The mechanisms of fast reactions in solution

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The Mechanisms of Fast Reactions in Solution by

E.E Caldin MA, D .Phil. Late Professor of Physical Chemistry University of Kent, Canterbury, United Kingdom

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lOS Press

:~iwi Ohmsha

Amsterdam • Berlin • Oxford.

Tokyo. Washington, De

© 2001, H.P. Caldin All rights reserved. No part of this book may be reprodueed, stored in a retrieval system, or transmitted, in any form or by any means, without the prior written permission from the publisher. ISBN 1 58603 103 1 (lOS Press) ISBN 4 274 90409 1 C 3043 (Ohmsha) Library of Congress Catalog Card Number: 00-109666 Publisher lOS Press Nieuwe Hemweg 6B 1013 BG Amsterdam The Netherlands fax: +31206203419 e-mail: [email protected]

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v

Preface Professor Edward Caldin, Emeritus Professor of Chemistry at the University of Kent at Canterbury, died peacefully in December 1999. Ted, as he was universally known, was bom in August 1914. He won scholarships to St Paul's School, and to study as an undergraduate at The Queen's College, Oxford, where he gained a reputation for his boxing prowess. He was one of the first students of R.P. (Ronnie) Bell, the distinguished Oxford physical chemist. His thesis was concemed with the mechanism of decomposition of nitramide, and experimental proof of the concept of non-classical proton transfer, or 'proton tunnelling' , whereby the wave properties of the proton could enable it to move from an acid to a base with less than the classical activation energy. Such tunnelling was predicted to be more evident at low temperatures. His developing academic career was interrupted by the advent of World War 11. Ted spent the war in various military research establishments, particularly in South Wales, working on pyrotechnic compounds for flares and tracer bullets, which, with characteristic deprecating humour, he referred to as 'fireworks'. It was there that he met his future wife, Mary, with whom he had a long and very happy marriage. He also found the time to research material for a book on Chemical Thermodynamics, published by Oxford University Press in 1958. After the war, Ted was appointed to a Lectureship in Chemistry at the University of Leeds. There was asevere shortage of research materials - an entire physical chemistry undergraduate practical course had to be devised with sodium sulphate as the sole compound under investigation. Ted at this time enjoyed the challenge of doing research with minimal resources, and he was a master of the 'string and sealing wax' school, which he had leamed from his mentor Ronnie Bel!. One Leeds research student, Geoffrey (now Sir Geoffrey) Allen tells how he studied the thermodynamics of hydrogen-bonding through carboxylic acid dimerisation equipped only with a carefully calibrated thermometer and some benzene. In the 1950's, Ted resumed his experimental studies on proton tunnelling. He realised the need to study reactions over a wide temperature range in order to detect deviations from the Arrhenius equation. Accordingly, he devised equipment to study reactions at very low temperatures (~ - 100°C) in non-aqueous media. This need to study rates over a wide temperature range led him into the study of fast reactions, and he was a pioneer in the development of stopped-flow equipment in the 1950's to study diffusion-controlled reactions. At Leeds, he also developed a novel micro-wave temperature-jump apparatus for studying a range of reactions in non-aqueous solvents. He taught a postgraduate course on fast reaction techniques at Leeds and this gave rise to his classic text on "Fast Reactions in Solution", published in 1964. He spent his retirement years on a sequel to this book, "The Mechanisms of Fast Reactions in Solution", which was finished just before he died. In 1965, he moved to the newly founded University of Kent at Canterbury, first as Reader, and then as Professor of Physical Chemistry. The University appealed to Ted, as it was intended to be mn in the Oxbridge tradition, but this was not to be. The profound social changes of the late 1960's and early 1970's were not really to his taste. At Canterbury, Ted masterminded the activities of a group interested in the rates and mechanisms of fast reactions in solution. A number of distinguished visitors spent

