition, the transfer always occurs in the direction from the higher to the lower pressure compartment. (c) C o n c e n t r a t i o n cells w i t h l i q u i d j u n c t i o n s . As an example in this category we list the cell Ag(s)[AgNO 3 (ml)[]AgNO 3 (mr)lAg(s). Consider an infinite copy of this cell in which one equivalent of silver enters into solution the left, and one equivalent is deposited on the electrode on the right, accompanied by the electron transfer of one Faraday through the external circuit. We analyze the overall process in several steps: nl and nr are the mole numbers of the species in the left and right compartments, and t+ and t_ represent transference numbers for Ag + and NO~- ions respectively. (i) Initially there are nl and nr moles of Ag + as well as of NO~- present in the left and right compartments respectively. (ii) The passage of 1 F of charge through the external circuit incurs a change in cation concentration, such that at the conclusion nl + 1 and nr - 1 moles of Ag + are present in the respective compartments. (iii) Simultaneously there occurs a cation transfer of t+ moles of Ag + across the liquid junction from left to right; nl + 1 - t+ moles of cations remain on the left and nr - 1 + t+ moles are found on the right. Actually, in a KC1 salt bridge normally used it is the K + compensating ions that move across the bridge. (iv) There also occurs an anion transfer of t_ mole of NO~- across the liquid junction in the opposite direction: now nl + t_ moles of the anion are present on the left and nr - t_ moles remain on the right. Actually, when a KC1 salt bridge is used it is C1- ions that move across the bridge. (v) With t+ -- 1 - t_, the final formal tally is as follows: nl § t_ moles of Ag + and NO~- are present in the left compartment and nr - t_ moles of Ag + and

NO~-are present on the right. From the above five formal steps (ignoring changes introduced by the functioning of the salt bridge) we then obtain t _ A g + ( a + ) r + t_NO~-(a-)r - t - A g + (a+)l + t_NO~- ( a - ) l . One sees that the left-to-right external electron flow is compensated for in part by transfer of Ag + ions past the junction in the same direction, and in

284

4. THERMODYNAMIC PROPERTIES OF ELECTROLYTES

part, by transfer of NO 3 ions in the opposite direction, in proportion to their transference numbers. All steps occur in proper synchronization so that electroneutrality is strictly preserved. The emf for this cell is thus given by .

RT lnF a+(1)a_(1) ]t. [. . J F a+(r)a_(r)

t_RT a+(r) 2 ~ In ~ . F a+(1)

(4.10.4)

Note how the transference number for the anion makes its appearance in this expression. In fact, the determination is usually cal~ed out in reverse: from an emf measurement one may determine the transference number. (c) Double concentration cells. An example is furnished by the cell Zn(s)l ZnSO4(m/), Hg2SO4(s)lHg(~)lHg2SO4(s), ZnSO4(mr)lZn(s), for which the sparingly soluble salt Hg2SO4 furnishes some Hg 2+ ions to the solution for transfer into or out of the Hg(e) compartment; the SO ] - concentration in both solutions remains essentially constant. Under normal conditions ml, m r >> m(Hg2+); thus, one may assume that m(Zn 2+) - m ( S O ] - ) . The left-hand operation is analyzed by the sequence" Z n ( s ) - Z n 2 + ( m t ) + 2e-, followed by 2Hg+(m ~) + 2e- - 2Hg(g), then, by Hg2SO4(s) -- 2Hg+(m ~) + SO]-(ml), which yields a net reaction Zn(s) + Hg2SO4(s) - Zn2+(ml) + SO]-(ml) + 2Hg(g). Similarly, on the right the reaction proceeds according to the scheme Zn2+(mr) + SO~-(mr) + 2Hg(e) - Zn(s) + Hg2SO4(s ). The overall reaction now reads Zn2+(mr) q- S 0 2 - ( m r ) - Zn2+(ml) -t- S02-(mt), for which the emf is specified by

This is clearly another example of a concentration cell. 4.11

T h e r m o d y n a m i c Information from Galvanic Cell Measurements

Measurements of the emf of galvanic cells can be used to advantage to extract thermodynamic information concerning the characteristics of chemical reactions. As already stated, corresponding to the symbolic chemical reaction ~ j uj A j -- O, one may specify an equilibrium parameter Kx -- I-Is(as) vs I - I j ( a j ) vj 9 If a cell can be devised in which the cell operation duplicates the reaction y~j l)jAj --0 then the relations -

r

AGo nF

AGd nF'

Rr ~lnKx nF

(4.11 la)

(4.11.1b)

THERMODYNAMIC INFORMATION FROM GALVANIC CELL MEASUREMENTS

285

may be applied. Thus, both the (standard) differential Gibbs free energies and the equilibrium constants can be determined once g and g0 are known. From the relation ASd -- --(0 A G d / O T ) T , x i one immediately obtains information concerning the differential entropies of the reaction, namely,

A Sd - n F

-ff--f

p ,x i '

(--~-lp,xi" 0g~

AS~

(4.11.2)

Finally, the Gibbs-Helmholtz relation may be applied to find the corresponding enthalpies, namely AHd = -nF

Ir - Z(~T)p,xi] ;

(or

A t-Iff -- - n F [~" ~ - T \ - ~ -

/ p ,x j

].

(4.11.3)

The above shows how important thermodynamic information may be obtained from convenient emf measurements of cells that undergo chemical reactions of interest. ASSIGNMENT

4.11.1. Derive equations relating A Cp ]d and A Vd for chemical reactions to emf measurements.

287

Chapter 5 Thermodynamic Properties of Materials in Externally Applied Fields

5.0

Introductory Comments

In this chapter we consider the thermodynamic properties of materials subjected to several external fields: gravitational, centrifugal, surface, radiation, electric, and magnetic. The analysis gives rise to several new effects that are of intrinsic interest; they also provide new insights on the systematics of thermodynamic analysis. The reader is urged to note these features in the subsequent discussion.

5.1

Thermodynamics of Gravitational Fields

To handle gravitational effects consider a tall vertical cylinder of height h and cross section A (Fig. 5.1.1), containing a fixed number of gas molecules of various chemical species at uniform temperature T in the earth's gravitational field. To cope with the nonuniform distribution of the gas molecules we divide the container into volume elements of cross section A and vertical extension dz. Material contained in each of the infinitesimal layers is characterized by an energy density u, an entropy density s, and a concentration ci for species i. The total energy U, entropy S, and mole number n i of species i is then given by U -

f0hu A d z ,

S -

f0hs A d z ,

ni --

/0 c i A d z .

(5.1.1)

Each element of height d z clearly constitutes an open system whose properties depend on z. The total energy includes contributions from both the internal energy and from the gravitational potential energy. Consider a typical volume element of cross section A and height dz, located at elevation z. Any change in its internal energy E is subject to the First Law: dE(z) - T dS(z) - P(z) dV(z) + Z

tzi(z) dni(z), i

(5.1.2)

288

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

z=h

A

/ f

z=0 Fig. 5.1.1. Cylinder of cross section A in the earth's gravitational field. A subsystem of thickness dz is shown at height z above ground level z -- 0.

in which allowance has been made for the variation of all properties except temperature with height. The total energy of the subsystem, including gravitational effects, is given by dU(z) - dE(z) + Z

Migz dni(z).

(5.1.3)

i

Here Mi represents the gram molecular mass of species i, and g, the gravitational constant. The expression shows explicitly how the total energy in the volume element changes through the influx or outflow of gaseous species through any particular volume element. On combining Eq. (5.1.2) with (5.1.3) we obtain the total energy of material at height z as dU(z) - T d S ( z ) - P ( z ) d V ( z ) + ~-'~[/Zi (Z) d- Migz]dni(z).

(5.1.4)

i

The combination lZi (Z) ']- Mi gz occurs so often in the subsequent analysis that it is convenient to assign it a new symbol, (i (Z) ~ [J'i (Z) ']- Mi gz,

(5.1.5)

THERMODYNAMICS OF GRAVITATIONAL FIELDS

289

which may be called the gravochemical potential; clearly, it is a generalization of the ordinary chemical potential. Thence,

dU(z) - T dS(z) - P ( z ) d V ( z ) + Z

~i(z)dni(z).

(5.1.6)

i In the present case equilibrium conditions are best dealt with by introducing the modified Helmholtz free energy function as

Fg (z) =- U (z) - T S (z),

(5.1.7)

from which one obtains the differential form

dFg(z) -- - S ( z ) dT - P(z) dV(z) + ~

(i(z) dni(z).

(5.~.8)

i Equilibrium conditions are enforced by subjecting the subsystem to a virtual displacement for which 3 Fg 1~,v --O, subject to the requirement that the number of moles of the various species i in the entire cylinder remain fixed: 8ni - O. For this purpose we introduce a Helmholtz free energy density fg by the relation Fg =

f : fg(z)A dz. We then use the properties introduced in Eq. (5.1.1) and consider a fixed element of volume, to rewrite Eq. (5.1.4) as

d[u (z)A dz] - T d[s (z)A dz] + Z

~i (Z) d[ci (z)A dz].

(5.1.9)

i The equilibrium constraint thus reads

o-

fo 6[fg(z)Adz][T, V- fo h [u(z)Adz]lw, v -- T f0h6[s(z)A dZ]lT, v -- f0h~ ffi(z)g[ci(z)A dZ]lT, v.

(5.1.10)

i

The auxiliary constraint is given by

~rti IT, V - 0 - fo h ~(ci(z)A dz).

(5.1.11)

The minimization is carried out using the Lagrangian multiplier technique: we minimize the enlarged function Fg - Z i )~ini, where the )~i are the undetermined multipliers. Thus, we write

fo

ffi (Z)~ [Ci (z)A dZ]lT, V--~,i

Z

i

fo

3[ci(z)A dZ]T,V -- O.

(5.1.12)

290

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

This condition can be met if and only if the integrands match for every i, so that ~I --

[Zi (Z) -4"- m i gz - )~i

(a constant).

(5.1.13)

The above requirement is remarkable" even though/Z i and Mi gz individually vary with z, their sum does not. We now examine the consequences. 5.1.1

The Barometric Formula

Consider a single chemical species obeying the perfect gas law. With T considered constant we write the chemical potential as d l x - ( V / n ) d P . We seek to determine the variation of P with z. For this purpose we write

d#

VdP

d~

dz

n dz

dz

Mg---Mg

(constant T),

(5.1.14)

where the right-hand side follows because ~" does not depend on z. For a perfect gas the above relation becomes

RTdP

dlnP = R T ~ =-Mg, P dz dz

(5.1.15a)

or, in integrated form, P-

P 0 e x p ( M gR z )T

'

(5.1.15b)

where P0 is the pressure at height z - 0. The above represents the well-known barometric formula for the dependence, at constant temperature, of the ideal gas pressure on distance z above ground level. 5.1.2

Systematics of Thermodynamics in the Presence of a Gravitational Field

A more systematic treatment of thermodynamic effects in a gravitational field is provided by rewriting Eq. (5.1.6) for the internal energy of a single species in a small volume element at location z as

dU-

TdS-

P d V + 7rdn + # d n ,

(5.1.16a)

in which we have set ~ - Mgz as the molar gravitational potential of the species under consideration, and where ( = ~ + #. We now introduce the conventional Legendre transformations: H -- U + P V, A = U - T S, G = H - T S to obtain

d H = T dS + V d P + ~/ dn + lzdn,

(5.1.16b)

dA = - S d T

- P d V + ~ dn + lzdn,

(5.1.16c)

dG = - S d T

+ V d P + ~ dn + # d n .

(5.1.16d)

THERMODYNAMICS OF GRAVITATIONAL FIELDS

291

In principle, one could generate four additional functions of state of the form Y~= Y - ~pn, whose differential forms d Y ~ would involve - n d ~ p in place of + ~ dn in the various resulting differential forms of the state functions. However, such a step does not lead to any useful results since gravitational fields normally considered in thermodynamic treatments are not subject to variation. Note that we have introduced a new set of conjugate variables, namely (~, n) that play a role in determining gravitational field effects. One may readily find expressions for T, P, V, S, ~, # by straightforward differentiation of the appropriate function of state listed in Eqs. (5.1.16), in analogy to the procedure of Section 1.13. Similarly, one may undertake a double differentiation in either order of the type o Z Y / O x i O x j -- o Z Y / O x j O x i to obtain the relevant Maxwell equations; only a small fraction of these produces useful relations. Here the condition of keeping ~p constant is equivalent to maintaining the system at constant elevation z. The above procedure again illustrates the methodological approach to thermodynamics. 5.1.3

Effect of Centrifugal Forces

We take up the effects encountered in a cylindrical vessel that rotates about a fixed axis with a steady angular velocity co. In this case there acts on a point of mass m, at a distance r from the rotation axis, a force f - mo)Zr in the direction along increasing r. The corresponding potential energy, i.e., the work involved in placing the mass at point r is given by I~ -- -

f0 "

mw2r dr - -lmw2r2

~

(5 1 17a) ,

.

and the differential increase corresponding to the placement of additional mass Mi dni of species i at that point is specified by di2V _

1 Mico2r 2 dni. 2

(5.1.17b)

This result may be compared with Eq. (5.1.3): it is seen that the relations derived for gravitational field effects may be carried over to the present case by replacing gz in the previous subsection with --lo)2r2. The subsequent portion of the derivation may be taken over without change; in particular, we obtain in place of Eq. (5.1.13) the relation ~i -- lZi (r) -

1

M i o ) 2 r 2 -- )~i

(a constant).

(5.1.18)

At constant temperature and pressure the generalized chemical potential is constant, but the chemical potential itself changes with r to the extent that pressure

292

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

and composition do. Let dlzi IT, P represent the change in chemical potential due to alterations in composition; Eq. (5.1.18) in differential form then specializes to (T constant)

d#i ]T,P at- Vi dP - Miw2r dr - O,

(5.1.19a)

where the second term derives from the partial differential of/z i with respect to P. Let xi be the mole fraction of i at location r; then Eq. (5.1.19a) may be rewritten as (T constant)

Zi xi dlzi

+ Z T,P

Xi Vi dP - ~ i

Xi Mi co2r dr - O.

(5.1.19b)

i

The first term in the above drops out because of the Gibbs-Duhem relation. The remainder may be recast in the form

d P = Z i xi Mi_ o92r dr - pw2r dr,

(5.1.19c)

Z i xi Vi

where p is the density of the constituent material at point r. We next substitute (5.1.19c) into (5.1.19a) to obtain d /Zi l T, p --

5.1.4

j jMj)

Mi - Vi ~ j -~j ~jj

co2r dr -- O.

(5.1.20)

Pressure of Gases in a Centrifugal Field

One may integrate Eq. (5.1.19c) after specifying how p varies with r. If this dependence may be neglected we obtain at a distance r from the rotation axis 1

P -- PO + =P w2r2, Z

(5.1.21)

where P0 is the pressure encountered at the rotation axis. However, for ideal gases we adopt the relation p - M P / R T. Then we obtain from Eq. (5.1.19c)

R T d In P -- Mw2r dr,

(5.1.22)

which assumes the integrated form

Mw2r 2 ln(P/Po) - ~ . 2RT

(5.1.23)

We see that the pressure of the gas relative to Po, its value at r - 0, rises exponentially as the square of the distance away from the rotation axis and as the square

THERMODYNAMICS OF GRAVITATIONAL FIELDS

293

of the angular rotation speed. All this, of course, holds only after the entire mass of gas has assumed the same state of uniform rotation as the container. For a mixture of gases we return to Eq. (5.1.19a) to write dltilT, P -- R T d l n x i and Vi - R T / P to obtain

Mi co2r

d In Pi = ~

dr,

(5.1.24a)

RT

or, in integrated form,

Mi oo2r 2 ln(Pi /Pio) -

~

(5.1.24b)

. 2RT

For a two component ideal gaseous mixture this may be recast as follows: since xl/x2P 1 / P 2 and x l o / x 2 o P l o / P 2 o we obtain

) c~ In XlX2O - (M1 - M2) X2Xl0 2RT '

(5.1.25)

which indicates that the separation effect becomes exponentially more pronounced with the difference in mass of the two components and with the square of the angular velocity and the square of its distance from the rotation axis. 5.1.5

Binary Ideal Liquid Solutions in Centrifugal Fields

Here we again set d # i IT, P -- R T d lnxi; Eq. (5.1.20) then assumes the following forms:

RT d lnxl -- [Ml - ~/l Xl Ml V1- -+-k- x2M2 x2 V2- ] c~

(5.1.26a)

RTdlnx2--[M2

(5.1.26b)

V2 xlMl-k-x2M2] _ x1V1 q-x2V2

-

co2rdr.

Multiplication with V,., followed by subtraction of these relations, leads to RT((/2dlnxl

-

Vl d l n x z )

-

(M1V2 - MzV1)coZrdr,

(5.1.27)

co2r 2 -- (M1 V2 - M2 V1)---~.

(5.1.28)

which may be integrated to yield

RT

V2 In x--L1- V~ In x2 Xl0 x20

)

Since (M1V2 - M291) - M 1 M 2 ( V 2 / M 2 - ( 7 1 / M 1 ) , one observes that there will be no separation in composition of the liquid unless the specific volumes of the two components differ. In that case the degree of separation rises exponentially as o02r 2.

294

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

EXERCISES 5.1.1. In contrast to the discussion of the text assume that adiabatic conditions apply to an ideal gas maintained in a tall vertical cylinder. Derive an expression relating d P and dT; then find an expression for dP in terms of dz. Next, determine d T / d z = ( S T / S P ) s ( S P / S z ) and integrate to determine T(z). Take C p / C v = 1.41, M = 29 g/mol for air, g = 980 cm/s 2 and determine the approximate change in temperature per km of elevation. 5.1.2. Determine all possible Maxwell relations based on Eqs. (5.1.16). 5.1.3. Prove that for a binary mixture involving mole fractions Xl and x2 the following relation holds: RT ln(xlx~ O) = ( M 2 - M1)gz, where the superscripts denote values of x at ground level. 5.1.4. Discuss the reasons why a gas, under the influence of a gravitational potential does not simply settle at the bottom of the tall vertical cylinder, where its potential energy would be least. 5.1.5. Determine the energy and entropy of an ideal gas distributed at constant T in a tall vertical cylinder of height h. 5.1.6. Compare the effect of placing one gram of material in the earth's gravitational field as against placing one gram at a distance of 6 cm away from the axis of a cylinder rotating at a speed of 40 revolutions per second. Comment on the ratio of the two forces.

5.2

Thermodynamics of Adsorption Processes

We next take up the topic of adsorption of gases on surfaces. This problem is not only of intrinsic interest but also provides valuable pedagogical insight on the systematics that obtain for the many choices for thermodynamic functions of state. These involve new degrees of freedom that are needed to characterize the adsorption process. Accordingly, we consider a system consisting of a very thin layer of atoms held on the surface of a material exposed to a gas phase. The bulk solid or liquid is termed the adsorbent, while the material held on its surface is termed the adsorbate. The process by which material is transferred from the gas to the surface phase is called adsorption. As an illustration one may consider the deposit of a thin film of moisture on a windshield of a car left overnight in cold weather. Here the windshield and condensed moisture may be regarded as adsorbent and adsorbate respectively. The condensation has occurred because at the prevailing temperature the saturation vapor pressure has been exceeded. The analogy is faulty because in the processes considered later we restrict attention to adsorbates consisting of at most a few atomic layers. The adsorbate forms a separate thermodynamic phase in its own fight, characterized by a volume Vs, an entropy Ss, mole numbers n s for each species, and a corresponding chemical potential/Zs. It is assumed that at equilibrium the temperature T and pressure P for the adsorbate matches that of the gas phase. To handle

THERMODYNAMICS OF ADSORPTION PROCESSES

295

adsorption effects a new thermodynamic variable must be considered, namely the surface area As. This is best understood by noting that an adsorbent in single crystal form differs in area from the same amount that has been crushed to a fine powder. Under a fixed set of conditions the latter state has a much greater surface area than the former and can thus accommodate a much greater quantity of adsorbate. We therefore must introduce a new control variable, the surface area As, which plays a role for adsorbates similar to that of the volume, Vg, for bulk materials. Associated with this new degree of freedom is its conjugate intensive variable, the two dimensional spreading pressure 4); for, according to Section 1.5, the element of work involved in an infinitesimal increase dA~ in surface area, is specified by -4) dAs. Moreover, the function 4~ = ~(Vs, As, ns, T) represents an equation of state for the surface phase analogous to the function P = P (Vg, ng, T) for the gas phase. For a more sophisticated introduction of the set of conjugate variables the reader should consult specialized sources. 1 5.2.1

Thermodynamic Functions for the Adsorbate

The above presentation shows that we must introduce 4~ and As as conjugate variables in setting up the thermodynamic functions of state of the surface phase. Accordingly, we write the First Law of Thermodynamics in the form dEs = T dSs - P dVs - ~ dAs + lZs dns.

(5.2.1)

From the above we can now derive several other functions of state, using the customary Legendre transformations. By strict analogy to Section 1.13 we may define an enthalpy H~ = Es + P Vs,

(5.2.2a)

a Helmholtz free energy (N.B. in this section we denote this quantity by Fs to avoid confusion with the symbol As for surface area) Fs = Es - T Ss,

(5.2.2b)

G~s - H ~ - TSs.

(5.2.2c)

and a Gibbs free energy

However, it is equally appropriate to introduce another set of functions of state, namely, another energy, !

E s -- Es + 4)As,

(5.2.2d)

Hs -- H~s + ~As,

(5.2.2e)

another enthalpy,

296

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

another Helmholtz free energy Fs -- E s! - T Ss,

(5.2.2f)

G s = Hs - T Ss.

(5.2.2g)

and another Gibbs free energy

No one set of state functions is to be preferred over any other; all physical predictions are ultimately independent of the particular choice, which is therefore merely a matter of convenience. A wise choice of state functions considerably simplifies the analysis, as we shall show. The differential forms are easily seen to be the following: d E s -- T dSs - P dVs - 49d A s + lZs dns,

(5.2.3a)

dHs = T dSs + Vs d P + As d ~ + lzs dns,

(5.2.3b)

dFs - - S s d T - P dVs - dp d A s + Its dns,

(5.2.3c)

dGs -- - S s d T + Vs d P + As dqb + lZs dns,

(5.2.3d)

dE~ -- T dSs - P dVs + As dqb + Its dns,

(5.2.3e)

d I 4 ; - TdSs + Vsde -r

(5.2.30

+ mdns,

dF~s = - S s d T - P dVs + as d ~ + lzs dns,

(5.2.3g)

dG~s - - S s d T + Vs d P - dpd a s + Its dns.

(5.2.3h)

The above expressions can be doubly differentiated in either order with respect to two different independent variables to produce 48 Maxwell-type relations, which are listed in Table 5.2.1. Only a small fraction of these turn out to be useful; they will be introduced below as needed. 5.2.2

The Gibbs-Duhem Relation as a Basic Expression for Gas Adsorption

We derive here the basic Gibbs-Duhem relation that is needed in our subsequent analysis. For a one-component adsorbate system the Gibbs free energy assumes the form Gs = lzs(T, P, dp)ns. Accordingly, we write (5.2.4a)

dGs = lZs dns + ns dlzs

and

(a.s)

dGs - lZs dns + ns \ OT ,] P,4,

+

d P + \ ( - ~lZs T,~

T, p d ~ ] '

(5.2.4b)

THERMODYNAMICS OF ADSORPTION PROCESSES

297

Table 5.2.1 Maxwell relations based on Eqs. (5.2.3)

From E s : ( a T / a Vs)Ss,as,ns -- --(a P / a S s )Vs,as,ns ( a T / a a s ) s , , Vs,n, = -(ar (aT/ans)Ss,Vs,a~ = (a#s/aSs)vs,as,ns

(Or = (OP/OAs)ss,Vs,ns (0 P/Ons)Ss, Vs,As = --(O#s /0 Vs)Ss,As,ns (Or Vs,As = --(O#s /OAs)Ss, Vs,n,

(T5.2.1)

From Hs : (aT/aP)ss,r = (aVs/aSs)p,r (aT/ar s = (aAs/aSs)p,r (aT/ans)Ss,P,r = ( a # s / a S s ) p , r

(OVs/OdP)Ss,P,n s = (OAs/OP)ss,r (OVs/Ons)S~,p,r = (O#s/OP)ss,r (OAs/Ons)Ss,P,r = (O#s/Or

(T5.2.2)

From Fs : (aSs/aVs)T, As,ns = (aP/aT)Vs,As,n s (aSs/aAs)v, Vs,ns = (ac/)/aT)Vs,As,n s (aSs IOns)T, Vs,As -- --(a#s /aT)Vs,As,n ,

(OP/OAs)T,V~,n, = (Oqb/OVs)T,As,ns (0 P/Ons)T, Vs,As = --(O#s /0 Vs)T, As,ns (Or Vs,As = - ( O # s /OAs)r, Vs,n,

(T5.2.3)

From Gs: (OSs /a P)T,r = - ( a Vs / a T ) p,r (aSs/ac/))T,P,ns = - ( a A s / a T ) P,r ( a S s / a n s ) T , p , r = - ( a # s / a T ) P,r

(OVs/Or = (OAs/OP)T,r (OVs/Ons)T,p,r -- (O#s/OP)T,r (OAs/Ons)T,p,r = (O#s/OC/))T,P,ns

(T5.2.4)

From Es~" ( a T / a Vs)ss,r = - ( a P/aSs)vs,r (aT/ar s = (aas/aSs)vs,r (aT/ans)s,,v~.,r = (O#s/aSs)vs,r

(O P/Or Vs,ns = - ( O A s ) / O Vs)Ss,r (O P/Ons)s~, g~,r = - ( O p s /O Vs)Ss,r (OAs/Ons)Ss,Vs,r = (Ogs/Or

(T5.2.5)

From/-/s~: (OT/aP)Ss,As,n, = (aVs/aS~)p,A,,,,, (OT/OAs)S~,p,,,, = - ( a r n,As,,,s (aT/ans)s,,n,A, = (a#s/aSs)n,As,ns

(O Vs /OAs)s~,P,n~ = --(Or P) ss,A,,n ~ (OVs/Ons)Ss,P,As = (O#s/OP)Ss,As,n s (Or = --(O#s /OAs)Ss,P,n,

(T5.2.6)

From Fs~: (OSs/OVs)T,r = (OP/OT)Vs,r s (aSs/Or gs,n, = - ( O A s /OT)g,,r s (aSs IOns)T, Vs,r = --(O#s /OT)Vs,r s

(0 P/Or Vs,ns = - ( O A s /0 Vs)T,r (0 P/Ons)T, Vs,r = --(O#s/0 Vs)T,dp,ns (OAs/Ons)T,V,,r = (O#s/Or Vs,ns

(T5.2.7)

From G~s9 (aSs/o P)T,As,ns = -(O Vs / a T ) P,As,ns (OSs/OAs)T,P,ns = (Odp/Or)p,As,n s (OSs /Ons)r,P,As = --(Ol,ts /OT) P,As,ns

(0 Vs /OAs)T,P,n, = --(0r P)T, As,n, (OVs/Ons)T,P,As = (O#s/OP)T, As,ns (Odp/Ons )T,P,As = --(O #s / a As )T,P,ns

(T5.2.8)

s s s

which shows how changes in G s may be achieved by alterations in T, P, and r This expression should be compared with Eq. (5.2.3d). Consistency requires that ns dps = - S s dT + Vs d P + As d e .

(5.2.5a)

We now define F _= ns/As as the surface concentration of the adsorbed species, whence the above equation reads 1 d#s - - S s d T + Vs d P + de.

7

(5.2.5b)

298

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

T3 T2

T1

P

Fig. 5.2.1. Schematic representation of several experimental isotherms for gases adsorbed on an adsorbent at a set of temperatures T 1 > T2 > T3.

The corresponding equation for the gas phase is given by

dlzg -- - S g dT + Vg dP.

(5.2.6)

On equating the two expressions at equilibrium one obtains the fundamental relation that generalizes the usual Clausius-Clapeyron equation: (Sg

5.2.3

--

Ss) dT

1

-

(Vg

-

Vs) dP + -~ d~ - O.

(5.2.7)

Adsorption Isotherms

We now cite the types of experimental data in the literature, by which an analysis of surface adsorption effects is carried out. One common experiment involves measuring adsorption isotherms. By weighing or by volumetric techniques one determines as a function of equilibrium gas pressure the amount of gas held on a given surface at a specified temperature. Usually this quantity varies sigmoidally with rising pressure P, as sketched in Fig. 5.2.1 for a variety of temperatures 7~. By standard methods that rely on the Brunauer, Emmett, Teller isotherm equation 2'7 one can determine the point on the isotherms at which monolayer coverage of the surface is complete; it is usually is located fairly close to the knee of the isotherm. From the cross sectional area of the adsorbate molecules and from the amount needed for monolayer coverage one may then ascertain more or less quantitatively the surface area of the adsorbent, As. A second set of measurements involves the heats of adsorption by calorimetric techniques under conditions specified below. These heats are reported as a function of the amount of adsorbate held on the surface. For both sets of measurements the fundamental experimental variable is the surface concentration of adsorbate,

F =--ns/As.

THERMODYNAMICS OF ADSORPTION PROCESSES

5.2.4

299

Adsorption at Constant Temperature

We now deal with Eq. (5.2.7) under a variety of special cases. At constant temperature one finds d~-

F(Vg - (ls)dP ~

r'f'gdP,

(5.2.8)

where the molar volume of the adsorbed phase has been neglected compared to that of the gas. For an ideal gas the above reduces to (5.2.9)

d~ = F(P, T)RTdlnP,

which is known as the (differential form of the) Gibbs adsorption isotherm equation. The integrated form of the above expression is qb(P, T) --

fo P F ( p ,

T)dlnp.

(5.2.10)

This relation shows how the two-dimensional pressure may be determined through measurements of F at a sequence of equilibrium gas pressures P at a fixed temperature, either by graphical integration, or analytically, by curve fitting procedures. 5.2.5

Adsorption at Constant Spreading Pressures

When 4~ is held constant Eq. (5.2.7) reduces to

oP) _ ,-

-

3s

-

3s

(5.2.11)

where we have again neglected the molar volume of the adsorbate on the fight. On introducing the ideal gas law we obtain OlnP) 3T

Sg-Ss ck

RT

(5.2.12)

In principle, knowing the molar entropy of the perfect gas (Section 1.17), and by measuring the change of equilibrium gas pressure as a function of temperature, one can determine the molar entropy of the adsorbed phase. The problem here is that the experiment has to be carried out at constant 4~, a problematic task. Methods for circumventing this difficulty are shown below. Meanwhile, for completeness, we observe that at equilibrium the chemical potentials of the gas and adsorbate must match; then H g - Hs - T ( S g - ~Ss), so that we obtain the alternative formulation 81nP) OT

4)

_ ISIg - ff-Is Q~ R T 2 = R T 2,

(5.2.13)

300

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

showing that under conditions of constant 0 an experimental determination of the left can be correlated with an enthalpy change involving the transfer of one mole of gas to the surface; the concomitant 'heat of transfer' is designated as shown on the fight. 5.2.6

Adsorption at Constant Surface Coverage

Experimentally it is much simpler to consider changes in adsorption at constant surface coverage rather than at constant spreading pressure. Unfortunately, the mathematical analysis now is also more complicated. When F ( T , P) is held fixed P and T are no longer independent. We set d P -- ( O P / O T ) F d T , and we rewrite Eq. (5.2.7) in the form

+

- L ) + r(9

- 9s) U

r (5.2.14)

On solving for (0 P / 0 T) r we obtain

r

(Vg - Vs) - (1/1-')(Oc)/OP)r,v _-

(I?Ig - t?Is) + (T/I-')(Oc/)/OT)r,p

.

(5.2.15)

T[(Vg - ("s) - ( 1 / F ) ( a q b / O P ) r , T ]

This rather formidable relation is generally simplified by noting that in the twodimensional equation of state 0 = 0 (F, T) the pressure P does not appear explicitly. Therefore, presumably, no significant error is introduced by neglecting this term. We also introduce the approximation Vs << Vg - R T / P . Then Eq. (5.2.15) reduces to (01nP)

~Sg-Ss

OT

C

RT

Rv2

+

1

(00)

FRT

-~

F,P

+ ?

U

,-',,'

=

RV2'

(5.2.16)

in which Qst is referred to as the isosteric heat o f adsorption. The left-hand side may be found by taking isotherms at a series of closely spaced temperatures and noting the change in equilibrium gas pressure P at fixed surface coverage F. Sg and Hg for the gas phase may be determined as shown in Chapter 1; ( O 0 / O T ) r , e is found either through the two-dimensional equation of state, or via Eqs. (5.2.8) and (5.2.10). Then Ss or Hs are accessible for any particular measured F value. ,..,

,..,

THERMODYNAMICS OF ADSORPTION PROCESSES

5.2.7

301

Differential Entropies of Adsorption

So far we have dealt with molar quantities such as Ss or/-)s. However, considerable interest also attaches to the corresponding differential quantities OSs/Ons and OHs/Ons that are to be determined under various conditions. A systematic analysis requires the use of four Maxwell relations:

0,s) O,s) (o,s) (O,s)

V, V s , A s

(O,s (O,s) (O,s) (O,s)

Ons r,P,ep

-

V2~s

~, ,,'s , O

-

-g2~s

~, ,,, As

-

(5.2.17a)

OT / Vs,r'

--

-~-

(5.2.17b)

P,4,,ns'

(5.2.17c)

,j

5 -f-

Vs , O,,,,

-5-i-

p, ,, "

(5.2.17d)

These relations were taken from sets (T5.2.3,4,7,8) of Table 5.2.1. One may now determine the right-hand side of the above equations using Eq. (5.2.5b). Due attention should be paid to the various experimental quantities that are to be held fixed in the partial differentiations. We find:

(~Ss

)

-

- (OP)

=Ss-Vs

1 (0qS)

~

r

(5.2.18a) 89

OSs) =~s, Ons r,P,4) ( o s,

)

- ~s - ~s

(5.2.18b)

(o,) -5-f

( OSs - Ss - -F - ~ \Ons T,P,As

'

"

(5.2.18c) (5.2.18d)

P,F

The above expressions furnish four interrelations between the molar entropy Ss and the various differential entropies (OSs/On~). Note that it is only when the three intensive quantities T, P, 05 are held fixed that the molar entropy of the adsorbed phase is equal to its partial molal counterpart. The terms involving I7"s are usually small and are generally neglected.

302

5.2.8

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

Other Partial Molal or Differential Quantities for the Adsorbate

Other interrelations between molar and differential quantities for adsorbates are furnished by use of Eqs. (5.2.2) and (5.2.3):

~ns/] T,Vs,As

~,,-~ns) T,Vs,As - Z ( ) ~ns T,Vs,As

Ons J T, P,4)

-~ns) T,P,qb - T ( o n s ]

T,P,~

= tts - Hs - T Ss, (5.2.19a)

(5.2.19b)

-~s-t~s-TL,

-

)

5219c)

(o,s

\ Ons r,e,as

\ ~ns ]T,P,as

_

~nsJr, P,as

The fight-hand side was determined through use of Eqs. (5.2.3c,d,g,h). Next, we eliminate the differential entropies between Eqs. (5.2.18) and (5.2.19). We then find

( ~ Es

~Wj~ ~,Vs,as= I4s - T Vs - ~ ( OHs )

Vs,~

r

5-f

= ffls,

Vs,_,-' (5.2.20b)

T,P,~

( OE~s~

- - T Vs - ~ = Hs

Ons J T, Vs,4~ OIlfs

= Hs - T

T,P,As

-~

, Vs,4~,ns

(5.2.20c)

"

(5.2.20d)

P,I-'

The above equations again correlate partial molal and molar energies and enthalpies; only when all intensive variables are held fixed is the partial molal and molar enthalpy the same. In most cases one may drop the term involving Vs. One may also use Eqs. (5.2.2) to access other thermodynamic functions of interest in terms of differential quantities. As a special application of the above we introduce Eq. (5.2.20d) into (5.2.16) to obtain N

OT

I-'

RT 2

,

(5.2.21)

which relates the quantity on the left, that can be readily determined experimentally by taking isotherms at narrow temperature intervals, to the differential enthalpy of the adsorbate.

HEATS OF ADSORPTION

303

REFERENCES AND EXERCISES 5.2.1. Consult T.L. Hill, Thermodynamics of Small Systems, Benjamin, New York,1963. 5.2.2. Consult S. Brunauer, The Adsorption of Gases and Vapors, Princeton University Press, Princeton, New Jersey, 1945. 5.2.3. The two-dimensional equation of state may be put in the following virial form: - RT[ns/As + n s2C2(T)/A2 -+-n s3C3(T)/A3s + .. .]. Derive the corresponding isotherm equation in the form P -- P(T, F). Derive expressions for the molar enthalpy and entropy of the adsorbate, assuming that the gas phase is ideal. 5.2.4. The Langmuir adsorption isotherm is represented by P = K F / ( F m - F), where K is a function of T only, and Fm represents the maximum concentration of adsorbed species that can be accommodated as a monolayer on the surface. Derive the corresponding two-dimensional equation of state. Determine the molar enthalpy and entropy of the adsorbate, using the perfect gas approximation for the gaseous phase. 5.2.5. Repeat Exercise 5.2.4 when the isotherm equation reads P = K ( T ) ( F / ( F m F))exp(C/FmRT), where the symbols were defined as above, and where C is a constant. 5.2.6. In the text the chemical potential of the adsorbed species was written as Gs = #sns. Explain carefully whether it is appropriate to introduce the defining relation as G1s - #sns. 5.2.7. Multilayer adsorption isotherms are usually analyzed in terms of the Brunauer, Emmett, Teller equation F/Fm = c x / ( 1 - x ) ( 1 - x +cx), where x =- P/Po, where P0 is the saturation vapor pressure of the liquid; c is a parameter, and the other symbols have been defined earlier. Find the equation for the spreading pressure for x < 1. Sketch plots of F/I'm and of q~ vs. x.

5.3

Heats of Adsorption

Aside from adsorption isotherm data one can use calorimetric techniques to obtain information on the thermodynamic properties of materials adsorbed on surfaces. The experimental techniques are now more involved but they do supply direct information on the heats liberated during the adsorption process. Here the use of partial molal quantities is imperative since increments of the heats of adsorption diminish with successive amounts of gas transferred to the adsorbed phase. Here we follow the systematic treatment furnished by Clark. 1

5.3.1

Heats of Adsorption

We begin by applying the First Law to the combination of adsorbed and gas phases:

dQ--dEs

4 - d E g + P dVg + P dVs + ~ d A s .

(5.3.~)

Before considering various special cases we introduce the differential heat o f adsorption 0_. through the definition dQ =_ -0_. dns.

304

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

The following particular experimental conditions are frequently reported in the literature"

(i) T, Vg, Vs, As constant. Eq. (5.3.1) now assumes the form

dQdlr, ve-- dEg + dEs.

(5.3.2a)

The energy of the gas phase may be written in natural coordinates as E g = Eg(Sg, Vg, ng); by use of the relation Sg - Sg(T, Vg, ng) we can reexpress the energy in unnatural coordinates Eg - Eg (T, Vg, ng). Under present constraints E g varies only with n g; thus,

(OEg --\~ng )r, vgdns ,

_

(OEg~

dEglr, v~- \~n~ )r,v~ dng

(5.3.2b)

where for the closed system normally employed in calorimetric measurements we have set dns - -dng. Precisely the same analysis can be carried out for the adsorbed phase where, in unnatural coordinates, we deal with the function Es = Es(T, Vs, As, ns); under the prescribed condition Es varies with ns alone. Thus,

dEslT'Vs'as=

( 0 Es ) dns. ~ T, Vs,As

(5.3.2c)

Putting these results into Eq. (5.3.1) yields

dns.

--O-'ddns----( oEg-~ng)T,Vgd n s + ( oEs)-~nsJ

(5.3.3a)

T, Vs,As

Next, we set E -- Egng and Es -- Esns and note that for an ideal gas J~giS independent of Vg or n g. Then Eq. (5.3.3a) reduces to -

Od -- Eg -

-

Ik~

T, Vs,as

-

(O

-

(O

s)

= Eg - Es -ns\-~ns ]

T, Vs,As

s)

(5.3.3b)

= Eg - Es - F --~-JT, Vs,As

The above quantity is known as the differential heat of adsorption. Although the measurement is simple, since volumes and surface areas are kept fixed, the interpretation of the experimental measurements is more complicated than for several of the cases cited below. (ii) T, P, 4~ constant. We rewrite Eq. (5.3.1) as D

dQe - d(Eg + PVg)+d(Es + PVs +qbAs) -- dHg-q-dHs =- -Qedns. (5.3.4a)

HEATS OF ADSORPTION

305

The above relation is clearly of the same form as Eq. (5.3.2a); moreover, T, P, q~ are the appropriate natural coordinates for Hg and for Hs; also, because (OHg/Ons)(Ons/Ong) -- -Hg, w e deduce that

-

(OHg)

Qe-

_(OHs)

Ong T,P

~

=iSlg_IYls '

(5.3.4b)

r,P,dp

which is known as the equilibrium heat of adsorption. Here we note that when all intensive variables are held fixed, the partial molal and molar enthalpies of the gas and adsorbate phases are identical. Obviously, this is the simplest relationship for the enthalpy of the adsorbed phase, but experimentally it is difficult to operate under conditions where the spreading pressure must be held fixed. (iii) T, Vg, Vs, 4~ constant. Here we rewrite Eq. (5.3.1) in the form

dQ - dEg + d(Es + ~As) -- dEg + de~ =_ - 0 ' dns,

(5.3.5a)

so that

~t

(OEg

--

\ Ons v, v~,4~

-- Eg - E s - n s

.

On~ } v, Vs,4,

(5.3.5b)

This quantity has not been assigned a name. (iv) T, P, As constant. Here we put (5.3.1) in the form

dQst - d(Eg + P Vg) + d(es + P Vs) -- dHg + dH~s =_ -Qst dns,

(5.3.6a)

whence

Ost-

(OHg) (OHms) = ISlg-tYI~s - F ( O H ~ ) . (5.3.6b) ~ng T , P - ~ T,P,As 1k 0I" /]T,P,As

This quantity is known as the isosteric (differential) heat of adsorption. The experimental requirements are easily satisfied, but the interpretation of the results is now more complicated than for Eq. (5.3.4b). (v) T, As constant. Under these conditions Eq. (5.3.1) reads

d Q t h - dEg + dEs + P dVg + P dVs =--Qth dns.

(5.3.7)

We must now use the entropy of the adsorbate, Ss -- Ss(T, Vs, As, ns) to eliminate the entropy in Es = Es(Ss, Vs, As, ns), so that at constant T, As

dEslT'as-- T \(OSs -~s

)

T, as,ns dVs + T Ons]

dns - P dVs + lZs dns. T, Vs,As (5.3.8a)

306

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

Now introduce Maxwell's relation (OSs/OVs)T,F = (OP/OT)v~,F as read off from Table (T5.2.3), and set lZs + r(OSs/Ons)T, Vs,As = (OE~/Ons)T,V,,As. This latter step follows since energy is a function of state; thus, dEslT, Vs,As = (OEs/Ons)T, Vs,Asdns, and this quantity must match the coefficient of dns in Eq. (5.3.8a). Under present restrictions we then find

dEs IT, As-- T - ~

(0 Es) r, Vs,Asdns.

Vs,F - P

(5.3.8b)

dVs+\~ns

By an exactly analogous procedure one obtains (dns - - d n g )

]

(OEg)

- P dV~ + \ ~ }

r,v~

dng.

(5.3.8c)

One should note that Eq. (5.3.8c) represents an extension of the ordinary caloric equation of state, Eq. (1.13.16); the above relation applies when mole numbers are allowed to vary. Equation (5.3.7), together with Eqs. (5.3.8b,c) and with dns - - d n g may now be rewritten as

Oth

-~ng}T, Vg

(O~ s)T,Vs,As

Vg,ng

T,P

(0 s) - T

-~

Vs,r\~sJr,

(5.3.9a)

As

which is known as the isothermal heat of adsorption. It is conventional to introduce the perfect gas law and to ignore terms involving Vs. Eq. (5.3.9a) then reduces to

-

O_,h ~ e ~ -

~,~s

~,Vs,As + P \ o n g }

T,P

(O s)

- I:I~ - Es - r \ - ~

}

T,Vs,As

(5.3.9b) The above shows again that under readily accessible conditions the interpretation of the experimental results is more complicated than for case (ii). One should also compare the present results to those displayed in case (iv). (vi) Adiabatic conditions, As constant. Here one must introduce the thermal properties of the calorimeter, denoted by the subscript c, since the measurements depend on the rise of the temperature of the entire system during the adsorption process. It is also simpler to adopt P rather than V as the independent variable. We thus rewrite Eq. (5.3.1) in the following form:

dEc + dEg + dEs + d(P Vg) - Vg dP -+-d(P Vs) - Vs dP - O,

(5.3.10a)

HEATS OF ADSORPTION

307

or as

dHc + dHg + dH~ - Vg dP - Vs dP - O,

(5.3.10b)

where the distinction between the energy and enthalpy of the calorimeter has been ignored. We next write

dHc -- Cp,cdT +

T dP,

-~

(5.3.1 la)

and we also set

dHg

(OSg Tk, OT

I

dT + T - ~ P,ng

dP + T Ong } T,P

T,ng

dng

+ V g d P + lzgdng = C p ,g d T +

+

Vg - r

--ff-~- P ,ng

(OHg) dng. Ong T,P

(5.3.1 lb)

In the preceding we have eliminated the partial entropy derivatives via the appropriate Maxwell relation and by use of the expression lZg -- Fig - T Sg. Eq. (5.3.1 lb) is a generalization of Eqs. (1.13.15) and (1.13.17) to the case of variable mole numbers. An analogous expression holds for the adsorbed phase, involving the function dH~s, with As held fixed. On inserting Eqs. (5.3.11a), (5.3.1 lb) and its adsorbate counterpart into (5.3.10b) one finds [Cp,c @ Cp,g nt- Cp,s] S,As Oad

~

~

+

\-~/]T

Ong

~

T

k,, OT

P,ng - T

(O.,s) T,P

~

-~

I(OP

P,As,ns

As

(5.3.12a)

T,P,As

m

where Qad is termed the differential adiabatic heat of adsorption. Ordinarily one may ignore the variation of He with P, neglect terms in Vs, and use the perfect gas approximation in the adiabatic limit. In that case the above reduces to

-

(O.,s)

RT VY-1 '

(5.3.12b)

308

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

where the last term is sometimes omitted as well. Note that the adiabatic experiment is a measure of H- 's, not of/ls. In summary, one observes that the various calorimetric measurements described above correspond to distinct differential energies or entropies or to their molar counterparts. The measurements differ in the experimental variables that are held fixed during the measurements. In cases where ~b is held constant the experiments are not readily carried out; then further manipulations are required to cast Eqs. (5.3.4) and (5.3.5) into a useful form for experimentalists; see the Queries below. In any case, it is important to examine closely the various assumed boundary conditions to obtain the correct interpretation of the experimental measurements. 5.3.2

Energetics of the Adsorbed Phase at Constant ns

We briefly consider how the energy of the adsorbed phase varies with temperature, pressure, and surface area at a constant density n s of the adsorbate. For this purpose we again express Es in unnatural coordinates by using the relations Ss = Ss(T, P, As) and Vs = Vs(T, P, As) at constant ns; this allows us to write

dEs -

3Ss - P --~ T - ~ P,F [

(

ST+ P,F

)

OSs T,e,ns - P + --ok+ T -~s

T

\ OP J T,F

- P ~

OP ] T,F

(OVs)]dAs. -~s T,P,ns

de

(5.3.13)

The various terms are interpreted as follows: T(OSs/OT)p,F represents the heat capacity, C p,F, of the adsorbate at constant pressure and surface occupancy F. The second term represents the mechanical work involved in the expansion of Vs on heating; here the coefficient of expansion is relevant: otp,c =-Vsl(OVs/OT)p,F. In the third term we invoke the Maxwell relation that is specified in Eq. (5.2.8) of Table 5.2.1" T ( O S s / O P ) T , F - - T ( O V s / O T ) p , F -TVsotP,F, which again relates to mechanical work associated with the alteration of surface phase volume induced by pressure changes. The fourth term describes the contraction in volume of the surface phase due to the application of pressure. This effect is described by the isothermal compressibility flT,C =--Vsl(OVs/OP)T,F. The product - ~ d A s obviously deals with the work of expanding the surface area. The sixth term is dealt with by use of the Maxwell relation from (5.2.8) from Table 5.2.1 T(OSs/OAs)T,P,ns - T(Odp/OT)p,c, which relates to the temperature coefficient of the surface tension. We may therefore recast the above equation in the form

dEs - (Cp,F - P Vsotp,i-,) dT + (-TVsotP,r, + P VsflT, r') dP [

(Ock)

_p(OVs)

-gTss ,P,ns

]das.

(5.3.14)

HEATS OF ADSORPTION

309

The above is considerably simplified under several conditions: if T and P are held fixed and the term in Vs is neglected we obtain for the energy per unit cross sectional area the expression

es -- -dp + T - ~

P,C

(5.3.15)

If, on the other hand, one imposes conditions of constant P and As, and if terms in Vs are neglected Eq. (5.3.13) reduces to

(O s) - ~ J P,C

-- T

(OSs) 3T ] P,C

-- Cp,y,

(5.3.16)

which involves the heat capacity. The above provides another route to the determination of entropies of the adsorbed phase, through heat capacity measurements under appropriate conditions. The equation also determines the energy Es. In the same vein, from Eq. (5.3.2f) one obtains at constant P

3Ss)

dT P,F

-Cp,rdTle,r,

(5.3.17)

which is frequently used to determine Hs~, since the constraints are readily applied experimentally.

5.3.3

Summary

Summarizing, an attempt has been made to provide a systematic account of the thermodynamic properties of the adsorbed phase. The Gibbs adsorption equation, as an extension of the Clausius-Clapeyron equation, has played a key role in linking experimental isotherm data to the determination of molar or differential entropies and enthalpies. Similarly, calorimetric measurements can be systematically applied to obtain the same type of information. REFERENCE AND EXERCISES 5.3.1. A. Clark, The Theory of Adsorption and Catalysis, Academic Press, New York, 1970, Chapter 1. 5.3.2. Rework Eqs. (5.3.4) and (5.3.5) to reexpress the heats in terms of experimental conditions under which F rather than ~b is held fixed. 5.3.3. The two-dimensional analogue of the van der Waals equation of state has the form (4) + ans/As)(As/ns - b) = kBT, where a and b are parameters and kB is Boltzmann's constant. Obtain the corresponding isotherm equation and determine the molar enthalpy and entropy for the case where the bulk gas is ideal. Find the various heats of adsorption and the differential enthalpies and entropies.

310

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

5.3.4. For an isotherm equation of the form P = [cF/(1 - bF)]exp[cF/(1 - bF)], where c and b are parameters, find the corresponding equation of state of the adsorbed phase and determine the molal entropy and enthalpy of the adsorbed phase relative to F -- 1. 5.3.5. Given the two-dimensional equation of state ~b= RTF/(1 - bF) + aF 2, where a and b are temperature-independent parameters, determine the corresponding adsorption isotherm. What is the three-dimensional analogue of this equation of state? 5.3.6. Verify the following relations: (a)

(OSs

[O(~Ss/ns)]

\ 3 As ,,I T,P,ns = (b) \ 3ns } T,P,As

0(l/F)

--~

T,P

T,P

OF

T,P

(3qb) ~ P,r

3T ]P,r'

(C) (O~ns)T,P,As =--~s - ~ (d) 1,2 31J,s

=_i_,2(O(Ss/ns))

=F

-~

T,P T,P

-- OAs,]T,P,ns=--n---~ O(1//-')T,P' -

31nF

T,P

Are any of the above relations useful?

5.4

Surface vs. Bulk Effects; Thermodynamics of Self-Assembly

Hitherto we studied properties of material adsorbed on an inert substrate that remained entirely passive; it merely provided a support for the deposition of a quasi-two dimensional adsorbed layer whose properties were of interest. We now study the characteristics of surfaces that envelop the bulk material, and correlate the surface properties with those of the bulk. These surface effects obviously become prominent in materials with very small physical dimensions, though still very large on the atomic scale. Whereas earlier the surface area could be varied at will through control of the extent of subdivision of the solid we are now constrained: the physical extension of the surface is directly linked to that of the bulk, as, for example, in tiny spherical bubbles of radius R, whose surface areas and bulk volumes are governed by R 2 and R 3, respectively. For illustrative purposes consider a semi-infinite rectangular block with a very thin surface layer that separates it from an inert gas phase. At equilibrium the pressure P is uniform perpendicular to the surface layer; the lateral two-dimensional spreading pressure q~ within the surface layer is constant as well. As the pressure on the block is decreased its volume increases; similarly, there is a change in surface volume Vs - As r, where r is the thickness of the surface layer. The element of work performed in slightly changing the surface volume is given by dWb = --P (As dr + r dAs); however, the lateral film enlargement is counteracted by the internal two-dimensional spreading pressure ~b that resists this process; see Eq. (1.5.8). This lateral effect thus contributes a term dWb = qbdAs to the work,

SURFACE VS. BULK EFFECTS; THERMODYNAMICS OF SELF-ASSEMBLY

311

Fig. 5.4.1. A two phase system separated by a curved interface, with appropriate inlets and outlets. After Levine, loc. cit.

for a total dW =-(Pr-~)dAs

- PAsdr =-PdVs

+ckdAs.

(5.4.1)

Note the change in sign of this surface work term relative to that in Eq. (5.2.1); work must be expended to enlarge the surface area. 5.4.1

Pressures on Curved Surfaces

We study properties of curved surface films by following the procedure of Levine. 1 Examine the two conical sections shown in Fig. 5.4.1 that form part of a sphere; an inner layer a is surrounded by an outer layer b. We are interested in the properties of the curved interfacial layer between the two phases. On the left we show more material a being forced through an inlet at the bottom, while some material b is forced out through the upper channel. During the process the curved interface moves upward, thereby expanding the area As of the interface. Clearly, work must be done on the system to increase the area. By contrast, work is released when diminishing As by pushing material b into channel B and removing material a through channel A. Thus, when dAs > O, Pa > Pb. The element of work involved in enlarging the surface area at constant temperature and volume of the system is given by dW --PadVa

- PbdVb + ~ d A s ,

(5.4.2)

with V = Va + Vb as the total volume of the system; we neglect the small contribution of the interface volume. Now, using the setup of Fig. 5.4.1 (b), we exert pressure Pb in channel B, which is also the pressure at the interface, thereby changing the total volume by - d V . The corresponding work is d W = - - P b d V = - P b ( d V a + dVb).

(5.4.3)

312

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

On equating (5.4.2) with (5.4.3) we obtain dAs Pa - Pb = ~ b ~ .

(5.4.4)

dVa

From Va - - 4 : r r 3 / 3 , A s - - 4 : r r 2 we obtain d V a - 4 7 r r 2 d r , and d A s - 8rrr d r , where r is the radius shown in Fig. 5.4.1(b). Thus, d A s / d V a = 2 / r , so that one obtains the fundamental relation (5.4.5a)

Pa - Pb = 2 ~ / r.

For nonspherical surfaces the pressure difference involves more complicated expressions; for example, for elliptical surfaces with principal radii of curvature rl and r2 one obtains Pa - Pb - ~

+

9

(5.4.5b)

r2

Note that the surface tension ~b can be determined if the equilibrium pressures can be measured experimentally. 5.4.2

Vapor Pressure of Small Drops

We compare the vapor pressure of a small droplet a of radius r with that of a large mass of the same material, b, both being in equilibrium with vapor at pressure P. Let the pressures inside these two phases be specified by Pa and Pb. Then, on account of (5.4.5) we find that Pb = P ,

P = Pa - 2 ~ / r .

(5.4.6a,b)

Thus, because of surface tension effects, the pressure inside a small droplet is greater than that of the bulk phase. Assuming the material to be incompressible the corresponding change of chemical potential at fixed T is given by # a - lzb -- V (Pa - Pb) - 2 ~ V / r.

(5.4.7)

If the vapor phase consists of an ideal gas we find

24~f'

ln(Pa/Pb) = ~. rRT

(5.4.8)

Lastly, setting Pb equal to the bulk vapor pressure P0, and dropping the subscript a we find to first order in ~p P-

Po 1 -F ~

prRT

'

(5.4.9)

SURFACE VS. BULK EFFECTS; THERMODYNAMICS OF SELF-ASSEMBLY

313

where M is the gram-molecular mass and p the density of the material. This relation shows explicitly the effect that a reduction in radius of a bubble has on its vapor pressure. This finding is particularly important in considering the environmental impact of tiny mercury droplets. 5.4.3

Functions of State for Surface Phases

Functions of state for surface phases of the type under consideration are handled in a manner analogous to those of the bulk and will only be studied cursorily. In view of the expression (5.4.1) for the element of reversible work we may write

dEs - T dSs - P dVs + ~ dAs + Z

l~is

dnis

(5.4.10)

i

for the energy of the surface phase. Here it is assumed that Vs and As may be altered independently, as by a change in thickness of the surface layer. Where this is not the case on must combine the two terms via the relation d Vs = q d As, where q is a factor that depends on the geometric configuration of the material. By use of standard Legendre transforms we find

dG;--SsdT

4- VsdP +dpdAs + Z # i s d n i s

(5.4.11)

i

for the differential of the Gibbs free energy considered as a function of T, P, As, and ns. This function of state is of immediate interest because experimentally one can control the surface extension, As, by adjustment of the bulk volume, Similarly, we may define an enthalpy H~s(Ss, P, As,ns) for the surface phase whose differential is given by

dH~-- TdSs + VsdP -FdpdAs 4- ~--~#isdnis,

(5.4.12)

i

which is consistent with the relation !

!

(5.4.13)

G s - H s - TSs.

By contrast, the surface analog to d G for the bulk phase is specified by

dGs - - S s d T + Vs d P - As d~ + Z

lzis dnis,

(5.4.14)

i

in which all independent variables are intensive quantities, but which is experimentally harder to handle, since surface pressure measurements are not necessarily easy.

314

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

We now compare Eq. (5.4.14) with the relation d G s ~-~i nisdl zis to find

- S s d T nt- Vs d P + As dq~ -

Z

nis dlZis

-- O.

~ i lzisdnis + (5.4.15)

i

For a closed system we obtain from Eq. (5.4.11) the relation

( OG~s) OT

-Ss,

(5.4.16)

= -Ss,

(5.4.17)

= -As,

(5.4.18)

As,P,nis

whereas from (5.4.14) we also find

OGs - i f - T - ) e , 4, , n i s

as well as

0r

r,P,nis

and -- lZis.

(5.4.19)

Onis ] T,P,dp,nisr

If we introduce (5.4.16) into (5.4.13) we obtain

G~s - T \ o r

As,Vs

- Hs~,

(5.4.20)

which is an adaptation of the conventional Gibbs-Helmholtz equation. The above represents but a small selection of properties amenable to thermodynamic analysis. Readers are invited to review the much more comprehensive undertaking in Section 5.2 and to adapt that scheme to the present situation. 5.4.4

Self Assembly of Monomer Units

We begin here a brief survey of thermodynamic principles that pertain to formation and self assembly of large scale aggregates of molecular units, such as polymers, micelles, vesicles, or similar macromolecules. We follow the exposition of Israelachvily. 2 As the guiding principle we use the thermodynamic equilibrium requirement that in a system of identical monomer units in equilibrium with its aggregates the chemical potential of the various r-mers in different states of assembly be the same. We further ignore the role of solvent in considering the stability of the

SURFACE VS. BULK EFFECTS; THERMODYNAMICS OF SELF-ASSEMBLY

315

aggregates relative to the fundamental constituent units. Let #1, #2, #3 . . . . be the chemical potentials of the monomers, dimers, trimers, ... relative to their standard values #l,* #2,* #3,'* .. and let x l, x2,x3, ... be the corresponding mole fractions. Then the equilibrium requirement may be reformulated as follows: # = #1 + R T lnxl

1 -~- -2 R T l n

--/~2

,

(~) x2

. --#3+- 3 RTln

( -ix3 ,)

....

(5.4.22a) or, more compactly, RT l z =~ ~ N

- - l.~N -Jr- - ~

ln(xN/N),

(5.4.22b)

for the N-mer aggregate. The 1 / N factor in the above serves the purpose of correlating the mole fraction and chemical potential of the N-mer with those of the N constituent units in the aggregate molecule. For use in our further deliberations we set Eq. (5.4.22b), for the N-mer, equal to its counterpart, involving an M-mer. This relation may be solved for

N

-~- exp

Rir

.

(5.4.23a)

For the particular case M -- 1 this reduces to XN

N

-

Xl exp

l'L1 ~

/~

N

"

(5.4.23b)

The above relations, together with the requirement ~ N X N - - l, define the properties of the system. We now consider a special case which is nevertheless very instructive. Assume that the interactions between molecules in the aggregated and monodispersed states are the same; then the standard chemical potentials are all equal" /Zl* --/Z2* - - //~3" - - . . . . - - /Z x.* In that event Eq. (5.4.23b) reduces to XN -- Nx u.

(5.4.24)

Since Xl < 1 it is clear that for large N, X N ~ X l . In this case only very small aggregates will be present in the solution, of which the monomer is by far the most stable. This trend is even more accentuated if we now allow the/Z N* increase with N. Thus, large aggregates cannot be expected to form unless at some stage the ~ U start to diminish with rising N and become smaller than #*1" In fact, the variation of/Z *N with N determines the characteristics of the solution with respect to the state of aggregation of the solute molecules, as we now investigate.

316

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

.

I

Fig. 5.4.2. Assembly of dissolved identical monomer units into rods, two-dimensional disks, or three dimensional spheres. After Israelachvily, loc. cit.

5.4.5

Geometric Effects; Linear Aggregates

The variation of #* with N largely depends on geometric considerations. Consider first a set of one-dimensional rod-shaped polymers, and let &IRT represent the molar bond energy for joining two monomers. Then the standard free energy of binding of the linear N-mer relative to an arbitrary reference state is found from (5.4.22b) by multiplication with N and discarding the constant term R T In N. Note that there is one fewer bond than the number N of units. We find that (5.4.25)

N/z~v -- - ( N - 1)c~IRT, so that, by a trivial rearrangement we obtain 6tl R T i~ N - - -

1 -

Oil R T

- - lzc~ +

N

(5.4.26)

Clearly, as the chain length increases the mean free bonding energy of the rod diminishes towards the 'bulk' value # ~ . From the above derivation it is evident that the last term on the right is proportional to the ratio 2 / N of the number of terminal atoms to the number of constituents in the chain. 5.4.6

Two-dimensional Sheets

For two dimensional structures (see Fig. 5.4.2) the number of N-mers in a disk is proportional to its area, re R~, whereas the number of monomer units on the periphery that bond only to interior molecules is proportional to the circumference, 2:r R0, and hence, proportional to N1/2. Thus, the ratio of the two quantities is

SURFACE VS. BULK EFFECTS; THERMODYNAMICS OF SELF-ASSEMBLY

317

given by N -1/2. In a derivation parallel to that of the one-dimensional case one is thus led to the result

dt2 R T

t~U -- ll~cx~~ N1/2 ,

(5.4.27)

where the multiplier differs in value from the one dimensional case (see below). 5.4.7

Three-dimensional, Spheroidal Aggregates

In a spherical aggregate the number of monomer units, N, is proportional to the volume 4rcR3/3 of the sphere, whereas the number of monomer units on the spherical surface is proportional to 47r R 2, that is, to N 2/3 . The ratio of these two quantities is N -1/3, whence, in a derivation analogous to the one-dimensional case one is led to the result

dt3RT

~N --/Zcx~ ~ N1/3 " :r

:r

_

~

(5.4.28)

This may be made more evident by considering the case of a small molecule with volume v, whereby in a spherical aggregate N - 47rR3/3v. The standard interfacial free energy of the sphere is specified by N # ~ + 47r RZy, where V is the interfacial free energy per unit area. We therefore write

* - - / z *~ + #U

4~g2ylv ~

* + --/zoo

N 14~y3( 4~N313)2/3

- - / z ~ -~ rT N1/3 ,

(5.4.29)

which shows how 63 may be obtained in this particular case. On examining the above derivations one notes that for the simple structures enumerated above the standard free energies are of the general form :r

:r

_

/Zu --/Zcx~~

dt R T

Nq

,

(5.4.30)

where the exponent q depends on the dimensionality of the array. This quantity need not be the reciprocal of an integer; nonintegral values for dimensions are commonplace among fractal aggregates. Note that in all the above cases, with ~ ~U a positive quantity, the standard free energy diminishes with rising N towards the value attained in an infinite aggregate, as a separate phase. This is a necessary condition for the formation of large scale structures. 5.4.8

Critical Concentrations

Once/x *U is known, as shown above, one can ask: what are the critical concentrations required for formation of aggregates? This may be established by employing

318

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

Eq. (5.4.30), in which the index is first set at N = 1 and then at N, and then inserting these expressions into Eq. (5.4.23b), so that we obtain

XN -

N{xlexpI&(1

Nq

~N[xl

]

(5.4.31)

For sufficiently low values of x l, such that x le ~ << 1, XN / N < Xl for any N > 1. In such a case, at these low mole fractions the solution will consist principally of monomer units dissolved in the solvent. However, this inequality no longer applies at higher concentrations: for, x N / N in Eq. (5.4.31) cannot exceed unity, which means that the quantity in braces is not allowed to exceed unity. In other words, we impose e x p { - ( ~ ( 1 - 1 / N q ) ) } - e x p { - [ ( # ~ - # * N ) / R T ] } as an upper limit on Xl. This cutoff value represents the critical aggregation mole f r a c t i o n (CAMF). Thus, when x u is specified by Eq. (5.4.31) we find that Xl ICAMF ~ e - a for all q; at these values of Xl any further addition of monomer molecules to the solution results solely in the formation of more aggregates. 5.4.9

Phase Separation vs. Finite Size Aggregates

In addressing this question we note that for the geometric shapes with q # 1 the exact form of Eq. (5.4.31) reads

XN -- N [ x l e S ] N e -SN/(N)q 9

(5.4.32)

Clearly, for any positive ~ of reasonable size, the usual case, one does not anticipate the presence of large scale combinations of monomers. What happens instead is that a separate phase forms, essentially an aggregate of infinite size. It is evident that even for any positive value of q < 1 in the above equation the formation of a separate phase cannot be avoided. Aggregates that fall into this category are those for which the intermolecular bonds are very flexible, so that symmetrically disposed polymeric disk-shaped or bilayer units may be formed. Hence, to develop large aggregates in solution it is necessary to seek conditions under which, beyond some finite N, the quantity x u of the system attains a minimum constant value. Such a case will now be considered. 5.4.10

Aggregates for which q = 1; Chain-like and Related Aggregates

We take up the case of systems for which one may set q = 1 in Eq. (5.4.32). This leads to the relation

XN _ U[xl e 8 ]Ne -8 ,

(5.4.33)

the factor in the right now being a constant. Note that above the CAMF the quantity x le ~ ~< 1, so that initially, for small N , X N is proportional to N. That is, the

SURFACE VS. BULK EFFECTS; THERMODYNAMICS OF SELF-ASSEMBLY

319

aggregates tend to grow in proportion to their size. This trend cannot, however, continue indefinitely; for, ultimately the [xle(~]N < 1 factor in Eq. (5.4.33) begins to dominate, thereby diminishing X N towards zero as N becomes very large. In short, under present circumstances one anticipates a distribution of aggregate sizes, with a maximum at intermediate values of N. By contrast, Eq. (5.4.32) shows that, with x l e a << 1 and for q > 1, X N very rapidly plunges toward zero, so that no large scale aggregates form as dispersed entities in solution. The case q = 1 appears to be very specialized; nevertheless, rod-like or chain-like structures, as well as cylindrical micelles, and microfilaments and microtubules fall into this category. Additionally, spherical vesicles and certain classes of microemulsion droplets are members of this class. More generally, it is aggregates for which the intermolecular bonding is very rigid that are members in this category. Thus, it is worthwhile to examine this special case in greater detail. The total mole fraction of all aggregates dissolved in the solvent is given by e ~ ] N e -c~

XA--ZXN--ZN[Xl N

N

= e-a[xle a + 2(Xl eCi)2 + 3(Xl eCi)3 + . . . ] -

Xl

(5.4.34)

( 1 - X l e a)2'

where the mathematical identity Z N N z N -- z / ( 1 - z) 2 (1 ~< N ~< cx~) had been introduced. We may solve the above quadratic equation for

Xl --

(1 + 2xAe a) --v/1 + 4 x A e a 2xAe2G

(5.4.35)

where the minus sign is retained to ensure that Xl does not exceed unity. For x A e c~ << 1 the expansion of the square root to second order terms leads to the relation Xl -- X A / 1 6 , whereas in the opposite limit x A e ~ >> 1,

X l - [ I - (xAea)-I/2]e-a <. e -a,

(5.4.36)

which indicates, as expected, that x] approaches the CAMF value. In the present approximation the distribution of aggregates of various sizes is found by inserting the above relation into Eq. (5.4.33) to obtain XN -- N [ 1 -

(xAea)-l/2]Ne

-~.

(5.4.37)

For large N this reduces to

XN -- N

exp{-G} exp

{ N }

V/XAec~ ,

(5.4.38)

320

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

as may be verified by carrying out a first order expansion on the second of the above exponential functions and on the multiplier of Ne -~ in Eq. (5.4.37). Eq. (5.4.38) has an extremum at OXN/ON = 0, i.e., at Nmax --

~/XA edt 9

(5.4.39)

Thus, Eq. (5.4.38) shows that the probability of encountering very large aggregates declines exponentially as exp{-N/Nmax}. The expectation value for N is specified by ( N ) = Y-~NXN/Y-~XN = NXN/XA (see Eq. (5.4.34)); one finds that (N) -- V//1 + 4xAe ~,

(5.4.40a)

which reduces to the limits (N) -- 1

and

(N) -- 2~/xAe (~ -- 2Nmax

(5.4.40b)

below and above the CAMF respectively. A survey of these findings shows that the distribution of aggregation is very broad: it initially rises with N, then peaks, and finally decays to zero for very large N. The distribution is strongly affected by the value of &. For much more detailed discussions of this topic a variety of specialized sources must be consulted. REFERENCES

5.4.1. I.N. Levine, Physical Chemistry, 5th edition. McGraw-Hill, Boston, MA, 2002. Chapter 13. 5.4.2. J.N. Israelachvily, Intermolecular and Surface Forces, 2nd edition. Academic Press, London, 1991. Chapter 16. 5.5

Pressure of Electromagnetic Radiation

In preparation for the thermodynamic analysis of radiation effects we study the pressure exerted by electromagnetic radiation, based on Maxwell's equations for electromagnetic fields. Readers not wishing to wade through the rather lengthy derivation may note the final result, Eq. (5.5.11), and proceed to the next section. 5.5.1

Pressure of Electromagnetic Radiation

As shown in Eq. (1.6.5), the force density acting on free charges in electromagnetic fields in a volume element d3r is specified by the relation d f (pfE, -t- c - l j x B)d3r, where pf is the density of free carriers in the system,

PRESSURE OF ELECTROMAGNETIC RADIATION

321

(not to be confused with the electrical resistivity of the material), J is the corresponding current density, E is the electric field, B is the magnetic induction (magnetic flux density vector); c is the speed of light. The instantaneous pressure exerted on a cross section dx dy of a wall by incident radiation that penetrates the surface to a thickness 0 ~< z ~ e is given by

df

P i -- d x dy

_

f0

(pfF_.-]- c - l j • ~ ) dz =-

f0

I dz.

(S.5.1)

The average pressure exerted over one period T of the incident radiation is then found to be

P -- -~ l f07- P i d t -- -~ lfoT-foe (lOf g -I- C- 1 J • B) dz dt =

lfo fo

T

(5.5.2)

Idzdt.

Now use the Maxwell relations V 9 E, - 4zrpf and V x J ~ - c -1 (OE,/Ot) - 4zr J / c to eliminate p f and J from Eq. (5.5.2). We rewrite the integrand I in the form I -- (4re)- 1 [V 9 E]E + [ ( 4 z r ) - 1V •

-

xj~.

(5.5.3)

+ g x [cV x g],

(5.5.4)

The last term on the fight is then recast as

OE

Ot

x B-

a(g x B)

Ot

-g

x

oB

8t

=

a(g x t3)

8t

where the third Maxwell relation V x g - - c - 1(0 J~/0 t) has been used on the fight. As the next step insert (5.5.4) into (5.5.3): I -- (4zr)- 1 [V 9 E, IE, - (4yr c)-1

-

(4zr)- 1g

0 (~ x t3) 0t

X

[V

X g]

_ (4Jr)- 1[B x (V x B)].

(5.5.5a)

It is expedient to recast the above as I -- E~ - (4yrc) -1

0(E x B)

Ot

+ B~.

(5.5.5b)

Next, substitute (5.5.5) into (5.5.2); without loss of generality we assume the incident radiation to propagate along the z axis. For monochromatic radiation along z, SkBl varies with time as cos(o)kt)cos(o)tt). When the time derivative is taken and the integration over time is performed over one period, the central term

322

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

averages to zero. One thus is left with the following expression for the average z-component of the radiation pressure:

P [z - -T

(Eckz + Bg~z) dz dt.

(5.5.6)

From the definition of E e one obtains by straightforward manipulations

F

I

E~z -- (4zr)-lgz L 0x + - ~ y + -~-z .] - (4zr

-t-(47r)-lgY[ OEzOy

)_, gx [ Oz

Ox

OgY '

(5.5.7a)

which may be recast as

Eg)z_ (8~)-10(E2 - Oz E2 - E2) +

(4zr) - 1

[ O(ExEz)+ O(EyEz)] . Ox

Oy

(5.5.7b)

An analogous procedure applies to B~. Here we also introduce the Maxwell relation V 9 13 = 0 as a trick to expand on the definition for B~ in Eq. (5.5.5b); namely, we set

(5.5.8)

B4~ - (4zr) -1 (V 9 /~)J~ - (4rr) -1J~ • (V • J~),

which has precisely the same mathematical form as does EO. Therefore we can immediately write

Bdpz-- (87r) - 1 0 ( ~ 2

2

-- 132 _ ~ y )

Oz

[

+ (4zr)

]

-10(]~x~z)

+

Ox

O(]~y]~z) . Oy

(5.5.9)

The next step involves substitution of Eqs. (5.5.7b) and (5.5.9) into Eq. (5.5.6). We omit all terms such as Byl3z ~ 182 cos(kx - cokt - q)) cos(/z - cott - 99) since integration over one period of the partial derivatives in (5.5.6) yields a null result. This reduces Eq. (5.5.6) to 2

Plz-

T1 (87r)-1 fO 7- dt

2

2

2

fo e dz O(~2--~2--~yq-]~Z--]~x--~y) Oz

.

(5.5.10a)

In the integration over z we note that, by definition, the electric and magnetic field vectors vanish at z - e and have their maximum impact values at the surface of the wall, z - 0 . Hence we find that P [z

1

~(8Jr)

~

1

f0

dt ( - C 2 + C2 + E2 - 132 + 132 + 13y). 2

(5.5.10b)

THERMODYNAMIC CHARACTERIZATION OF ELECTROMAGNETIC RADIATION

323

In the time integration over one period we replace a term such as (1/7-) x fo7- dt E 2 by its average, (C2), and we subsequently set (E 2) - (C2) + (g 2) + (E2), and similarly for (/32). This leads to the result

PIz -(gYt')-1 [(~2) -

2(E2) + (/32) - 2(/32)] .

(5.5.10c)

At equilibrium we assume that the radiation is completely isotropic. With no distinction in direction the average values, all the components of ~2 are the same, so that (C2) -- (E2) - (E~) - (E2)/3, and similarly for the components of (/32). This leads to the final expression

el z -

1 3[ (8~)-1 ((~2)-Jw (~2))]

U = -~-.

(5.5.11)

As is well established, the quantity in square brackets in the above equation represents the energy density of an electromagnetic field in vacuo, designated here as u. This establishes the expression on the right. The result P lz = u/3 will be used below.

5.6

Thermodynamic Characterization of Electromagnetic Radiation

Here we discuss the thermodynamic description of electromagnetic radiation in equilibrium with the walls of an evacuated cavity that contains a tiny porthole for experimental observations. If the walls are opaque and form an ideal black body the nature of the radiation is independent of the quality of the walls and depends solely on the temperature. To prove this assertion, consider two cavities, made of different materials, which are connected through a tube that permits radiation to be passed back and forth. Let the cavities be initially at the same temperature and assume that by virtue of its superior composition cavity B is capable of "attracting" more radiation than cavity A. Then the uneven flow of radiation would cause the walls of chamber B to receive additional energy and to heat up. This rise in temperature causes an increase in the transfer of energy to the surroundings. Ultimately a steady state sets whereby the increase in its temperature is offset by the increasing rate of energy transfer to the external universe. In short, heat would flow spontaneously from a cooler to a warmer body, in contravention to the corollary to the Second Law. Thus, the radiation intensity can depend only on the temperature T of the walls; for, it is an experimental fact that as T is raised the nature of the radiation inside the cavity is observed to change. The thermodynamic characterization of this effect is actually rather limited; one must turn to quantum and statistical mechanics for a full description, but certain predictions are within the realm of thermodynamics.

324

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

<

-...S/ o ,/7

Fig. 5.6.1. Figure illustrating the radiation inside a cylinder of cross section d A and length c dt impinging on a wall in time dt and exerting a force d F • perpendicular to the plane.

5.6.1

Pressure of Impinging Radiation

Electromagnetic radiation impinging on a wall exerts a pressure Pc that will be determined via a classical argument. Referring to Fig. 5.6.1, consider radiation contained in a cylinder of length c d t and cross section d A whose long axis is at an angle 0 with respect to the axis perpendicular to the wall. In the time dt all electromagnetic radiation traveling with velocity c will emerge from the cylinder to hit the slab. The corresponding incident momentum is given by dA pc dt, where p is the momentum density vector of the radiation. If that radiation is totally absorbed by the surface then the corresponding force exerted on the plate is found from Newton's Law as the time rate of change of momentum: (5.6.1)

dF(O,~)=dAp(O,r

where ~b is the azimuthal angle about the perpendicular axis, taken to coincide with the z direction. As long as the radiation is isotropic one obtains the above contribution at every angle 0 and ~b. In integrating first over ~b, the F components parallel to the surface sum to zero, and only those along the z axis are additive. The perpendicular component is thus given by

dF•

= d A p(O, ~b)ccos O.

(5.6.2)

The cross sectional area on the surface, d A• being larger than that of the cylinder, is specified as d A = d A• cos O. Thus, the pressure exerted by the radiation may be written as

Pr(O) --

dFx dA•

= p(r

cos 2 0,

(5.6.3)

THERMODYNAMIC CHARACTERIZATION OF ELECTROMAGNETIC RADIATION

325

which is to be attributed to all radiation incident on the slab at an angle 0. When averaging over all angles we obtain fir

f Jrl22re sin 0 dO --

pc

dO

f

2zr COS 2 0 sin 0 dO.

(5.6.4)

dO

On evaluating the integrals one obtains

1

1

fir -- -~ p c -

-~u .

(5.6.5)

On the fight we have introduced Einstein's formulation for the relation between energy density u and m o m e n t u m density p of electromagnetic radiation, namely p c -- u. A classical derivation of the above expression was furnished in the preceding section.

5.6.2

Thermodynamics of Electrodynamic Radiation

For the thermodynamic analysis we begin with the standard expression m

T d S - d U + Pr d V , which may be rewritten as

(0s) rSV

--

-[- fir-

(5.6.6a)

(5.6.6b)

On invoking the appropriate Maxwell relation we find

-ST]v

--

5V r

+ Pr.

(5.6.6c)

Recall that at fixed T, u -- u ( T ) is constant and Pr - u/3. Moreover, U = u V; then 1TdU (OuV) - ~ -3 dT OV

1 4 + - u -- - u 7~ 3 3 '

(5.6.7)

which simplifies to

du

T~ -- 4u. dT

(5.6.8)

The solution of this differential equation is given by In u - 4 In T + In cr, where cr is an arbitrary constant. This may be rewritten in the form u -- o"T 4.

(5.6.9)

326

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

The above relation is known as the S t e f a n - B o l t z m a n n L a w (1879); the constant is empirically determined to be a - 5.67 • 10 -8 W/K4m 2. One should note the very strong dependence of the energy density of the electromagnetic radiation on the temperature. We obtain the entropy for electromagnetic radiation from the expression m

dU + PrdV dS -

On setting

uV -

U - a T4 V

(5.6.10)

.