Preface

vi

sabbaticals with the group, and Ted was the prime mover in setting up the very successful series of Royal Society of Chemistry meeting s on mechanisms of reactions in solution, the first of which took place in 1970, the last one being three years ago. He was also the First Chairman of the Fast Reactions in Solution (FRIS) Discussion Group, when it was founded in 1976, and he regularly attended its meeting s even when well into his eighties. In addition to his commitment to chemistry, Ted had a lifelong interest in the history of science and the link s between science and religion. He gave an unforgettable 'open' lecture on the former topic in the 1970's. Ted was a devout Catholic who insisted on the fundamentally rational nature of Christianity. Among his philosophical writings are to be found "The Power and Limits of Science" (1949) and "Science and the Christian Apologetic" (1953), as wel1 as numerous articles. Ted was a scholar and philosopher in the old academic tradition. His breadth of other interests, such as music, art, literature and history was vast, as evidenced by his library, and provided him with an apparently limitless source of reflection and humour; Milton and Samuel Johnson may have been among his favourites, but he was equal1y as likely to quote lines from Lewis Carrol1 and A.A. Milne. A research student who lodged in his house was amazed to find a set of the works of Thomas Aquinas, careful1y annotated in the margin in Ted's characteristic minute scrawling hand. Ted always took his pastoral duties very seriously, and was always concemed for the welfare of his students. He gave much time to university activities, especial1y in his later years, and he carried out these duties with conviction and enthusiasm. Mary Caldin died in 1997 after a long illness. They leave two sons, Hugh and Giles, to whom we extend our deepest sympathy.

Brian Robinson

John Crooks

vii

Foreword Ted Caldin read chemistry at The Queen's College, Oxford and started his research career under the supervision of R.p. (Ronnie) Bell. This was his introduction to fast reactions, for at that time Bell was preoccupied by the recently predicted isotope effect on proton exchange. Later, during World War 11,Ted was assigned to work on explosives in Ministry of Supply laboratories. In 1945 a lectureship at Leeds University brought him back to academic research and to teaching. The physical and inorganic department was led by the charismatic and forwardthinking M.G. Evans who was bent on creating a modem chemistry course. Ted's role was to follow M.G.'s all-pervading first year 'introductory' undergraduate course with a secondyear course on Chemical Thermodynamics - not an easy task. He was a shy man with a waspish sense of humour, his slight frame topped with a shock of black hair, who bustled fram laboratory to lecture theatre in a nervous frenzy. He taught thermodynamics with an infectious enthusiasm for the originalliterature and scathing criticism of the textbooks of the day. Above all, he maintained the momentum established by the suave head of the department and was equally popular with the undergraduates. Ted Caldin's interests ranged more widely than chemistry. Philosophy and especially its interface with the scientific method attracted him. He lectured in the Department of Philosophy and in 1949 published "The Power and Limits of Science". Extramural sessions at his home were par for the course. His research programme was really launched in 1949 with two projects. One was on the colligative praperties of non-aqueous solutions (carbox ylic acids in benzene) with pressure and temperature as variables. The other was the seed com for studies of fast reactions. Acid-base reactions involving a coloured reactant or product were followed by measuring the time-dependence of the UV spectrum of the mixture. In order to slow down the reaction, measurements were made at around 193 K. The reactors used were of 0.5 to 1.0 litre capacity with 10 cm vacuum-sealed double windows to prevent misting. So began Ted's magnum opus.

Sir Geoffrey Allen, Chancellor ofthe University of East Anglia

viii

Acknowledgements Grateful acknowledgements are due to the authors, editors, and publishers concemed for permission to reproduce certain figures and tables. References are given in the texto It is a pleasure too for Ted's family to thank those who have contributed so much to this book: Janet Pitcher, who patiently typed and retyped the manuscript; Peter Brown and his colleagues at lOS Press; Ted's colleagues who assisted with general research, in particular Ceri Gibson, and those who kindly read and commented on the text, in whole or in parto Grateful thanks go to friends such as Dom Cyprian Stockford OSB, Maurice and Joan Crosland and Derek and Christiana Crabtree, who provided so much support and encouragement throughout the writing process. Special thanks are due to Professor Brian Robinson who so generously and selflessly undertook almost the entire burden of producing the book after Ted's death. Finally, this book is published in memory of Ted and Mary themselves.