T

and/Sr - u/3 we obtain V d u + (Pr + u) d V

dS -

,

(5.6.1 la)

V) ,

(5.6.1 lb)

T

so that in view of (5.6.9), dS--

4o" (3 VT 2 d T 5

T 3d

+

which, at fixed temperature T, becomes (OS/O V _= s) 4 T3 , 3

4 3

s -- - a

which is the desired relation. Other functions of state are obtained by writing A-

U -

(5.6.12)

S -- -aT3V,

1

TS-

--a

3

U -- a V (3S/4a

T4 V

V) 2/3

(5.6.13)

and m

G--A

+ PrV--O.

(5.6.14)

One should note that the Gibbs free energy of electromagnetic radiation vanishes identically. This apparently peculiar result is correlated with the fact that electromagnetic radiation consists of energy packets---quanta. These discrete units, called photons, are present at densities that depend on the intensity of the radiation according to the Planck distribution law, given by [ e x p ( h v / k B T ) - 1] -1. Here h represents Planck's constant, v is the frequency of the radiation, and kB is Boltzmann's constant. The quanta, h v, are not real particles, since their density can be changed through temperature adjustments. Thus, one cannot allow terms of the type/Zphdnph to appear in the expression for d G ; nph is not independent of T. To guarantee consistency in the theory one must therefore require that/Zph vanish, which in turn invokes the requirement G - 0 as applied to electromagnetic radiation.

EFFECTS OF ELECTRIC FIELDS ON THERMODYNAMIC PROPERTIES OF MATTER

327

EXERCISES

5.4.1. Derive an expression for the heat capacity of electromagnetic radiation and determine its value for unit volume at 1 K, 300 K, 30,000 K. Compare these values with those of a monatomic ideal gas ate room temperature. At what temperature do these quantities become equal? 5.4.2. In the early history of the cosmos temperatures of 106 K were common. What was the radiation pressure at that time? 5.4.3. Contrast the relation Pr = u/3 with a comparable expression for a monatomic gas.

5.7

Effects of Electric Fields on Thermodynamic Properties of Matter

The treatment of thermodynamic characteristics of matter subjected to electromagnetic fields requires considerable care. The next few Sections are devoted to a discussion of this topic; for a more complete and systematic exposition the reader is referred to a review article. 1 In this section we deal with several preliminaries that will be generalized later. We begin with the formulation for the element of work exhibited in Eq. (1.6.8); in the absence of magnetic fields

--

f d3r s

9 ds

- f

T:' . ds

(5.7.~)

Here E0 is the applied electric field prior to insertion of the sample, and "P is the polarization vector. The system is assumed to be totally surrounded by empty space. In that event E0 is the applied electric field in free space arising from a distribution of generating charges placed in confines beyond that space. The disposition and magnitude of these charges is changeable and so designed that the electric field remains unaltered when a test system is immersed in the electric field. Now, 7:' - - 0 in the vacuum outside the sample; hence the second integration may be limited to the confines V of the system, whereas the first integral must be taken over the entire space over which E0 extends. This term obviously relates to the work of establishing the electric field in vacuum, which normally is of no immediate interest; hence, we frequently drop this term and refer all thermodynamic properties of materials to the work involved in placing the sample in the field. Also, we introduce the constitutive relation 7:' = oe0g0 that specifies the polarization dependence of an isotropic material on the applied electric field; tensorial notation is required in case the material is anisotropic. The quantity oe0 is termed the polarizability of the material.

5.7.1

Isothermal Responses

We begin with the First Law of Thermodynamics. Including the energy required to establish the electric field in vacuo the differential form of the total energy of

328

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

the system may be written as

dU - T dS - P dV + (4yr)-l f d3r s 9 dEo - fv d3r T:' . d~.o.

(5.7.2)

Next, we switch directly to the differential form of the Helmholtz free energy,

A=U-TS: dA--SdT-

PdV +(4Jr)-~ f d3rEo.dEo- fvd3rT'.dF.o.

(5.7.3)

Clearly, A is here considered to be a function of T, V, and E0. Now let V and T remain fixed. Then

dAl ,

- (4yr)-l f d3rEo.dEo- fvd3rT:'.dgo

(5.7.4) T,V

We next define the Helmholtz free energy density a through the relation A -fv d3r a; comparison with (5.7.4) establishes that

dalT, v--

(4zr)- 1E0- dE0 - 7:'.

d~r01 T, v"

(5.7.5)

At this point we reintroduce the constitutive relation 7:' = ct0(T, V)E0; and henceforth assume that the polarizability does not depend on the electric field. Phenomena such as ferroelectricity or hysteresis are thereby excluded from consideration. Then Eq. (5.7.5) can be readily integrated with respect to Eo; one obtains C~ a(T, V, E,o) - a(T, V, O) -- 8yr

c~o(T,V) ~

~ = 8yr

Eo" T' ----~"

(5.7.6)

As deduced from (5.7.1), the first term in the middle represents the Helmholtz free energy density of the free space associated with the field E0; the remaining term exhibits the response of the matter to the field, in terms of its Helmholtz free energy density. The entropy density is found from the relation S

Oa OT v,~0

(5.7.7)

which, on differentiation of Eq. (5.7.6) with respect to T, leads to

s(T, V, go)- s(T, V, O)-

\ OT / v

_

2

0P) 9 - ~ - V"

(5.7.8)

Here one should note (Ooto/OT)v and (OT'/OT)v for all materials are negative, because a rise in temperature tends to destroy the degree of order implicit in the

EFFECTS OF ELECTRIC FIELDS ON THERMODYNAMIC PROPERTIES OF MATTER

329

polarization of the dipoles in the field. Therefore, the imposition of an electric field always lowers the entropy of the material: the field produces at least a partial alignment of the elementary dipoles in the system and thereby increases the degree of order. Note further that in vacuo ~o and "7' both vanish; empty space permeated by an electric field does not carry any entropy. The energy density is obtained from the relation e - a -t- T s: e(T, V , C . o ) - e ( T ,

V,O) - g2 87r

g2[oto-T(OOt~ 2 \ OT

_ E~ _T' [1 _ T( olna~

v

(5.7.9)

Eq. (5.7.6) is sometimes labeled as the energy density of the system in an electric field, but in light of Eq. (5.7.9) this is clearly an erroneous statement. The radiation pressure exerted by the field may be determined from the relation Pr = - ( O A / O V)T. The partial differentiation process called for here is not straightforward because the variable volume occurs in the limit of integration in Eq. (5.7.4). For the present we circumvent this difficulty by assuming that 79 (not to be confused with P) is homogeneous throughout the sample, so that it varies solely with the temperature and density of the sample. This cannot be strictly correct, because, minimally, surface regions differ in their polarization from the bulk; in fact, in electrodynamic theory surface regions are treated as separate entities. A more sophisticated treatment for addressing this matter is developed later. For now we rewrite Eq. (5.7.6) in the approximate form A ( T , V, EO) - A ( T , V, O) -- (4yr) -1

f

d3r Eo" dEo - V a o ( T , V)E-~~. (5.7.10)

The radiation pressure is obtained by differentiation as P r - P (O) + oto( T, V)

-k- V

-~

T

(5.7.11)

where the first term on the fight represents the pressure in the absence of the field. 5.7.2

Adiabatic Response

We have so far considered isothermal conditions. To handle the adiabatic response we consider S, V, E0 to be the fundamental variables that relate to the energy U, whose differential form is specified in Eq. (5.7.2). Under isochoric and adiabatic conditions and with d U - f d r 3 duls, v we find that duls, v - (4zr)-1Eo. dEo - 79. dEo[s ' v

= (4yr) - l E o . dEo - ao(T, V ) E o . dEols, v.

(5.7.12)

330

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

From here on we dispense with vectorial notation, on the assumption that the material is isotropic. Note that on integrating over the field at constant entropy the temperature of the system necessarily increases, so that, in contrast to the earlier treatment, ~0(T, V) now changes during integration. In fact, since the entropy has the functional dependence S = S(T, V, g0) we must first invert this relation to specify T -- T (S, V, g0), so that the polarizability is written out as oto(S(T, V, go), V). We then integrate (5.7.12) by parts to find

u(S, v, &) - u(S, v, o) =

(s, v,

8Jr

+ fv

(

) s, vd&. (5.7.13)

The entropy differential is given by

dS-

( ) V,Eo aS

dT +

-~

dV+

dgo.

T,&

(5.7.14)

V,T

We now specialize to isochoric, isentropic conditions, d V = d S = 0 and we replace the first partial derivative with C e / T , where Ce = Cv is the corresponding heat capacity. We use lower case letters to denote the densities and we set Ce - f v Ce d3r. Then the above reduces to Ce d T -T s,v

Os

dgo V,T

L

S,V

--Eo\

OT ,1 S,

I

,

(5.7.15)

S,V

where Eq. (5.7.8) has been applied. The integration can be carried out after the dependence of Ce and of or0 on T has been inserted. This procedure furnishes the interrelation between the temperature and electric field under adiabatic conditions, T = T (V, go). Eq. (5.7.15) is actually deceptively simple; for, at constant S the temperature changes with the application of the electric field. Thus, one must rewrite oto(S(T, V), V, g0) so as to involve temperature as the variable and then introduce the quantity T = T (V, g0) wherever T occurs in or0(T, V) before proceeding. It is therefore easier to introduce this step in Eq. (5.7.12) prior to carrying out the integration. 5.7.3

Electric Field Effects at Constant Temperature and Pressure

When T and P are held fixed it is appropriate to return to the Gibbs free energy function analogous to Eq. (5.7.3), dG--SdT

+

VdP + (4Jr)-f d3rEod&- fv d3rT)dCo.

(5.7.16)

Assume constant P and T, and define dGe ~ d G - (47r) -1 f d3r godgo, so that the new differential relates solely to the materials properties. Then, introduce the

EFFECTS OF ELECTRIC FIELDS ON THERMODYNAMIC PROPERTIES OF MATTER

331

Gibbs free energy density g(r, T, P, go), so as to write the differential form as

dGelT'P -

[ 0 J~ d3rg(r,T,P, Eo)] dCo -~o (T, P,CO) T,P

=- f Jv

(T,P,go)

d 3r 7adgo

.

T,P

(5.7.17)

One must now proceed with caution because V in the limits of the integral is no longer constant. However, if we introduce the simplifying assumption that the integrands in the middle and on the right are independent of r, then Eq. (5.7.17) may be replaced by

['

-~o (V(T, P, Co)g(T, P, go))

]

T,P

dCo

= - V ( T , P, go)ao(r, v ( r , P, 8o))8odgolT, P,

(5.7.18)

so that on integration we obtain formally

V(T, P, Co)g(T, P, go) - V(T, P, O)g(T, P, O) -- G(T, P, go) - G(T, P, O) -- - f V(T, P, C0)ot0(r, V(T, P, C0))Co dC0 d

T,P

.

(5.7.19)

The integration over E0 on the right requires that we specify the variation of volume with applied electric field. In the event that we are allowed to ignore this dependence we find for linear materials

g(T, P 8o) - g(T, P, O) -- -oto(T, V(T, P)) C2 ' 2 '

(5.7.20)

which, in the present approximation, coincides with Eq. (5.7.6). However, if we wish to continue using Eq. (5.7.19) without further approximation we must determine how V and c~0 change with E0. This is accomplished by using Maxwell equations, as shown below. 5.7.4

Maxwell's Equations

Here we illustrate how Maxwell's equations may be utilized. From Eq. (5.7.16) we obtain

02Ge - - ( 0~~0) - 02Ge -O -- - [ f v d3tT) OgoOT T,P OTOEo OT (r, P,E0)

.

(5.7.21)

T,7a

We replace S by the entropy density s so as to write

~0

T,P

d3r s(r, T, V,

c0)J

(5.7.22)

332

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

The differentiations called for in the last two equations again involve integrals with variable limits. For simplicity, we now assume that the integrands s and 79 are uniform throughout the sample; then on equating (5.7.21) and (5.7.22) we may write

OgO T,P

8T

p,&

which, in the approximations introduced so far, may be integrated to yield (for linear materials)

[

S(T, P , Eo) - S(T , P , O) -- oto(T , V(T, P))

-~

P +V(T,P)\oT}

P

9

(5.7.24) Once again we simplify by assuming 79 to be independent of location r within the sample. With this simplification a second relation of interest is derived from Eq. (5.7.17) by the cross differentiation

~2Ge - (~~~0) O8oOP -- -

~

-

~2Ge

T,P

OPO8o

(V (T,

P, 8o)T'(T, P, 8o)

.

(5.7.25)

T,80 Integration of the simplified expression leads to V (T, P, 8o) - V (T, P, 0) -- -

f

V (T, P, 8o) -ffff T,Eo

V ) T,8oE~176 - f o t o ( T , V ( T , P , 8o)) ( O-ff-fi (5.7.26) Actually, Eq. (5.7.26) is an integro-differential equation since the unknown V appears on the left-hand side as well as under the integral sign; techniques for solving such an equation are available in standard mathematical treatises. In the present case, however, an iterative method should work well because the changes of V with g0 are likely to be small. A more systematic approach to the entire subject will be presented next. REFERENCE AND EXERCISES

5.7.1. J.M. Honig and L.L. Van Zandt, J. Franklin Inst. 323 (1987) 297. 5.7.2. Derive expressions showing how the heat capacity of a material at constant pressure and constant volume changes with application of an electric field.

SYSTEMATIZATION OF ELECTROMAGNETIC FIELD EFFECTS IN THERMODYNAMICS

333

5.7.3. Establish Maxwell relations based on Eqs. (5.7.3) and (5.7.2) and discuss their utility. 5.7.4. Compare the two expressions for entropy density variations with an applied electric field and discuss the implications of your findings.

5.8

Systematization of Electromagnetic Field Effects in Thermodynamics

We now generalize and systematize the discussion of the previous section by including the thermodynamic properties of magnetic fields. The starting point, as usual, is the First Law; it now contains the elements of work associated with both electric and magnetic fields, as specified in Section 1.6. At the outset we discard the contributions Ez/8zc and 7-/~/87r concerning the electromagnetic field effects in free space. The remaining contributions involve the integral shown below in the equation for the differential of the total energy, namely

dU' - T dS - P dV + fv d3r (-79" dEo + 7-s d,A/l).

(5.8.1)

This particular formulation includes the magnetization ,Ad as the independent field variable. The above equation is perfectly acceptable; however, the magnetization is not subject to direct experimental control. The experimentalist would greatly prefer a version in which 7-t0, the applied magnetic field prior to insertion of the sample, is the appropriate independent variable; this quantity can be adjusted at will. To achieve the required change we introduce a Legendre transformation as follows:

d U = d U t - d ( f v d3ra~vl 9"l-to/V).

(5.8.2)

Here we encounter the persistent problem of having to deal with differentials or derivatives of integrals with variable limits. Procedures for dealing with this problem were detailed in Section 1.3. On applying this method, using Eq. (1.3.36), one may rewrite the above as

dU-dU'-fvd3r(,Ad.dT-to+7-s163 (5.8.3) In the last term the integral is a multiplier of d V and therefore represents a contribution to the overall pressure that is generated by the magnetic field. We thus introduce the symbol

Pm =- P + f v d 3r (,AA . 7"(,0/V)

(5.8.4)

334

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

for the generalized pressure appropriate to the present formulation. The differential of the energy involving the independent variables S, V, E0, and 7"r therefore takes the form d U - T d S - Pm d V - f v d3r (79. dE, o + .A,4 . d7"r

(5.8.5a)

Application of the standard Legendre transformations then yields the remaining functions of state: d H - T d S + V dPm - f v d3r (79. dE, o + .A4 . d7-s

(5.8.5b)

d A - - S d T - Pm d V - f v d 3r (79. d E,o + .A/I . d7-s

(5.8.5c)

d G - - S d T + V dPm - / v

(5.8.5~)

d3r ('p " dEo + . M . dT"~o).

The above equations represent differential forms properly adapted to describe electromagnetic phenomena in terms of applied fields. In what follows we introduce constitutive relations, assuming that 79 and .A4 are collinear with E0 and 7% respectively, namely 7:' = o~oEo and

.hal = Xo7"r

(5.8.6)

Here c~0(T, V) and x0(T, V) represent respectively the modified scalar electric polarizability and magnetic susceptibility. 1 As before, we assume that these parameters do not depend on the intensity of the applied electromagnetic field; such materials are said to be linear in their response to electromagnetic fields. Phenomena such as ferroelectricity and ferromagnetism are thus excluded from the current formulations. Clearly, when E0 and 7-r are used as independent variables all thermodynamic functions that earlier involved the pressure P must now be reformulated in terms of the generalized pressure Pm. In particular, under equilibrium conditions where d G - O , it is not just the mechanical pressure, but the magnetic component as well, that must be uniform throughout the sample. In examining the fundamental relations (5.8.5) it should be evident that another twelve functions of state may be defined, in which U, H, A, G are specified in terms of (79, 7-s or (E0, .h,4), or (79, .A4) as the independent electromagnetic variables. None of these sets is less fundamental than are the functions (5.8.5); which set is adopted depends solely on the usefulness of the expressions in analyzing theoretical or experimental work. We adopted Eqs. (5.8.5) because g0 and 7-r are subject to direct experimental control. Henceforth we drop the vector notation, which limits further derivations to the case of isotropic materials.

SYSTEMATIZATION OF ELECTROMAGNETIC FIELD EFFECTS IN THERMODYNAMICS

5.8.1

335

First Order Partial Derivatives

We begin a systematic exploration of the thermodynamic functions by partial differentiation of Eqs. (5.8.5) as follows:

_

(OG) -

~

_ Pm---

~

, Pm,gO,~O

(OU)

T,go,7-[O

--

-~

(5.8.7c,d) S, go , ~ o '

OH) OPm S,Eo,no 9

V --

ONto

T--

-~

T,Co,~o

V,go,~o

(5.8.7a,b)

--~

(5.8.7e,f)

Pm , gO, 7-[O'

P--

(0.) ----

-~0

0~o

S , V, ~o

T, V,eo

__(OU) 07% s, V,eo

--

~0

07-[0

(5.8.7i,j,k,1) S, Pm , 7-{.0

T, Pm,CO

0~o S,Pm,CO

(5.8.7m,n,o,p)

Here we defined/5 _ .Iv d3r 7~ and M -- fv d3r A4. Eqs. (5.8.7a-h) are straightforward extension of the results cited in Section 1.13. The remaining equations serve as starting points that indicate how the electric and magnetic polarizations change in applied electric or magnetic field. These relations are particularly useful if the Helmholtz or Gibbs free energies are known (e.g., from theoretical considerations) in their dependence on these fields. 5.8.2

Maxwell Relations

Equations (5.8.5) may be used to obtain twenty four Maxwell relations by cross differentiation; these are listed in Table 5.8.1. Several of these arise as trivial modifications of those specified in Section 1.13. The new expressions involve partial derivatives of either fv d3r 79 or of fv d3r 3A with respect to independent variables. A number of the interrelations are useful for starting further derivations; they show, for example, how Pm varies with g0 or ~0 under a variety of fixed

GO GO

r

"I" m 2O

Table 5.8.1 Maxwell relations based on the (E 0, H 0) set of thermodynamic functions

From Eq. (5.8.5d) (a) (OS/OEo)T, Pm,Ho = (OP/OT)Pm,E__o,Ho (b) (OV/OEo)T, Pm,H__o=--(OP/OPm)T,E__o,Ho (c) (OP/OHo)r, Pm,E_E_o= (OM/OEo)r, Pm,H_H_o (d) (OS/OPm)T,Eo,Ho=--(OV/OT)Pm,Eo,Ho (e)

(OS/OHo)T, Pm,E__0 -= (01QI/OT)Pm,E__o,H0

(f)

(OV/OHo)T, Pm,Eo=--(OM/OPm)T,E__o,H_H_H_H_~

From Eq. (5.8.5c) (g) (OS/OEo)T,V,H__o = (OP/OT)v,E__o,N o (h) (OPm/OEo)T,V,H_H_o = (OP/OV)r, Eo,H_H_o (i) (OP/OHo)r,V,Ko = (OM/OEo)r,V,H__o (j) (OS/OV)T,E__o,Ho = (OPm/OT)v,E__o,Ho (k) (OS/OHo)T,V,E_o = (01fi/OT)V, e o,H__.o (1) (OPm/OHo)T,V,E__o = (01~/OV)T,E__o,Ho

0

From Eq. (5.8.5b)

(m) (n) (o) (p) (q) (r)

Z >

(0 T~ 0 E O)S, Pm,Ho = --(0 ['/0 S) Pm,E__o,tto (0 V/ OEo) S, Pm,Ho = --(0 ['/0 Pm ) S,E__o,H__.o (OP/OHo)S, Pm,E o = (01V1/OEo)S, Pm,H____o (OT/O em)s,E__o,tI o = (O V / O S ) Pm,E__o,Ho (o r / OHo) S, PmEo -- --( 01~ / OS) PmE , ___oH , _H_H_H_~ (a V / OHO) S, Pm,E o = -(aM/a Pm ) S,E_.o,H__o

:D

0

m :13 -t m s 0-n

From Eq. (5.8.5a)

(s) (t) (u) (v) (w) (x)

0

(0 T~ 0 EO) S, V,H o = --(0 [' / 0 S) V,E__o,Ho (OPm/OEo)s,V,H o = ( O P / O V ) s Eo,H__o _

~

(Ofi/OHo)s,V,Eo = (OM/OEo)s,V,H___o ( a r / a V)S,~,H_o = - ( a Pm/aS)v,E__o,H___o ( a r / a Ho)s, V,Eo = --( a i(4/ a S) v,E.E_o,Ho (O Pm / OHo) s, V,Eo = (O I1/I/ OV) S, E o,H__o

m :13 r-" m

Z

m X m

::13

Note: t5 = f v d3r~p, ~ = f g d 3 r M , Pm ~ P + f v d3vMT-lO/V" Underlined quantities represent fixed electromagnetic variables. E0 and H0 are applied electric and magnetic fields prior to insertion of sample.

Z r-

-0 -0 rm

0 -I"1 m i'0 co

SYSTEMATIZATION OF ELECTROMAGNETIC FIELD EFFECTS IN THERMODYNAMICS

337

conditions (lines h, l, t, x), or how the entropy depends on electromagnetic fields (lines a, e, g, k). Still others pertain to interrelation between electric or magnetic polarizations in applied fields under various constraints (lines c, i, o, u). Other interrelations specify electrostrictive or magnetostrictive effects (lines b, f, n, r). We shall later reexamine some of the above expressions. The reader may readily construct three other sets of 24 Maxwell relations, for a grand total of 96, that are based on the differential forms for the various thermodynamic functions involving (79, 7-/0) or (s A/l) or (79, A/I) as the electromagnetic variables. These determinations are left as exercises. 5.8.3

Thermodynamic Equations of State

Further useful information is obtained from Eq. (5.8.5a) in strict analogy to the method used in Section 1.13. We consider the entropy to be a function of T, V, g0, and 7-/o; accordingly, we write

( ) d~o - T (o,) -~

dV + dU--(O~T)v,go,~o dT + -OU ~ T,go,~o

( OU)

at-

0']/0

+

T ~

T, V,go

T,Eo,~o

dgo T,V,~o dT

V,s162

- Pm]dV -fvd3rTg]dgo

+ T 07-[o T,V,go

-fvd3r.hd]dT-{o.

(5.8.8)

On matching coefficients and introducing Maxwell relations j, g, k from Table 5.8.1 one obtains

(or) -~

OS ) - Cv,Eo,7%, = T -~ V,Eo,~o

(5.8.9a)

--

(5.8.9b)

T, No, ~ o

-- P m ,

OT V,~o,~o

(075)v,E0,~0 _fvd3r7) '

(5.8.9c)

(O,g4)

(5.8.9d)

= T -~

-~o T,V,s

=T

-~

V,Eo,~o

_fvd3rAd"

338

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

Precisely similar methods are applied to determine the thermodynamic equations of state for the enthalpy. One obtains

( ) ( OH

= T

--~

Pm , g.O, 7-(.O

-~m

T, EO, ~O

=

OH

0"7-{'0

T, Pm , ,f.O

-~

(5.8.10a)

=-- C Pm,gO,7-[o ,

Pm , gO , 7-{O

(5.8.10b)

+ v,

~ Pm , s , 7-[O

fv =r (a.M)~ _fvd3rM.

= T T, Pm , 7-f.O

( )all

(") -~

-

d3 r 79,

(5.8.10c)

Pm , g.O, 7%

(5.8.10d)

Pm , gO , 7-LO

In principle the above partial differential equations may be integrated to find how U and H change with their respective variables of integration. Eqs. (5.8.9a,b) and (5.8.10a,b) are handled as described in Section 1.13. The remaining relations must be approached carefully. Assuming linearity of response we write or0 = ot0(T, V) and X0 = x0(T, V), thereby neglecting their electromagnetic field dependence. Since the O/OT operation in Eqs. (5.8.9c,d) is to be carried out at fixed V there is no problem in the interchange involving 0/0 T and the operation f v d3r" Also, the integration over g0 may be carried out independently of that over space. From Eq. (5.8.9c) we therefore obtain

U(T, V, go)- U(T, V, O)-

iv{

I

d3r T \ OT I v - a o ( T , V) ,

(5.8.1 la)

and correspondingly, -

xo(T,

-~/ v

V)}. (5.8.11b)

Eq. (5.8.1 la) duplicates Eq. (5.7.10); Eq. (5.8.1 lb) is its obvious magnetic analog. To provide an explicit example we introduce Curie's Law through the temperature dependence (condition b~) ol0 = A e / T , XO = A h / T ; where necessary we also set (OV/OT)Pm,EO,~o ~ (OV/OT)p = ocV. Here, oc (with dimensions of reciprocal temperature, set in Roman lettering) represents the isobaric expansion coefficient, not to be confused with or0; Ae a s well as Ah are parameters. We ignore any possible temperature variation of o~. Where these conditions apply Eqs. (5.8.11) simplify to U-Uo---~

Ae V~ 2

T

and

A h VJ-[.g

U - U o - - ~ . T

(5.8.11c)

SYSTEMATIZATION OF ELECTROMAGNETIC FIELD EFFECTS IN THERMODYNAMICS

339

One should note the dependence of U on temperature and field variables for this particular case. The corresponding procedure involving Eqs. (5.8.10c,d) is more complicated because now it is Pm rather than V which is to be held fixed; moreover or0 and X0 depend on V rather than on Pm directly. One must therefore invoke Eq. (1.3.6) and the procedure associated with Eq. (1.3.34) to handle the mathematical niceties; however, these are perfectly straightforward though somewhat messy in detail. We find that

H(T, Pm,EO)- H(T, Pm,O)-- E~2[TI- fvd3rao(T, V) ],

(5.8.12a)

in which

I==_fvd3r[(8~~ (OV)

OT

v) )

v,8o,~o {uo(T, V(T, Pm, CO,~O)) +

OV

V )}1

T,Co,~o " (5.8.12b)

This is clearly more complicated than having to deal with the function U. An analogous relation holds for the dependence of H on ~0; this requires the replacement of g0 and or0 respectively by 7-/o and X0. The above equation is greatly simplified by introducing Curie's Law ~; we obtain H

-

Ho

H-H~

=

V Ae 0% 2 [or T - 2]

T 2 VTAh g22 [~T -

2]

(5.8.12c)

which should be compared with Eq. (5.8.11 c); oe is the isobaric expansion coefficient. 5.8.4

Electromagnetic Effects Under Isothermal-Isochoric Conditions

Variations of several thermodynamic functions with electromagnetic fields under conditions of constant T and V are readily handled. Eq. (5.8.5c) is rewritten through the string of relations in the form (no problem arises in the interchange between integration and differentiation because V is kept fixed):

dA[T,V,~~ = d fv d3ra T, V,7-[o

-fvd3rda]

- - i v d3r ao(T, V)EodCo

--fvd3r79dgo]

T, V, 7-[o

T, V, 7-[o

(5.8.~3) T,V,~o

340

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

which specifies the Helmholtz free energy density differential as dalT, V,7% = -c~o(T, V)Co dEoIT, v,7%, whose integration yields V)C-~~ 2 .

(5.8.14a)

a(T, V, ~o) - a(T, V, O) -- - g o ( T , V) 7~~.

(5.8.14b)

a(T, V, g o ) - a(T, V, O ) - - u o ( T ,

In precisely the same manner it may be shown that

On adoption of Curie's Law b~ we obtain Ae~ - a(T, V, O) . . . . 2T

a(T, V, s

(5.8.14c) '

Ah T-[2

a(T, V, 7-[0) - a(T, V, O) --

(5.8.14d)

2T

We next determine the entropy density via

S --

d 3r s - -

-~

V,Co,7-to V,Co,7-/o

--fvd3r( ~ V,Eo,~o -if-T)

(5.8.15a)

or

S

m

(5.8.15b)

m

Using (5.8.14a) one readily finds

s(T, V, go)- s(T, V, 0 ) -

\ 0T 1 v

(5.8.16a)

In the same manner s(T, V, 7-[0) - s(T, V, O) -

-~

v

(5.8.16b)

Again adopting condition b~ the above expressions reduce to

s-so-

AeC 2 2T 2 ,

s-so-

AhT-[,~ 2T 2 .

(5.8.16c)

SYSTEMATIZATION OF ELECTROMAGNETIC FIELD EFFECTS IN THERMODYNAMICS

341

This approximation should be compared with Eq. (5.8.1 l c). It has already been mentioned that the entropy of materials is less in an electromagnetic field than in the absence of such a field. Another quantity of interest based on Eq. (5.8.7c) is the generalized pressure; here one must be careful about the variable limit of integration. Using the methodology of Section 1.13 we obtain

Pm - _ ( OA

O-V)T,EO,~O

--fvd3r[v+(

T,C0,~0 Oa

(5.8.17a)

Substitution from (5.8.14b) then yields the expression of interest. For an explicit relation, assume that X0 is independent of position within the material; one then finds

Pro--P+

{

( xo)

xo(T, V) + V - ~

(5.8.17b)

T

2

If we adopt condition ~ the partial derivative drops out and

Pm

Ah~

-

-

P + ~.

(5.8.17c)

2T

This expression shows the magnitude of the magnetic contribution to the mechanical pressure under the indicated conditions. In Eq. (5.8.12) we already specified H in terms of T, Pm, ~0. The determination of G in the same variables is more involved: we base our derivation on Eq. (1.13.19) adapted to the present situation. This is actually an ordinary differential equation of standard form since all variables save T are fixed. Invoking Eq. (1.3.27) as the solution to the first order differential equation (1.13.19) one obtains the expression

G(T' Pm C~o)- G(T ' Pm' O) - -TE~ --2 I I

- fvd3r~176 V) }dT] ' (5.8.18a)

which looks deceptively simple: however, I is specified by Eq. (5.8.12b), so that the required integration over T is not necessarily trivial. But if again condition is assumed to hold then the above simplifies to

. . . .

G(T Pm CO)-G(T Pm O)-

-

E~VAe(1 2

-~

-

ol

)

.

(5.8.18b)

342

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

An analogous expression applies to the magnetic field dependence. These expressions should be compared with Eqs. (5.8.14c); throughout, the limitations imposed by the assumptions inherent in condition ~ should be clearly kept in mind. 5.8.5

Adiabatic Conditions

In Section 5.7 the adiabatic response of a system to an electric field was discussed in detail. We handle magnetic field effects, in the customary manner by rewriting Eq. (5.7.16), using (5.1.16b): TdT

I S, V

-- -

( os )

d~o

-

0~-~0 V,T

S,V

-~o(OX~ g-i-)

(5.8.19)

S,V

S,V

For further progress we revert to Eq. (5.8.1), set s = 0, and determine -

+

O~-~O T,V

r 7%

O~-~O T,V

.

(5.8.20)

O~-~O T,V

Next, introduce the Maxwell relation, line (k) of Table 5.8.1, as well as the energy density f v d3r u' =_ U'. We then obtain

()Ou ' -~0

T, V

-- T

(Oj~f) -~

V, ~O

-+- ~-~0

(O.A4)

O~-~O T , V

_0.

(5.8.21)

This expression is of intrinsic interest: for many classes of materials the magnetic moment is found to be an entire function of 7-/o/T, in which case, as is readily verified, (Ou'/OT-go)r,v = 0. Then u' and U' do not depend on 7-/0, in which case the heat capacity is a function of T and of V alone. Now introduce the defining relation U - U' - ,Iv d3r 7-loA/l, and differentiate with respect to T at constant V and 7-/o to obtain ; CV,~o -- CV,~o

- 7-{0

(5.8.22) V,7-/0

For a large class of materials it is found empirically that Cv,~o' - A 1/ T2, and that ,All = A 2 ~ o / T is an entire function of 7-/o/T; here A1 and A2 are independent of 7-/o and T. On introducing these results in (5.8.22) we obtain A1

cV,~o -- ~

+ A2(7-~0/T

)2.

(5.8.23)

This example shows how the heat capacity varies with temperature in this particular instance.

ADIABATIC DEMAGNETIZATION AND TRANSITIONS TO SUPERCONDUCTIVITY

343

COMMENTS AND QUERIES 5.8.1. We use the term modified electric polarizability and modified magnetic susceptibility because in the standard literature these quantities relate 79 to E, rather than to g0, and M to ~ , rather than to ~0. 5.8.2. Set up twelve additional functions of state whose independent variables involve (79, A4), (79, ~0), (.AA,E0). Discuss their utility. Derive Maxwell relations based on these state functions and identify those that you deem useful. 5.8.3. Derive relations for the physical properties of materials whose magnetization follows the Curie-Weiss Law: A4 = ,,427-/o/(T + 69), where 69 is a parameter, called the Weiss constant. 5.8.4. Show that when Curie's Law holds (a) U and CA,[ are functions of T alone, (b) S - So + V2.A/[2/2C ', (c) C~o - CA/[ -- V2.A/[2/2C ', also, that (d) (OCT-to~ 07-(O)T : 2C'7-~0/T 2. Here C' is the Curie constant. 5.8.5. Show that (OU/O~O)V,T = T[(O(VM)/OT)v,7% + (O(VM)7%/O~-[O)V,T]. Derive an analogous relation for (OH/OT-[O)Pm,T. 5.8.6. Derive an expression relating the constant volume heat capacity at constant 7-[0 to that at constant A4.

5.9

Adiabatic Demagnetization and Transitions to Superconductivity

We discuss the further manipulation of Eq. (5.8.19) via (5.8.23) in the context of adiabatic demagnetization. The experiment consists in allowing a material to equilibrate with surroundings held in a liquid helium bath maintained at a low vapor pressure corresponding to roughly 0.3 K. The sample is subjected to an intense magnetic field and is then adiabatically isolated, after which the external field is switched off. As repeatedly emphasized, any change of state under such conditions incurs a change of temperature, whose difference is determined by Eqs. (6.8.19) and (6.8.23). With M = A 1 7 - / 0 / T , as before, we find that (dT)__

_--

s,v

A2~0

which is recast in the form

d ln T ls' v -

A1 -~2 + ~ 2

(5.9.1a)

dT-[o

A1 + A27-[2

s,v

)-l

(5.9.1b)

~o d~o s,v

On integrating from an initial value (Ti, 7-/o) to a final value (Tu, 0) at constant S and V we find for the class of materials that is subject to (5.8.23)

Ti

'

(5.9.2)

which is the solution to the problem. One concludes that for a Curie-type paramagnetic material with A/[ = A27-[.o/T, the above demagnetization lowers the

344

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

temperature, since 7-[.2A2/A1 is positive. The effects is the larger the greater the I ~A1. Curie constant A2 and the smaller the heat capacity density cV,~o One may also determine the amount of heat transferred in this process: start with Eq. (5.7.15), set d V = 0, exchange magnetic for electric field variables, and introduce the heat capacity via Cv,7%/T = (SS/ST)v,7%. Then use the Maxwell relation, line (k) of Table 5.8.1, and multiply both sides by T. This yields

T d S ] v --Cv,~o d T Iv + T

d 3r -if-T- v, 7-to

(5.9.3) v

If now the field is changed in a reversible manner at constant temperature we obtain

dQrlr'v- T

d3r - - ~

t

(5.9.4a)

V,~o

V,T

Since the partial derivative is negative, a rise in magnetic field at constant T will result in a transfer of heat from the sample to the surroundings. On assuming that the integrand does not vary with position within the sample we obtain

OrIT, v - 5.9.1

V A17-[2 2-----------T~

(5.9.4b)

Transitions to the Superconducting State

The preceding material may be used to characterize the thermodynamics of transitions from the normal to the superconducting state. This transformation takes place for a limited class of materials at a particular temperature Tc, currently below 140 K. For soft superconductors of type I this state is marked by a complete disappearance of electrical resistivity and by the fact that at moderate values the magnetic induction 13 -- AA 4- 4:r7-/vanishes within the bulk of the sample, so that for such materials AA = -47rTJ. However, as the field is increased a critical magnetic field 7-{c is reached beyond which the material reverts back to its normal state. In first approximation 7-/c depends only on temperature according to the relation J-~c -- ~"/1

1-

,

(5.9.5)

where Tc is the normal-superconducting transition temperature in zero field. The transformation for fixed T and 7-{1 is an ordinary phase transformation that is most easily handled by returning to the relation d G = - S d T + V d Pm f v d3r .AA dT-{o. At constant Pm ~ P we characterize the equilibrium between

ADIABATIC DEMAGNETIZATION AND TRANSITIONS TO SUPERCONDUCTIVITY

345

the normal (n) and superconducting (s) state by the condition dGsIPm - dGnlp, so that

-SndT- fvd3rMnd~-[o---SsdT- fvd3rMsd~-[o (7-{.0-~-[c,PIn's'P) 9 (5.9.6a) For simplicity, replace the integral fv d3r M by VaT/ and rearrange the above relation to read OHo)

_

S,-Ss

-

VM,-

(5.9.6b)

v#,

If the magnetization of the material in its normal state is essentially zero, and on setting Ms --7-{c/4rr, one obtains

S n - S s - - 4 - - - ~ ~c - ~

p

9

Several points are to be noted. (i) First, the obvious analogy to the ClausiusClapeyron equation. (ii) According to the Third Law, the entropy of any material must reach a minimum value with zero slope; hence, as T --+ O, (07-Lc/OT)p --+ 0 as well (cf. Fig. 5.9.1). (iii) Experimentally it is found that (07-[c/OT)p < 0; therefore, in a magnetic field the superconducting state has a lower entropy than does 1000

I

I

I

I

I

I

I Pb

B

Ho

normal state

m "O

500 0

superconducting

l

\

--3

o

I

i

I

!

I

!

I\T~ t

1

2

3

4

5

6

7

8

T (K)

Fig. 5.9.1. Variation of 7-Lc with temperature for elemental lead in the superconducting and normal state.

346

5. THERMODYNAMIC PROPERTIES OF MATERIALS IN EXTERNALLY APPLIED FIELDS

the normal state. (iv) The enthalpy of the transition is given by A H = T (Sn - Ss); this quantity vanishes for ~ c = 0. Eq. (5.9.7) may be differentiated with respect to T and then multiplied by T; one obtains an equation for the heat capacity as follows:

CS ~ Cn

VT 0 ----4rc3 T { ~ C [ ( 37-[c3T) P ] } -- -~VT [7_[c(32~C3T2 ) p

+

3T

p" (5.9.8)

At T =

Tc the condition ~ c = 0 holds; for this case P,T=Tc

- ~c (Cs - Cn)Tc

9

(5.9.9)

The above expression is known as Rutger's equation; it determines the slope of the plots of ~ c vs. T at T -- Tc. This forms the basis of the plot shown in Fig. 5.9.1. Lastly, we may integrate Eq. (5.9.8) to obtain

/o

"E

( C s - C n ) d T - 4 rc

T-d-f

4re

He

OT

3T Tc

p

p V

l o

47CfHoEHc( 37-[c

v~g

(5.9.10)

8zr where the term on the fight, top line, vanishes at T - - 0 and expression may be inverted to read ~o -

--~

f0c

(Cs - Cn) d T

],,

,

T -- Tc. The above

(5.9.11)

which serves as a means of finding the maximum value of the critical field.

347

Chapter 6 Irreversible Thermodynamics

6.0

Introductory Comments

We begin here the study of thermodynamics in the proper sense of the word, by exploring a variety of physical situations in a system in which one or more intensive variables are rendered nonuniform. This results in the establishment of irreversible phenomena that will now be explored. So long as the variations in T, P, p or other intensive quantities are 'small' relative to their average values, one can still apply the machinery of equilibrium thermodynamics in a manner discussed below. It will be seen that the identification of conjugate forces and fluxes, the Onsager reciprocity conditions, and the rate of entropy production play a central role in the subsequent analysis.