The Estate of Professor E.F. Caldin

ix

Contents Preface Foreword Acknowledgements Prologue

v vii viii xi

PartOne Chapter 1. Introduction: Origins, Methods, Mechanisms, Rate Constants 1.1. Introduction 1.2. The structure of this book 1.3. Theory of rate constants for diffusion-controlled reactions References

3 10 12 18

Chapter 2. The Rates of Diffusion-Controlled Reactions 2.1. Introduction 2.2. Application of the diffusion law to rates of encounter and of chemical reaction: the Smoluchowski equation 2.3. A molecular model for translational diffusion and diffusion-influenced rates of reaction: random-walk theory 2.4. Encounter in solution: the solvent 'cage' 2.5. The course of an encounter 2.6. Alternative theoretical approaches: refinement of theory References

31 33 36 45 48

Chapter 3. The Mathematical Theory of Diffusion and Diffusion-Controlled Reaction Rates 3.1. Introduction 3.2. Theory of rates of diffusion, in terms of concentration gradients 3.3. Calculation of diffusion-controlled rate constant 3.4. Diffusion coefficients: theory and experiment References

51 54 59 65 79

21 21

Chapter 4. Flash Photolysis Techniques 4.1. Introduction 4.2. Experimental flash techniques 4.3. Some applications of flash techniques References

81 85 98 115

Chapter 5. Initiation by High-Energy Radiation: Pulse Radiolysis 5.1. Introduction 5.2. Experimental techniques

119 122

x

Contents

5.3. Some applications of pulse radiolysis References

125 136

Part Two Chapter

6. Fluorescence Quenching and Energy- Transfer from Excited Molecules 6.1. General principIes 6.2. Theory of fluorescence quenching 6.3. Experimental investigation of fluorescence quenching 6.4. Kinetic and mechanistic applications of fluorescence measurements 6.5. Physical mechanisms for non-radiative energy transfer between molecules References

Chapter 7. DUrafast Processes: Sub·Picosecond and Femtosecond Techniques 7.1. Introduction 7.2. Femtosecond studies of the entire reaction path 7.3. Some systems studied by sub-picosecond and femtosecond techniques References

141 142 151 156 167 187

191 192 197 221

Part Three Chapter

8. Proton· Transfer and Group- Transfer Reactons in Solution: Marcus Theory (1) 8.1. Marcus theory. Introduction 8.2. Application of Marcus theory to proton-transfer reactions in solution 8.3. Application of Marcus theory to group transfer 8.4. Refinements and extensions of the Marcus treatment References

227 228 256 261 261

Chapter 9. Electron- Transfer Reactions: Marcus Theory (11) 9.1. Introduction I. Electron-transfer in outer-sphere reactions of metal ions 9.2 11.Electron-transfer reactions in organic systems 9.3. Tailpiece: a brief re-capitulation References

265 266 292 292 317 318

Epilogue

323

List of symbols used frequently Subject Index

in the Marcus theory of reactions

325 327

xi

Prologue Mechanisms of Fast Reactions in Solution This book is a sequel to the author's Fast Reactions in Solution (1964). It is not a revised edition of that text, but an entirely new book, with a different approach. Fast-reaction techniques are now well established and widely used, and it is timely to consider what they contribute to the study of reaction mechanisms, i.e., to our understanding of events on the molecular scale during reactions. They have in fact transformed our notion of what can be expected in mechanistic explanations. Experiment and theory alike have shown great progress. On the experimental side, techniques of monitoring and recording the course of chemical reactions have proliferated and become more sensitive and more accurate. Laser devices capable of producing very short light-pulses have led to remarkable advances. It has be come possible to detect and record the motions of atoms and electrons in reacting systems at intervals of les s than a million-millionth of a second, thus making possible direct monitoring ofbond-stretching and bond-formation. On the theoretical side, several interrelated developments call for remark. Moleculardynamics calculation (1inked to high-speed computers) have made possible the rapid testing of molecular models of the course of fast reactions; often it tums out that a simple classical model gives a reasonable approximation to the full quantum-mechanical calculation. Such models can be derived from 'Marcus theory', which is, like transition-state theory, a general framework relating reaction rates to energetics; it takes systematic account of the solvent reorganisation due to the changes of electron distribution required by reaction. Advances in spectroscopy have led to a much fuller knowledge of reaction energetics. These interlocking developments have greatly improved and facilitated the theoretical understanding of the kinetic data produced by experimentalists, who in tum have been enabled to devise new experiments likely to produce other important results. A formidable alliance of experimental and theoretical methods has thus developed. The course of events at molecular level for a considerable number of diverse types of reaction, can now be described in previously unheard-of detail. Our conception of chemical understanding has been enlarged. This book is intended to present these advances to the attention of a wider audience.

Part One

3

Chapter 1

Introduction: Origins, Methods, Mechanisms, Rate Constants 1.1. Introduction 1.1.1. Whatdoes

'fast' mean?