6.1

Generalities

Any nonequilibrium phenomena necessarily involve states in which different portions of a given system display different physical characteristics. To handle this situation we subdivide the system into tiny subunits and allow all variables of interest to become functions of their position within the sample, and also, to become functions of time. Thus, each thermodynamic variable t~i of interest must be specified in terms of its position r at time t: t~i = t~i (r, t). We render such systems subject to the scrutiny of thermodynamics by establishing the Principle of Local State that is based on two assertions: (i) The instantaneous values of all thermodynamic quantities ~bi centered on a small region about the location r satisfy the same general thermodynamic relationships as do the corresponding quantities for a large uniform copy of the same small region at that instant of time. (ii) The local, instantaneous gradients of 4~i, and their rates of change, are to be ignored within the volume element, when setting up the specification of the thermodynamic configurations. This addresses the fact that at any position r all relevant thermodynamic coordinates are likely to be characterized

348

6. IRREVERSIBLE THERMODYNAMICS

by different values in the contiguous regions, and hence, by gradients directed to neighboring locations. Nevertheless, as long as the variation in ~bi is 'sufficiently small' over the small region under study (what 'sufficiently small' means can generally only be decided by experiment) it may be left out of account. When the quantities ~bi vary with time one must specify a corresponding velocity function v(r, t) to describe the change in properties of 4~i. However, if it so happens that the entities ~bi remain unaltered and that only the surroundings change in such a process, then the system has reached a steady state with respect to changes in these particular properties. A more precise specification of this state is furnished in Section 6.4. 6.1.1

The General Balance Equations

Consider a quantity R which is distributed at a density PR over a system with a particular volume V; R -- fv PR d3r. We wish to specify the time rate of change of this quantity. Changes in R may be brought by two processes. The first involves an influx vs. outflow of R across the boundaries of the system. The rate of flux of R per unit time in a direction perpendicular to a unit cross section is given by theflux vector JR. Hence, the net rate of accumulation (or, depletion) of R within V is given by - fa J R 9h dZr. Here h is the unit normal vector perpendicular to the element of the surface through which the flux passes; a minus sign is inserted because the vector h points outward. The dot product involves the component of JR that is perpendicular to the surface element d2r, and the integration extends over the entire surface that surrounds the system. The second mechanism for altering R involves the generation or dissipation of this quantity entirely within the volume of the system; such processes generally take place via chemical reactions. Let Pv represent the time rate of net production of R per unit volume; then the total rate of change of R through this mechanism is given by f v Pv d 3r. We now put these results together: The overall time rate of change of R within the system is given by 1

dRdt -- fv OPR d 3 r - - - f a J R ' h d 2 r +

(6.1.1)

We apply Gauss' theorem, Table 1.3.1, line (k) to the first integral, converting it into an integral over the volume of the system, to obtain

fv OPR d3r - - fv v JR d3r + fv Pv d3r"

(6.1.2)

For such a relation to hold it is sufficient to demand that the integrands match on both sides. One then obtains a balance equation of the form

OpR - - - V . JR + Pv. Ot

(6.1.3)

GENERALITIES

349

This expression is known as the equation of continuity. It clearly differentiates between processes that involve transport of R across boundaries and the local generation of R. However, in some cases, considered later, R cannot be generated locally; in that event Eq. (6.1.3) specializes to Opn Ot

= -V.

JR,

(6.1.4)

known as a conservation equation; here R can only be changed through transport across the boundaries of the system. In line with Eq. (6.1.4) we may write the fundamental flux vector as J n -- PR V, SO that Op Ot

=-V

. (pv)--pV

. v-

v. Vp.

(6.1.5)

= -pV.

v,

( 6 .1 .6 )

A trivial rearrangement leads to the result Op Ot

-+- v . V p =

Dp Dt

where D p / D t is the convected derivative, i.e., the derivative with respect to time inside a volume element that moves relative to the center of mass of the system with velocity v. A proof of this statement is furnished in the appendix to this section. In many treatises this particular quantity is written out as d p / d t . We next consider several special cases that apply to Eq. (6.1.3). 6.1.2

Balance E q u a t i o n for Concentration

Here we study processes that affect the (molar) concentration ck of a chemical species k; for this purpose we set up a flux vector CkVk(r, t) for the rate of transport of k at concentration ck past location r at time t. The amount of k may be altered either through chemical reactions that change its concentration locally at a rate (OCk/Ot)[c, or by convective or diffusive transport processes that take place at a rate (OCk/Ot)lt. Convection relates to the motion of the center of mass of the system that carries all constituents with it, while diffusion relates to the motion of species k relative to the center of mass. The velocity of the center of mass of the system of intermingling species k is given by v --

vkm ck

=

VkP pk

,

(6.1.7)

where Mk is the gram molecular mass of species k. Let Vk be the velocity of motion of species k. To determine the rate of entry of this species into a small volume element that itself moves with velocity v we

350

6. IRREVERSIBLE THERMODYNAMICS

now introduce the (molar)flux vector relative to the center o f mass for species k across perpendicular unit cross section per unit time as (6.1.8)

J k -- Ck(Vk -- v).

The time rate of change of material of type k in a local volume element, other than through diffusion or convection, occurs when this species participates in r distinct chemical reactions. We therefore write r

n

j6v - Z

E

(6.1.9)

ckvklWl,

/=1 k = l

where, as before, the 1)kl are the stoichiometry coefficients (positive or negative) for species k participating (as products or reagents) in each of the r distinct chemical reactions, and O)1 :~ d)~i/dt is the rate of advancement of t h e / t h chemical reaction. Putting these results together we write a balance equation for species k, based on Eqs. (6.1.3) in line (a) and (8) in line (b), as OCk

OCk

Ot

Ot

F

OCk t

+Vi-

-- -- V " (Ck Vk) + Z

Ck 1)kl O)l

(6.1.10a)

l=l r

= -V.

J k -- V . (CkV) + ~

CkPklO)l 9

(6.1.lOb)

/=1

This leads to the obvious identifications OCk

OCk Ot

-- E c

Ck Vkl gOl ,

l=l

Ot

--V

. J k -- V . (CkV).

(6.1.11)

t

In the event that there is no convection in the system (v - 0) we find r

0 Ck _-- _ V 9 J k + Z Ck vkt col Ot /=1

(6.1.12)

to be the basic equation o f conservation o f mass for species k.

6.1.3 Energy Balance Equation The total energy density of a system involves contributions from the internal energy density e, the kinetic energy density/C, and the potential energy density ~;.

GENERALITIES

351

We consider first 1C - 89Y]k MkCk v 2. It is a trivial matter to verify the identity 1

-~ Z

1

Mkck(Vk -- v)Z -- -~ y ~ MkckvZ -- Z

k

k

1

MkCkVk " V + -~ Z

k 1

1

- -2P

k 1) 2

1

+ -2 Z

1)2

k

1 k

Mkck v2

k

(6.1.13)

M k Ck Vk

k

where p - Y-]k Pk is the total density of the system, and where Eq. (6.1.7) has been introduced. Inserting Eq. (6.1.13) into the kinetic energy e x p r e s s i o n / C 1 Y]k MkCk V2 leads to the result 2 1 ]~_,-- ~ p v 2 -q- -~ Z

mkCk(Vk -- V)2

(6.1.14)

k

which breaks/C up into a term involving the motion of the center of mass, and the kinetic energies of the various species relative to the center of mass. Henceforth we neglect motion of the center of mass (setting v = 0) and we now consider the time derivative of the internal energy density, given by

Oe(T, ck)

dT

OCk

+ Ek

=

0-7

dT

(6.1.15) k

l

k

in which ~v is the heat capacity density at constant volume, and e k - (Oe/ OCk)T,P,ct~k represents the partial molal energy density of species k. Here Eq. (6.1.12) had been introduced to obtain the expression on the right; the middle term relates to the 'heat of reactions' dissipated or absorbed as the various chemical species participate in all of the chemical reactions in the system. 2 It remains to consider the contribution arising from the passage of material in externally applied fields (potentials) ~, including external pressure, that are presumed to be time-invariant. We begin with Eq. (6.1.10), set v -- 0, and operate on both sides with Y]k 7tk, whence the potential energy variation with respect to time is given by

k

k

= -- Y ] V " ~ k J k + ~ k

k

Jk " V ~ k . k

1

(6.1.16)

352

6. IRREVERSIBLE THERMODYNAMICS

The double sum in the middle vanishes because the potential energy of all constituents remains unaltered during the various chemical reactions within the system. Also, we used line (d) of Table 1.3.1 in arriving on the right-hand side. Putting the various results together we now find an energy balance equatton of the form

OT

Obl

=cv-~-+ZZekCkVklOOl--ZekV'Jk--V'Z~kJk Ot k l k

k

dE + Z

J~" V ~ + d---t"

(6.1.17a)

k

In accordance with the First Law the total energy of a system can only be changed by energy transport across its boundaries. We therefore introduce a total energy flux vector J v by the conservation law 0u

Ot

=-V.

Ju.

(6.1.17b)

In looking ahead to the theoretical development in later sections it is expedient to separate out from the right-hand side of Eq. (6.1.17a) the last three terms and to sum the remaining contributions as a time rate of change of the internal energy density, e. Thus, we set

3e Ot

_

~vOT - ~ -+-~

Z

k

ekck VklCOl-- ~ l

ek V . Jk,

(6.1.18)

k

so that Ou _ ae

Ot

V .Z

Ot

J k " v ~k + dE dt - - V .

~kJk + Z k

~

Ju

~

(6.1 19) 9

k

where we inserted from Eq. (6.1.17b) on the right. It is evident that for consistency the first, third, and fourth terms in the middle must be regarded as equivalent to and expressible as the divergence of a flux vector that we shall call J Q. Thus, we set

3e -v

. jQ

-

dlC

57 +

"

+

d--i-

k

OT k

k

l

k

dE dt

(6.1.20a)

GENERALITIES

353

We must digress briefly at this point. Let us write ~-~'~J~. V 7r~ = )--~kV- 7r~ Jk ~-~k~PkV 9 J~. Then it is seen that the negative divergence of the flux vector J Q m u s t match not only the contributions -y~'~ e k V . J~, ~-~k V . ~ J k , and - ~--~k~P~V 9J~, but also the remaining terms that do not have the form of a divergence function. This physically means that energy in terms of heat transfer must be supplied to or withdrawn from the system to keep pace with the rate of energy changes ~v(OT/Ot), Y~k ~-~l ekckv~lcol, and dlC/dt that are specified in (6.1.20b), i.e., the rate of change in temperature, the rate of change of energy internally through the occurrence of chemical reactions within the system, and the rate of change of the kinetic energy of the system. These effects cause a change in internal energy of the system that must be matched, as already stated, by the flow of heat across the boundaries of the local system. In particular, it is important to distinguish the rate of energy change arising from chemical reactions, in which the various species k have different internal energies ek, from the effects of internal chemical reactions discussed in Section 1.21 that were altered by transport of chemical species across the boundaries. For future use we rewrite Eq. (6.1.20a) as -

_

Oe _ - V . Ot

d/C J~" VTrk - ~ " dt

JQ - Z k

(6.1.20b)

Then the energy balance equation (6.1.19) reads OH

Ot

= - V 9 JQ - V . ~ , ~PkJk -- - V .

Ju.

(6.1.21a)

k

We therefore interrelate the various flux vectors by J u - JQ + ~

7~,Jk.

(6.1.21b)

k

We may term J Q a heatflux vector; for, we have consistently viewed the net total energy change as arising from the flow of heat and from the performance of work. The latter involves the term Y~k 7rk Jk. Note that J Q includes all the contributions shown in Eq. (6.1.20a), except terms involving Jk. 6.1.4

Entropy Balance Equation and Rate of Local Entropy Production

For a system at constant volume the Second Law is handled via the relation T ds - d e Y-~k #~ dck, for the entropy density, where/z~ is the chemical potential of species k. Accordingly, Os Oe T~ = Ot at

~ Oc~ 2.., # k ~ . at

(6.1.22)

354

6. IRREVERSIBLE THERMODYNAMICS

We now introduce Eqs. (6.1.12) and (6.1.20b); we also assume that the kinetic energy contribution remains time-invariant. We then find Os = T - 1 - V . JQ + Ot

Jk " f k -k=l

lZkV . Jk + k=l

.AlO)l

,

(6.1.23)

/=1

in which Jgtl ~ - Z k ls is known as the chemical affinity; we have also introduced the force density as f k = - V 7tk. The next step consists in an extensive rewriting of the above equation; standard mathematical manipulations acting on Eq. (6.1.23) lead directly to the following equivalent form: Os --V . T -1 0-7 = 1

JQ -

lZkJk

n

Z ~

--

k=l 1

Jk"

k=l

J

VT +

e

lZkJk 9V T k=l

r

fk)+ --~ Z

(VlZk-

T

(6.1.24)

"AlO)l"

l=l

While this expression looks much more complicated it is actually readily interpreted. Note that the quantity involving square brackets in the first term on the right-hand side represents the divergence of a set of flux vectors. Now, Eq. (6.1.3), specialized to the entropy balance, has the form Os

Ot

= - V.

J s + O,

(6.1.25)

whence it clearly makes sense to correlate the entropy flux vector J s with the term in square brackets. Thus, in (6.1.24) we set Js-

T-1

JQ -

lZkJk

9

(6.1.26)

k=l

The remaining terms on the right-hand side of (6.1.24) must then represent source terms 0 also known as the dissipation function; accordingly, 1

0 - --~Js"

1

VT

n

T Z k=l

1

r

Jk" V(k + -~ Z ~ 1 0 9 I >1 O,

(6..1 27)

/=1

in which we have now set V/zk -- f k = V (/zk + 7tk) = V (k. Thus, (k may be regarded as a generalized chemical potential for species k. Thus, in Eq. (6.1.25) the term - V 9 J s represents the net increase of entropy density in the system from flux across the boundaries, whereas Eq. (6.1.27) refers to the generation of entropy density through irreversible processes occurring locally within the volume element. In subsequent discussions Eq. (6.1.27) will play a cardinal role.

GENERALITIES

355

The specification of 0 is not unique. We provide an alternate relation by combining Eq. (6.1.2 l b) with (6.1.26) to write 3 t/

Ju - TJ s + ~ ~J~.

(6.1.28)

k=l

Now substitute this relation into Eq. (6.1.27); a slight rearrangement leads to the dissipation function of the form 1

1

n

O----Ju.VT T

1

k=l

1

k=l

n = Ju-v(

n

r /=1

1 r

/r) -

+ T k=l

o.

(6.1.29)

/=1

The form of Eqs. (6.1.27) and (6.1.29) is highly significant. In both cases the rate of entropy density generation through irreversible processes taking place locally is specified by a sum of products of the general form 0 - ~ J~ 9X~ ~> 0, wherein the Jc~ represent either generalized fluxes or reaction velocities, and the Xc~ represent generalized forces. As explained in Section 2.2, this nomenclature is appropriate because 0 - 0 only occurs when both the forces and the fluxes vanish. In this sense the flux Jc~ is a response to the imposition of an applied force X~. Of great importance in our future development are the particular forces and fluxes that occur pairwise in the expression 0 - y-~ Jc~ 9Xc~. The partners in these pairs are said to form conjugate variables: In Eq. (6.1.27) they are respectively ( J s / T , - V T ) , (J~/T,-V~'~), and (COl,.A1/T); in Eq. (6.1.29) they are specified respectively by the set (Ju, V ( 1 / T ) ) , ( J ~ , - V ( ~ / T ) ) , (COl,.Al/T). Which of the two sets of conjugate force-flux pairs are to be used is simply a matter of convenience. No flux can occur without being 'driven' by a force field; in its absence 0 vanishes and Eq. (6.1.25) then becomes a conservation equation for entropy. For many applications (see, e.g., Section 6.8) it is expedient to recognize that the gradient V ~'~ may be split up into a component that involves T and a remainder involving mole fractions, both of which are functions of positions r within the sample: ~(T,r) - #(T(r),x(r)) + O(r) - ~(T(r),x(r)), so that

5 r --

=

-~

x --~r -+- -~x

T -~r -+- Or

_~OT (0#)Ox Ogr ~Or + -~x r-~r + ~'Or

(6.1.30a)

or, in abbreviated notation" V~- - - ~ V T + V~-IT ,

(6.1.30b)

356

6. IRREVERSIBLE THERMODYNAMICS

where the T subscript indicates that the partial derivative is to be taken at constant temperature. On introducing (6.1.30b) into (6.1.27) we combine the two V T contributions to find 1

0 -- - ~ f f s "

1

VT

n

T Z

1

Jk"

r

v(klr + ~ ~

k=l

~tlO)l ~ O,

(6.1.31)

l=l

wherein n

(6.1.32) k=l is the entropy flux arising from sources other than matter flow. The conjugate force-flux pairs now differ from those of Eq. (6.1.27). 6.1.5

Appendix

We elaborate here on Eq. (6.1.6): consider a function F that pertains to a volume element d V which at time t is at position r and which moves with velocity v to a new position r + A r at time t + At. Then take the total time derivative of F, D F~ Dt, with respect to the moving center of mass of the volume element d V, and compare this to 0 F lOt, taken with respect to the external laboratory axes. Given the position vectors x(t), y(t), z(t) we then find that

DF OF OF Ox OF Oy = -~ ~ Dt Ot Ox Ot Oy Ot =

OF Ot

+ v. V F.

OF Oz Oz Ot (6.1.33)

The two terms on the right correspond, respectively, to those on the left-hand side of Eq. (6.1.6). The quantity on the left is often written out in the form d F/dt. NOTES AND QUERIES 6.1.1. As shown in Chapter 1, one cannot with impunity interchange differentiation and integration operators. The steps shown below apply only for volumes that do not vary with time. If the total volume is time-dependent then procedures of the type described in Section 1.3 must be employed, which leads to the presence of additional terms in the equation of continuity. 6.1.2. In fact, one may reformulate the First Law as e = ~vdT + ~k(lZk + Ts)c~, and recognize that/z~ + T s = h is an enthalpy density, which in this case may be regarded as a 'heat of reaction'. 6.1.3. For a single species the summations drop out; then the ratio Jv/J1 may be regarded as the partial molal energy density carried by species 1, U*; likewise, Js/J1 represents the partial molal entropy carried by species 1, S*. Eq. (6.1.28) then specializes to U* - TS* = ~, and (6.1.26) to E* - T S* = / z , which is an analog of

H-TS=G.

SHOCK PHENOMENA

357

6.1.4. How are all of the above relationships changed if, instead of dealing with extensive variable per unit volume, one were to switch to 'specific' quantities, i.e., extensive variables per unit mass? 6.1.5. Construct equations of continuity that explicitly account for the presence of pressure gradients in a system. Consult major treatises on irreversible thermodynamics for an extensive treatment of this subject.

6.2

Shock Phenomena

6.2.1

Introductory Considerations; Small Departures from Equilibrium

As our first illustration of nonequilibrium phenomena we consider the case of shock effects in conjunction with Fig. 6.2.1. (i) Let a piston be suddenly accelerated to a velocity u, traveling to the right in the shock tube depicted in Fig. 6.2.1. A s s u m i n g steady-state conditions, the material to the fight of the piston moves along in the same direction as the piston and with the same velocity. The compressed material at pressure P0 + A P is preceded by a "shock front", ahead of which is the undisturbed material at pressure P0 and at rest. The shock front moves with sound velocity co and extends over the region where the pressure changes from P0 to P0 § A P. The density of the undisturbed material is p0 and that of the region behind the shock front is P0 + Ap. (ii) Let an observer ride along the shock front down the tube. He would see material 'entering' on the right with velocity co at a density P0 and 'leaving' on the left with velocity co - u at a density P0 + Ap. The mass of material processed in this manner must be conserved; per unit time, the mass crossing unit area into the shock front, and the mass leaving unit area at the back of the shock wave, is given by rrtt : pOCO :

(6.2.1)

(PO -Jr- Ap)(c0 - u).

u=0 u

u

Po

Po+ Ap

~'~] Piston

Mediummoving at speed u

Co ~

Medium at rest

Shock front

Fig. 6.2.1. Illustration of the motion of a piston and of a shock front in a shock tube.

358

6. IRREVERSIBLE THERMODYNAMICS

This relation may be solved for

coAp . PO + Ap

u= ~

(6.2.2)

(iii) Next, we invoke Newton's Second Law of Motion: Per unit time and cross section a mass mr of material changes its momentum from 0 to m tu; according to Newton's Law, this rate of change in momentum must be accounted for by a force per unit area which changes from P0 to P0 + A P. Hence,

A P = pocou.

(6.2.3)

We eliminate u from Eqs. (6.2.2) and (6.2.3) and solve for c 2 = (Po + A p ) A P .

(6.2.4)

poAp (iv) Under the further assumption that all disturbances are small, we set Ap << P0 and A P / Ap = d P / d p , so that

dP

dp

=Co2.

(6.2.5)

Assume next that the compression occurs so rapidly that the material has no time to respond before it is transformed from the undisturbed state to the steady state behind the shock front. In this event the transformation occurs adiabatically. (v) We now specialize to the case where the material under study is an ideal gas. For the small disturbances envisaged in (iv) the temperature is assumed to remain constant, and under adiabatic conditions, P V • = constant, where y = C p / C v . Since V ~ po 1 ~ p-1 the adiabatic condition may be reformulated to read P - Ap~ ~ Po. Then

dP ?' Po , dp - y Ap~ 1 = ~Po

(6.2.6)

co - V/V Po/Po.

(6.2.7)

and in view of Eq. (6.2.5),

For an ideal gas, P0/P0 = R To/ M, where To is the temperature of the undisturbed medium and M is the gram molecular mass. Accordingly,

co - v/y R T o / M .

(6.2.8)

Note that Eq. (6.2.8) may be inverted to determine g from a measurement of the velocity of sound. Inasmuch as g - (Cv + R ) / C v , both Cv and (~e are directly available from sound velocity measurements.

SHOCK PHENOMENA

6.2.2

359

S h o c k P h e n o m e n a in L a r g e D e p a r t u r e s from E q u i l i b r i u m

So far, we have assumed only infinitesimal departures from equilibrium. We generalize considerably by allowing for steady state conditions extensively removed from equilibrium; this forces us to take into account severe excursions of T, P, p, _ -1 or v = p 1 from the equilibrium properties To, P0, P0, and v0 = P0 9 The situation may be visualized with the diagram shown in Fig. 6.2.2. As before, we invoke the conservation law for matter, m t being the mass of material that is being overtaken by unit area of the shock front per unit time. Then, in analogy to Eq. (6.2.1), mt -- poc--

(c-

(6.2.9)

u)p,

where c is the velocity of propagation of the shock front. We will later relate this quantity to co; the two differ because with rising temperatures the propagation velocities increase. As was done in conjunction with Eq. (6.2.3) we can set up an equation based on Newton's Second Law of Motion: P -

Po -

pocu

-

(6.2.10)

mtu.

Next, we introduce the First Law of Thermodynamics: Let e0 and e be the energies of the material per unit mass of material being overtaken in unit time by unit area of the shock front (for which m t = 1); the difference e - e0 in energy, before and after the shock wave has hit, must reflect any chemical reactions initiated by the shock. The change in kinetic energy acquired by this quantity of material is u2/2; thus, we write H

2

Ae -- e - e0 + ~ . 2

(6.2.11)

The work performed by the piston on the material per unit time and transmitted across unit cross section is P u . A s s u m i n g adiabatic shock conditions, the First C ----ii~

T , p , p = v -1

To, Po, Po = Vo-1 u = 0 , eo

u,e

Medium moving at speed u Piston

Medium at rest Shock front

Fig. 6.2.2. Illustration of the motion of a piston and of a shock front in a shock tube under severe departures from equilibrium.

360

6. IRREVERSIBLE THERMODYNAMICS

Law of Thermodynamics then states that P u -- m t

6.2.3

(

u2)

e - eo + - ~

.

(6.2.12)

Algebraic manipulations

We have at hand now all the laws needed in our further development; the rest is algebra. First, solve Eq. (6.2.9) for c: /9

vo

c -- ~ u p -- po

-- ~ u . v0 -- v

(6.2.13)

Second, divide (6.2.12) by (6.2.10), Pu

e - eo + u 2 / 2

=

P - Po

,

(6.2.14)

u

and solve for 1 P - + - Po e-e~

~ u P-

2

Po

U2

--+ ~

2

for P >> P0.

(6.2.15)

Third, eliminate c in Eq. (6.2.9) by use of (6.2.13) and simplify. This yields mt

--

U ~ . VO ~

(6.2.16a) V

Fourth, eliminate m t via Eq. (6.2.10) and solve the resultant for U2 -

(P-

Po)(vo-

v).

(6.2.16b)

Finally, use (6.2.16b) in (6.2.15) to obtain e - eo -

(P + Po)(vo - v)/2,

(6.2.17)

which is known as H u g o n i o t ' s e q u a t i o n . If we set h - e + P v, ho - eo + P r o , we may write h - ho -

6.2.4

( P - P o ) ( v 4- vo)/2.

(6.2.18)

Use of Ideal Gas as a Working Substance

We now specialize considerably by dealing with an ideal gas as a working substance. Then PV--nRT--

m ~RT, M

(6.2.19)

SHOCK PHENOMENA

361

or

Pv -

RT

(6.2.20)

M

and

e -- Cv T + constant --

Cv T + constant. M

(6.2.21)

Use Eq. (6.2.21) on the left and (6.2.20) on the right of Eq. (6.2.17)"

Cv --~- ( T - To) - -~ --~ ( P + Po )

Po

(6.2.22)

P

Note how e has been eliminated in favor of 6'v. We reformulate the above by defining a shock strength b y / 7 - P / P o , in terms of which we rewrite Eq. (6.2.22) as

.

(6.2.23)

R) T 2Cv+R+-~ --(2Cv+R+HR)To,

(6.2.24)

Cv(T-To)---~(H+I)

T0-~

Then collect terms in T and in To:

or

T

=

To

2Cv + R + R H _ . 2Cv + R + R/FI

(6.2.25)

F o r / 7 >> R this relation reduces to T

--+

To

R _ H. 2Cv + R

(6.2.26)

The factor on the right appears so frequently that we introduce for it a new symbol, IZs = R / ( 2 C v + R) -- R / ( C p + Cv). We then obtain

T To

=

l+tzsH 1 + lzs/H

~ #s/7

f o r / 7 >> 1.

(6.2.27)

At high T, Cv - 3 R / 2 for a monatomic gas, and Cv - 5 R / 2 for a diatomic gas. Hence, T~ To --+/7//4 o r / 7 / 6 for monatomic or diatomic gases respectively. Note the route we took to obtain information on the rise in temperature when an ideal gas is shocked, and note that the asymptotic limits for T/To differ for monatomic and diatomic gases.

362

6. IRREVERSIBLE THERMODYNAMICS

6.2.5

Interrelations Involving Shock Phenomena

From the above we can now establish a considerable number of interrelations using various algebraic manipulations. For instance: (i) We can find the ratio P/Po from p Po

=

P To Po T

--+ 1/#s

/7 +/Zs

=

1 +#sf/

for/7 >> 1.

(6.2.28a) (6.2.28b)

Thus, there exists a distinct upper limit on p/po, of 4 and of 6 for monatomic and diatomic gases, for very large shock strengths. (ii) Information on the mass flow velocity is obtained by first using Eq. (6.2.21) to determine -r ~ ( T - To)

e-eo--

(6.2.29)

and then using this result in (6.2.15), eliminating T through Eq. (6.2.27), and reintroducing/7 = P/P0. This yields

tt2 -- 2 Cv To (/7

- 1)2 #s, 17+lZs

M

(6.2.30)

which shows that in (an ideal) gas there exists a connection between shock strength and mass flow velocity. (iii) We may eliminate To for the undisturbed medium from Eq. (6.2.30) by recalling (6.2.8) and noting that lZs - R / C v ( 1 + y); on carrying out the indicated operation and taking square roots of the resultant we find:

2 ( / / - 1 )12/]2 _ y(y-+1) H ~ # s

u_[

co

(6.2.31b)

--, v / 2 f l l y ( y + 1)

0.716CH, ~ 0.890~-H,

(6.2.31a)

monatomic gas, /7 >> 1

(6.2.31c)

diatomic gas, /7 >> 1.

(6.2.31d)

Thus, for H sufficiently large, u/co > 1; i.e., the mass flow velocity becomes supersonic. (iv) To examine the ratio c/co we begin with (6.2.13)" c

( )1 P0

u

co

7

co

1

H + lzs u (/7 - 1)(1 - #s) co

(6.2.32)

SHOCK PHENOMENA

363

where Eq. (6.2.28) had been used. Now substitute for u/co from (6.2.31) to obtain

C J

c0

2(/7 + #s) y ( y + 1 ) ( 1 - / Z s ) 2"

(6.2.33)

This relation may be simplified by noting from the definition of #s and g that #s = (g - 1 ) / ( g + 1); then C

co

= v/(g + 1)/2V)(17 + #s),

- - -~ v/(• + 1 ) 1 2 •

(6.2.34a)

f o r / 7 >> #~

CO

(6.2.34b)

-+ 0.895x/-H,

for a monatomic gas, /7 >> #~

(6.2.34c)

-+ 0.926x/-H,

for a diatomic gas, /7 >> #~.

(6.2.34d)

A comparison of (6.2.34) with (6.2.31) establishes that c > u; the shock wave will always outrun the mass velocity of the gas. The ratio c / c o --= M is called the Mach

6.2.6

number.

Equation of State

We can write a shock equation of state by defining P/Po = vo/v =- ~s. Then Eq. (6.2.28) may be rearranged to read

/7 =

~S m ]-~S

1 -

.

(6.2.35)

#s~s

Compare this to the case of the reversible, adiabatic equation of s t a t e / 7 = ~'sy and to the reduced isothermal equation of state H = ~'s. EXERCISES 6.2.1. Derive the following relations involving shocked material: (a) m 2 -- p p o ( P - P o ) / ( P - Po), (b) u 2 = - ( P - P o ) ( P - P o ) / P P o , (c) #~ = (• - 1)/(• + 1).

6. IRREVERSIBLE THERMODYNAMICS

364

6.3

Linear Phenomenological Equations

6.3.1 SimplePhenomenologicalEquations We cite here again the expression for the local rate of entropy production in an isotropic medium, n

?"

- - - T - ] J s 9V T - T-I ~ Jk" V~k + ~Ogl(,Al/T) ~ O. k=l

(6.3.1)

l=l

Consider first the case where no chemical reactions take place and where no particle fluxes occur. Then 0 - - T - 1 j s . V T >~ 0. In the absence of any particle flux J s - J O / T . Then Eq. (6.3.1) reads 0 - - T -2 J O. V T -- J a . V (1 / T). According to our standard interpretation J Q and V (1 / T) are conjugate flux-force pairs, so that the heat flux J Q is 'driven' by the gradient of 1/T. For small departures from equilibrium one may assume a linear dependence which is homogeneous, so that no additive constant prevents the flux from vanishing simultaneously with the force V (1 / T). In short, we write J a - LV (1 / T),

(6.3.2a)

where L ( T ) is a scalar function of temperature; a tensor would be inappropriate since this would imply a set of preferred directions. The above may be rewritten as

L J Q - - -- T--~ V T -- - K V T,

(6.3.2b)

which represents Fourier's Law of heat conduction; tc is the thermal conductivity. We have thus recovered a well-established law. We now repeat the argument for the case where no temperature gradient prevails, no reactions occur, and where flux of one type of particles takes place. Eq. (6.3.1) then specializes to 0 - - T - 1 J i 9V ~i ~ O. This suggests that a gradient in generalized potential, V (i is a driving force, to which the quantity T - 1 J i is a response. Assuming small departures from equilibrium a linear relation of the type Ji - L ' ( T ) V ( i

(6.3.3)

is then set up; the extra T -1 factor has been absorbed into the L ~ coefficient. In the absence of any external fields ~'i may be replaced by the chemical potential ~i -- ~ i + R T In ci, whereby, at fixed T, we obtain Ji - - D ( T , ci)Vci.

(6.3.4)

LINEAR PHENOMENOLOGICAL EQUATIONS

365

Here Ci is the concentration of species i, and the quantity R T / c i has been absorbed into the diffusion coefficient D. Eq. (6.3.4) is known as Fick's Law of diffusion. The coefficient is clearly concentration dependent. In Eq. (6.3.4) the concentration gradient serves as the 'driving force', but in actuality it is the gradient in chemical potential that activates the particle flow, as shown in Eq. (6.3.3). If the particle flow involves electron with charge - e we recast Eq. (6.3.4) for the charge flux as Je = L ~'(T)V (~"/e). Under conditions where the concentration of charge carriers in the volume element remains uniform we may set V (~"/e) = -V~b, where 4~ is the electrostatic potential. We then obtain

Je = L " ( T ) V ( - ~ b ) = L " ( T ) E =_ erE,

(6.3.5)

where E = -V~b is the electric field vector and cr is the electrical conductivity. The above equation represents one formulation of Ohm's Law. The above examples illustrate the fact that when only one driving force is present one recovers well established phenomenological relationships cited in the literature.

6.3.2

Linear Phenomenologieal Equations

Whenever more than one driving force is operational one assumes that the linear superposition principle holds; that is, every force Xj influences every flux J j, in a linear manner according to the relations

J1 = L l l X 1 + L12X2 if-'-'-}- LlnXn, J 2 -- L21X1 -+- L22X2 if-'"-}-

L2nXn,

Jn = Ln 1X 1 at- Ln2X2 -at-... -at- Lnn Xn.

(6.3.6)

Expressions of this type are generally known as phenomenological or macroscopic equations. They are based on the dissipation relation 0 - ff-~j J j . X j, with 1 <~ j ~< n sets of conjugate variables. As shown above, every flux is accorded its own phenomenological equation, which additively involves every one of the prevailing forces. Thus, every force linearly affects each flux. The various Lij are commonly known as phenomenological or macroscopic coefficients. Those for which j -- i are known as proper coefficients; the remainder provide cross coupling effects between forces of one type and fluxes of another kind, and are known as interference coefficients. The validation of the phenomenological equations ultimately rests on the success with which such relations provide an interpretation of experimental results. In particular, these equations hold only for 'sufficiently small' departures from equilibrium conditions; what 'sufficiently small' means can only be established by noting the experimental conditions under which the use of these equations fails.

366

6. IRREVERSIBLE THERMODYNAMICS

In conjunction with Eq. (6.3.6) we introduce the Onsager reciprocity conditions (OCR) which will be derived below under a restricted set of circumstances. The OCR read as follows:

Lij -- Lji.

(6.3.7)

It is stressed that these relations apply only if the phenomenological equations are based on conjugate force-flux pairs. If other types of pairs are employed the coefficients with i ~ j are functionally related but not equal. In the presence of magnetic fields B and or in systems undergoing angular rotations at rates to the above expression is replaced by the Casimir-Onsager reciprocity relation

Lij(l~, 6o) -- -+-Lji(-l~, -09)

(j =/=i),

(6.3.8)

where the negative sign prevails if the applicable forces change in direction as the field or angular velocity direction is reversed. This relation has been derived on the basis of statistical mechanics.

6.4

Steady State Conditions and Prigogine's Theorem

Steady state conditions obtain when the fluxes and forces giving rise to irreversible phenomena in a system remain time-invariant, whereas the properties of the surroundings change. We now render this idea more precise by introducing Prigogine's Theorem: Let irreversible processes take place through imposition of n forces X1, X 2 , . . . , Xn that result in n fluxes J1, J 2 . . . . . Jn. Let the first k forces remain fixed at values X ~ X~ ..., X~ then it is claimed that the rate of entropy production 0 is minimized when the fluxes J~+l, Jk+2,..., Jn all vanish. We first prove the theorem and then discuss its relevance to steady state conditions. As before, we set 0 -- Z j J j " X j, to construct the phenomenological equations //

Ji - Z LijXj,

Lij - Lji

(i - 1, 2 , . . . , n)

(6.4.1)

j=l

and the expression n

g/

-- Z ~ LijXi "Xj ~ O.

(6.4.2)

i=1 j = l

An extremum is found by differentiating 0 with respect to the nonfixed forces; (we ignore niceties involved in differentiating vectorial quantities 1). This leads to the result n

0 XJr i

j=l

ONSAGER RECIPROCITY CONDITIONS

367

Note that the local rate of entropy production is a minimum, since 0 is nonnegative. On introducing the OCR and Eq. (6.4.1) one obtains ?/

2Z

Lij X j - 2Ji - 0

(i = k + 1,k + 2 , . . . , n ) ,

(6.4.4)

j=l

which proves the theorem. In light of the above the relations specifying Ji with i -- k + 1, k + 2 , . . . , n (all of which vanish) may now be solved for the various X j, as unknowns, (where 0 thereby rendering all X j timej - k + 1, k + 2 . . . . . n) in terms of the fixed X j, invariant. 2 Accordingly, all of the fluxes Ji either are zero or assume fixed values. But these circumstances are precisely the hallmark of steady state conditions. We have thus established that a steady state prevails when the local rate of entropy

production is at a minimum with respect to the prevailing constraints. Steady state conditions tend to be inherently stable: Consider a case where all forces except one, namely Xm, k + 1 ~ m <~n, is held fixed. Now apply a perturbation 3Xm to this force; this engenders a nonzero flux Jm = L m m t ~ X m that had previously vanished. Since 0 - Lmm((~Xm) 2 > O, Lmm > 0. Thus, Jm6Xm > O. Now, any flow Jm of a given magnitude brings about a change of opposite sign in the associated conjugate force; this matter is explored in the comments section. Therefore, the flow Jm cannot be sustained; the system ultimately returns to the quiescent condition. We see that this leads to an application of Le Chgttelier's principle to steady state phenomena. Steady state conditions remain stationary. C O M M E N T S AND Q U E S T I O N S

6.4.1. Examine more closely the proof of Prigogine's theorem and make due allowance for the fact that one must deal properly with vectorial quantities. This may be done by putting the relations into component forms. 6.4.2. This statement is best verified by use of a live example, such as setting n = 4 and assuming that the first two forces remain fixed in time. Then carry through the various steps to check the correctness of all assertions. 6.4.3. Discuss the appropriateness of the statement that a flux always occurs in a manner that under steady state conditions causes a reduction in the conjugate force that maintains it. You may consider as illustrations the flow of a current from an isolated battery, or the passage of matter from an isolated reservoir into the system.

6.5 Onsager Reciprocity Conditions We provide here a simplified derivation of the Onsager Reciprocity Conditions (ORC) that is applicable only when steady state conditions hold. A full derivation, which is based on the machinery of statistical mechanics, is beyond our purview; the derivation, based on Tykodi's work, 1 is satisfactory for present purposes. In following the derivation the reader is advised to write out the steps in

368

6. IRREVERSIBLE THERMODYNAMICS

full, using three forces and fluxes as an example. We also dispense with vectorial notation. The phenomenological equations will be written in the form shown below, with

j--l,2,...,r. L lj X j,

(6.5.1 a)

Ji -- Z Lij X j,

(6.5. li)

J1 - ~ J

J

Jk -- Z Lkj Xj,

(6.5.1k)

J Jr -- ~

Lrj X j .

(6.5. lr)

J

Now solve Eq. (6.5.1k) for

Xk-- Jk Lkk

~ Lkj X-. ~r Lkk J

(6.5.2)

Substitute this quantity in Eq. (6.5.1i) to obtain

Li k Jk Ji -- ~ Lij Xj --} Lkk j#k

, ~ LikL kj L.--..d

j#k

Lkk

X j,

(6.5.3)

whereby all Ji have been expressed in terms of Jk and all X jr We now take partial derivatives of the dissipation function 0 - y-~j Jj Xj to obtain

00

= Xk -'}-Zi~_kXi O' ~ Ji -fk _-- Xk + ~.

LkkLi--~~,Y" k

(6.5.4)

Next, rewrite Eq. (6.5.1k) as Jk -- LkkXk + ~-~i=/=kLkiXi, and use this form to eliminate Xk from Eq. (6.5.4). This leads to

O0 __-- Jk -t- Z

OJk

Lkk

i--/:k

tik -- Zki Si.

(6.5.5)

Lkk

Referring to the previous section we note that when no constraints are imposed the left-hand side vanishes under steady state conditions, and all fluxes Jk do likewise.