The term 'fast reaction' is not a precise one, but it is non e the less serviceable. It is commonly used to mean, broadly, a reaction that is more than half completed within a time comparable with that required to initiate a reaction and make a measurement of its progress. For reactions in solution, initiation is often done by mixing two reactant solutions, an operation which takes a few seconds unless special devices are used. If the course of reaction is monitored by withdrawing samples for analysis, or by taking readings of some kind, each such operation will add a few more seconds. Consequently, when these 'conventional' methods are used, accurate rate constants cannot be determined for reactions with half-times much less than 10 seconds. An individual reaction which is half completed in one second might, in a rough-and-ready way, be taken as marking the borderline between 'fast' and 'conventional' rates. But the observed rate will depend on the experimental conditions; a second-order reaction may have a high rate constant and yet take place comparatively slowly if the concentrations are small enough, or if the temperature is lowered. The term 'fast reaction' is therefore commonly used to inelude reactions which would be too fast for 'conventional' methods if conducted at 'ordinary' temperatures and concentrations. These are not precise definitions; they serve only to indicate, in a preliminary way, the range of rates for which the methods described in this book are required. Most of the fast-reaction techniques can measure half-times down to 10-7 s, several of them to 10-9 sor below, and a few to 10-12 sor below. The range of first-order rate constants amenable to these techniques is thus from about 1 s-1 to well above 109 s-I, so the accessible range has been extended by over ten powers of ten (see Figure 1.1).1 1.1.2. Origins The development of fast-reaction studies may be dated from investigations begun in the 1920's on bimolecular reactions using flow techniques. Reaction was initiated by rapidly 1 The time for half-change t1/2 of a first-order reaction is related to lhe first-order rate constant k by ktl/2 = In2 "" 0.7. For a second-order reaction, with each reagent at concentration a, lhe corresponding relation is ktl/2

= l/a.

4

Chapter 1

mixing reactant solutions by forcing them into a mixing chamber and thence down a glass tube of uniform bore at a velocity of several metres per second; thus distances along the tube corresponded to times from initiation, and measurements of (e.g.) optical absorbance at a series of points gave a reaction/time plot. Reactions with half-lives of the order of a millisecond could be studied; the limiting factor was the rate of mixing, determined by physical factors such as the viscosity of the solutions. The techniques were steadily developed and thoroughly tested, largely on biochemical problems [1], but they were not widely known to chemists until the 1950's. Apart from some investigations on fluorescence quenching and on photostationary states, and on the use of low temperatures to reduce rates to the conventionally-measurable range, no other special techniques had been developed for fast reactions in solution by 1939 when war broke out. By 1954, when the first international conference on fast reactions was held, by the Faraday Society [2], a range of radically new techniques had emerged, in which initiation is not dependent on mixing (see Section 1.1.4). Relaxation techniques, both singledisturbance and periodic-disturbance, had been developed, largely at Gottingen, and were unveiled in a paper by Eigen [2,a]. Porter and Norrish [2,b,c] at Cambridge were working on flash-photolysis techniques (see Chapters 4 and 7), and their apparatus was described. (Pulse radiolysis carne not long after; see Chapter 5.) Fluorescence-quenching techniques (see Chapter 6), although well established, were not represented. Electrochemical techniques had been developed, mainly by Czech workers, since the 1940's; the possibilities were here discussed. Nmr methods were coming to be applied to solutions; the kinetic uses of analysis of line-broadening measurements were illustrated by some measurements on proton-transfer in liquid ammonia. All the main strategies for fast-reaction research were exemplified at this meeting. In 1960 the publication of the papers presented at an international meeting at Hahnenklee in Germany [3] showed that these new methods were coming into wider use and had produced results of great interest. The more-developed methods (flow, relaxation, flash photolysis) were already being applied to quite complex systems (photosynthesis, enzyme reactions). Review volumes began to appear, making available to wider audiences the results published in primary research journals. The first volume of the Weissberger series [4] was published in 1963; it contained authoritative articles on various methods, with particularly full treatments of flow, relaxation and flash techniques. Porter's series Progress in Reaction Kinetics contained review articles and summary tables of data [5]. A singlevolume book was published in 1964 [6]. In 1967 the Nobel Foundation held a high-level symposium of leading scientists in the field of fast reactions [7], largely those in solution, and the Nobel Prize for chemistry was awarded jointly to Eigen, Norrish and Portero All the main techniques, whatever mode of initiation (small-perturbation or large-perturbation), had been developed and tested, and some had been widely used. Major themes that could be noticed [7] were the newly-realised possibility of studying directly the fast elementary steps in reactions (e.g., proton-transfer [7,e], electron transfer [7,b,f], the role of electronically-excited states [7,b,c,f], and the applications to reactions of biological interest (e.g., photosynthesis [7,b] enzyme reactions, [7,g] base-pairing of polynucleotides [7,e]). Reaction half-lives of a few nanoseconds had been measured, and some diffusion-controlled rate constants had been determined. This symposium is a landmark in the development and spread of fast-reaction techniques. The scientists concerned were mostly chemists interested in elucidating the mech-