THERMOMOLECULAR MECHANICAL EFFECTS

369

This forces the second term on the right to vanish as well, thereby establishing the ORC in this special case. Again, the present derivation is highly restrictive; however, it is germane to the present and later derivations, and it does furnish a simple proof of a theorem that has been shown to hold under far more general conditions. REFERENCE

6.5.1. R.J. Tykodi, Thermodynamics of Steady States, MacMillan, New York, 1967, pp. 31-33.

6.6

Thermomolecular Mechanical Effects

By now we have set up the basic machinery which permits the principles of irreversible thermodynamics to be applied to problems of interest. We next illustrate the procedure by an elementary example. The same approach will be used in later sections, with appropriate variations on the basic theme. 6.6.1

Experimental Conditions

The system under study consists of two vessels at constant volume filled with a single type of fluid and connected by a small opening; the vessels are maintained at two different, uniform pressures and temperatures. We wish to examine the heat and mass flows between the two portions of the system. Attention is focused on the quantity 0 representing the rate of change of entropy density due to processes occurring totally within the system. This permits the identification of pairs of conjugate fluxes and forces as prescribed in Section 6.1. We can use one of two formulations, namely Eqs. (6.1.27) or (6.1.29); other formulations have also been specified in the literature. Select Eq. (6.1.29) as the basis of further operations: The fluxes (presently on a molar basis, dropping vectorial notation) are then taken to be Ju and J1, and the corresponding conjugate forces are V (1 / T) and - V (/z 1/ T) respectively. Let us temporarily replace Ju by J0 and set V ( 1 / T ) = X0, and - V ( / z l / T ) -- X1. The following phenomenological relations now result, valid for fluxes and forces operating along one dimension: Jo -- LooXo + L01X1,

(6.6.1a)

J1 = L10X0 -Jr-L l l X 1 .

(6.6.1b)

Inasmuch as conjugate force-flux pairs have been selected, Onsager's reciprocity conditions apply: L01 -- L10.

370

6.6.2

6. IRREVERSIBLE THERMODYNAMICS

Reformulation of Driving Forces

For further progress it is desirable to recast (6.6.1) in terms of experimentally measurable driving forces: We set X o - - T - 2 V T and X1 - - T -1V/z1 + ( l z l / T 2 ) V T -- - T - 1 [ - S 1 V T 4- 17'1VP] + ( / 1 1 - TS1) T - 2 v T -- - 9 1 T - 1 V p 4(/41/T2)VT; thus, V1

LOl H1 - Loo T2 VT,

(6.6.2a)

V1 L 11/-)1 - LOl J1 -- - L l l - - ~ V P + VT. T2 1

(6.6.2b)

Ju - - L o l -~--VP +

In the present system nonuniformities in P and T are encountered only at the junction between the vessels; accordingly, XTP and V T may be replaced by the pressure and temperature differences at the junction, A P and AT, respectively; the thickness of the connecting unit may be absorbed in the coefficients L. 6.6.3

Steady State Conditions

The next step consists in imposing a variety of steady-state conditions on Eq. (6.6.2), to endow the coefficients with physical interpretations and to arrive at a variety of predictions. Consider first the special case where the temperature is maintained at a uniform value. The sole driving force is now the pressure difference between the vessels. Setting V T = 0 and dividing (6.6.2a) by (6.6.2b) yields a relation of the form

Ju )

= VT=0

Lol

= U~.

(6.6.3)

Lll

Here Ju is the rate of energy density transfer across unit cross-section in unit time arising from the flux in moles of species 1 across unit-cross section in unit time. This ratio is clearly the energy transported under isothermal conditions per mole of species 1, denoted by U~ in Eq. (6.6.3). We see then that a thermomechanical effect is predicted: for a fixed pressure difference across the junction, A P, and at constant temperature, a particle flux J1 gives rise to a proportional energy transport Ju -- U~ J1. This is a very sensible conclusion. A second special case is now invoked, namely the stationary state under which no mass transfer occurs, but heat flux is permitted. We now set J1 - - 0 in Eq. (6.3.2b) and solve for the ratio /-)1 - Lol/Lll

VP) VT

Jl=0

V1T

U~ _ Q~ V1T V1T

1711 -

(6.6.4)

THERMOMOLECULAR MECHANICAL EFFECTS

371

where the quantity on the fight results from use of (6.6.3); Q~ is a molar 'heat of transfer', defined by H1 - U~. We thus encounter a second physical prediction: Under conditions where mass flow is blocked, a difference in temperature between two vessels, which are allowed to interchange energy, necessarily results in the establishment of a pressure difference A P -- (Q~ / V1 T) A T between the communicating vessels. This is a physically sensible prediction. As a third special case, consider the mass flow resulting from a pressure difference between the two vessels maintained at a uniform temperature. According to Eq. (6.6.2b) this yields J1 -- - ( L 11 V1/ T) V P, which is an analogue of electric current flow arising from a difference of electrical potential. Accordingly, it is sensible to introduce a hydraulic permittivity, Z , for mass flow, defined as [Why include the minus sign as part of the definition?] (J~p) VT=0

_ Lll 1 ) 1 - - Z " T

J1 -- - Z V P .

(6.6.5)

Lastly, it is instructive to determine Ju under conditions of no net mass flow. Accordingly, we set J1 = 0, solve (6.6.2b) for V P, and use this relation to eliminate V P in (6.6.2a). This yields

JglJl-O-

--

1 (LooLll - L21)VT. T2Lll

(6.6.6)

The above represents an energy flux arising from the temperature gradient, in the absence of any net particle flow; also, at constant volume no work is performed. The resulting Ju thus is a heat flux; the proportionality coefficient in (6.6.6) is equivalent to the thermal conductivity, tc. This leads to the identification tc --

6.6.4

LooL 11

-- L~I

.

,

T2Lll

JQ -- - K V T .

(6.6.7)

Phenomenological Equations

The analysis may now be completed by collecting Eqs. (6.6.3), (6.6.5), and (6.6.7) and solving these three equations for the three unknowns L00, L01, Lll in terms of to, S , U{ or Q*I. This yields ZT

Lll -

91 '

(6.6.8a)

Lol -- U~ ---~-1 ZT '

Loo -

+ (uf)

(6.6.8b)

91

(6.6.8c)

372

6. IRREVERSIBLE THERMODYNAMICS

and when these results are introduced into (6.6.2) one obtains a complete phenomenological description of the form .

.

.

.

.

VT

(6.6.9a)

V1T _ VT. J1 - - r V P + Q'~r

(6.6.9b)

VlT

Equations (6.6.9) show explicitly, in terms of phenomenological coefficients that may be experimentally determined, how the effects of pressure and temperature gradients superpose in the system to produce concomitant fluxes of energy and of material. All prior information is contained in these relations: If a difference in T is established while no net mass flow is encountered one recovers the effect predicted by Eq. (6.6.4), and the energy flux is given by Eqs. (6.6.6) and (6.6.7). If uniform temperature is maintained the mass flux is given by Eq. (6.6.9) as J1 - - r V P and the energy flux, by Ju - - E U{ V P. If the pressure is held uniform one encounters a temperature-driven particle flux J1 - (~' Q'~/V1T)VT and an energy flux Ju - - I x - E U ~ Q T / T V 1 ] V T . The superposition effects established by both forces are formulated through the entire set of Eqs. (6.6.9). A complete analysis of the experimental results has now been furnished.

6 .7

Electrokinetic Phenomena

Here we consider the case depicted in Fig. 6.7.1 of a charged membrane (with appropriate counter-ions in solution) separating two identical solutions maintained at fixed temperature. An electric field or a pressure gradient is now applied, as a

O I

Fig. 6.7.1. Illustration of apparatus for carrying out electrokinetic experiments. Pressure is applied via movable pistons P and P' on liquids in compartments R and S. Electric fields are generated via condenser plates C and C'. Solvent and positive ions may move through a membrane M separating the compartments. Fluids may be added or removed via stopcocks I and O mounted on the pistons.

ELECTROKINETIC PHENOMENA

373

result of which both the solvent (water, designated by 0) and positive ions in solution (designated by ' + ' ) move through the membrane unit until a new steady state has again been achieved. Under the action of the pressure differential an electrical potential difference is established across the membrane; alternatively, because of the imposition of a potential gradient, a pressure difference is established between the two solutions. The physical situation may be analyzed as shown below.

6.7.1 Phenomenological Equations As emphasized earlier (see Eq. (6.1.27), for example), any flux of charged particles J+ (on a per mole basis) arises in response to the establishment of a gradient Vr in electrochemical potential. For one-dimensional flow we may write J+ = LV~" = L ( V # + Z+FV4~) = L ( V + V P + Z+FVq~), where the contribution - S d T has been dropped because constant-temperature conditions were adopted; similarly, J0 = L' V0V P. Actually, the compartments R and S in Fig. 6.7.1 are assumed to be uniform in their properties, so that the changes in P and q~ occur only across the membrane M. In this case V4~ and V P may be replaced by the discontinuities A4~ and A P across the membrane, the constant thickness of the membrane being absorbed into the phenomenological coefficients. The total flux of solvent (J0) and of ions (J+) is thus given by J0 -- (Lll I7"04- L12V+)VP -+- L12Z+FV(]),

(6.7.1a)

J+ -- (L21Vo nt- L 2 2 V + ) V P --F L 2 2 Z + F V ~ ,

(6.7.1b)

where we have set L ll = L t, L22 = L, and where we have taken care of the cross interactions by introducing the coupling coefficients L 12 and L21 that link Jo and J+ to Vq~ and to V P, respectively. In Exercise 6.7.1 the reader is asked to show that L 12 = L21. The preceding phenomenological relations may be rendered symmetric by considering instead of J0 and J+ the total volume flow Jv - ("oJo + ('1+J+ and the total current density I+ = Z+ F J+: J v -- (Lll Q2 + 2L121,5+Qo + L22~r2)Vp + Z+F(L12(/'O + L22I~+)VqS,

I+ -- Z+F(L12(/o + L22V+)VP + (Z+F)2L22V~,

(6.7.2a) (6.7.2b)

which may be abbreviated to read

Jv = L v v ( - V P )

+ LvI(-V4~),

(6.7.3a)

I+ = L v I ( - V P )

+ Lii(-Vck).

(6.7.3b)

374

6. IRREVERSIBLE THERMODYNAMICS

Eqs. (6.7.3) satisfy the Onsager reciprocity condition, showing that ( J r , - V P ) and (I+,-VqS) are sets of conjugate variables. Eqs. (6.7.3a) and (6.7.3b) are the phenomenological equations of interest. In the subsequent analysis it is convenient to generate an inverted set of phenomenological equations, by solving Eqs. (6.7.3) for the gradients in terms of the fluxes:

- V P = Rvv Jv + RVI I+,

(6.7.4a)

- V ~ = RvI Jv +RIII+.

(6.7.4b)

In Exercise 6.7.2 the reader is asked to determine the various R coefficients in terms of L 11, L 12, and L22. Again, these particular relations hold only for constant temperature conditions. Suppose that, in addition, no current flow is permitted. Then I+ = 0; according to (6.7.3b) this imposes the constraint

LvI LII

(6.7.5a)

whereas, if no pressure gradient is allowed to develop, i.e., with VP --0, one finds by division of (6.7.3a) with (6.7.3b) that

Jr) +

L VI = fit

vP=O

(6.7.5b)

LII

where/3 ~ is the is the so-called electro-osmotic transfer coefficient. The quantities on the left of Eq. (6.7.5a) and (6.7.5b) are termed streaming potentials and electro-osmosis respectively. It is immediately evident that (6.7.6) VP=O

which relationship is known as Sax&'s Law. In Exercise 6.7.3 the reader is asked to prove that (6.7.7) 0

Here the left-hand side is known as the electro-osmotic pressure, and the righthand side as the streaming current. The relations developed here point up an interesting feature: The streaming potential (Vdp/VP)I+=O cannot readily be experimentally determined, since it forces imposition of a change in electrostatic potential in the absence of a net responding current. However, this quantity is also given by the ratio

ELECTROKINETIC PHENOMENA

375

- ( J v / I + ) v P = O ~ - f i ' , which can readily be determined experimentally. Here one measures the volume flux and current in response to the imposition of a gradient in electrostatic potential when the pressure in the two compartments is identical.

6.7.2

Transport Coefficients

The remainder of this section is devoted to the specification of phenomenological equations (6.7.3) and (6.7.4) by which the coefficients L or R are eliminated in favor of experimentally measurable quantities. As a first step, solve Eq. (6.7.3b) for -V~b and substitute the result in (6.7.3a); this yields

Jv --

L v v - L2I ( - V P ) + LII

L v/I+. -LII

(6.7.8)

Then, for conditions under which no current flow occurs,

[Jv] (-VP)

L vv - L2I I+=0

LII

= L p,

(6.7.9)

where Lp is the hydraulic permeability of the membrane; note that Lp >/ 0 [Why?]. This quantity is readily determined experimentally. With f i ' - L v I / L I I Eq. (6.7.8) now reads

Jv - L p ( - V P ) 4- fi'I+,

(6.7.10)

which is known as the first electrokinetic equation. In conjunction with Eq. (6.7.3b) we now define the membrane conductivity as a --

['+] (-vr

= LII,

(6.7.11)

vP=0

so that with the aid of Eq. (6.7.5b), LVI - - a f t ' .

(6.7.12)

Introduction of Eqs. (6.7.11) and (6.7.12) into Eq. (6.7.3b) yields the second

electrokinetic equation I+ - ~ / ~ ' ( - v P ) + ~ ( - v ~ ) ,

(6.7.13)

which is simply a reformulation of the second phenomenological equation, Eq. (6.7.3b), in terms of readily measurable quantities. In Exercise 6.7.4 it is to be shown that

Jv - (Lp + otfi'2)(-VP) + a/3'(-V~b),

(6.7.14)

376

6. IRREVERSIBLE THERMODYNAMICS

which is a reformulation of the first phenomenological equation, Eq. (6.7.3a). Note how Eqs. (6.7.9), (6.7.11), and (6.7.12) have been used to solve for the individual L's in terms of experimental parameters. In addition to the preceding quantities, the following transport coefficients are in common use: The steady state electrical resistivity

P lJv=O ~

(--V(~ ) -- RII, I+ Jv=O

(6.7.15)

where (6.7.4b) was used to arrive at the relation on the fight. To realize this condition a difference in pressure must be established between the fight- and left-hand compartments of Fig. 6.7.1 such as to oppose the volume flux Jv normally accompanying the ion flux I+, which itself responds to the imposition of the potential gradient -Vq~. In the steady state the electro-osmotic flux from left to fight is counterbalanced by the hydraulic flux from fight to left. The hydraulic resistance is defined by

RH --

Jv

= Rvv,

I+=O

(6.7.16)

where Eq. (6.7.4a) was used to establish the equation on the fight. Finally, in view of (6.7.4) and (6.7.5b), the electro-osmotic flux may be rewritten as

fl~ (Jr) -~+ v P=O

_-- Rv___~i.

(6.717)

Rvv

On introducing Eqs. (6.7.15-17) into Eq. (6.7.4) one obtains final phenomenological equations of the form = RHJv - ~RHI+,

(6.7.18a)

- - 7 ~ = --~' RH Jv + pI+,

(6.7.18b)

-7P

which again involve a set of measurable transport coefficients. All the necessary information relating to electrokinetic phenomena is contained in the phenomenological equations (6.7.13) and (6.7.14) or in the equivalent set (6.7.18a) and (6.7.18b). The former set is especially useful if one inquires about state conditions under which either Jv or I+ is held fixed. The latter set is useful to characterize operating conditions at constant pressure or constant electrostatic potential. The preceding discussion illustrates the flexibility of phenomenological equations that permit either fluxes or forces to be used as dependent variables.

THE SORET EFFECT

377

EXERCISES

6.7.1. Prove that the phenomenological coefficients in Eq. (6.7.1) satisfy the ORC. 6.7.2. Express the various coefficients R of Eq. (6.7.3) in terms of the various L in Eqs. (6.7.3). 6.7.3. Derive Eq. (6.7.7). 6.7.4. Derive Eq. (6.7.14). 6.7.5. From Eqs. (6.7.4b) and (6.7.9) obtain a relation between RVI and Lp. 6.7.6. Discuss the physical mechanism that gives rise to the first electrokinetic equation. 6.7.7. Characterize the steady state of the system (a) when there is no net current flow; (b) there is no net volume flow; (c) the pressure is uniform; (d) the electrostatic potential is uniform. 6.7.8. Provide an explicit relation for the rate of dissipation of entropy for the general operation of the system and for each of the four cases cited in the preceding exercise.

6.8

The Soret Effect

As the third application of irreversible thermodynamics we consider the Soret effect (1893) for a two-component system: a flow of particles under the influence of a temperature gradient produces a gradient in concentration. We are ultimately interested in the magnitude of this effect under steady state conditions. In the present case it is expedient to adopt Eq. (6.1.27) as the starting point because we are ultimately interested in determining the variation of the mole fraction under the influence of a temperature gradient, whence it is necessary to distinguish between temperature T and mole fraction x as independent variables, while noting that both of them depend on position r within the sample. Accordingly, we consider J0 = J~, J1, J2 be the entropy and particle fluxes in response to three generalized conjugate forces, namely X0 -- - T - 1V T and X 1 , 2 - - - - T - 1V~r/z 1 , 2 as prescribed by Eq. (6.1.27). The phenomenological equations in the molar representation then assume the form Jo = LooXo + L01 X1 -q- Lo2X2,

(6.8.1a)

J1 = LloXo-+- L l l X 1 -Jr-L12X2,

(6.8.1b)

J2 - L20Xo + L21 X1 + L22X2.

(6.8.1c)

Note first that even in the absence of a temperature gradient a flux of entropy and matter can occur. For, when X0 -- 0, J1 - Lil X1 -+-Li2X2, where i = 0, 1, 2. For purposes of identification we first consider the constant temperature case, where we define (S')~ and (S')~ as the entropy intrinsic to (i.e., exclusive of entropy transport) one mole of species 1 and 2. Then at constant T the entropy flux is given by the postulated form J0 - (S')TJ1 + (S')~J2

(T constant).

(6.8.2)

378

6. IRREVERSIBLE THERMODYNAMICS

On insertion of the appropriate phenomenological equations, this yields J0 - -

((S')ILll * + ( S ' ) 2*L 2 1 ) X 1

~- ( ( S t ) l *L 12

-+-(S')~Lzz)X2.

(6.8.3)

Comparison with J o - LolX1 + Lo2X2 at constant T allows one to identify the coefficients of X1 and X2 and to solve the resulting linear equations for (S')* -

LOl L22 - Lo2L 12

1

L 1 1 L 2 2 - L 212

(S')~ =

(6.8.4a) '

L0zLll - L01L12 Lll L22 - L22

(6.8.4b)

Now apply the steady state condition under which J1 - J2 - 0 , eliminate the constraint of fixed T, and allow X0 to have a fixed, nonzero value. On eliminating X2 between (6.8. l a) and (6.8. lb) one may solve for the ratio X1

=

L 12L20

X0

-

L 11 L 22

L 10L22 --

.

(6.8.5)

L 22

On introducing the representations for X0 and X1 and Eq. (6.8.4a) we find that -(S')~VT.

V/Zl[ T --

(6.8.6)

Here V#IIT is the gradient of the chemical potential of species 1 with respect to position coordinates, which must be evaluated at constant temperature. Under this restriction,/~1 can only depend on changes in mole fraction that vary with p o s i t i o n : V / ~ l l T - - (OlZl/OXl)T(OXl/Or)T;insertion into (6.8.6) yields

r \ Or l r which is the expression for the Soret effect. This is a new, perhaps unexpected prediction based on irreversible thermodynamics: In a closed system at constant temperature a heat flow arising from a temperature gradient must produce a gradient in chemical potential under steady state conditions. For an ideal gas system Eq. (6.8.7) may be reformulated as

dlnxllr =

(S')~ RT dT.

(6.8.8)

This analytic relation shows how the mole fraction for component 1 in a twocomponent ideal gas system is changed by temperature differences prevailing under the assumed steady state condition. For the special case considered here integration leads to

ln(xl/x~ - -

-~7 dr'. fr~ (S'l*(r'))

(6.8.9)

THERMOELECTRIC EFFECTS

379

If the dependence of (S ~)*1 on T is sufficiently weak, one finds m

ln(xl/x~

-

(S')~ R In(T/To),

(6.8.10)

with (S')~ the entropy that is intrinsic to one mole of species 1, suitably averaged over the temperature interval To to T. One thereby determines the relative change in gas composition as a function of the relative temperature. S1 for this case is specified by Eq. (2.4.10). EXERCISES 6.8.1. Provide a physical mechanism which explains on a microscopic level the thermodynamic result of Eq. (6.8.7). 6.8.2. Specialize the derivation of this section to a single gaseous species. Show that under steady state conditions a temperature gradient produces a pressure gradient and express the magnitude of the latter in terms of the former. 6.9

Thermoelectric

Effects

In this section irreversible thermodynamics will be used to establish the interrelation between heat flow and electric current in a conductor. The field of thermoelectric effects has been treated elsewhere in great detail. 1 Consider a rectangular bar (Fig. 6.9.1) that is connected to two thermal reservoirs maintained at different temperatures. Provision is made for adiabatic insulation of the sample, if needed. Charge may be made to flow through the bar in

Fig. 6.9.1. Experimental setup for thermoelectric measurements. A bar is clamped between two reservoirs maintained at different temperatures T1 and T2. S and S p represent two removable strips used for thermal insulation of the bar. Current is caused to flow in the bar by continuous charging of condenser plates C and C r. (See discussion in text.)

380

6. IRREVERSIBLE THERMODYNAMICS

the same direction as the flow of heat (or in the opposite direction) by charging an external set of condenser plates. This cumbersome method is introduced here to avoid the use of electrical wire connections; these lead to distracting complications at junctions between the bar and the current leads. We are interested in the flow of charge and of heat along the bar, and in any resulting interference effects. 6.9.1

Phenomenological Relations

According to Section 6.1 an appropriate choice of conjugate fluxes and forces for the present situation is based on the dissipation function, Eq. (6.1.27), 0 - T - 1j s 9V T - T - 1j i " V ffi, where Ji is the particle flux vector ffi is the electrochemical potential acting on the particle flux, and where J s represents the total entropy density flux vector. We choose this expression, rather than a version based on Eq. (6.1.29), because we wish to treat separately the effects of temperature and of electrochemical potential. The latter involves all of the contributions associated with temperature gradients, electron density gradients, and the naturally occurring electrostatic field. It is expedient to replace the particle flux vector J i by the current density J + according to J + - ( z i e ) J i , where e is the charge on an electron and zi - • depending on whether one deals with a flow of positive charge (p, zi - + 1) or of negative charge (n, zi - - 1 ) . This is a situation generally encountered in extrinsic, one-band p-type (hole) or n-type (electron) semiconductors. The case of intrinsic semiconductors will not be considered here; interested readers are referred to Ref. 6.9.1. For current flow along one dimension we therefore set 0 - Js" ( - T - I V T ) + J+" [ - T - l z i V ( ~ i / e ) ] 9 This expression identifies the conjugate fluxes and forces in a unidirectional flow pattern and leads immediately to the phenomenological relations

Js=

J+--

Lss V T T

-

zi

L s / v (~i /e) T

Lii Lis V T - zi V(~i/e) T -~ "

(6.9.1a) (6.9.1b)

Here we have assumed that the fluxes and generalized forces are collinear and are oriented along one dimension only, which allows us to drop the vector notation. We also adopted the definition ~i = ll~i -1- zieq9 for the electrochemical potential. Note the signs of the second set of phenomenological coefficients that depend on the sign of the charge that is being transported. 6.9.2

Identification of Phenomenological Coefficients; Ohm's Law

To determine the phenomenological coefficients we now consider the special case VT - 0 . On eliminating V ( ~ i / e ) between (6.9.1a) and (6.9.1b), we obtain J s / J + - L s i / L i i ; now since Js/Ji =- S*e - ( z i e ) J s / J + is the total entropy Se*

THERMOELECTRIC EFFECTS

381

carried per particle (including both the intrinsic and the transported portion) at constant temperature we can set

J~ Lsi S* Lis = -= . J~ VT=0 Lii zie Lii

(6.9.2)

Next, examine Eq. (6.9.1b); when VT = 0, J+ = - z i ( L i i / T ) V ( ~ i / e ) . For a homogeneous sample at constant temperature this latter relation reduces to J+ -( L i i / T ) E , where g = -Y'~b is the electrostatic field, independent of the sign of the charge carriers. This represents a particular case of Ohm's Law (1826), J+ = ai g, whence (6.9.3a)

L i i / T -- a i ,

where

O"i

is the electrical conductivity of the specimen. The more general version, (6.9.3b)

( 7 T --0),

J:t: -- - z i a i V ( ~ i / e )

involving current flow in the direction of the decreasing electrochemical potential, is an elaboration of Ohm's Law that applies more generally to chemically inhomogeneous samples at constant temperature. 6.9.3

Heat Transport; Fourier's Law

Next, examine the case where no current flows: Set J+ = 0 in (6.9.1b) and then substitute for V ( ~ i / e ) in Eq. (6.9.1a). This yields

I [ Lss - L2s ] V T Js -- - -~ Lii

(J+ -- O).

(6.9.4)

Entropy flux in the absence of a net particle flow is equivalent to J q / T where Jq is the heat flux. Thus, Eq. (6.9.4) is a formulation of Fourier's Law for heat conduction, Jq = -tc V T, thereby identifying the thermal conductivity associated with the transport of charge carriers as

tc =_ Lss

- L2s/Lii

(6.9.5)

9

In the more general case J ~ 0, one may again eliminate V(~i/e) between (6.9.1a) and (6.9.1b) to obtain, in view of (6.9.5) and (6.9.2),

Js - zi

S*

e J+

K

T

V

T

, -- S e

K

Ji -

-- 7

T

T,

(6.9.6)

which shows how the total entropy flux is composed of contributions associated with the temperature gradient and with the particle flux or the current flow of either positive or negative charge. To the thermal conductivity K of the charge carrier response we should also add the lattice contribution KL, but we will not do so here.

382

6.9.4

6. IRREVERSIBLE THERMODYNAMICS

Thermoelectric Effects

For another physical prediction, return again to Eq. (6.9. lb) and set J+ - 0. One then obtains V((i/e)

VT

d((i/e) ~,

dT

Lis = -zioti

--

Lii

(J --0).

(6.9.7)

This expression shows that the imposition of a temperature difference d T in the absence of any current produces a difference d(i in electrochemical potential; i.e., d(~i/e) = --ZiOti dT. This effect is known as the thermoelectric effect, and the ratio d((i/e)/dT, or ~ A ( ( i / e ) / A T - - - z i o l i i s known as the Seebeck coefficient (1823), or thermoelectric power. 2'3 Experimentally, the difference of electrochemical potential may be measured by a voltmeter under open circuit conditions, and dT, measured by means of thermocouples; Ct i is thereby experimentally determined. As defined here for p-type (n-type) material the measured Seebeck coefficient is a positive (negative) quantity. For, (p and (n both increase in the direction of increasing hole or electron concentration, which is in a direction opposite to the increase in temperature. 4 Comparison with (6.9.2) shows that Oli : Zi S*e / e . Then Eq. (6.9.6) becomes K

Js - - oti J + -

-- V T,

T

(6.9.8)

which shows the additive effects involving the contributions from the current and from the heat flow. 6.9.5

Phenomenological Equations; Generalizations of Ohm's and Fourier's Laws

Finally, we may rewrite the phenomenological equations as follows: Since L i i / T = t y i and L i s / L i i = o t i , Eq. (6.9.1)becomes J+ -- -criotiVT

- ziaiV(~i/e);

(6.9.9a)

use of this in (6.9.8) yields

Js -- -(o'iot2 -t- T ) V T

- zioticriV(~i/e ).

(6.9.9b)

The phenomenological equations (6.9.1) have thus been reexpressed 4 solely in terms of the measurable transport coefficients cri, Ki, and c~i. Note how the distinction between carriers of opposite sign arises in the final relations. The Seebeck coefficient is equal to the entropy carried per electronic charge. Eq. (6.9.9a) represents a further generalization of Ohm's Law, showing how the current density depends on both the gradient of electrochemical potential and on the temperature gradient. Eq. (6.9.9b) specifies the entropy flux under the joint action of a gradient in electrochemical potential and in temperature; this represents a generalization of Fourier's Law.

IRREVERSIBLE THERMOMAGNETIC PHENOMENA IN TWO DIMENSIONS

383

COMMENTS AND EXERCISES 6.9.1. See, e.g., T.C. Harman and J.M. Honig, Thermoelectric and Thermomagnetic Effects and Applications, McGraw-Hill, New York, 1967. The presentation given here is of limited (but nevertheless, didactic) utility since it applies only to a metal that is modeled in the free electron approximation or to extrinsic semiconductors. For more complicated models, particularly those involving charge transport by electrons and holes in multiband materials, the more elaborate analysis presented in advanced treatises is required. 6.9.2. This step requires some care; under an external gradient positive and negative charges move in opposite directions, but the particle flux vectors are multiplied by e and by - e , respectively. Hence, in both cases the currents point along the direction of the electric field vector, but the latter points in a direction opposite to that of the increasing electrochemical potential gradient for holes. 6.9.3. This particular appellation is highly undesirable and should be eliminated in favor of the designation as Seebeck coefficient. 6.9.4. Note that the cross coefficients in the phenomenological equations as written differ in sign for n-type material. This reflects a corresponding sign difference in the original phenomenological relations (6.9.1). However, if we had adopted V(~n/(-e)) as the force conjugate to Jn, a s well as the defining relation V ( ~ n / ( - - e ) ) / V T : Otn, then the ORC conditions would have been satisfied and the phenomenological equations for both sets of charge carriers would have been identical. The proof of this statement is left as an exercise. 6.9.5. How (if at all) must the above approach be modified to deal with transport of positive and negative ions in solution? 6.9.6. Provide a physical interpretation for the origin of the thermoelectric effect by noting the difference in kinetic energy of the charge carriers at the two ends of the sample. 6.9.7. Introduce E* as the internal energy carried by an electron moving under the influence of an external electrostatic field. Relate this quantity to transport coefficients Lss, Lis, Lii, and thence, to the transport coefficients introduced in this section. 6.9.8. Consider thermoelectric measurements that are carried out under adiabatic conditions for which Js = 0. Relate the resulting electrical conductivity and Seebeck coefficient to the quantities introduced above.

6.10

Irreversible Thermomagnetic Phenomena in Two Dimensions

In this section we consider effects arising in conjunction with the geometry illustrated in Fig. 6.10.1. A rectangular slab is aligned with the x and y axes of a Cartesian coordinate system, and a magnetic field Hz is directed along the z axis. Provision is made for flux of current and of heat along x and y. One is interested in the possible interference effects that may be encountered in such a system. This leads to a consideration of what are termed thermoelectric and thermomagnetic phenomena; the magnetic field will be shown to give rise to a host of new cross interactions between processes occurring along the x and y directions.

384

6. IRREVERSIBLE THERMODYNAMICS A

/x

Fig. 6.10.1. Parallelepiped geometry for current/heat/entropy flux along the x and y directions in the presence of a magnetic field Hz aligned with the z direction.

To facilitate the exposition, a somewhat different approach will be used relative to the methods introduced in the earlier sections. As in Eqs. (6.9.1), we select (Js, VT) and (J, V ( ( / e ) ) as the conjugate set of variables but will absorb the minus signs and the T - 1 factors in the phenomenological coefficients. Since we deal with electron flow effects we consider J to be the electron current (designated by J _ in Section 6.9) and ( as the electrochemical potential for electrons (designated as (,, in Section 6.9). Three new points are introduced at this time: (i) Since fluxes may occur in two orthogonal directions, the conjugate flux-force pairs now are: (jx, VxT), (JY, VyT), (Jx, Vx((/e), (Jy, Vy((/e)). The appropriate geometry is depicted in Fig. 6.10.1. (ii) For later convenience we shall select as independent variables from this particular set the quantities Vx T, Vy T, Jx, Jy, so that the phenomenological equations appear in partially inverted form as follows: jx _-LllVxT

- L 1 2 V y T -Jr-L13Jx -k- L14Jy,

(6.10.1a)

JY -- L 1 2 V x T - L l l V y T - L14Jx + L13Jy,

(6.10.1b)

V x ( ( / e ) -- L13VxT -t- L14VyT nt- L33Jx nt- L34Jy,

(6.10.1c)

V y ( f /e) - - L 1 4 V x T

(6.10.1d)

-k- L 1 3 V y T - L34Jx -Jr-L33Jy,

where the Lij a r e appropriate phenomenological coefficients. (iii) For later convenience we have arbitrarily selected the minus and plus signs in the indicated sequence in Eq. (6.10.1a); the other signs in Eq. (6.10.1) are then governed by the Casimir-Onsager reciprocity conditions, Eq. (6.3.8), as required by the presence of a magnetic field H - lcHz. We now engage in a systematic treatment of the thermodynamics of irreversible processes in the above configurations. Consider first the isothermal case summarized by the constraints: (a): Jy = Vx T = VyT = 0. Isothermal conditions are maintained along x and y, and no current is allowed to flow along y. Then

IRREVERSIBLE THERMOMAGNETIC PHENOMENA IN TWO DIMENSIONS

385

Eqs. (6.10.1) reduce to jx = L 13Jx,

(6.10.2a)

JY = - L 1 4 J x ,

(6.10.2b)

V x ( ( / e ) = L33Jx,

(6.10.2c)

Vy ((/e) = -- L 34Jx.

(6.10.2d)

According to Eqs. (6.10.2c) and (6.10.2d) current flow along x generates a gradient in electrochemical potential along both x and y. The first effect is simply a manifestation of Ohm's Law Jx -- plVx ( ( / e ) , wherein P I = L33 is the isothermal resistivity. The second is an example of the isothermal Hall effect, characterized by L34 Vy ((/e) -- - ~ J x H z =_ R~ Jx Hz,

(6.10.3a) (6.10.3b)

wherein, for convenience, the magnitude of the applied magnetic field has been introduced explicitly. As Eq. (6.10.3b) shows, a flow of current longitudinally induces a transverse gradient in electrochemical potential. The magnitude of this effect is specified by the Hall coefficient, defined a s R I = - L 3 4 / H z . We next consider the constraints (b)" Jy - Vx T - Jsy -- O. No current flow is allowed along y and no heat flow is tolerated in this direction. Isothermal conditions are maintained along x. This represents a (transverse) adiabatic set of operating conditions. Equations (6.10.1) now reduce to Jsx - - L 1 2 V y T

+ L13Jx,

(6.10.4a)

0 -- - L l l V y T - L14Jx,

(6.10.4b)

V x ( ( / e ) - L 1 4 V y T + L33Jx,

(6.10.4c)

V y ( ( / e ) -- L 1 3 V y T + L34Jx.

(6.10.4d)

Equation (6.10.4b) shows that current flow along x produces a temperature gradient along y; this is the so-called Ettingshausen effect, specified by

VyT

L14

--

~ Jx Hz LllHz

=-- T J x Hz,

(6. lO.5a) (6.10.5b)

in which T - Vy T~ Jx Hz is the Ettingshausen coefficient. On inserting (6.10.5a) into (6.10.4c) one finds ( Vx(f/e)-

L24) L33 - ~ l l Jx,

(6.10.6)

386

6. IRREVERSIBLE THERMODYNAMICS

which is of the form of Ohm's Law under adiabatic conditions, with an adiabatic resistivity PA -- L33 - L24/L11. When (6.10.5a) is introduced in (6.10.4d) one obtains the expression

Vy((/e)--

L13L14) L34 + ~ (Jx Hz), Lll

1 Hz

(6.10.7a)

which represents the adiabatic Hall effect, with

RA =

1

Hz

L34

d--

L13L14) Lll

~

9

(6. lO.7b)

We next consider conditions (c): Jx = Jy = Vy T = 0. No current flow is permitted, but a temperature gradient is established along x, while isothermal conditions are maintained along y. The phenomenological equations reduce to Jsy -- - L l l VxT,

(6.10.8a)

jX=L12VxT,

(6.10.8b)

Vx (( /e) = L13 Vx T,

(6.10.8c)

L14 V y ( ( / e ) -- -L14VxT -- - ~ H z V x T .

(6.10.8d)

Under the postulated conditions T J x and T J y represent heat fluxes. Then Eq. (6.10.8a) leads directly to the definition of an 'isothermal heat flux' (a contradiction of terms!): T J x --- - L l l TVx T, whence we may write tcI-- T L l l ,

(6.10.9)

where KI is the thermal conductivity when no transverse temperature gradient is allowed to exist. According to Eq. (6.10.8c), a longitudinal temperature gradient produces a longitudinal gradient in electrochemical potential. This represents nothing other than the 'isothermal' Seebeck effect, introduced in Section 6.9. Thus, with Vx ((/e) = L 13Vx T one finds the relation OtI = L13,

(6.10.10)

where O~I is the Seebeck coefficient for electrons in the absence of a transverse temperature gradient. Next, according to Eq. (6.10.8d), a temperature gradient along the x direction produces a gradient of ( / e along the y direction; this is a manifestation of the transverse Nernst effect: the relation V y ( ( / e ) = - ( L 1 4 / H z ) H z V x T suggests the definition of a corresponding transverse Nernst coefficient as NI = - L 1 4 / H z .

(6.10.11)

IRREVERSIBLE THERMOMAGNETIC PHENOMENA IN TWO DIMENSIONS

387

Another set of operational conditions frequently encountered is specified by (d)" Jy - Jx - JY - O . Here, no currents are allowed to flow, and adiabatic conditions are imposed along the y-direction. Then the phenomenological relations (6.10.1) reduce to

jx = - L l l VxT - L12VyT,

(6.10.12a)

0 - - L12VxT - Lll VyT,

(6.10.12b)

Vx((/e) = L13VxT + L14VyT,

(6.10.12c)

Vy(~/e) = - L 1 4 V x T + L13VyT.

(6.10.12d)

Equation (16.10.12b) shows that the establishment of a longitudinal temperature gradient gives rise to a transverse one. This interrelation is known as the RighiLeduc effect. It is convenient to rewrite (16.10.12b) as L12 Vy T -- ~ Hz Vx T, LllHz

(6.10.13a)

whence the Righi-Leduc coefficient becomes

VyT

L12 ----Mr = ~ . Hz Vx T L I ~Hz

(6.10.13b)

Another relation of interest is found by inserting Eq. (6.10.13a) into (6.10.12a), and multiplying through by T; this yields

rJx--r

Vxr,

(6.10.14a)

which gives rise to the definition for the 'adiabatic' thermal conductivity.

KA--T Lll +~111 "

(6.10.14b)

Use of (6.10.13a) in (6.10.12c) yields

Vx(~/e)--

L134-

L14L12]VxT, Lll

(6.10.15a)

which is the Seebeck effect when adiabatic conditions are maintained in the transverse direction. The corresponding coefficient reads OtA-

L14L12 L13 + ~ . Lll

(6.10.15b)

388

6. IRREVERSIBLE THERMODYNAMICS

Finally, if (6.10.13a) is combined with (6.10.12d) and the magnetic field is explicitly introduced, one finds

,[

V y ( ( / e ) - -~z

- L 1 4 -Jr- L 13L 12 ] Hz T, Lll 3

(6.10.16a)

which gives rise to the adiabatic transverse N e r n s t effect, with a corresponding coefficient of the form 1 NA--~z

[

-L14+

J

~L13L12 . Lll

(6.10.16b)

Many more effects may be treated on an analogous basis, as is suggested by Exercises 6.10.1 and 6.10.2. The physical basis on which these effects rest is to be explored in Exercise 6.10.3. One should note that the various coefficients listed in this section are all measurable experimentally according to the prescriptions imposed by the boundary conditions (a)-(d) and by the indicated definitions for each coefficient. Note that, having set up phenomenological equations in partially inverted form, the phenomenological coefficients L jj in Eqs. (6.10.1) assume a particularly simple form: Lll = K I / T ,

(6.10.17a)

L 12 - - H z K . A ~ r / T,

(6.10.17b)

L13 =otI,

(6.10.17c)

L14 = - H z N I ,

(6.10.17d)

L33 = PI,

(6.10.17e)

L34 = - H z R .

(6.10.17f)

On inserting these relations into (6.10.1) one thus obtains a complete description of irreversible processes for the system under study. This in turn, permits an analysis to be made of the 560 possible galvano-thermomagnetic effects that can be achieved in the rectangular parallelepiped geometry of Fig. 6.10.1. EXERCISES 6.10.1. Develop phenomenological relations for the set of conditions Jy - Vy T - j x = 0 and prove that a temperature gradient is set up along x as a consequence of current flow along that direction. What is the resultant heat flux along y ? Express the resistivity and Hall coefficients in terms of the various Lij. 6.10.2. Impose the conditions Jy - j x _ 0 on the phenomenological equations. Express Vy T and Vx T in terms of Jx. Express the resistivity and Hall coefficient in terms of appropriate Lij and compare your results to those in the text. 6.10.3. Provide physical mechanisms that show how transverse interference effects arise under the various boundary conditions that were taken up in the text.

CHEMICAL PROCESSES

6.11 6.11.1

389

Chemical Processes Chemical Reaction Rates and Affinities

Irreversible phenomena pertaining to chemical processes may be handled by the same techniques as previously employed. At uniform temperatures and constant electrochemical potentials Eq. (6.1.27) becomes 0 - T -1 ~-~r COrAr >/O, which then leads to a set of reaction velocities (fluxes) COr that respond to the corresponding driving forces Ar, the chemical affinities introduced in Section 6.1. In what follows we closely adhere to the treatment provided by Haase. 1 Note that at equilibrium COr - Ar -- 0; however, situations may arise where (i) COr = 0, Ar #: 0, corresponding to inhibited reactions that may be remedied by introduction of a suitable catalyst, (ii) COr -%=0, Ar = 0, as in thought experiments in which a reaction is carried out under near-equilibrium conditions. If only one process is considered (r = 1), then coA > 0, so that co and A must have the same sign. If two processes occur simultaneously, colA1 § O92A2 ~ 0; thus, for example, it is possible to have col ~A1 < 0 if O)2J42 > IcolA1 [. 6.11.2

General Phenomenological Equations

Consider two reactions of the type A Z B and B Z C. If the third process A Z C is not feasible the number of elementary reactions is the same as the number of linearly independent reaction equations; the reactions are said to be uncoupled. If A ~ C represents a feasible reaction the three processes are said to be coupled; generally, coupling occurs whenever there is a redundancy in the number of reaction steps. For situations not far removed from equilibrium (what this implies will be fully documented later), one postulates the usual linear relations between fluxes and forces. In the present context (As should not be confused with the generic chemical symbol A) we set up R phenomenological equations of the form R

(-Or-- Z arsAs

(r - 1,2 . . . . . R).

(6.11.1)

s=l

Coupled equations are characterized by nonvanishing cross coefficients: ars 7A 0 for r :~ s. The dissipation function is given by R

R

O - ~-~ Z ars~ArAs ~ 0. r----1 s--1

(6.11.2)

390

6.11.3

6. IRREVERSIBLE THERMODYNAMICS

Two Coupled Reactions

It is instructive to specialize to the case of two reactions (R - 2)" 091 = aliA1 + al2A2, a21 = a 1 2 .

0)2 = a21.,41 + a22A2,

(6.11.3)

Then, noting that the T -1 factor was absorbed in the phenomenological coefficients, we obtain 0 -- all.A 2 + 2a12.A1.A2 + a22A 2 ~> 0,

(6.11.4)

which requires a l l /> 0, a22 /> 0, and a l l a l 2 -- a122/> 0 (see Section 2.2). Where there is no coupling, al2 - a21 - 0; in that event, o91,,41 -- a l l , / [ 2 /> 0 and w2A2 -a22A~ ~> 0.

6.11.4

Applicability of Linear Approximation

We next inquire as to the range of validity of the linear approximation. For this purpose note that if a system is characterized by n + 1 deformation coordinates x i , then, in general, o9 = co0(xl, . . . , Xn+l) and ,4 = .Ao(xl . . . . . Xn+l); one may thus eliminate Xn+l between the two functions to obtain 09 = W ( X l , . . . , X n , ,4) and ,4 = A0(Xl . . . . . X n + l ) . But as A --+ 0, o9 --+ 0 as well, so that the deviation of ,4 from zero may be taken as a measure of the deviation of the system from the equilibrium conditions at which the xi assume their equilibrium values xi - x ~ It is therefore reasonable to expand o9 as a Taylor's series in ,4 while setting all xi - x ~ on retaining only the term of lowest order, one obtains 0o9)

,,4 + . . .

(i - 1, 2 . . . . , n).

(6.11.5a)

x/-xO On writing co = aA, one finds that 0o9)

a--

(6.1 1.5b)

x/-xO

identifies the coefficient a. To check on the adequacy of the linear approximation we now introduce the l a w o f m a s s a c t i o n in the form (_D -- K I-I c~ i -- KI I - I cy j , i

j

(6.11.6a)

CHEMICAL PROCESSES

391

corresponding to the schematic reaction K

Z viAi = Z vj A j, KI

i

(6. l l.6b)

j

where, as usual, the v's are stoichiometry coefficients and the A's are reacting species; K and tc~ are reaction rate constants for the forward and reverse process as written in Eq. (6.11.6b). Now rewrite Eq. (6.11.6a) as (6.11.6c) in which o f is the rate of the forward reaction, tc l-Ii cVi, and k = K~/K. Referring back to Section 2.10 one notes that the affinity may be reformulated as (6.11.7) where K is the equilibrium constant appropriate to the reaction (6.11.6b) when #l is referred to the standard chemical potential. Thus, K e x p ( - A / R T ) - lie cVee; when this expression is introduced in (6.11.6c) one finds oJ- cof(1-kKe-A/Rr).

(6.11.8)

Now at equilibrium, co - A - 0; according to (6.11.8) this means that k K - l, so that one obtains the final expression in the form co -- ~of(1 - e - A / R T ) ,

(6.11.9)

which involves an exponential dependence of co on A. It is now clear that the postulated linear dependence in Eq. (6.11.3) may be justified only if I A / R T I << 1; in which case one may approximate 1 - e - A / R r by A / R T . This specifies what one means by "small departures from equilibrium" as a prerequisite for the application of the linear phenomenological equation co = aA. Comparing co = o)f jA/R T with (6.11.5a) one notes that (_Of//R T = (Oco/OA)xi=xO i . On the right-hand side the subscripts clearly refer to equilibrium conditions; thus o f may be computed from the equilibrium values of the various concentrations, denoted by c o in the present context. Thus, one may set -

I-I (c ~

=

This

nally lead

co -

to

co~ ,,4 - a A RT

(6.11.10)

as the phenomenological expression for the rate of a process; the phenomenological coefficient a is also identified in this procedure.

392

6. IRREVERSIBLE THERMODYNAMICS

REFERENCE

6.11.1. R. Haase, Thermodynamics of Irreversible Processes, Addison-Wesley, Reading, MA, 1968.

6.12

Coupled Reactions: Special Example

Consider the following case of three coupled reactions denoted by A ~ B (I), B ~ C (II) and C ~- A (III), and compare this situation to the linearly independent reactions A ~ B (1) and B ~ C (2). The rate of disappearance of the various reagents is related to the reaction rates, co, by

dnA dt dnB dt dnc dt

--

O91

--

091

--

09111,

:

0)2 -- 601 - - (-OII -- 0)1,

- - --0)2 - - 0)III -- 0911.

(6.12.1)

Comparing the co's with Arabic and Roman subscripts, one obtains CO1

--

091

--

60111,

(6.12.2)

0)2 - - O)II -- WIII.

The chemical affinity for reaction r is given by ~4r ~ E g 1)gr/Le, where e refers to the various species that are involved. In the present case ~AI - - .,41 - - / L B - - / / , A , ./411 - - A 2 - - / Z C - - / Z B , AIII-

-(,A1

d- ,,2[2) - / Z A

--/ZC,

(6.12.3)

and the dissipation function is given by 0 - - O) 1 A 1 4- 092.,42 - - w I . A I + coII.AII q- (.oIIIAIII.

(6.12.4)

At equilibrium the co and A vanish for the linearly independent reactions: Wl w2 - 0 and A1 -- A2 -- 0. From (6.12.2) it now follows that for the linearly dependent case o9I AI-

--

o911

--

AII-

(6.12.5a)

o)III, AIII-

0

(equilibrium conditions).

(6.12.5b)

COUPLED REACTIONS: SPECIAL EXAMPLE

393

Note that the co's in (6.12.5a) have been shown to be equal, but at this stage they do not necessarily vanish. When Eq. (6.11.7) is adapted to the present case one finds .AI = R T l n ( K I C A / C B ) , AII -- RT ln(KiicB/cC), AIII = RT ln(Kiiicc/cA).

(6.12.6)

At this state we introduce the laws of mass action and use v to denote the rate of the forward reaction. Also we set k =_x'/K, where tc and tc' are the rate constants for the forward and reverse reactions as written. Then o91 -- KICA -- KICB -- VI

OaII - - KIICB - - / r

1-

- - VII

ki~ CA

,

1 - - kII

(

o)III -- KIIICC - xiiiCA -- viii 1 -- kIII cc

"

(6.12.7)

When (6.12.6) is introduced in these relations one obtains the expressions ~oi -

vi [1

-

ki KIe - A I / R

r ],

~Oli -

vii [1 - klI Kiie - A " / R

r

], (6.12.8)

OaIII - - VIII[ 1 - - kIII K I I I e - ' A I I I / R T ] .

At equilibrium where the co's are equal (see Eq. (6.12.5a)), and when .A vanishes, VI(1 - - k I . A I ) = v i i ( 1 -

kII~AII) = v i i i ( 1 -

klli.Aili).

(6.12.9)

The principle of detailed balance or microscopic reversibility is now invoked. It states that at equilibrium each elementary process proceeds as readily in one direction as in the other. According to this principle, one must demand that o)I O)II - - O)III - - 0 ; note that this enlarges on requirement (6.12.5a). It is an immediate consequence of this requirement that ki K i ---- 1 in Eq. (6.12.8), i = I, II, III. It then follows that O)I-

vI(1 -

e-AI/RT),

(.Oil -- VII ( 1 -- e-'AII/R T ), 09111 - - VIII ( 1

-- e - J4iii/R

T ).

(6.1

2.10)

These are the fundamental equations of interest. Close to equilibrium, where << l, one obtains the linearized forms

IA/RTI

v0

kI -- ~ . A I ,

0 vii I

v0

klI - ~-~ AII,

klII

-- ~-~ AIII,

(6.12.11)

394

6. IRREVERSIBLE THERMODYNAMICS

where the equilibrium values of the v's have been introduced in accord with the discussion of Eq. (6.12.10). Now on account of (6.12.2) and (6.12.3), Eq. (6.12.10) assumes the final form M1

+

0 M2

+

(6.12.12)

.IRr,

0 M1 0 A2 0)2 -- VIII ~ - ~ + (V 0 + VIII) R T"

(6.12.13)

These expressions are in the form of linear phenomenological equations wherein 0)1 -- a l l A 1 + a l 2 A 1 and o92 -- a21A1 -q--a22.A2, with a12 -- a21 -- v ~ all + /RT, and a== + oI /RT. This example illustrates the fundamental principle that if one describes coupled reactions in terms of a set of linearly independent steps, then sufficiently close to equilibrium the reaction rates may be formulated in terms of linearly phenomenological equations involving the chemical affinities as driving forces and the reaction rates, as responses. -

6.13

-

Coupled Reactions, General Case

The preceding argumentation will be briefly extended to cover the case of more complex reactions of the type vimAi ~

( m - 1 2 . . . . M),

vjmAj

K/m

i

(6.13.1)

~

j

involving M elementary reactions indexed by m, of which R, indexed by r, are linearly independent; M ) R. The notation used below conforms to the scheme pem

I - I C'e f.

Vjm

17

l l j cj

(6.13.2)

Vim 9

YIi C'i

The rates of the various reactions are specified by the law of mass action 0)m ~ Km

I--I~Pim ci ~ K mlH~l)Jm cj i

j

~ Vm

(1H -- km

~l)g'm) ,

ce

(6.13.3)

f.

~l)im represents the forward rate of reaction m and km :where Vm - Km I-Ii ci l ~Vf.m Km/g m. Using A m - - ~Qe 1)f.m~m -- R T l n ( K m / Fie ce ) one obtains 0)m -- Vm(1 -- k m K m e - A m / R T )

(m -- 1, 2 , . . . , M ) .

(6.13.4)

COUPLED REACTIONS, GENERAL CASE

395

As already discussed, at equilibrium only the reaction rates mr of the R linearly independent reactions can initially be assumed to vanish. To handle the correspondence between the linearly dependent and independent reactions we relate the . A m and J 4 r affinities through linear equations of the form R

Am -- E

brmAr,

(6.13.5)

r=l

wherein the brm a r e appropriate combination coefficients. Since the rate of entropy production has to be independent of the manner in which one writes out the reaction sequences, one must h a v e Z m ~ -E r ('OrAr ; using (6.13.5), one obtains R

M

R

E Zbrmo)mJtr-ZO)r.Ar. r=l m=l

(6.13.6)

r=l

This implies that M

Or ~ Z

brm(-~

( r - - 1, 2 , . . . , R).

(6.13.7a)

m=l

For the set of mr in (6.13.7a), one may specify at equilibrium that R

Or-- Z

brm~

( r - - 1,2 . . . . ,R),

-- O

(6.13.7b)

m=l or

Ar - 0

(equilibrium conditions).

(6.13.8)

However, by the principle of detailed balance, one can further specify that 6_Om - - 0 (m -- 1, 2 . . . . , M); also, , A m - - 0 at equilibrium. From Eq. (6.13.4) it now follows that kmAm - 1, i.e., O)m - -

Vm

(1

-

(m - 1, 2 , . . . , M),

e -Am/RT)

(6.13.9)

so that we have once more recovered an exponential dependence of O) m on Am, even in the case of M coupled equations of any degree of complexity. For o ~ Km Vii ci ~Vim (m -- 1 . .2,. . . . M ) and Eq. (6.13.9) then ] A m / R T ] << 1, 09m -- o)m reduces to the following linear form 0

O9m = Vm A m , RT

(6.13.10)

396

6. IRREVERSIBLE THERMODYNAMICS

where the superscript 0 again refers to equilibrium conditions. Because of the interrelations between coupled and uncoupled reactions, one also obtains 1 (-Or : e z

M

1 brm YmAm 0 - ez A

~

M

R

I

m=l

Z

Z

brmbsm Ym OAs

(r -- ] , 2, . . . . R ) "

m=l s=l

(6.13.11) This expression gives rise to a set of linear phenomenological equations R

(.Or -- Z

arsAs

( r - 1 , 2 , . . . , R)

(6.13.12a)

s--1

with M

ars -- ~

0

brmbsm 1)m.

(6.13.12b)

m=l

This set of equations, obeying the Onsager reciprocity conditions, obtains only near equilibrium. Ordinarily the linear approximation for the reactions is valid over a far more limited range than is the linear approximation for the types of processes discussed in Sections 6.7-6.11. This therefore restricts the usefulness of the present analysis.

397

Chapter 7 Critical Phenomena

7.0

Introductory Remarks

In this chapter we provide a heuristic introduction to scaling procedures that characterize properties of systems close to their critical point. The objective is to provide some general insights and to convey the flavor of the methodology. Much of the treatment of critical phenomena falls into the province of statistical mechanics, well outside the confines of classical thermodynamics. However, certain illuminating aspects that can be discussed without having to resort to statistical approaches will be taken up below. We center the discussion on critical exponents and the Landau theory of critical phenomena. For a proper exposition of the subject the reader is referred to several sources in the literature. 1

7.1 7.1.1

Properties of Materials Near Their Critical Point Scaling Procedures

The problems arising in the thermodynamic approach to critical phenomena were briefly alluded to in Section 2.3, in conjunction with the discussion of the van der Waals equation of state. The difficulty here is that all standard thermodynamic treatments rest on an averaging procedure: a given molecule moves in and interacts with the average field of all the other material in the system, leading to a mean field systematization of experimental observations. In normal circumstances this works well for macroscopic systems~witness the presentation of all the material in this book in the preceding chapters. However, as we will see shortly, near the critical point of any system one begins to encounter large scale fluctuations. These affect the physical properties of the material that cannot be quantitatively characterized by the standard procedures. Hence the need for a new approach that comes under the heading of scaling laws. Scaling procedures are prompted by the failure of commonly used functions of state to deal with the properties of materials close to criticality Consider again

398

7. CRITICAL PHENOMENA 1.0

!

0.9 0

0.8 0.7 0.6 Z 0.5 0.4 1.1

0.3 0.2 0.1

....1

0

Water Nitrogen

x

Methane

o Ethylene x n-Butane

o

_

9 +

......... I ........

i ......

1

i

......

I

.......... 1 ......... I . . . .

2

3

i

4

I 5

P/Pc Fig. 7.1.1. Plot of the compressibility Z for five different fluids against reduced pressure P/Pc at a series of reduced temperatures T/Tc, as indicated by the numerical figures next to each graph. Note the remarkable superposition of the data points and the significant deviation of the actual curves from the ideal gas law Z - 1.

the theoretical d r o s o p h i l a for fluids, 2 the van der Waals equation of state. As was established in Section 2.3, in its reduced formulation the equation of state reads ~r + ~5

(3v - 1 ) - 8v,

(7.1.1)

where Jr = P / P c , v = V / V c , z = T / T c , P is the pressure, V the volume, T the temperature, and where the subscript denotes their values at the critical point. This relation is independent of the van der Waals parameters a and b, and should thus be applicable to any fluid; it is termed the L a w o f C o r r e s p o n d i n g S t a t e s . Similarly, the c o m p r e s s i b i l i t y r a t i o Z = P V / R T = 3/8 for a van der Waals fluid is a universal constant. The extent to which real fluids obey the van der Waals equation of state is shown in Fig. 7.1.1, in terms of the dependence of the ratio Z on reduced pressure at different reduced temperatures for five different systems near their respective critical points. Obviously a certain universality scaling law is well obeyed at each T / T c , but the observed numerical values for Z differ greatly from prediction; hence, the need for a different approach that leads to a theory in agreement with experiment. For a second illustration of the quantitative failure of the van der Waals equation close to criticality consider small deviations t = r - 1 = ( T - T r and q9 = v - 1 = ( V - V c ) / V c from the critical point, at which Jr = r = v = 1. In

PROPERTIES OF MATERIALS NEAR THEIR CRITICAL POINT

399

T9

T8 '~~T7 ~T6

I

~"'Q

I

T1

v --------~ Fig. 7.1.2. Schematic diagram for the pressure-volume relationship for a van der Waals fluid

at a series of temperatures. The Maxwell equal area rule applies in each case.

terms of these parameters the reduced equation of state then becomes

8(1 + t) 7r-

3

3(1 -+-99)- 1 - (1 --k q~)2'

(7.1.2)

which, near the critical point may be expanded as 3 ~3 27 3 rr -- 1 + 4t -6~ot - ~ + 9~o2t + -~-~o t.

(7.1.3)

We may discard the last two terms. The reduced coexistence volumes vl (P) and Vg(P) for the liquid and vapor phase are in equilibrium at T < Tc; equivalently, so are the corresponding quantities ~0l and ~0g. We now connect the properties of the liquid and vapor phases: with reference to Fig. 7.1.2 (where, however, the pressure vs. volume is plotted in a highly schematic manner for purely illustrative purposes) we apply Maxwell's equal area rule to require that, for a fixed value n

t < O, f~g V dP - 0 - Vc f~g (V/Vc - 1 ) d P . The last integral follows since f;gdP vanishes over the path indicated in the figure. Since by (7.1.3) dP = -Pc(6t + 9q92/2)d~0 the vanishing integral may be reformulated as of

~Tg Pc~p[6t + ~g~ d~P - 0 .

(7.1.4a)

400

7. CRITICAL PHENOMENA

This equation must be satisfied for all fixed t for which the approximation (7.1.3) holds. On integration we find that we must enforce the requirement 9 3t (q92 _ q92) + g(~p4 _ q)4) _ 0.

(7.1.4b)

The only way to satisfy Eq. (7.1.4b) is to require that qgg = +q)t. We discard the positive root since this leads to an identity. On applying the condition qgg = -qgl to Eq. (7.1.3) we obtain

7"(g -- 1 + 4t - 6tqgg -

33

7rl -- 1 + 4t + 6tq)g

-~qgg,

+

33

-~g.

(7.1.5)

On subtraction and solving for q)g o n e predicts that (fig

--

2~-t

~

I Tc - T ] 1/2

Tc

(T

<

Tc),

(7.1.6)

showing that the reduced volume should change as the square root of the small deviation of T from its critical value. This is in serious disagreement with experimental observation: Fig. 7.1.3 shows a plot of reduced temperature vs reduced density for eight distinct fluids. 3 The collapse of almost all these data onto a single curve is striking. However, when these data are replotted as P / P c - 1 vs. IT~ Tc - 1 I, they fall on a curve with a critical exponent close to fi ~ 1/3, rather different from the value 1/2 cited in Eq. (7.1.6). Measurements on many other systems have shown that in general q ) ~ ( - t ) ~, with/~ in the range 0.31-0.33. Similarly, at T = Tc, t = 0 in Eq. (7.1.3) one finds that Jr- 1-

P-Pc Pc

=--

3IV-Vc] ~

2

Vc

3 '

(7.1.7)

which also disagrees with observation: experimentally the reduced pressure varies with reduced volume as q)~, with ~ in the range 4.6 to 4.9. These findings again reflect the fact that the van der Waals equation of state has been applied to a situation where mean field theory fails because fluctuations dominate. Therefore, some other means of characterizing critical phenomena in fluids must be found. Many other physical phenomena are subject to similar problems. Among the most commonly treated of these is the ferromagnet. Fig. 7.1.4 shows how the magnetization of nickel changes with rising temperature. 4 The initially gradual falloff accelerates as a critical temperature Tc is reached, beyond which the material remains essentially unmagnetized. The decline of magnetization A4 near the critical temperature follows the same power law dependence on temperature as density measurements for fluids: 34 "-~ IT - Tel ~ with fi again in the range 0.33. This similarity in characteristics may be understood on the basis that the elemental units in a fluid are the constituent atoms, whereas those of the ferromagnet are the electron spins residing on lattice sites. As the critical point is

PROPERTIES OF MATERIALS NEAR THEIR CRITICAL POINT . . . .

! .....

I

!

i ...........

l ............

!

i

'1

401 !

s

1.00 o

0.95

~

~176

0.90 0.85

T/-I'c

;} 0.75 -i

t 0.70

,

0.65

[r

+ Ne 9 Ar

,~ N2 v 02

,, Kr x Xe

oCO o CH4 o

0.60 !! 0.55 0.0

,

~

I

i

i

i

l

,

i

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

, 2.0

~ 2.2

~ 2.4

,

2.6

P/Pc Fig. 7.1.3. Data near the critical point of eight fluids involving the reduced temperature T~ Tc plotted against the reduced density P/Pc. Note the remarkable superposition of the data points.

d

M/Mo

0

1 T/-I-c

Fig. 7.1.4. Dependence of the reduced magnetization .AA/.A4O on reduced temperature T~ Tc for Ni metal. The solid points indicate experimental measurements. 4 The reduced magnetization represents an order parameter of unity at T = 0 and of zero at the critical temperature Tc.

402

7. CRITICAL PHENOMENA

approached from above, magnetized domains with perfect spin alignments grow as islands within the unmagnetized matrix, and their size ultimately becomes very large compared to the atomic dimensions. Similarly, as the critical temperature of a fluid is approached from above, liquid droplets of increasing size begin to float in the vapor phase and ultimately combine to form a larger liquid phase. Thus, one can conceive of an analogy between magnetization and liquid-gas transitions near the critical point. However, as in the case of liquids, the mean field theory leads to an incorrect specification of the critical exponent for magnetization: Using the average field theory approach outlined in Section 3.17 (here the constituents A and B correspond to 'spin up' and 'spin down' states) A4 is found to vary as IT - Tel ~, with/3 - 1/2. Again, a different approach must be used to obtain results consistent with experiment. 7.1.2

Thermodynamic Properties near the Critical Point

Preliminary to such a search we examine several thermodynamic properties of fluids at or close to criticality, that clearly show why and how fluctuations dominate under such conditions. (i) Consider first the isothermal compressibility, X l = --(OV/OP)T/V. At the critical point the isotherm ( O P / O V ) T has zero slope; thus, x i grows indefinitely as T --+ Tc. (ii) Using Eq. (1.3.13) and the definition for KI one finds that ( O V / O T ) p = - ( S V / O P ) T ( O P / O T ) v = KIV(OP/OT)v, wherein ( O P / O T ) v does not vanish. Therefore, the coefficient of thermal expansion, /37- = ( 1 / V ) ( O V / O T ) p also grows without limit as the critical point is approached. (iii) According to the Clausius-Clapeyron equation in the form A H = T ( V g - V l ) ( d P / d T ) , the heat of vaporization of the fluid near the critical point becomes very small, since Vg - Vl --+ O, whereas d P / d T remains finite. Thus, in circumstances peculiar to criticality, just below Tc the liquid state becomes highly compressible, and can readily be converted to the vapor with expenditure of very little enthalpy. This, in turn, produces significant fluctuations in the density and related properties of liquid and gas. Nevertheless, since the overall density of the closed system is fixed at any particular temperature, small regions of high density must be compensated for by small regions of lower than average density. These types of large scale variations, which attest to the growing indistinguishability between liquid and vapor near the critical point, are ignored in the thermodynamic mean field theory. The peculiar properties cited above are reflected in the following experimental findings. (i) As already mentioned, near the critical point the densities of the liquid and gas converge toward each other according to the relation IPl - Pg[ ~ ITc - TI ~,

where/5 is termed a critical exponent.

fl - 0.326 4- 0.002,

(7.1.8)

PROPERTIES OF MATERIALS NEAR THEIR CRITICAL POINT

403

(ii) The chemical potential difference near criticality is specified by # - #~ ~ IP - P~l s-1 (P - P~),

(7.1.9)

3 - 4.80 + 0.02.

(iii) The heat capacity at constant volume obeys the relation

Cv ~" IT - Tel -'~,

(7.1.10)

oe = 0.110 + 0.003,

which explicitly shows the power law divergence of the heat capacity at the critical point. (iv) The isothermal compressibility for a wide variety of substances obeys the following relation:

M_l(Oft)

_

p_2X

I ~

IT - Tel - y ,

y - 1.239 4- 0.002,

(7.1.11a)

where M is the gram-molecular mass. 5 This again shows the power-law divergence of XI. Equivalently, we may write for the pressure-volume dependence

(P -

I V - Vcl -l(vc- v).

Pc)"~

(7.1.lib)

The truly remarkable feature is that the values of the various critical exponents are universal, i.e., independent of the chemical constitution of materials whose properties are being studied near their respective critical points. As already stated, for fluids this feature arises because the range of fluctuations in properties such as density greatly exceeds atomic dimensions. NOTES AND REFERENCES 7.1.1. N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Perseus Books, Reading, MA, 1992. H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, London, 1971. S. Ma, Modern Theory of Critical Phenomena, W.A. Benjamin, Reading, MA, 1976. 7.1.2. I have adopted this characterization as a direct quote from N. Goldenfeld, loc. cit. 7.1.3. E.A. Guggenheim, J. Chem. Phys. 13 (1945) 253. 7.1.4. E Weiss and R. Forrer, Ann. de Physique 15 (1926) 153. 7.1.5. This relationship is established as follows (v =_ V/n): =

vZ(oP)

v2 (0~) -

p

-

-

-

-

-

U

M

M - -

v2 - -

7x,

404

7.2

7. CRITICAL PHENOMENA

Homogeneity Requirements, Correlation Lengths, and Scaling Properties

Because of the failure of the van der Waals equation of state near the critical point we look for a different characterization of fluids. We note that with diminishing T close to Tc droplets of liquid begin to form in the vapor; there is obviously a distribution of droplet 'sizes', but it is possible to specify an average 'diameter' ~, known as a correlation length, that shows the physical extension of the droplet floating in the vapor. It also specifies the range over which fluctuations in physical properties tend to be uniform. This macroscopic quantity is expected to depend on the experimental conditions, and to grow without limit as the critical temperature is approached from above. Here again the sizes of the homogeneous regions of materials near criticality become very large compared to atomic size, whence critical phenomena should be independent of the microscopic characteristics of the system. Indeed, this is verified by the collapse of all data onto a single curve in Fig. 7 . 1 . 3 ~ a very commonly quoted piece of evidence.

7.2.1 ScalingRevisited A theoretical treatment emphasizing the above feature is found in scaling properties. Under ordinary circumstances we impose the homogeneity requirement on the extensive variables that occur in the Gibbs function G as expressed by (see Section 1.3) G ( T , P; )~nl, ~.n2, . . . , ~.nr) = ~.G(T, P; nl, n2, . . . , nr).

(7.2.1)

As a generalization it is now proposed to introduce the following scaling hypothesis for the chemical potential of pure materials near their critical point: /z()~Ctt, )~r p) -- )~/z(t, p),

(7.2.2)

where ~'t and ~'p are scaling exponents for the scaling p a r a m e t e r )~, all to be determined by experiment. Here t = T~ Tc and p =_ P / P c represent the reduced temperature and pressure. Note that the scaling now involves the intensive variables of the state function/z. At the end of this section we present a somewhat loose rationale for adopting the present procedure. We now write tz(t, p) - - S T c dt § V Pc dp,

(7.2.3a)

and tz(~.r162

-- -STc)~ r d t + VPc)~ Cp dp,

(7.2.3b)

and carry out partial differentiations with respect to p at constant t, Olz(t, p) Op

= PcV(t, p),

(7.2.4a)

HOMOGENEITY REQUIREMENTS, CORRELATION LENGTHS, SCALING PROPERTIES

405

and

O[Jb(~~tt, ~P p) 0(~ ~pp) = e c ~ p ~ (i~ t t, i~~pp). O(~,~Pp) Op

(7.2.4b)

Then, after differentiating Eq. (7.2.2) with respect to p at constant t, and inserting the above expressions one obtains (7.2.5) No restriction has been imposed on the parameter )~. We take useful advantage of this freedom by setting )~ - ( - t ) -1/c' or )~Ct _ ( _ t ) - I (t < 0) and then take the limit P --+ Pc, i.e., p --+ 0. This is consistent with the requirement that correlations become very large near the critical point. Eq. (7.2.5) may then be solved for W(t,

0) --

( - t ) (1-ffp)/~t 9 ( - 1 , 0 ) .

(7.2.6)

According to the above it is predicted that close to the critical point the molar volume of the fluid changes with temperature deviations from criticality as - t raised to some power, which function is multiplied by a constant that is of no interest; this is a first step toward characterizing processes near criticality. As a second step, return to Eq. (7.2.5) and set t = 0, so that V(0, p ) = )~r (t,(0 ' )~Cpp) and then select )~ - p-1/Cp; thus,

9"(0, 1),

f'(o, p) -

(7.2.7)

which indicates that near criticality the molar volume changes as p raised to some power, multiplied by a constant of no interest. We can link the above results to experiment: Eqs. (7.2.6), (7.2.7) hold separately for the liquid and vapor phases, hence the difference, ~ - Vg (-t)(1-r162 may be compared to the experimental result A V ~ ( - t ) ~, which leads to the identification fl = (1 - (p)/(t. Eq. (7.2.7) may then be reformulated to read V - V0 ~ M ( p o - p ) / p 2 ~ A p ~ p(1--~p)/(p, where p represents the density. Comparison with the experimental finding p - Pc ~ p l/~ shows that 8 = ( p / ( 1 - (p). The above relations for/3 and 8 may be solved for

(t - - / 3 (8 + 1)'

8 ffp =

--------~.

0--1-1

(7.2.8a)

(7.2.8b)

These expressions specify the scaling exponents in terms of experimentally accessible parameters.

406

7. CRITICAL PHENOMENA

A second important result is based on use of the relation 1 0 # / O p (V)2xI / M I ; tel is the thermal conductivity and XI 1/KI. When the same scaling hypothesis is applied to the fight-hand side of Olx/Op that has been adopted in setting up Eq. (7.2.2) it follows that --

~

,2 2

t,

p) x,,

t,

p) -

f 2 (t,

x1 (t,

(7.2.9)

When Eq. (7.2.5) is inserted one obtains (7.2.10)

Xi(~.~tt, )~pp) - )~2~p-lxi(t , p).

Since there is no restriction on the choice for )~ we now introduce the convenient identification )~ -- ( - t ) -1/c~ and p --0; then XI(t, O) -- ( - t ) (2~p-1)/~t

X(-1, 0).

(7.2.11)

Based on experimental observation we cite the relation XI Itl -y, which shows how XI diverges near the critical point. We can thus identify "~

1 -

), -- ~ .

2~'p

(7.2.12)

fit We have thereby specified the scaling exponents in terms of several experimentally observed critical exponents and have succeeded in providing a thermodynamic description of the properties of materials near their critical points. The above procedures are not self evident; they may be justified principally by their success in quantifying experimental observations on materials near their respective critical points. A further heuristic argument proceeds as follows: In a regime where the correlation length ~ is very large compared to the atomic size the idea of scaling invariance may be invoked: it should not matter whether the fundamental unit of length L adopted for use is chosen to be ten times or a hundred times the fundamental scale, so long as L remains appreciably smaller than ~. This should be reflected in relating the Gibbs potential G(ts, ps) for a relatively small volume L 3 to the same potential that pertains to the larger size G(tl, Pl), where t =- (T - Tc)/ Tc and where p =_ (P - Pc)/Pc. It is reasonable to posit that G(tl, Pl) -- L d G(ts, Ps), where d is the dimension of the system (3 in this case). We next postulate that p should change linearly with size: as the cell area increases, so does the force required to maintain a steady pressure, but in a manner that accounts for the smaller effect of the surface area as the cell size is increased. This effect is addressed by assuming that pl = Up (L)ps. A similar argument is adopted to write tl - .Tt(L)ts; this ensures that t for the cell goes critical when the system does. To specify the multiplying factor we can reasonably assume that f p (L) ~ L q and f t (L) ~ Lr, with q and r unspecified. We are thus led to the relation G ( L r t, L q p) -- L d G(t, p), which is equivalent to Eq. (7.2.2).

DERIVATION OF GRIFFITH'S AND RUSHBROOKE'S INEQUALITY

407

In any event, adoption of the above scheme leads to a framework within which the divergence and other effects near the critical point can be rationalized. However, even after studying the foregoing rationalization the reader may still be excused for wondering whether all this seemingly artificial machinery really explains anything or has any relation to reality. There is a simple experimental test: Insert Eq. (7.2.8) into (7.2.12) to obtain the following interrelation among scaling variables: V =/3(6 - 1).

(7.2.13)

The above is known as Widom's equality. Its applicability has been tested and verified under many different conditions, thus vindicating the entire approach. In early work in this area there was no reason to suspect the existence of any interrelation between critical exponents. Several other relations may be cited; among them: ot +/~(6 + 1) = 2,

(7.2.14)

which will be derived in the next section, as will be ot + 2/3 + V = 2,

(7.2.15)

V(6 + 1 ) = ( 2 - or)(3 - 1).

(7.2.16)

Note again that the critical exponents do not depend on the physical characteristics of any material. Therefore, all data pertaining to their properties near the critical point should collapse onto a single curve, as is exemplified for fluids by Fig. 7.1.3. Other types of verification and tests are quoted in the literature, lending strong support to the basic soundness of the approach. In other words, we may assume the appropriateness of using relations such as (7.2.6), (7.2.7), (7.2.11) in interpreting the properties of materials near their critical points. NOTE

7.2.1. See the derivation in Note 7.1.5.

7.3

Derivation of Griffith's and Rushbrooke's Inequality

Here we derive Eq. (7.2.14) on the basis of somewhat tortuous thermodynamic arguments. An alternative, simpler procedure is provided in the next section, but the present method is instructive and included for that reason. We base our considerations on the portion of the liquid-gas phase diagram very close to the critical point, shown in Fig. 7.3.1. One of the two isotherms is taken at the critical temperature Tc; the other, at a lower temperature T1 very close to Tc. We investigate

408

7. CRITICAL PHENOMENA

I

r~

p I

L

P#

M

"

R

S~~~T

I

I

I

vl

vo

vg

t

Fig. 7.3.1. Pressure-volume diagram (schematic) for a fluid near its critical point. See text for details.

the energy changes in going around two closed loops: MNSRM and LMRQL, so as to include the critical point in both traverses. To prevent excess proliferation of symbols we will, in the present section assume that all extensive quantities are molar variables, thus omitting the tilde symbol here. For the M --+ N segment the energy change is given by AENM-

Tc(SN-

(7.3.1a)

P(Tc)dV.

SM)-

Similarly, for N ---> S we find (7.3.1b)

AEsN -- fTT ~Cg dT, and for S ~ R,

P(T1)dV,

A E R s - 7'1 ( S R - S s ) -

(7.3.1c)

while for R ---> M, A EMR --

Cc d T.

(7.3.1d)

Next, write (l and g stand for liquid and vapor respectively) (7.3.2a)

SN -- SM ~ (SN -- Ss) + (Ss - SR) -1- (SR -- SM),

SR -- Xl SQ + xgSs,

gc = x l gl -+-x g gg,

Xl --~-Xg =

1,

(7.3.2b)

DERIVATION OF GRIFFITH'S AND RUSHBROOKE'S INEQUALITY

Ss-SR(dP)

Wg- Wc

~

409

(7.3.2c)

TI"

Then Eq. (7.3.1 a) may be rewritten as

AENM--Tc

[fT,T`--~ Cg dT + (Vg -

-

Vc)(dP Cc dT] --~ ) T1 + fT~ 1 --i-

P(Tc)dV,

(7.3.3a)

and Eq. (7.3.1 c) becomes A ERs -- - TI ( Vg - V~ ) - ~

~,

--

F

P(T1)dV.

(7.3.3b)

Now sum around the closed loop MNSRM, for which the energy changes add to zero, and rearrange to find (P (T1) = P, a constant)

T r~ dT -- fI ~C~ C g

T c - T

~Cc T

,IT 1

--

dT + (Tc - T1)(Vg - Vc) - ~

T,

T

P(Tc)dV + P(T1)(Vg- Vc).

(7.3.4)

To the above add and subtract P1 (Vg - Vc) and rearrange to find

~

T~ T c - T ~CcdT 1

T

_--~ ~c T c - T ~Cg l

dT +

(Pc - P) dV

T

+ ( T c - T1)(Vg- Vc)

-~

T1

PCTc - _P T~ ( T1) 1"

(7.3.5)

Now carry out the same steps for the LMRQL loop, with the proviso that a minus sign enters at the LM and RQ stages relative to the MN and SR stages. We obtain

~

rc T c - T ~CcdT l

T

_ fT T~ ~T Cc -l T 1

-(r~-

dT +

F

T 1 ) ( V c - Vl)

( P - Pc)dV

-~

r,

Pc - _P T1 ( T1) ]

(7.3.6)

410

7. CRITICAL PHENOMENA

Next, multiply Eq. (7.3.5) by (Vg - V c ) respectively; then add the two. We obtain (

-1

and Eq. (7.3.6) by ( V c -

~/~)-1

1)fTIt~Tc-TCcdT

1

v~- Vc + V~- V~

-----T---

1 f T ~ ~~TCc g- Td T Vg - Vc T 1

+Vc-Vl +v~-v~

fT Tc T c - T C~ d T +

1~

1

V~ - Vc

fviZg(pc_P)TcdV

(P - Pc)Tc dV.

(7.3.7)

Now, the first two terms on the right are positive definite. Therefore, on deleting them the left-hand side becomes greater than the truncated right-hand side; which leads to the following inequality:

(1 1)fTy~Tc-TCcdT v ~ - V c + Vc-V~ ~ >

1

vs - Vc

( P c - P)T~ dV +

if

v~ - v~

( P - Pc)r~ dV.

(7.3.8)

Into the above we now insert the following scaling relations:

IV - VcI -- B(T - Tc) ~

(B > O),

P - Pc-- - D I V - Vcl a-I(V - Vc) Cc= A(Tc - T) -a

(D > O),

(A > O).