Introduction:

Origins, Methods, Mechanisms,

Rate Constants

5

anism of chemical reactions by showing them as combinations of simpler 'elementary' reactions, Le., as rearrangements of the atoms concerned. There were also physicists and chemical physicists concerned with the physical aspects of (for example) exchange of energy between molecules or spectroscopic determination of energy levels, and also theoreticians working on improvements in molecular dynamics or wave-mechanical calculations. Developments of two kinds ensued. There were many among these scientists who were able to devise new types of apparatus and experimental procedures, seeking to improve the speed, sensitivity, versatility, reliability and ease of operation of the techniques. There were many others, especial1y those already interested in particular classes of reactions, who saw how the new kinetic techniques would permit systematic investigation of wider ranges of reactions. Some of these were concerned with inorganic reactions [8], such as complexing between ligands and transition-metal ions in aqueous solution. Advances in the preceding two decades in studies of structure and thermodynamics were recognised as a 'renaissance' of inorganic chemistry. There had been thorough investigation of the complexation reactions of octahedral Co(III), whose kinetics could be studied by mixing the reactant solutions manual1y and making measurements on a spectrophotometer at intervals no shorter than a few seconds. The corresponding reactions of Ni(I1) were too fast for this technique, with half-lives typical1y in the millisecond region, but they could be studied with a stopped- flow apparatus. The reactions of Fe(I1I) were too fast for this, but could be studied with a temperature-jump apparatus; those of Cu(I1), faster still, by the ultrasonic-absorption method. Physical organic chemistry was similarly enlarged. For example, proton-transfer reactions of carbon acids and bases could be studied by conventional kinetic measurements, but those of oxygen and nitrogen acids and bases were inaccessible until relaxation techniques eould be applied (cf. Seetion 8.2.3.1). The historieal distinetions between inorganie, organie and physieal ehemistry had therefore beeome less applieable. One result of this reeognition of eommon ground between chemists segregated in traditional departments and institutes was that eonferences flourished. A eommon interest in keeping up with developments in teehniques, and a realisation of eommon purposes, brought together meehanistie ehemists with a wide range of preoeeupations, from the simplest systems to the most eomplieated. Natural1y these conferences were sometimes foeussed on a particular topie or teehnique [9], but wide-ranging conferenees have remained popular [10,11]. An excel1ent survey of the teehniques was published in 1992 [12]. 1.1.3. Reaction rates and reaction mechanisms Rate measurements are important because they enable us to test hypotheses about the molecular ehanges involved, i.e., the reaetion meehanism. They eontribute to an understanding of how bonds are made and broken, how the eorresponding shifts of distribution of electrons are brought about, and how energy is transferred from bond to bond within a reacting moleeule or between a reaetant molecule and the solvent. The maeroseopie observed rates are interpreted in terms of events at the mieroseopie moleeular leve!. The rate law wil1 involve one or more rate constants, and the ways in which these change with temperature and other adjustable parameters describing the initial state of the system will constitute the evidence against which a mechanistic hypothesis is to be tested. The ultimate goal is to calculate from the fundamental wave-mechanical properties of the reactants a predicted value for the rate constant in agreement with experimental results. For example, consider the