(7.3.9a) (7.3.9b) (7.3.9c)

To satisfy the inequality (7.3.8) the integrals on the right must be so formulated that the integrands remain positive. These two integrals then are required to have a form that may be evaluated by standard methods to yield

1 1 - V gD - Vc fviZg( V - Vc)~ dV = D ( V1g+- V6c ) ~

(7.3.10a)

D fvc D ( V c - Vl) ~ . 12 - V c - V1 Jr, ( V c - V) '~d V = 1+ ~

(7.3 10b)

and

DERIVATION OF GRIFFITH'S AND RUSHBROOKE'S INEQUALITY

411

Then,

11+12-- 1 +D6 [ ( V g - V c ) ~ + ( V c - V l ) ~ ] -

]~

1--~c

. (73" "11)

In introducing the last step we recognized that Vg and Vl are variables, whose place is taken by V in Eq. (7.3.9a). In the integral on the left of (7.3.8) T1 is supposed to be extremely close to Tc; hence, no significant error is introduced in replacing T in the denominator of the integrand by Tc. We thus write

I3 -- A

/TTo'(Zc--Z)l-~ dT -- ATc1-~ /T1 ( 1 - ~cc)1-0
=

2-o<

Tc

1/r,.

1 - -Tc

.

(7.3.12)

Here T1 is the variable quantity T very close to Tc. On introducing this relation on the left of (7.3.8) and using (7.3.11) on the fight we finally obtain (~c) 1-

2-~

>

DBI+S( 2 - ~ ) A(1 + a )

Tff +t~

(1+8)-1

-- L > 0.

(7.3.13a)

Lastly, we take logarithms on both sides to find [ 2 - o< - fl(1 + 6 ) ] [ l n ( 1 - T/Tc)] > lnL.

(7.3.13b)

This must be examined carefully. T~ Tc is very close to unity; hence, the logarithmic factor becomes a hugely negative quantity. Thus, to preserve the inequality two possibilities must be considered: (i) Either L > 1; so that the right-hand side is positive; then, for the inequality to remain effective the multiplier of the logarithm on the left must be negative, so that necessarily oe + fl(1 + 6) > 2; (ii) or 0 < L < 1. In that event the right-hand side is negative. But we can make ln(1 - T~ Tc) as negatively large as we please by moving T ever closer to Tc. There is then no way of satisfying the inequality except by demanding that the multiplier of the logarithmic term on the left vanish. Then o< + fl (1 4- 6) = 2. We conclude that o< + fl(1 + 3) ~> 2,

(7.3.14)

which is Griffith's inequality. When numerical values are inserted into Eq. (7.3.14) one obtains oe + fl(1 + 3) = 2.000 + 0.022, equal to the value 2 within experimental error. In fact, by adopting the static scaling hypothesis in common use in statistical mechanics one finds that the inequality (7.3.14) is replaced by an equality, as specified by Eq. (7.2.14). We also prove this assertion below by a different approach.

412

7. CRITICAL PHENOMENA

Rushbrooke's inequality is far easier to obtain: By comparing the analogous relations d E - T dS - P d V and d E - T dS + ~ 0 dA4 one readily converts Eq. (1.14.7) into the form

CT-t

CM --

T(O.A4/OT)~

( O.A4 / O~-~O) T

.

(7.3.15)

On the left we omit the positive term C M, so that the equality is replaced by an inequality. We also introduce the conventional relations

M-B

07-[0 T

--xo--D

(

1-~

(B > O, T < Tc),

(D>0,

1-

T )-c~ 1 - - ~ cc

CT~--A

T < Tc),

(A > 0),

(7.3.16a)

(7.3.16b)

(7.3.16c)

which reflect how the magnetization approaches zero and how the magnetic susceptibility and heat capacity diverge as the critical temperature is approached from below. On inserting (7.3.16a-c) into the inequality (7.3.15) we obtain

( ~1-c )

-ct-2(fl-Y 1-)

Tfl2B 2 -- Q, TZAD

(7.3.17a)

and on taking logarithms we find

[(

-[ct + 2(/3 - 1) + ?,] In 1 - -~c

> In Q.

(7.3.17b)

This inequality is handled precisely in the same manner as Eq. (7.3.13b). If Q > 1 then the left-hand quantity in square brackets has to remain positive; if 0 < Q < 1 then this quantity must vanish. We therefore find that ot + 2/5 + ~, ~> 2,

(7.3.18)

which is known as Rushbrooke's Inequality. Again, one finds that, given the numerical values for the critical exponents, this relation holds as an equality. This will also be verified below by considering magnetic properties of materials near their critical points. In addition, the reader is invited to establish, starting with Eq. (1.14.7), that Eq. (7.3.18) also holds for fluids. 1 Eq. (7.2.16) is now obtained by combining Eqs. (7.2.13), (7.2.15).

DERIVATION OF GRIFFITH'S AND RUSHBROOKE'S INEQUALITY

7.3.1

413

Magnetic Properties Near the Critical Point

To drive home the point that critical phenomena for different physical situations are analyzed in the same manner, and to convert the above inequalities into equalities we now consider the magnetic characteristics of material close to the temperature at which magnetic order gives way to disorder. We begin with the Gibbs free energy in the form dG = - S d T - 3A d ~ ; to avoid encumbering the notation we here omit the subscript 0 f r o m 7-[. We have also quietly dropped the pressure term Pm that includes the magnetic contribution, as specified by Eq. (5.8.4). For small magnetic fields this condition may be approximated by keeping the mechanical pressure fixed. Note the parallelism with the case of fluids. The starting point is the usual scaling relation; we set H =_ (7-{ - ~c)/7-[c to obtain

G()~att,)~ann)--~.G(t,n),

(7.3.19)

where t =_ (T - Tc)/Tc. Now differentiate with respect to H" Xa/-/

OG()~ at t, Z aI-IH )

= X

O()~aI-IH )

OG(t, H )

OH

.

(7.3.20)

On executing the required partial differentiations one obtains - M = -(.AA A4c)/.AAc, which leads to the scaled equation

LaI4M(Zatt, zaI4H) - - Z M ( t , H ) .

(7.3.21)

We now set H - - 0 and note that the above relation must hold for any 3~ of our choice, and in particular, for )~ - ( - 1 / t ) 1~at . Then M ( t , O) -- (--t)(1-aH)/atM(--1, 0).

(7.3.22)

This may be compared to experimental observation" As t ~ 0 - it is found empirically that AA ~ ( - t ) ~, whence 1 ~

o H

/3 - ~ . at

Next, we set t - 0 in Eq. (7.3.21) and adopt the choice )~ -

(7.3.23)

H-1~all, whereby

M(O, H ) -- ~ (1--aH)/aH M(O, 1).

(7.3.24)

For small fields one finds empirically that 17-/1 ,-~ AA~, whence we obtain

aH 8- ~ . 1 -a/-/

(7.3.25)

414

7. CRITICAL PHENOMENA

These two last relations may be solved for

at=

/3(1-+-6)

an-

and

I+6'

(7.3.26)

thereby linking the critical exponents to experimental parameters. We now carry out a second differentiation of the Gibbs potential, Eq. (7.3.19), with respect to H to obtain the magnetic susceptibility X (we again omit the 0 subscript). This leads to the result

~.2an x (~.at t, )van n ) -- ~,X (t, H ) .

(7.3.27)

Then set H - - 0 and take ) ~ - ( - t ) - l / a t , this leads to

X(t, O) -- ( - t ) -(2aH-1)/at X ( - 1 , 0).

(7.3.28)

This formulation may be compared to the empirically determined relation X "~ ( - t ) - • 2 1 5 for T < Tc, T > Tc respectively. We then see that ,

2all

y -- ~

, at

-- 1

(7.3.29)

so that when we insert (7.3.26) we obtain y' --/3 (~ - 1),

(7.3.30)

which is Widom's equality. The same argument may be made to obtain 9/. We may return to the relation specifying G(T, H) and differentiate twice with respect to T to obtain - C / - / / T . Translating to the scaled variable equation we find that

~2at CH(~att, )yaH H ) -- )vf H(t, n ) .

(7.3.31)

Now set H - O , ~ - (--t) -1~at , which converts the above to the form

CH(t, O) -- (--t)(1-2at)/atCH(--1, 0).

(7.3.32)

Experimentally it is found that the variation of heat capacity with deviation from the critical temperature proceeds as C / - / ~ ( - t ) -c~'. This immediately establishes the correspondence , 1 - 2at ot = ~ . at

(7.3.33a)

When substituting from (7.3.26) one obtains c~' +/3(1 + 6) = 2,

(7.3.33b)

which is Rushbrooke's relation as an equality. Many other interrelations between various types of critical exponents have been derived, but we refer the reader to other sources 1 for an exhaustive listing.

415

SCALED EQUATION OF STATE

REFERENCE AND EXERCISE

7.3.1. See references in Note 7.1.1. 7.3.2. Show that Eq. (7.3.18) is recovered when considering fluids as a working substance. What do you conclude from your analysis?

7.4

Scaled Equation of State

It is possible to establish a scaled equation of state for magnetic materials near their critical points along the following lines: we return to Eq. (7.3.21) and set X = (-t) -1~at, such that

(' ")

M(t, H) --It] (1-an)/atM Itl ItlaH/a' ~

o

(7.4.1)

Now introduce (7.3.23) and rewrite the above in the form

M(t,H) itl/~

(tH) - M

ItS' Itl ~

"

(7.4.2)

It is expedient to introduce a scaled magnetization as m = it l-~ M(t, H), and a scaled magnetic field as h - I t l - ~ H; then the above relationship becomes m = m(+l,h),

(7.4.3)

or in inverse form, = h(=i=l, m).

(7.4.4)

According to this analysis magnetization data plotted in this manner should produce two branches of a single curve, depending on whether T > Tc or T < Tc. This is found experimentally to be the case, as long as one is 'sufficiently close' to the critical temperature.

7.5

Landau Theory of Critical Phenomena and Phase Transitions

In this Section we study an approach to critical phenomena and phase transitions that was pioneered by Landau. It is based on the use of the order parameter, q, as the characteristic variable to describe the degree of ordering in a binary system, analogous to the quantity s that was introduced in Section 3.17. We concentrate on the thermodynamic characteristics of a system whose properties depend on the degree of its deviation from a phase change. As an example one may cite the gradual change in lattice structure resulting from a so-called soft mode, a decrease in a particular lattice vibration frequency and concomitant increase in bond length,

7. CRITICAL PHENOMENA

416

that precedes the change in lattice symmetry at a transition temperature Tc. Alternatively, it might describe the degree of approach of the system to the critical point, discussed at length in the preceding parts of this chapter. In each instance one assumes with Landau that for sufficiently small departures from the relevant critical point the thermodynamic properties of the system can be expanded in ascending powers of r/. We thus write 12 -- ao + al r/+ a2r/2 -q- a3r/3 -k- a4r/4 -k- ...,

(7.5.1)

where s is a functional that becomes the Gibbs free energy of the system only after the quantity 77has been optimized. We may ordinarily discard the irrelevant constant a0. The extent to which the phenomenological equation (7.5.1) serves as a basis for describing phase transitions can only be decided by experiment. Nevertheless, the utility of this relation is perhaps best gauged by examining the generic phase diagram of Fig. 7.5.1. Entered as inserts in the various regions are the forms assumed by the molar Gibbs free energy G(V) when the volume of the phase is caused to deviate from its equilibrium value. Away from the T-C phase boundary and for T < Tc these Gibbs potentials exhibit two asymmetric minima (1, 2, 3) associated with the liquid and vapor phases. At the boundary separating the phases (4) these minima are symmetric, reflecting the equality of the chemical potentials in the two-phase configuration. At the critical point Tc (5) one encounters a shallow, essentially fiat bottom, and away from that condition (6) one may find a skewed configuration. The Gibbs potential for the one phase fluid (7) is U-shaped. Reference to Fig. 7.5.2 shows that these various potentials are all properly mimicked by Eq. (7.5.1) on assignment of different values to the parameters ai. It is therefore evident that Landau's approach should be useful in dealing with the phenomenological aspects of the phase transition. This is reinforced by the fact that curves of the type shown in the figure also properly reproduce the molar Gibbs free energies derived from the regular solution theory, as specified by the parameters A(T) and r of Eq. (3.14.24). Somewhat similar results are obtained by setting a - - 0 and allowing c to assume various values. Obviously, enormous leeway is attained when all four coefficients are allowed to vary. To make contact with physical properties it is assumed that at least some of the coefficients depend parametrically on temperature (and, if needed, on other variables such as pressure). We now examine several operating conditions that correspond to the various curves displayed in Fig. 7.5.2. As a guiding principle, in the absence of external forces we may optimize s by imposing the equilibrium constraint

0s

= 0 - al + 2a2q0 + 3a3r/2 + 4a4r/03 + ' " .

(7.5.2)

LANDAU THEORY OF CRITICAL PHENOMENA AND PHASE TRANSITIONS

417

Liquid

~ (v) G (V)

5

~ (v)

~ (v) . ~

\,d

c 7

G (V)

~J

3

Gas

Fig. 7.5.1. Generic phase diagram for one-component system; the inserts show in various locations of the diagram how the molar Gibbs free energies change when the volume is displaced from its equilibrium value. These various curves are well reproduced by the Landau functions displayed in Fig. 7.5.2.

At equilibrium the Gibbs free energy is a minimum when r/ vanishes, so that a l - 0 as well. This leaves Eq. (7.5.1) in the form s -- a2q 2 -+- a3r/3 + a4r/4 Jr- . . . . 7.5.1

(7.5.3)

Expansion in Even Powers of the Order Parameter

In applying these concepts we first consider the case where/2 is an even function of r/, which implies the absence of external constraints that impart a direction to the manner in which the transition point is approached. On setting a3 - 0 one obtains a2 -+- 2a4r/2 - - 0 .

(7.5.4a)

The above coefficients a2 and a4 may themselves be expanded in terms of the deviation of the system from the transition. If temperature is the relevant control

418

7. CRITICAL PHENOMENA -f(x)

-f(x) k

0.07[

\ k

~176 ~176 o.o,I ~176 ~176

-\

.

.

.

.

.

.

.

(b) t

/

/

/

0.0 0.02

o~

/ -

-0.4 -0.2

/

(a)

0.2

.

0.0

.

.

.

.

.

.

0.4

.

.

x

_

-0.4

_

.

X

--

0.2

-0.2

0.4

-f(x)

-f(x)

0.08

0.03

0.06

(C)

0.02

(d)

0.04 0.01

.....

0.02 .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

-o.~ -0.2

-f(x)

-f(x) o.o3 j ( e l

0.03

t

0.02

(0

9

0.02

0.01

0.01

......

:oy .

.

",-,,"

.

.

.

.

.

.

.

x

-0.01 -0.02

-f, x) 1

(

0.8 0.6 0.4

=0~s-o-~--o~,~.o2 ~ ~ 5 ~

==x

Fig. 7.5.2. Plots of f ( x ) = a x + b x 2+cx 3+dx 4. (a) a = 0 , b=-0.3, c=d=0, (b) a = b = c = O , d = - 5 . (c) a=O, b=0.5, c=O, d = - 3 . (d) a = 0 . 0 5 , b=0.5, c=O, d = - 3 . (e) a = - 0 . 0 2 , b = 0.5, c = 0, d = - 4 . (f) a = 0.02, b = 0.5, c = 0, d = - 4 . (g) a = 0.7, b -- 0.5, c - 0, d - - 3 . Similar curves are obtained by setting a = 0 and allowing c to vary.

variable we may set a2(T)-

a 0 -t-

T-Tc Tc

t a2 + ' ' "

(7.5.4b)

LANDAU THEORY OF CRITICAL PHENOMENA AND PHASE TRANSITIONS

419

where the prime indicates a partial derivative of a2 with respect to T, evaluated at T=0. a4(T)

--

a~ +

T-Tc

l

Tc

a 4

+....

(7.5.4c)

In the approximations used below we may neglect the expansion term in a4 because their presence leads to second order effects that are ignored; thus we set a4 equal to a constant. Also, without loss of generality we may set r/-- 0 for T > Tc, and 77 ~ 0 for T < Tc. The solution of Eq. (7.5.4a) reads q0-0

or

Oo---+-v/-a2(T)/2a4.

(7.5.5)

But if r/0 is to be nonzero in the range T < Tc, we must also set a ~ -- 0, to exclude a2~ - - [ ( T - Tc)/Tc]a~2 as a possible solution that causes a2, hence r/o, to vanish. We see then that r/o varies as ~ / T c - T. Note the two symmetrically placed equilibrium values of the order parameter. As an example of how the above result is applied consider the phase transformation in a solid that is the result of a gradual displacement of atoms from their original position in a 'soft mode' transition. For simplicity we adopt a onedimensional model and suppose that the gradual shift in atomic location may be modeled in terms of anharmonic terms in the lattice vibration of the participating atoms. This is quantified via the relation V - Vzx 2 - V4x 4 for the potential energy in terms of the displacement x of the oscillator from its equilibrium position. The above represents a first order expansion for the anharmonic term. In the present approximation we replace the x 4 term by (X2)X2, where the quantity in angular brackets is an averaged displacement variable, given by kB T~ V2, where kB is Boltzmann's constant. Thus, we find that V = Vzx 2 - V4x 4 = [V2 k B T V 4 / V z ] x 2, from which one deduces that the vibration frequency varies as v ~ [ V z - k B T V 4 / V 2 ] 1/2, which is ofthe form v ~ v0(1 - A T ) 1/2 that agrees with the formulation (7.5.5). Surprisingly, soft mode transitions do indeed conform to this very simple model. The transformation is complete at T - Tc, where v = 0. As an alternative, it is obvious that the result (7.5.5) coincides with the mean field approach to describe the critical phenomena of fluids. It is evident that this model corresponds to the formation of Gibbs free energy curves such as shown in panel (c) of Fig. 7.5.2. It relates to the boundary at which a system executes a first order transition; the minima correspond to the 77o values given by Eq. (7.5.5). 7.5.2

Effects Encountered in External Constraints

In a second application of the above concepts consider the magnetization of a material in an external magnetic field 7-{ for which the relevant order parameter is the magnetization A4 = r/and for which the (free) energy is - ~ A / / p e r electron

420

7.CRITICAL PHENOMENA

TH
TIc T

H=O T>Tc~-

T:~~///

/ I ]~-"/_ T
H>O ~ m>Tc~l~

T< c l./~..J

Fig. 7.5.3. Variation of the Landau free energy density with the order parameter at temperatures above, at, and below the critical temperature and for changes in direction of the magnetic fields. The heavy dots indicate the value of ~/for which the Landau free energy density is minimized. The central row represents a discontinuous first order transition as the temperature is dropped past the its critical value. After Goldenfeld loc. cit.

spin. We now consider the functional s

1

2 -Jr- -~bq 4,

(7.5.6)

where we have introduced trivial notational changes relative to Eq. (7.5.1); the temperature variation in the second order term is explicitly introduced through the factor t = (T - Tc)/Tc, and the t dependence of the fourth order term is ignored. Obviously, the subsequent analysis is not limited to magnetization effects; any free energy change that in part depends linearly on an order parameter rl is treated in the same manner. Several possible variations of s with r/ in conformity with Eq. (7.5.6) are sketched in Fig. 7.5.3. These correspond to several of the diagrams shown in Fig. 7.5.2. The three rows depict the variations encountered when the temperature is altered while keeping the field at values 7-/< 0, 7-/= 0, and 7-/> 0 respectively; the heavy dots locate the minimum, equilibrium value of 7"/.The three columns left to fight depict the s variation with ~ for the cases T > Tc, T = Tc, and T < Tc respectively. Note that in going down the left-hand and central columns

LANDAU THEORY OF CRITICAL PHENOMENA AND PHASE TRANSITIONS

421

one encounters continuous transitions: the change in ~ from negative to positive values is accompanied by a concomitant continuous shift from negative to positive values in the minimum of the function 12. By contrast, the right-hand column, for which T < Tc, is related to a first order transformation: at 7-/= 0 one encounters two equivalent equilibrium values for 77, with a discontinuous change in r/ as the direction of the magnetic field is reversed. Such first order changes were characterized via thermodynamic considerations in Chapter 2. 7.5.3

Continuous Transitions

We next examine further the case of a continuous phase transformation. Consider first the situation where 7-/= 0. As long as T > Tr and T -- Tc, the global minimum of the order parameter occurs at r/0 -- 0; cf. Eq. (7.5.5). But for T < Tc the equilibrium values of the order parameter are specified by

qo -- -+-(-at~b)1~2

(t < 0),

(7.5.7)

according to which the power law variation is characterized by the mean field value fi - 89 The Gibbs free energy, Eq. (7.5.6), is given by 1 a2t 2

G(r/0) = 0

(t > 0)

and

G(r/0) =

2 b

(t < 0),

(7.5.8)

from which, by a double differentiation, one obtains the contribution to the total heat capacity as a 2

Cp-O

(t>0)

and

Cp=bTc

(t<0),

(7.5.9)

respectively. Thus, in contrast to G, the heat capacity does exhibit a discontinuity at the transition point. This is in accord with the results obtained by different methods in Section 3.1 7. We next take up the case 7-( ~ 0. Equilibrium is characterized through the requirement 0 s 77= 0, which leads to the general condition

at~ + brl 3 -- ~ ~ .

(7.5.10)

The critical isotherm is characterized by t - 0 and 7-{- 2b773. This immediately shows that the critical exponent for the magnetic field variation with 77-- A//is 8 = 3, in conformity with the standard mean field value encountered in the literature. Returning to Eq. (7.5.10) and differentiating with respect to AA one obtains the magnetic susceptibility as

X (~) -

-~

- { 2[at + 3br/2 ( ~ ) ] }-1,

in which r/(~) is a solution to Eq. (7.5.10).

(7.5.1 1)

422

7. CRITICAL PHENOMENA

/2

1/

0

o

rl

t
!o

o

LF

Fig. 7.5.4. Depiction of a first order transition in terms of the variation of the Landau free energy density with the order parameter at several reduced temperatures. For details see text. After Goldenfeld loc. cit.

In zero field and for t > 0, 7 / = 0 and Z = 1~2at. In the limit of zero field and for t < 0, rl = + ~ / - a t / b and X = - 1 ~ 4 a t . In either case one finds that the critical exponent both above and below the critical point is given by V = 9/' = 1, which is the ordinary mean field value cited in earlier sections. 7.5.4

Reprise to the First Order Phase Transitions

We next consider the Landau approach to first order phase transitions by including a cubic term and omitting the linear term (7-L - 0) in the expansion for s whereby we set

- - a t o 2 + ~bq 4 + qrl 3. Optimization via ( 0 s

(7.5.12)

- 0 yields brl 2 + ~qo + at - 0,

(7.5.13)

which is a quadratic equation in rl whose solution is q0 -- 0

and

~o -- --C i g/c 2 - at~b,

(7.5.14)

with c = 3q/4b. The nonzero solution is viable so long as the argument of the square root remains real, i.e., so long as t < bc2/a =_ t*. Since this latter quantity is positive the condition can be satisfied over some range in the regime T > Tc. By contrast, for continuous transitions the condition rl0 # 0 becomes acceptable only for T < Tc.

LANDAU THEORY OF CRITICAL PHENOMENA AND PHASE TRANSITIONS

423

To visualize what happens we plot in Fig. 7.5.4 part (a) the variation of/2 versus q, in the regime t > t*, as specified by Eq. (7.5.12); without loss of generality we selected c < 0. In conformity with Eq. (7.5.14) we note that the minimum in the Gibbs free energy is found at q0 = 0. No transition is encountered for the range t > 0. However, if one now sets t < t*, Eq. (7.5.12) yields a secondary minimum at a positive value of q, as shown in panel (b). Eventually, a value t = tl is reached where s = 0 at two values of q, as shown in panel (c); at this point one encounters two coexisting phases. On reducing t to values below tl, as sketched in panel (d), the global minimum in/2 occurs at a value q0 > 0, specified by Eq. (7.5.14). It is important to recognize that the order parameter that minimizes s jumps discontinuously from zero to a positive value. Thus, with a reduction in temperature, one passes through a first order transition. The above indicates that the presence of the cubic term in the Landau expansion will in general lead to first order transitions that have been considered in earlier chapters. One should be aware of the limitations of the Landau theory, chief of which are the fact that it can be applied only where the expansion of/2 in a limited power series in q is tenable; also, in common with all other mean theories, fluctuation effects have not been taken into account.

425

Chapter 8 A Final Speculation About Ultimate Temperatures--A Fourth Law of Thermodynamics?

It is perhaps not inappropriate to close by introducing a speculative idea, namely that a Fourth Law of Thermodynamics might be set up, to the effect that there exists an upper limit Tu to the temperature scale. These arguments are modeled after the presentation by Kelly 1" consider a mole of H2 gas that is being heated continuously in an appropriate enclosure. The accompanying Table 8.1 shows a record of events as various temperatures are reached. One notes that the heat Table 8.1 Record of events encountered on heating H 2

Temperature (K)

Event

300-600

Heat capacity nearly constant: Cp ~ (5/2)R Dissociation begins; C p rises Dissociation nearly complete. System now consists of 2 moles of H; Cp ~ 5 R Ionization of atoms noticeable; Cp rising Ionization nearly complete. System contains 2 moles of protons and 2 moles of electrons; C p has leveled off close to C p = 6R C p remains nearly constant Collisions produce electron-positron pairs, Cp rising Proton-proton collisions produce pions, Cp rising Pions abundant. Energetic proton-proton collisions produce proton-antiproton pairs; Cp rising Temperature has reached an upper limit. Addition of more energy increases number and variety of fundamental particles. Cp rising (see text)

1000

5000 104 105

10 5-10 8

109

1011 1012

~1.2 x 1012

426

8. A FINAL SPECULATION ABOUT ULTIMATE TEMPERATURES

capacity always rises strongly as those temperatures where particle production takes place, such as the dissociation of H2 into atoms, ionization processes that form H +, electron-positron production, and so on. In those ranges where Cp does not change much the added energy becomes manifest as a temperature increase in the system, as governed roughly by the relation A E -- CpAT. The point of introducing this table is to suggest that beyond a temperature of roughly 1012 K it becomes more economical to dissipate the input of additional energy through the generation of larger numbers of elementary particles, at roughly constant temperature, than to compensate for the internal energy increase of a fixed number of particles via a rise of their temperature. The upper limit of 1012 K is arrived at by noting that particles whose kinetic energies exceed the rest mass of pions tend to generate new particles through collision processes, in which the excess energies are converted into additional mass. The detailed consideration of such events are clearly beyond the confines of the present book; readers are encouraged to search the literature for further insights. On this note it seems appropriate to close the discussion. REFERENCE 8.1.1 D.C. Kelly, Thermodynamics and Statistical Physics, Academic Press, New York, 1973, Chapter 18.

427

Chapter 9 Mathematical Proof of the Carath6odory Theorem and Resulting Interpretations; Derivation of the Debye-Hiickel Equation

9.1

Fundamentals

In Chapter 1 we had introduced functions of state that are mathematical analytic functions of the general form R -- R(xl

.....

Xi,...,

(9.1.1)

Xn),

in which the function R is uniquely specified by the values of the n independent variables xi for the thermodynamic properties of the system. The corresponding linear differential form is given by

n(o.)

d n-i~l. ~

dxi

~j~i

(9.1.2)

,

and in Chapter 1 was labeled an exact differential. In thermodynamic studies we have frequent occasion to examine linear differentials of the form t/

dL -- Z

Xi(xl,

. . . , xi,

. . .Xn)

dxi

(9.1.3)

i=1

that resemble Eq. (9.1.1); these are called Pfaffian forms. Here dL is simply a short-hand notation for the function on the right whose values depend on the selected path, in the sense that the Xi can no longer be determined by differentiation of the single function R in the manner shown in Eq. (9.1.2). The Xi are no longer partial derivatives of a function of state. The entities L occur very frequently in the description of thermodynamic processes but should be replaced, wherever possible, by functions of state, so as to avoid the necessity of having to deal with path-dependent quantities.

428

9.1.1

9. MATHEMATICAL PROOF OF THE CARATHr

THEOREM

Integrable Functions

If a function L does not admit of an exact differential of the form (9.1.2) it may nevertheless be possible, under conditions established below, to set up functions q (x 1 . . . . , xi . . . . , Xn) such that the ratio d L / q = dR does constitute an exact differential. Pfaffian forms of this genre are of special interest; they are said to be holonomic or integrable. For obvious reasons q is said to be an integrating denominator and 1/q, an integrating factor. The particular equation /'/

d L -- E

X i ( x l . . . . . xi . . . .

, xn) dxi =-- X . d x - 0

(9.1.4)

i=1

is of special interest. If it happens to represent an exact differential, then Eq. (9.1.2) applies: dL - - d R , and an algebraic equivalent exists, that has the form R(xl ..... xi,...,Xn)

--C,

(9.1.5)

where C is a constant. Eq. (9.1.5) specifies an interrelation between the independent variables xi. In the n-dimensional hyperspace that is spanned by these particular variables the particular function R ( x l , . . . , xi . . . . . X n ) - - C is representable as a hypersurface of n - 1 dimensions in that space. As discussed below, the special relation (9.1.4) represents a curve on that surface. If E i x i dxi is not an exact differential but is nevertheless holonomic then it has an associated integrating factor such that q dR - E i x i dxi, whence

Xi(xl,..., Xn)--q(xl

. . . . .

Xn)(O~iXi)xjr i

(9.1.6)

Further, Eq. (9.1.4) then implies that ( i / q ) E i X i dxi = 0, hence, dR = 0, which establishes the existence of an algebraic equivalent, Eq. (9.1.5), to Eq. (9.1.4). If the Pfaffian form is holonomic then the Xi, that are functions o f (OR/Oxi)xj#i by Eq. (9.1.6) may be thought of as components of a vector X which is proportional to the gradient V R of the function R, as indicated on the right of Eq. (9.1.6). Similarly, the various dxi may be thought of as components of a displacement vector dx = {dxi }. By its very definition the gradient of R is everywhere orthogonal to the surface specified by R = C. Therefore, X (with components X i - - ( O R / O x i ) ) likewise points in the direction normal to the surface 8. Moreover, the requirement X 9d x = 0 is met by requiring dx to be perpendicular to X, that is, tangent at all points to the surface 8. Therefore, the solution curves C formed by adjoining all the dx segments must lie entirely on the surface 8 defined by Eq. (9.1.5). It should be self-evident that there exists a throng of points in the hyperspace, outside the surface 8, that are not accessible via the solution curves C.

PROOF OF HOLONOMICITY

9.1.2

429

Carath~odory's Theorem

We have so far demonstrated that if (9.1.3) is holonomic and if (9.1.4) applies these conditions are sufficient to guarantee that in any neighborhood of a given point x0 in the hyperspace there exist other points, corresponding to (9.1.5) that are not accessible from x0 via solution curves that are subject to the relation X.dx =0. Is the converse also true? That is to say, from the assumption of nonaccessibility can one deduce that Z i Xi dxi is holonomic? The answer is in the affirmative and is furnished through Carathdodory's theorem: If every neighborhood o f an arbitrary point xo in a hyperspace contains points not accessible from it via solution curves o f the equation ~ i Xi dxi : O, then the Pfaffian f o r m dL = Z i Xi dxi is holonomic.

The proof of this important theorem is provided in the next two sections. 9.2

Proof of Holonomicity

We digress here to specify necessary and sufficient conditions to establish whether or not the Pfaffian form d L -- ~ i Xi dxi is holonomic. 9.2.1

Integrability of Linear Forms of Two Variable

We first prove that any linear form involving two variables is integrable: given (9.2.1)

d L = X1 d x l nt- X2 dx2 ,

we seek a function q(xl, X2) such that this form obeys the relation (9.2.2)

d L = q d R,

where d R is an exact differential of the form dR=

OR

OXl

OR d x l -~- ~ dx2.

(9.2.3)

OX2

Because of (9.2.1)-(9.2.3) we require that OR

OR

X 1 dxl -Jr-X2 dx2 - q O-2X1 d x 1 + q O-2X2 dx2,

(9.2.4)

which for arbitrary dxl and dx2 means that

X1

-

-

OR q -~x l dxl

and

OR

X2 -- q 0-2X2dx2.

(9.2.5)

430

9. MATHEMATICAL PROOF OF THE CARATH#ODORY THEOREM

In principle these last two relations can always be solved for q and for R, since X1 and X2 are established functions of the xi. We can then construct the ratio X1

=

X2

(OR/Ox1)x2

__

_|Ox2il-

(OR/Ox2)xl

L

-!

.

(9.2.6)

JR

One should note how the displacement vector components are related through the Xi. We conclude that the Pfaffian form (9.2.1) is always integrable since Eq. (9.2.5) represents a set of simultaneous relations that may be solved for q and R. However, this scheme fails if a function of three or more variables, such as dL = X1 dxl + X2 dx2 + X3 dx3 were to be examined. One would then have to adjoin to (9.2.5) the additional relation X3 = q(OR/Ox3)dx3. It is no longer clear whether this is possible; in fact, we now address precisely this question.

9.2.2

Integrability of Linear Forms of Many Variables

In the more general case one seeks a function q (Xl . . . . . Xn) such that ( 1 / q ) d L dR, where R (x l, . . . , Xn) is a function that (in contrast to dL) has an associated differential form n

OR

n

dR -- ' ~ dxi =--~ Yi dxi. ~ OXi i--1

(9.2.7)

We now examine the necessary conditions for this scheme to work: if indeed a function qi with the desired properties has been found then comparison of dL -q d R with (9.2.7) shows that

OR Xi -- q Yi -- q Oxi

(9.2.8)

must hold. For the special case n = 3 one then finds

OR

OXl

X1

OR

q

OX2

-- Y 1 - ~ ,

X2

OR

q

Ox3

-- Y 2 - ~ ,

X3

---- Y 3 - ~ .

q

(9.2.9)

Since the order of differentiation is immaterial we may find a relation between any of the two functions listed in (9.2.9) according to the following procedure:

O2R

OY1

1 OX1

X1 Oq

OXZOXl

OX2

q OX2

q2 0X2

02R OXlOX2

OY2

1 0 X2

OXl

q OXl

X2 Oq q2 0Xl (9.2.10)

The central and last term of the above sequence may be combined to obtain

OXl q Ox2

OX2 ] _ X I ~Oq _ X 2 ~Oq OXl Ox2 OXl

(9.2.11 a)

PROOF OF HOLONOMICITY

431

Using similar procedures for the remaining partial derivatives of R one obtains

[

OX3 q OXl

OX1 Ox3

0 X2

0 X3

q Ox3

] ]

(9.2.11b)

_ X3 061 - X I ~ , OXl Ox3

Oq

_ X2 ~

OX2

-

Oq

X3

OX3

(9.2.1 lc)

9

~X2

Next, multiply (9.2.1 l a,b,c) by X3, X 2 , X1 respectively and add the resultants. The sum is found to vanish identically, whence the common factor q may be dropped. This leaves

X1

E

OX2 Ox3

] E

OX3 OX3 Ox2 + X2 OXl

J E

OX1 -+-X3 Ox3 Ox2

OXl

]

--0.

(9.2.12)

We see then that if an integrating factor 1/q exists that converts the Pfaffian dL into the exact differential (9.2.7), then the coefficients in the relation dL - X1 dxl + X2dx2 + X3dx3 are subject to the requirement (9.2.12). For the more general case of n > 3 we obtain very similar relations, with 1, 2, 3 in (9.2.12) being replaced by indices i, j, k that are cyclically permuted, in the manner shown in Eq. (9.2.20) below. Next, we seek the inverse: given a Pfaffian form n

dL -- ~

Xi dxi,

(9.2.13)

i=1

in which the X i a r e subject to the requirement (9.2.12), with cyclic indices i, j, k, we now establish that there will always exist an integrating factor 1/q that converts the inexact differential into an exact differential of the type displayed in (9.2.7). To prove this, turn to Eq. (9.2.13) and hold the quantities Xl . . . . , Xn-2 fixed. Eq. (9.2.13) then reduces to the form Xn-1 dxn-1 -+-Xn dxn, which is certainly integrable. Thus, one must be able to find a function H with a differential form analogous to (9.2.3), as well as an integrating denominator co such that

Xn- 1dxn- 1 -Jr-Xn dxn -- cod H.

(9.2.14)

Since Xn-1 and Xn a r e both functions of all Xi, 09 and H will likewise depend on these variables. Now relax the constancy requirement and write n

d H _ ~--~ OH dxi. Oxi i=1

(9.2.15)

432

9. MATHEMATICAL PROOF OF THE CARATHr

THEOREM

In order to match up (9.2.15) with its more restrictive counterpart (9.2.14) we rewrite the above equation in the form

X n - 1 dxn- 1 + Xn dxn - ~ d H _

~0H dxi 9

(9.2.16)

i=10Xi

At this point reformulate the Pfaffian as n-2

Xi dxi -Jr-Xn- 1dxn- 1 + Xn dxn -- dL

(9.2.17)

i=1

and substitute from (9.2.16). Also set dL - O. Then

OH] dxi + d H = dL = 0 .

~[Xi i=1

60

OXi

(9.2.18)

09

At this stage let us switch variables: we set Yi =- xi for i - 1, 2 . . . . , n - 2, n and set Yn- 1 -- H (Xl . . . . . Xn). This renders Yn- 1 a function of all independent variables. Lastly, set Yi = X i / c o - (OH/Oxi) for i - 1, 2 . . . . , n - 2. Then Eq. (9.2.18) takes the form

n-2 dL Z Yi dyi + dyn- 1 = i=1 6.0

0.

(9.2.19)

This expression contains one fewer variable than the corresponding form (9.2.17). Unfortunately, the Yi would seem to depend on all the variables Yl, Y2, . . . , Yn. To show that this is not the case we proceed as follows" Since we passed from (9.2.17) to (9.2.19) by purely algebraic transformations the former equation will be integrable if the latter is. By hypothesis, Eq. (9.2.12) applies to (9.2.17); hence, a corresponding relation will have to hold for (9.2.19), namely,

[ OYj Yi Oyk

OYk ] + yj [ OYk Oyj Oyi

OYi ] + yk [ OYi Oyk Oyj

OYj I _ o"

(9.2.20)

Oyi

Apply the above relation to the special case j - n - 1, k - n. But, according to (9.2.19), Yn-1 - 1 and Yn - O . Hence Eq. (9.2.20) reduces to

OYi = 0 for all i. Oyn

(9.2.21)

This demonstrates that in fact none of the Yi in (9.2.19) depends on Yn. Thus, Eq. (9.2.19) involves only the n - 1 independent variables Yl, . . . , Yn-1. We now repeat the entire process, beginning with Eq. (9.2.14), and ending with (9.2.19), which contains one fewer variable than its equivalent, Eq. (9.2.17). In the final iteration the analogue of (9.2.19) contains only two variables and thus is integrable, so long as Eq. (9.2.20) continues to hold.

NECESSARY CONDITION FOR ESTABLISHING THE CARATHI~ODORY THEOREM

9.3

433

Necessary Condition for Establishing the Carath~odory Theorem

In Section 9.1 we had established sufficiency requirements that relate to Carathdodory's theorem. We now attend to formulate the necessary statement by proving the following: If in every neighborhood o f any point in the space spanned by the set {xi }, i = 1 , . . . , n, there are points inaccessible along solution curves Z i Xi dxi = O, then the corresponding Pfaffian f o r m d L = Ei Xi dxi is necessarily integrable.

As is well established, a curve C in a hyperspace may be defined as a collection of all points generated by equations of the form xi

-- fi(u)

(i = 1, 2 . . . . . n),

(9.3.1)

where the f/represent continuous, differentiable, single-valued functions involving a relevant parameter u, which frequently is simply time. Let points Pa and Pb correspond to two specific choices for u, with Ua < u < Ub. Let C a be another curve that intersects C at Ua and that otherwise differs from C only infinitesimally. Then C a is specified by an equation of the form

Xi -- fi(u) + Sqgi(U)

(i -- 1, 2 . . . . , n),

(9.3.2)

in which s is a small quantity, and (/9i (U) is another function analogous to fi (u), with the constraint that ~oi(Ua)- 0

(all i).

(9.3.3)

Moreover, since e and (~a are solution curves they are subject to the requirement (9.1.4); that is, we set /,/

ZXi(xi)f/(u)--0,

(9.3.4)

i=1

where the prime indicates the partial derivative with respect to u. In what follows we also will set Yi(u) =-- X i ( f i ( u ) ) whenever we wish to emphasize the dependence of X i on u. The same requirement must be met by the displaced curve: /2

E { X i [ f i ( u ) Jr-8(fli(u)]l[f/(u) + 8 ( f l ; ( u ) ] - - 0 .

(9.3.5)

i=1

We expand the argument of

X i in a Taylor's series to first order in s. This yields

Z Y(u)i(f[(u)+ 8(fli(u)) + Z Z 8(flJ(U)-~xj i (U)i=1

i=1 j = l

(9.3.6)

434

9. MATHEMATICAL PROOF OF THE CARATHr

THEOREM

On account of (9.3.4) the above simplifies to

n n n OX i Y~ Xiqg; -+- i~l Z -~xj qgj fit -- O. i=1

(9.3.7)

j=l

"

This expression may be satisfied by arbitrarily selecting n - 1 functions and requiting the remaining one, say, r to conform to the condition set by Eq. (9.3.7). To do this, rewrite Eq. (9.3.7) in the form n

n

n

Xkqgtk.qt_Zt(OXi~ ' -- ~ t , i=1 Ik--~Xk J f i qgk -X j goj j=l

n

ZtztOXi

, -~xj f i (flj .

(9.3.8)

i=1 j = l

The prime symbol on the summation sign indicates that the term i - k or j - k is to be omitted from the summation. This first order differential equation (9.3.8) is ultimately solved for qgk by the standard technique of finding a multiplier ,I.(u) for both sides of (9.3.8) that converts the left into a differential function of the form n

d()~Xkqgk) , z, du =-- XXkCPk + )~

OXk , ~ f i r + X' XkCPk,

(9.3.9)

i=1

where Eq. (9.3.2) was invoked, while retaining only terms to first order. Comparison with Eq. (9.3.8), multiplied by ~., shows that if an identity is to be reached then the function )~ must satisfy the relation

n(

~t__ ~" Z t

Xk i=1

OXi

OXk

Oxk

Oxi

)

t fi.

(9.3.10)

Next, multiply both sides of (9.3.8) by ~. and introduce (9.3.9) to eliminate Xkcpk. Then eliminate U in the intermediate step by using (9.3.10). Integration of the resultant and subsequent use of Eq. (9.3.3) leads to

XXkq9k - - - f u UX[ ~

; - ~ X j f[qgj ] du. tXj~o) + Znt ; n~---~,OXi

a

j=l

(9.3.11)

"

We treat the first integral on the right by integration by parts:

--

fuU)~ a

t X j qgj du

-- -X

j=l

+

~X

j=l .qL.

yuU ,X j (flj d u )~ a

fuU xJ a j=l

!

!

i=1-~Xi

j=l !

fi(fljdu'

(9.3.12)

NECESSARY CONDITION FOR ESTABLISHING THE CARATHE~ODORYTHEOREM

435

where Eq. (9.3.3) was again used to obtain the first term on the right, and Eq. (9.3.1) was introduced to write out the last term. At this point eliminate U by use of Eq. (9.3.10); this yields --

tXjqgj

~

fu u ~ a

'

du

j=l ?/

-- ~ ~tXjqgj j=l

+

)~ a

t~-~t(fi, qgj)

OXi OXk

i=1 j = l

OXk ) Xj Oxi

OXj ]

--~ + Oxi

du. (9.3.13)

This resultant may now be substituted in (9.3.11); on simplifying one obtains /7 (19k - - _

~-~t X j j=l

n )~Xk i=1

n fu u I j=l a Xk

( Oxk OXi

OXi) OXk

( Oxi OXj

OXj)] OXi (9.3.14)

Lastly, observe that on account of (9.3.4) one may add the term

1 f Uy-~, n Zz ?l fit99j)~Xil OXj )~Xk a i=1 j = l Xk OXk

0 Xk ~ du -- 0 Oxj J

(9.3.1 5)

without changing (9.3.15), since solely the s u m m a n d f/99j is involved in the summation over i and this sum vanishes. When (9.3.15) is adjoined to (9.3.14) one finally obtains

Zn 'x; j = l ~k99J(U)

l

n

~,Xk i=1

n

fu u

j=l

a

~

Fijk du,

(9.3.16)

Xk

in which Fij k is defined by the left-hand side of Eq. (9.2.20), with the interchanges Y ++ X and y ++ x. Now examine (9.3.16) after replacing the upper limit on the integral by Ub. Observe that 99k(Ub) on the left depends not only on all qgj(Ub) in the first term on the fight, but also on the values that these qgj (U) take over the entire interval Ua <<,u <~Ub when the integration in the second term is carried out. Hence, one may choose the arbitrary functions qgj (U) in such a manner that from point Pa one may reach any other point in its neighborhood. For example,

436

9. MATHEMATICAL PROOF OF THE CARATHI=ODORY THEOREM

one may select Ub sufficiently close to Ua and permute the indices so that all n functions q)j are successively assigned the index k. However, this is contrary to the initial hypothesis; we assumed at the outset that in the neighborhood of Pa there were points inaccessible from Pa. The only way to avoid this impasse is to demand that all integrals in (9.3.16) vanish. We cannot allow )~, qgj, or 1/X~ to be equal to zero, for then all else would be lost. The only viable option is to require that t/

Fij f i' - 0;

(9.3.17)

i=1

for, in that event the integral drops out and q)n(Ub) then depends only on the specific values that the n - 1 arbitrary functions g)j a s s u m e at the location u = Ub. This clearly limits the number of curves that can be constructed to pass through Pa, and hence restricts the number of points Pb that can be reached from Pa. According to (9.3.17) we then require either that all f j be zero or that all Fijn vanish. The first alternative cannot be correct since all the functions ~ except fk may be chosen arbitrarily and fk is absent from the summation over i. This leaves only the alternative that all Fij k = O. From the earlier discussion involving Eq. (9.2.20) it follows that the Pfaffian dL = Z i x i dxi is integrable. We have thereby established the necessary condition for the Carath6odory theorem of Section 9.2 to hold. Given the fact that in the neighborhood of a point in phase space other points are inaccessible via solution curves of the f o r m Y~i Xi dxi = 0 , the Pfaffian form is integrable.

9.4

Relevance to Thermodynamics

We now utilize the machinery of the preceding Sections to rationalize the Second Law of Thermodynamics as specified by Carath6odory: In the neighborhood of any state of an adiabatically isolated system there exist states that are inaccessible from that state. According to the previous line of argument this implies that when the condition dQ = 0 is met the linear form dr Q = Z i Xi dxi may be converted into the differential of a function of state, the metrical entropy, ds, through an integrating factor )~, by which ds = dr Q/)~. By the reasoning of Section 1.9 this relation may be converted to the standard form dS = dr Q / T that is then directly linked to the Second Law. In the present approach this function of state is directly derived from a mathematical construct, rather than resting solely on an empirical foundation, as set forth in Section 1.9. To individuals favoring a mathematical development of the subject matter, instead of an assertion resting on purely experimental investigations, this may be a preferred alternative.

DERIVATION OF THE LIMITING FORM FOR THE DEBYE-HUCKEL EQUATION

437

By use of the Carath6odory approach one may also draw a parallel between the First and Second Laws. Briefly repeating the remarks of the last paragraph, we adapt the above postulate to the performance of work. We note that from the absence of any infinitesimal heat exchange for the linear form dr Q -- Z i Xi dxi = 0 in an adiabatic system one may infer the existence of a function of state s (hence, the function of state S) that is a constant under this condition. In a similar manner one may apply Carath6odory's theorem to the existence of the linear form dr W -- Z j Zi dzj for performance of work. On specializing to an adiabatic system and then imposing the requirement dW = 0, the same theorem necessitates the existence of another function of state, called the internal energy E, which is a constant. This provides another perspective on the introduction of the First Law, with obvious implications concerning the conservation of energy. Such a development may again appeal to those who wish to emphasize the mathematical development of the subject matter. Beyond this point one must be aware of important differences between the two laws. The performance of work is directly linked to changes in energy of a system, so that the integrating factor q relevant to the First Law is unity. Furthermore, changes in S are tracked by the reversible transfer of heat across the boundaries of the system. Other changes in S are incurred when irreversible processes occur; this subject was treated in detail in Sections 1.12, 1.13, and 1.20. By contrast, alterations in E are tracked by performance of work, whether reversibly or irreversibly, under adiabatic conditions. Different changes in E are incurred when these processes take place under non-adiabatic conditions, as discussed in Section 1.7.

9.5 9.5.1

Derivation of the Limiting Form for the Debye-Hiickel Equation Fundamentals

We present here a simplified derivation of the Debye-Hfickel equation as specified by Eq. (4.2.2a). The presentation is adapted from Ref. 9.5.1; for more advanced treatments that provide better insights on the various approximations inherent in the Debye-Hfickel theory the reader may consult standard references. 2 The present exposition is also of intrinsic interest, as will be detailed below. We consider a collection of positive and negative ions in a solution in appropriate concentrations so as to preserve electroneutrality. We single out a particular cation, 3 termed the central ion, located at position r l within the volume element dr 3. The electrostatic potential 7r that it generates is spherically symmetric: at a given position r2 away from r l, ~ has the same value as at any other point on the surface of a sphere of radius r = ]r2 - r 1] centered on r l. As the central ion moves through the solution the sphere moves with it; hence, we may dispense with the use of angular coordinates all electrostatic interactions depend solely on

438

9. MATHEMATICAL PROOF OF THE CARATHr

THEOREM

the separation distance between the ions. These interactions are, however, responsible for a nonuniform, time-averaged distribution ionic configuration: the co-ions and counter-ions on average are spaced further apart from and closer to the central cation as compared to the average, even separation that prevails in the absence of interactions. The Debye-Htickel theory specifies deviations from thermodynamic ideality that are generated by the electrostatic interactions. The chemical potential for a given central ion of type i at location r is given by

lzi(r) - lZ* + k T lnni(r) + z i e ~ ( r ) .

(9.5.1)

Here we explicitly introduced the electrostatic potential 7r which is associated with the ionic charges in solution that contribute to 7t. As implicit in our earlier discussion, this potential, as well as the number density n i of species i, is a function of position r within the solution. The corresponding energy for the central ion is specified by zie~, where zi is the valence (positive or negative) of the ion, while - e is its electronic charge. In citing the above equation we have switched from molar to atomic units and have set k _-- R/A/', where A/" is Avogadro's constant. Until later we drop the subscript i and note that at equilibrium the chemical potential of an ion under consideration must be the same at two different locations rl and r2. On imposing this condition on Eq. (9.5.1) we may rearrange the resultant to read In

n (r2) ap (r2) -- ~ (rl) = --ze , n(rl) kT

(9.5.2)

where we dropped the vectorial notation as superfluous. Now specialize to the case where the central ion interacts with another one of its kind, and let this second ion then be moved to infinity where ~r(rl) - 0, and where n(rl) - no, which is the normal number density of the species in the absence of ionic interactions. We then find that for positively charged ions

n+(r)--n+exp(-ze~(r)) kT

'

(9.5.3a)

while for negatively chargedcounter-ions zeTt(r) )

n_(r)

-- n_

e x p ( + ~k T \

"

(9.5.3b)

The charge density from the two types of ions is given by

p(r) - e[z+n+(r) - z_n_(r)].

(9.5.3c)

We suppress the r dependence, and note that Eq. (9.5.3c) is to be interpreted as a local rather than as a global relation. In the event that more than one set of co-ions

DERIVATION OF THE LIMITING FORM FOR THE DEBYE-HOCKEL EQUATION

439

and counter-ions exists in solution their presence must be included in the above summation, whereby Eq. (9.5.3a) is generalized to the form p ( r ) -- e E

zini -- e Z i

(9.5.3d)

lO-3ziciJV" i

in which gi is positive or negative according as it refers to the valence of cations and anions respectively. Here we set ni - 10-3.A/'ci to convert to concentrations. We next introduce Gauss' law in the form V. E -- 4zoO~e, where e is the dielectric constant of the medium and E is the prevailing electrostatic field. On setting E = -V~p one obtains

-~,4rrP

V 2 lp _

(9.5.4)

which is Poisson's equation. After introducing Eq. (9.5.3d) one finds that V2 ~

--

47re Z 8

zini exp i

(-zieO) kT

,

(9.5.5a)

which is a transcendental equation that can only be handled numerically. To continue the analytic treatment we adopt the first order expansion for the exponential function to write, for ze~/<< k T, V2 ~

--

4rre E 8

zini

(zie~) 1

-kT

i

47re2~ Z z2ni, kTe i

(9.5.5b)

where, on account of electroneutrality constraints, the summation ~ i zirti vanishes. The above relation may then be rewritten in the condensed form 1 d2(rO) V 2 ~ __ K2~ __ _ ~ r dr 2 '

(9.5 5C)

z2ni ekT

(9.5.5d)

where K2 ~ 4rre2 Z i

87r e22~S

103ekT '

where S represents the ionic strength introduced in Eq. (4.2.1). The right-hand side of Eq. (9.5.5c) represents the standard formulation for the second derivative taken in spherical coordinates when the use of angular coordinates is irrelevant. We thus need to solve the differential equation d2(r~) dr 2

-- K2(r~),

(9.5.5e)

440

9. MATHEMATICAL PROOF OF THE CARATHi~ODORY THEOREM

whose solution (as may be verified by direct substitution) for the potential associated with the central ion is given by

(r) _ ~C1eX r + ~C2 e_Xr, r

(9.5.6)

r

where C1 and C2 are two constants. Their determination requires two boundary conditions. The first is based on the requirement that the electrostatic potential at an infinite distance r - e~ from the central ion must vanish. Since e x p ( K r ) / r increases indefinitely with r we require that C1 = 0. The second condition rests on the model adopted for a representation of the central ion. We assume it to be spherical in shape, with a radius a. At that particular distance from its center its potential has the value C2

~p(a) = ~ e a

-xa

,

(9.5.7)

whose substitution in (9.5.6) leads to ~ ( r ) - a ~ (?-a ) e -K(r-a)

9.5.2

(r > a).

(9.5.8)

The Electrostatic Potential at r - a

To determine ~p(a) we invoke the requirement that the electrostatic potential, which is to be a solution to a second order differential equation, must be continuous and vary smoothly across the boundary that separates the ion at r = a from its surroundings. For ease of further mathematical manipulations we now adopt a representation for an ion as a sphere whose entire charge, Qe, is uniformly distributed inside its surface~the balloon model of an ion. This may be justified by appealing to a consequence of Gauss' law: a uniform charge distribution within an enclosed sphere has the same effect on the outside world as the entire charge would have if it were concentrated at the center. The balloon model simply represents a very convenient special uniform charge distribution within the sphere. For continuity considerations we require the first derivative of Eq. (9.5.8), which is then to be evaluated at r = a. Standard differentiation, followed by setting r = a, leads to dO ] -E;r

= -~(a) r=a

1 + Ka ---T--

(9.5.9) "

Since the electrostatic field outside the sphere is given by g is the distance from the center, the corresponding potential is

~o(r) -- -- f ~ ~Qe dr , -- Qe er

(r > a).

Q e/ er 2, where r

(9.5.10a)

DERIVATION OF THE LIMITING FORM FOR THE DEBYE-HOCKEL EQUATION

441

On the other hand, in the interior of the balloon the electrostatic field vanishes by Gauss' law, since no charge is enclosed. Then E = 0 for r < a, whence ~i

---

C3,

a constant for r < a.

(9.5.10b)

First, since 7t must match at the boundary, we may equate (9.5.10a) with (9.5.10b) to find C3 ~ i ( a ) = Qo/ea. Second, at the boundary =

[d~]

_ r=a --

Qe ca2"

(9.5.10c)

Since the potential must change smoothly across the boundary this derivative must match the one in Eq. (9.5.9). This enables us to solve for 7t(a) --

Oe

(9.5.11)

ca(1 + Ka)'

which, in turn, may be applied to Eq. (9.5.8), for the final desired relation

~(r) -9.5.3

Qe

er(1 + Ka)

e-X(r-a)

(r > a).

(9.5.12a)

Discussion

The above expression is a fundamental result of importance; for, it shows how the potential associated with an ion in the presence of others differs from the conventional form 7r(r) = Qe/er, for an isolated ion in solution, to which it reduces when tc = 0. As is seen, the potential (9.5.12a) drops off exponentially with distance r beyond the radius of the sphere, and the effective charge acting on the outside world is given by Qe/(1 + xa). The type of exponential decay shown here is characteristic of systems of interacting charges, and is encountered in many other areas in science. One generally refers to this formulation as a shielded potential. The degree of reduction of the potential relative to the isolated case is governed by the magnitude of the exponential decay constant, Eq. (9.5.5d), which thus varies as

W/Zi z2ci/ek T.

For a particular medium at fixed temperature the rate of decay

depends on the square root of the ionic strength. Eq. (9.5.12a) also shows that one may consider 1/K as representing an average interaction distance: for a 0.1 M solution of ions in water at room temperature this distance is approximately 10/k, beyond which the shielding is fully effective. In many cases of interest the product Ka is small compared to unity; so that Eq. (9.5.12a) reduces to

7t(r)-- Qee-x(r-a)

(r > a).

(9.5.12b)

442

9. MATHEMATICAL PROOF OF THE CARATHr

THEOREM

Lastly, in physical problems where electrons rather than ions act as carriers of negative charge, they are considered to be point charges, with a -- O, which leads to the commonly quoted result Qe

~p(r) -- - - e

-Kr

(r > a)

(9.5.12c)

sr

that is cited in the literature for the electrostatic potential of interactive electrons. The origin of the above findings may be traced to the solution to the Poisson equation. When all of the ions or electrons are far removed from each other in an infinite copy of the system, tc vanishes; one then obtains the standard expression lp = O e / e r .

9.5.4

Relation to Thermodynamics

To connect the above findings to thermodynamic principles we imagine a hypothetical situation in which all ionic species in the solution are placed in their final positions, but initially are devoid of their charges. We then initiate a reversible charging process that converts each ion from a value Qe - - 0 t o the value Qe - z i e . This action may be monitored by use of a parameter )~ that increases from zero to unity during the charging process; ~. is thus the fractional charge on the central ion. The process requires work that must be executed reversibly at constant temperature and pressure. Hence, in accord with the discussion in Section 1.12, the charging process results in a change in the Gibbs free energy of the system. Furthermore, as described in the introduction, this step is precisely the operation responsible for departures from ideality due to the concomitant electrostatic interactions. Therefore, on introducing the relation for the chemical potential of species i, terms of number density [Zi -- lZi -Jr-R T l n n i

+ R T l n Yi,

(9.5.13)

where Yi is the activity coefficient for species i, as defined in Section 3.4, we may set 1

R T l n gi -

f0

zi+e~()~) d~..

(9.5.14)

Here the differential of work performance is expressed by the integrand. The further development now hinges on the specification of the appropriate potential in Eq. (9.5.14). Here we are guided by the fact that the charging process in part requires work to establish a particular central ion in final form in its aqueous milieu, which quantity is independent of concentration and may therefore be included in the constant term #*. The second contribution deals with the redistribution of ions that is the result of the interactions that arise during the charging. This requires the specification of an

DERIVATION OF THE LIMITING FORM FOR THE DEBYE-HUCKEL EQUATION

443

electrostatic potential of the central ion with its co-ions and counter-ions relative to its potential in the pure solvent, ~0. We approximate the integration process by recognizing that the value of the potential is governed by the average separation distance b between the i o n s ~ a parameter in the theory. Hence, we may use Eq. (9.5.11), with a replaced by b, for the corresponding potential. The reference value is that for which the ion resides in the pure solvent, whereby K = 0 and ~ i --- Zi+ e / e b . Thus, for the central cation at a given state of charge )~ zi+e

~(1.) -- - ~

[

1

]

- 1 k

1 + Kb

zi+e

--

tc

eb 1 + tc--------~k"

(9.5.15)

Now perform the integration over k to find for cations of species i ~2+ e 2

In ~ / i +

--

---/~f"

K

2 e R T 1 + tcb'

(9.5.16a)

which is the central result of interest. For the anion species we obtain similarly Z 2_ e 2

In Yi_ = - N ' ~

K

2 e R T 1 + tcb

(9.5.16b)

However, the only experimentally accessible quantity is the mean activity coefficient, specified by v In Yi+ = 1)i+ In Vi+ -Jr-1)i- In Yi_,

(9.5.17)

wherein vi+ and vi+ are the stoichiometry coefficient for the cationic and anionic species in solution, and v =_ vi+ + vi_. On introducing appropriate substitutions one obtains the expression

r]ie2K In Yi+ -- -Af2~RT(1 + z b ) '

(9.5.18a)

in which r]i ~ (Vi+Z2+ Jr- Vi_Z2)/V. The parameter tc is defined via Eq. (9.5.5d), which involves the quantity ~ / S / e T . In most cases the product Kb is negligible compared to unity. Thus, other factors being equal, the logarithm of the mean activity coefficient for species i (which encompasses the cations and corresponding anions) depends principally on the square root of the ionic strength of the solution. This prediction is experimentally verified for very dilute ionic solutions. More sophisticated derivations are required for better insights on the characteristics of the Debye-Htickel theory.

444

9. MATHEMATICAL PROOF OF THE CARATHr

THEOREM

COMMENTS 9.5.1 Ken A. Dill and Sabrina Bromberg, Molecular Driving Forces, Statistical Thermodynamics in Chemistry and Biology, Garland Science, New York, 2003, Chapter 23. 9.5.2 See, e.g., A.W. Adamson, Physical Chemistry of Surfaces, 3rd edition, Wiley, New York, 1976); EH. MacDougall, Thermodynamics and Chemistry, Wiley, New York, 1939; R. Stephen Berry, Stuart A. Rice, John Ross, Physical Chemistry, 2nd edition, Oxford University Press, New York, 2000. 9.5.3 What changes, if any, would be required if an anion were selected as the central ion?

445

Index

Acetic acid, dissociation of 263,264 Activity 166-168, 173 chemical 166 from lowering of freezing point 210-213, 254, 255,259 of electrolytes 249-251,443 pressure variation of 162, 188 product 265 pure condensed phases 184, 185 temperature variation of 162, 189 Activity coefficient 166-168, 173 binary mixtures 201-205,207, 208, 215, 216 determination of 201-205 for gases 160 interrelations 172-174 mean, for electrolytes 250, 251 measurement by vapor pressure 201-203, 259 numerical example 206, 207 pressure variation of 162, 188 pure condensed phases 184, 185 temperature coefficient of 162, 189 Adiabatic demagnetization 343,344 processes 86, 87 Adiabatic systems, defined 2, 32 Adsorbate 294 thermodynamic properties of 295ff Adsorbent 294 Adsorption 294ff at constant spreading pressure 299, 300 at constant surface coverage 300 at constant temperature 299 heats of 299, 300ff isotherms 298 Affinity, chemical 116, 117, 157, 354, 389 Algebraic equivalent 428 Ammonia, synthesis of 199, 200 Amalgam electrode 277 Anisotropic deformations 70-73

Anode 269 Arrhenius theory 4-1 Barometric formula 290 Balance equation for concentrations 349, 350 for energy 350-353 for entropy 350, 353 general 348, 349 Berthelot equation of state 70 Binary solutions, nonideal 215ff Body, defined 2 Boiling point, elevation of 142, 143 Boundaries, defined 2 diathermic 2 permeable 2 semipermeable 2 Boyle's law 7 Bragg-Williams approximation 240ff Brunauer, Emmett, Teller equation 298, 303 Calcium carbonate, dissociation of 198, 199 Calomel electrode 278 Caloric equation of state 66, 96, 97 Calorie, defined 80 Calorimetry 80 and functions of state 80ff Canonical form 163, 165, 169, 172 Carath6odory's statement of the Second Law, necessary conditions for 433-436 sufficient conditions for 429 Carbon dioxide phase diagram, schematic 124 Carnot engine 45, 46 efficiency of 45, 46 Carnot's first theorem 46 Carnot's second theorem 46 Casimir-Onsager reciprocity conditions 366 Cathode 269 Celsius scale 7 Centigrade scale 7

446

Centrifugal effects 291,292 ff composition variation due to 293 pressure variation 292 Chemical cells 282 Chemical change internal ramifications of 103,104 irreversible 156-158, 389ff Chemical equations, representation of 143 Chemical equilibrium in ideal solutions 150, 151 in nonideal solutions 178-182 Chemical potential 92, 94, 97, 102 at equilibrium 113 canonical form 131, 163, 165, 166, 169 gauge invariance of 170 ff generalized 354 of dissolved species 131,134, 135 of gases 129, 159, 160 of ionized species 254 of mixing 221 of open systems 92, 94 reference 159, 167,171 specification of 165-167, 169-173, 177 standard 129, 131,136, 165, 166, 173 standard, for electrolytes 250 temperature dependence of 138, 139 Chemical reactions, 143 ff, 389 ff coupled 390, 392, 394 irreversible 389 phenomenological equations 330 reaction velocities 389, 392, 393-396 schematic 143 unit advancement of 144 Clausius inequality 44, 45 Clausius-Clapeyron equation 122, 123 Closed system, defined 2 Common tangent construction 227-230 Compliance coefficients 72 Components, number of independent 4 Compositional changes within a system 103-106 Compressibility ratio 398 Concentration cells 282ff double 284 electrode concentration 282, 283 with liquid junctions 283, 284 Conductivity, electrical 381 Configuration space 3 Conjugate variables 116, 118, 355

INDEX

Conservation equation 349 Conservation of mass, equation for 350 Convected derivative 349, 356 Conventions, galvanic cell 274-276 Constraints, effects of on functions of state 54-56 Coordinates, natural 67 thermodynamic 2 unnatural 67 Correlation length 404 Corresponding states, law of 125, 398 Critical phenomena 397ff, 408ff point, thermodynamic properties of 398-403 state, criteria of 218 Criticality, materials at 397, 398,408ff Curie's law 338 Curved surfaces, pressure on 311 Cyclic processes 42-44 Daniell cell 268-271 Debye-Htickel equation 256 if, 437 ff chemical potentials in 438, 442 electrostatic potential in 438, 442 extended 258 limiting 256, 257, 443 Deficit function 48, 49, 60if, 64, 99ff lower bound on 52 Degrees of freedom 3 number of 4, 112-113 Derivatives, interrelations for 10, 11 Detailed balance, principle of 393 Dieterici equation of state 70 Differentials, exact 14 Differential equations, 15 Dilution, heats of 195, 196 Dissipation function 354-356, 364, 389, 392 Dissociation of salts 263-265 Divergence 14 Duhem-Margules equation 137 Efficiency, Carnot engine 46 Electric displacement vector 23 field 23 polarization vector 23 Electrical conductivity 381

TH E RMODYNAM ICS

current density 380 resistivity 385,386 Electrochemical potential 270, 380 Electrode concentration cell 282-284 Electrodes, types of 277-278 Electrolytes 249ff Electromagnetic effects adiabatic 329, 330, 342 isothermal 327 Electromagnetic radiation, pressure of 320-325 thermodynamic properties of 325,326, 329-331,333-337 Electromotive force 271 dependence on activities 273, 279, 280 standard 273 Electroneutrality, law of 251 Electrokinetic equations 373-375 resistivity 376 Electro-osmosis 374 flux by 376 pressure 374 transfer coefficient 374 Emf of half reactions 275 Emf, relation to Gibbs free energy 272, 273, 284 Energy balance 350-354 conservation of 33 flux vector 118, 352, 353 function of state, 32, 58, 59 in irreversible processes 58, 98, 99 of adsorbed phase 308, 309 of electromagnetic field 328, 329, 333,334, 338, 340 Enthalpy as a function of state 55, 50, 61 determination of 85 flux vector 352, 355 from emf measurements 285 of adsorption, differential 302 of adsorption, integral 302 of formation of H + ions 266 of electromagnetic fields 334 of irreversible processes 60, 101, 102 of mixing 214, 221,222, 243-245 of reactions 191, 192 standard 190-193

447

Entropy analogy 47 as a function of state 39, 40 balance equation 353, 354 changes in reactions 158, 193 deficit function 49 ff determination of 82-84, 158 empirical 39 flux vector 350, 353,354 from emf measurements 285 generation by irreversible processes 60 ff, 99 if, 353-355 integrated form in irreversible processes 64 metrical 40 of adsorption, differential 301 of adsorption, integral 299-301 of electromagnetic fields 328, 329, 334 of ideal gas at 0 K 88 of materials at 0 K 88 of mixing 215,220, 222 of superconductors 344 radiant 326 rate of production 115-118, 353-356 Equation of continuity 348, 349 Equation of state, 6 caloric 66, 67, 96, 97 for materials 66-70 for shock effects 463 in electromagnetic fields 337, 338 in nonequilibrium processes 98ff near the critical point 398-400 scaled 415 virial 69 Equilibrium 5, 56 between different phases 153 calculations 197ff constant, for gases 145 constant for ideal solutions 151,152 constant for nonideal solutions 179 ff, 186 constant, pressure variation of 150, 151, 154, 155 constant, temperature variation of 147, 148, 150, 151,154, 155 for electrolytes 261,262 in galvanic cells 273,274, 284 in gas phase 145 in heterogeneous systems 153 if, 184 ff in reactions 144 thermodynamic 3 Ethyl acetate, hydrolysis of 200, 201

448

Ettingshausen coefficient 385 effect 385 Euler's theorem of homogeneous functions 12 Eutectic 234 temperature 234 Exact differentials 12, 13,427 Faraday 270 Ferromagnetism, near criticality 400-402 Fick's law of diffusion 365 First law of thermodynamics 31 ff as a parable 37, 38 Flory-Huggins model 221,222 Flux 116, 349, 350, 352-354 generalized 355 Forces, electrodynamic 29 generalized 355 thermodynamic 18, 19 Fourier's law of heat conduction 117, 364, 381 Free energy changes from emf measurements 272, 273, 276, 277, 284 in chemical reactions 147,150, 182, 183, 185, 186 in gaseous reactions 146, 147, 153 in solutions 147, 148, 158, 182-184 ionic 267 of mixing, asymmetric 222, 223 of polymer solutions 223 relation to equilibrium constants 146, 147, 155, 183, 187 standard 194 Freezing point depression 139-142, 210-213, 259, 260 Fuel cell 281 Fugacity determination of 160-162, 219, 220 of gases 159 pressure variation of 162, 163 temperature variation of 162 Functions for work performance 54, 55 of state 1-54 of state, differentiation of 93ff, of state, extremal 53ff of state for adsorbed layers 295,296 of state for energy 55, 58, 59, 94, 96, 98, 99 of state for enthalpy 55, 94, 97, 101,102

INDEX

of state for entropy 53 of state, Gibbs 55, 63, 92, 94 of state, in electromagnetic fields 327-331, 333,334 of state in gravitational fields 287-291 of state, Helmholtz 54, 62, 94 of state for radiation 325,326 of state for surface layers 310-312 potential 27 Galvanic cells 280-284 examples of 280-284 Gases, expansion of 33ff equation of state for 69, 70 Gauge invariance, of chemical potentials 171 ff Gibbs adsorption isotherm 299 Gauss' theorem 15 Gibbs adsorption isotherm 299 Gibbs Duhem relation 88 ff, 92, 259 for adsorbed layers 296-298 Gibbs free energy, as a function of state 54, 62, 63 change during mixing 214, 217, 220-222, 224-230 differential change in a reaction 144 for irreversible processes 100, 101 for mixing, in the Bragg-Williams approximation 240-242 for the H + ion 255 in electromagnetic fields 330, 331,334, 341 Gibbs-Helmholtz equation 67 Gibbs phase rule 2-1 ff Glass electrode 278 Gradient theorem 15 Gravitational field effects 287-290 Gravochemical potential 288-290 Griffith's (in)equality 411 Grtineisen law 70 Hall coefficient 385 effect 385, 386 Heat capacity, ideal gas 74 capacity from experiment 66, 67, 73-75 capacity of mixing 245 deficit function 49ff electromagnetic 330, 342 flux vector 116, 353

THERMODYNAMICS

in second order transitions 244 in transit 31 isosteric, of adsorption 300 of adsorption 299, 300, 303-308 transfer, at different temperatures 50, 115, 116 transfer, irreversible 51, 52, 116 transfer, measurement of 80, 81 transfer, reversible 39, 50-52 Heat engine 45 operation of 45, 46 Helmholtz free energy, as a function of state 61, 62 in irreversible processes 99, 100 for electromagnetic fields 329, 334, 338 Henry's law 164, 202, 220, 253 as a standard state 202, 203,253 departures from 202 for homogeneous solutions 226 Homogeneous systems, defined 2 Hooke's law 26 Hotness levels 6 measurements of 7 Hugoniot's equation 360 Hydraulic permeability 375 permittivity 371 phenomenological equations for 376 resistance 376 Hydrolysis reaction 263 Ideal gas adiabatic changes of 130 chemical potential of 129 energy of 126 enthalpy of 126 entropy of at 0 K 88 entropy of at 0 K 1-83 free energy of 128 heat capacity of 126 standard state of 128 thermodynamic properties of 127-129 use in thermometry 7 Ideal solutions chemical potential of 131 criteria for 131,132 thermodynamic properties of 131-133 Integrals with variable limits 16, 17 Interference coefficients 365 Internal energy, change of 351

449

Inversion temperature 77, 78 Ion-specific electrode 278 Ionic strength, defined 256 Ionization process 249, 250 Isenthalps 78 Isolated system, defined 2 Isosteric heat of adsorption 300 Joule, definition of 80 Joule-Thomson coefficient 76 Joule-Thomson experiment 75ff Kelvin statement of the second law 46 temperature scale 7 Kinetic energy of system 350, 351 Lambda anomaly 245 point 245 Landau theory 415ff constraints in 416, 419 continuous transitions 420-422 expansion in order parameter 417-4 19 first order transitions 420, 422, 423 Langmuir adsorption isotherm 303 Laws of thermodynamics first 31 ff fourth (?) 425,426 second 38ff third 86ff Law of corresponding states 125,398 at the critical point 125 Lead storage cell 281,282 Legendre transformations 94, 106, 107 Lever rule 225 Line integrals, path independent 27 Liquid junction potentials 278, 279 Liquidus 232 Local states, principle of 347 Macroscopic coefficients 118 interference 365 proper 117,365 Magnetic induction 24 Magnetic moment 24 Magnetic properties, near criticality 400, 401, 453, 454 Magnetic susceptibility 334 Magnetic induction 24 Magnetic moment 24

450

Magnetite-zinc ferrite free energy change in 208, 209 fugacity of system 216 oxidation of 209 solid solutions 208 standard chemical potentials 209 Margules equation 215, 216 Mass action, law of 390 Matter flux 118 Maxwell relations for adsorbates 297 for electromagnetic fields 331,332, 335-337 general 66, 95 Mean activities for electrolytes 250-252 activity coefficients for electrolytes 251-253, 262 concentrations for electrolytes 251,252 molalities for electrolytes 251,252 mole fractions for electrolytes 251,252 membrane conductivity 375 Metal ion electrode 277 Metal-insoluble salt electrode 277, 278 Methane, adiabatic combustion of 196 Microscopic reversibility 393 Miscibility gap 232 Mixing functions asymmetric 214, 215, 220-223 symmetric 214ff Mixtures gaseous, nonideal 159, 160 solutions, nonideal 214ff temperature variations of 231 Molality 134, 170 Molarity 133, 134, 170 Nemst effect, transverse 386, 388 Nemst equation 271,273 Nonideal solutions, formation of 214 Ohm's law 365, 381 Onsager reciprocity conditions 366-369 Open circuit voltage 271 Open system, defined 2 thermodynamic relations for 94 Operational cells 280-282 Order-disorder phenomena 246, 247 Order parameter 244

INDEX

Partial molal entropies 95 functions in adsorption 302 volumes 88, 89, 95 volumes, determination of 90-92 Paths, defined 3 Peritectic 235 temperature 235 Pfaffian form 428 Phase, defined 2 Phase diagram construction of 230 ff correlation with free energies 230 ff interpretation of 230ff of carbon dioxide, schematic 124 of water schematic, 120-122 Phase separation, criteria for 216, 218 Phase space 3 Phenomenological coefficients 117, 118, 365 Phenomenological equations chemical 389 coefficients for 117, 118 electrokinetic 376 in two dimensions, electromagnetic 384 linear 365 simple formulation 365 thermoelectric 382 thermomagnetic 383 thermomolecular 372 Planck's statement of the second law 47 Point, representative 3 Polarizability, electric 334 Poisson's equation 439, 440 Porous plug experiment 75, 76 Potentials, in galvanic cells 269, 273 Pressure, magnetic 333, 341 Prigogine's theorem 366, 367 Processes cyclic 42, 44 quasistatic 3 reversible 3 steady state 4 Properties thermodynamic 2 transitive 5 Quinhydrone electrode 277 Radiation pressure 320, 329 Raoult's law 131, 132, 201,202, 220

THERMODYNAMICS

Reciprocal theorem 11 Reciprocity theorem 11 Redlich-Kwong equation of state 70 Redox electrode 277 Reference standard 127, 159, 160 Reservoir, defined 3 Resistivity, electrical 385,386 Righi-Leduc coefficient 387 effect 387 Rushbrooke (in)equality 412, 414 Rutger's equation 3464 Sax6n's law 374 Scaling exponents 404 interrelations 407 methodology 404ff parameter 4046 parameters experimental determination of 404-406 Second law of thermodynamics 38 ff corollary 40 relation to heat transfer 39, 40 Second order transitions 245 Seebeck coefficient 382, 386, 387 effect 382, 386, 387 Self-assembly critical concentration 317, 318 linear aggregates 316 of chain-like units 318-320 of monomer units 314, 315 three dimensional aggregates 317 two-dimensional aggregates 316, 317 Small drops, vapor pressure of 312, 313 Solidus 232 Shock adiabatic 358 effects far removed from equilibrium 359 effects in ideal gas 358, 360, 361 effects, large departures from equilibrium 359 effects near equilibrium 357 effects, small departures from equilibrium 357 equation of state 363 front 357, 358 front, speed of 357, 358 tube 357, 359 variables, interrelations of 360-363

451

Solubility measurements 261 Solutions composition dependence on temperature 138, 139 elevation of boiling point 142, 143 entropy of mixing 133 free energy of mixing 133 lowering of freezing point 139-142 Solution curves, algebraic 428 Sommerfeld, electronic contribution to heat capacity 70, 83 Soret effect 377-379 compositional changes in 378, 379 Specific heat, see heat capacity Specific heat, electronic contribution to 83 Spreading pressure 295 Stability, of systems 107ff Standard emf 271-273 Standard enthalpy of compounds 190, 191 of elements 189, 190 Standard state 166 Standard states of electrolytes 255, 266, 267 State space defined 3 Steady state conditions 3,366, 367 Stefan-Boltzmann law 325,326 Stiffness coefficients 72 Stokes' theorem 15, 28 Streaming current 374 potentials 374 Subsystem, defined 2 Superconductivity 344, 346 Surface layers, properties of 310-312 Surroundings definition of 2 participating in processes 58, 98 System defined 2 one component 120 Temperature, empirical 6 in systems out of equilibrium 59, 98 measurement of 7 Temperature scale absolute 40, 68, 69 Celsius 7 centigrade 7 thermodynamic 7, 40, 68, 69

452

Thermal conductivity 117, 381,386, 387 Thermal stress coefficients 73 Thermochemistry 189ff of ionic solutions 265 Thermodynamic properties near criticality 7-4 of adsorbate 295, 296, 302 of surface layers 313, 314 Thermodynamics first derivatives in 65 first law 3 lff fourth law (?) 425,426 near the critical point 402, 403 of electromagnetic fields 327 ff of gravitational fields 290, 291 of open systems 93 ff second law 38 ff second law, corollary to 40 third law 86-88 zeroth law 5, 6 Thermomagnetic effects 383ff Thermoelectric effect 379ff effect, phenomenological equations for 380 Thermoelectric power 382 Thermometers different types of 8 gas 8 Thermomolecular effects fluxes 369 forces 369 mechanical effects 369 phenomenological equations for 370, 372 steady state conditions for 370 Third law of thermodynamics 86 ff Tie lines 232 Transference number 283 Triple point, of water 8 van der Waals equation of state 124 ff near criticality 125 reduced form 398

INDEX

van't Hoff equation 147ff, 187 Vapor pressure of small drops 312, 313 Variables conjugate 116, 118, 355 extensive 3 intensive 3 thermodynamic 3 transformation of 1-11 ff Vector analysis 13 ff cross product 14 dot product 13 electric displacement 23 electric field 23 gradient cross vector 14 gradient dot vector 13 magnetic induction 24 magnetic moment 24 polarization 23 table of operations 15 Virtual processes 110 Volume, partial molal 88, 89 Water dissociation of 3-40, 4-15 phase diagram, schematic 120-122 Weak electrolytes at equilibrium 261 ff Widom's equality 407, 414 Work adiabatic 31 electric 23, 24, 327 electromagnetic 23-25, 29, 30, 327 elements of 19ff gravitational 22 in extension of springs 25-27 magnetic 24, 25 mechanical 21, 22 performance 27, 33-37, 51, 52 pressure volume 21, 22 Zeroth law of thermodynamics 5, 6