6

Chapter 1

recombination of iodine atoms after dissociation by a powerful flash of iodine dissolved in an inert solvent: h -+ 21. The recombination process (2 I -+ h) can be monitored by fast-detection techniques that enable us to record the value of the optical absorbance of a sample solution over short periods of time. Plots of these values against time give us the rates of the reaction in the chosen solvent at a series of iodine concentrations at a particular temperature. Plots of rate against concentration can then be made; they indicate that (minor deviations apart) the rate is proportional to the square of the iodine concentration; thus we can write the equation rate = k[I]2. This second-order rate law is readily interpreted by a molecular model in which reaction occurs only when two iodine atoms collide: 1+1 -+ h. This is an instance of a relation in chemical kinetics. The next step is to vary the conditions of the experiment, particularly the temperature, and interpret the results in terms of our knowledge of the forces involved. In the present case, for instance, we may expect that two iodine atoms will always recombine if they collide, so we can use our knowledge of diffusion to relate the rate constant for such a diffusion-controlled reaction (k3) to the viscosity of the solvent (see below, Section 1.3); for water and other common solvents, the predicted value is ~ 1010 dm3 mol-1 s-l, which is normally close to the experimental value. The ultimate goal is to calculate rate constants in agreement with experiment from the forces involved, both between the atoms and between the atoms and the solvent.2 We can now give precise definitions of some words and phrases that are often used loosely. A reaction rate is arate of change of the concentration of some chemical species at a particular moment; it is derived from a set of observations, in which the course of a reaction is monitored over a period of time. Such observations are the basic experimental data. By analysing the relation between these rates and the corresponding concentrations, one obtains arate law that fits both the data and (often) some standard form (e.g., first-order, in which the rate is proportional to the concentration of one of the reactants). These rate laws, along with known structural data, may be given some interpretation in terms of reaction kinetics; one would describe a scheme of molecular motions that would explain the rate law. Often there are several alternative schemes that would be consistent with a particular set of data; further experimentation would then be needed in order to choose between them. Such schemes in terms of chemical kinetics describe events on the microscopic scale, involving atoms and molecules, as distinct from rate laws which are expressed in terms of macromolecular quantities (time and concentration). The schemes may in turn be interpreted in terms of a reaction mechanism, which relates them to chemical dynamics, i.e., to theories of how molecules behave, in terms either of some particular model with limited scope (such as collison theory, or transition-state theory) or of the more fundamental body of theory based upon quantum mechanics. Few types of reaction are simple, however. Most require a combination of 'elementary' steps - reactions assumed to be single irreducible acts at the molecular level OCCUfring concurrently or in a sequence. The objective of an investigation of a reaction from a mechanistic point of view is to construct a scheme of elementary reactions that predicts correctly the observed behaviour. The closer the predictions are to the observations, the

2 Such a calculation is always a difficult task, and is even more difficult for reactions in solution than for gas reactions, but it has been achieved in the case of electron transfer between transition-metal ions in aqueous solution, a type ofreaction unique in its mechanism (see below, and Section 9.1.3.4).

Introduction:

Origins. Methods. Mechanisms,

Rate Constants

7

better supported is the mechanism. It remains, however, a theoretical construction, subject to correction or improvement.3,4 In parenthesis, we must note an ambiguity in the term 'fast'. A 'fast' reaction is, in common parlance, one whose observed macroscopic rate constant (k) approaches that calculated for a diffusion controlled reaction (kD). A 'fast' molecular process, however, is one which is soon over, i.e., requires little time (e.g., vibration within a molecule); it may usefully be called a 'short-time' or 'short-lived' process. Such a process does not necessarily lead to a fast reaction. For instance, the time required for a proton to migrate fram a carbon atom to an oxygen atom is around 10-13 s, but if it is coupled to slower concomitant changes (e.g., of configuration or solvation) the macroscopic rate constant may be far below the diffusion-controlled value, while approaching kD when such changes are minimal (see Chapter 8, Section 8.1.3). Conversely, however, an observed rate constant near kD requires that the molecular changes be short-lived, so that in all or most encounters between reactive molecules they are completed within the short lifetime of the encounter complex (cf. Section 2.5.1.1). Reaction rate measurements are essential to a full account of any reaction mechanism. One at least of the elementary reactions making up a given reaction must be at least as fast as the overall reaction. All reactions in solution, except rearrangements or configurational changes, necessarily involve diffusion: in bimolecular reactions the reactant molecules must meet each other in the course of random diffusional motions, and if there are two products (A + B -+ C + D) these willlikewise separate by diffusion; this will occur also in unimolecular reactions with more than one product (A -+ B + C, etc.). We therefore need as good a theory as possible of the process of diffusion and of the properties of any short-lived intermediate complexes. We must also take account of the fact that three of the main techniques - flash photolysis, pulse radiolysis, and fluorescence quenching - differ fram the rest in that they involve initiation of the reaction by absorption of energy from, for example, pulses of visible or ultraviolet light. This has two consequences: (i) since such pulses can be made very short-lived, 'ultrafast' phenomena can be studied, and (ii) initiation often produces excited states or free radicals, which may be highly reactive.5 These techniques are therefore particularly valuable in building up a body of knowledge about mechanisms. 1.1.4. The range of rate constants of fast reactions in solution Most of the fast-reaction techniques can determine half-times down to 10-7 s, several of them to 10-9 s or below, and a few to 10-12 s or below. The range of first-order rate 3 An excellent introductory account of these fundamentals is contained in the first four pages of D.N. Hague, Fast Reactions [6,b]. 4 An instance of such a revision occurred in connection with the reaction H2 + 12 --+ 2 HI in the gas phase. It was long thought to be a bimolecular, reaction between H2 and 12 molecules, but was shown in 1967 to occur mainly by a combination of two parallel steps each involving atoms, such as 12 ~ 21 and H2 + 21 --+ 2HI (cf. l.H. Sullivan, J. Chem. Phys. 46 (1967) 13). 5 The great majority of the second-order rate constants in solution recorded in Tables 01 Chemical Kinetics: Homogeneous Reactions (N.B.S. Circular 510; 1951) lie in the range 10-7 to 1.0 M-I s-l. The lowest directlydetermined value recorded in these standard tables is 5 x 10-10 M-1 s-l.

8

Chapter 1

Fast-reaction methods

Conventional methods

.1

I~ I

1

I

1 1

1

1.0 1

I

First-order rate constant (S-I) (logarithmic scale)

Conventional methods

Fast-reaction methods

-1--

14 I

_1 1

I

I

I

I

10-6 I

10-4 !

10-2 I

1.0 !

102 I

Second-order rate constant (M-I

4 10 I

S-I)

6 10 I

108 I

10 10 I

I

(logarithmic scale)

Figure 1.1 Ranges of rate constants accessible by conventional and by fast -reaction techniques.

constants amenable to these techniques is thus from about 1 s-I to above 109 s-I, so the accessible range has been extended by over ten powers of ten.6 The range of rate constants accessible only by means of special techniques is greater than the whole of the 'conventional' range. The longest half-time that is commonly convenient to measure is of the order of one day, or 105 s; by measuring initial rates it is possible to extend thus by a factor of perhaps 102. The range of first-order rate constants that are accessible by ordinary methods is therefore about 10-7 to 10-1 S-1; for second-order rate constants it is about 10-7 to 1 M-I s-l. These ranges are shorter than the ranges where measurement requires fast-reaction techniques (Figure 1.1). Since there is no evidence that reaction rates are grouped in any way, we may expect very many reactions to have rates in the 'fas!' range. 1.1.5. Strategies and methods for determining the rates of fast reactions in solution Methods for the determination of the rates of fast reactions exemplify four possible strategies or guiding principIes; these are as follows. 6 The corresponding range of second-order rate constants depends also on the rate measurements can be made, and therefore on the sensitivity of the detection 1011 M-I s-I have been measured by several techniques, and the range extends range, for which the maximum rate constant may be taken somewhat arbitrarily sponding to a half-time of 10 s at concentration 0.1 M. or 100 s at 0.01 M; thus range is again in the region of 1010.

lowest concentration at which system. Values up to 1010 or down to meet the conventional as around I M-I s-I, correthe extension of the accessible

Introduction:

Origins. Methods, Mechanisms,

Rate Constants

9

(i) The principIe on which some of the simplest methods are based is to bring the rate down into the 'conventional' range, where the course of reaction can be monitored without the use of special techniques. Examples of methods based on this strategy are the use of low concentrations or of low temperatures. (ii) Another principIe that leads to some relatively simple methods is to reduce the mixing time so that it becomes small compared with the half-time of the reaction. This is commonly done by flow techniques. The reaction may be monitored either (a) by some means that avoids the use of fast detection techniques, such as the continuous-flow technique, or (b) by some fast measuring device, as in the stopped-flow method. (iii) External initiation of reaction without mixing. The shortest time that can be measured by any of the preceding methods is determined by the least time of mixing of the solutions, which is not easily reduced much below a millisecond. There are, however, ways in which mixing can be avoided altogether, by initiating fram outside the reaction solution. Two different principIes are available, as follows. (a) In relaxation methods, the principIe is to disturb a system which is in chemical equilibrium, and monitor the resulting concentration changes as the system 'relaxes' towards the new equilibrium. In the temperature-jump method, for example, an equilibrium is perturbed by imposing a sudden change of temperature. In response to this, reaction must take place, in one direction or the other, until the new equilibrium concentrations have been attained, and the course of the concentration changes is monitored with the aid of a fast oscillographic or digital method. Pressure-jump and electric-field-jump methods are also well established. Altematively, a periodic (sinusoidal) variation of temperature and pressure can be effected by means of ultrasonic vibrations; when the half-time of the reaction is comparable with the period of the disturbance, there is a sharp increase in the power absorbed, and fram the power/frequency spectrum the rate constant can be deduced. An oscillating electric field may be similarly used. In all these techniques the displacements of equilibrium are kept small, since this greatly simplifies the mathematical treatment. (b) In photochemical and related methods the initiation of chemical change is achieved by irradiation. The absorption of light, or of high-energy radiation, or electrans, by a solution can lead to drastic changes, by praducing atoms or free radicals or excited states of molecules, which then set off further changes. In the jlash-photolysis and pulseradiolysis techniques, a single powerful pulse of energy is used, and the subsequent reactions may be followed by fast recording techniques. (iv) A fourth strategy is to make the fast chemical reaction compete with some other fast process whose rate can be separately determined. There are three methods which, while otherwise differing widely, depend on this principIe. (a) Fluorescence-quenching methods depend upon competition between reaction and fluorescence emission. When a substance fluoresces in solution, the excited molecules have a certain mean lifetime before they emit light; this can be determined, and is commonly of the order of 10-8 s. If a solute is added which reacts rapidly with these excited molecules, so that an appreciable number of them are destroyed before emission can take place, the average lifetime and the fluorescence intensity are reduced. Fram their variation with concentration, the rate constant of the reaction can be found.

10

Chapter 1

(b) Electrochemical processes can be arranged in which (for example) a current which would normally be controlled by the rate of diffusion of some species is affected also by the rate at which that species is produced by a reaction in the solution. The first technique to be so used was polarography; others are the rotating-disc, potentiostatic and galvanostatic techniques. (c) Nuclear magnetic resonance and electron-spin resonance (paramagnetic resonance) can be adapted to rate measurements. The width of a line in (for instance) a proton nrnr spectrum is related to the lifetime of the spinning pro ton in a particular spin state. If this lifetime is cut short by a reaction involving that proton, the corresponding line in the nmr spectrum is broadened, to an extent depending on the rate of the reaction. The same principIe can be applied to determine rates of electron-transfer reactions from epr spectra. 1.1.6. Reaction rates accessible by the various methods The reaction half-times and rate constants accessible by the various techniques are summarised in Table 1.1. The smallest half-times, from about 10-9 s down to 10-13 or even 10-14 s, have been determined by the flash-photolysis, fluorescence-quenching, ultrasonicabsorption, and esr methods; next come the temperature-jump and electric-impulse techniques (approximately 10-6 s). In terms of rate constants, for first-order reactions the upper limit for a given technique is approximately the reciprocal of the least half-time that can be measured (k ~ O.7t1/~); for second-orderrate constants, however, the upper limit depends also on the concentration at which the reaction can be observed, and hence on the sensitivity of the technique; for most of the techniques the maximum observed value is of the order of 109 M-I s-I or above, even for stopped-flow for which the smallest accessible half-time is little less than 10-3 s. The usefulness of a technique is not, of course, related solely to the maximum time resolution or observable rate constant; other factors such as versatility, precision, convenience of operation and availability can all be important. The stopped-flow method, for instance, which is the most widely used of all, owes its popularity to its adaptability, speed and convenience, robustness, and wide availability; it is suited not to the fastest reactions but to those with rate constants less than about 105 M-1 s-l. Conversely, fluorescencequenching methods are applicable only to very fast reactions, for which however they are very powerful; and flash techniques, uniquely, can follow changes down to 10-14 s. The availability of commercial equipment, and its cost, may also be important, especially where sophisticated electronic instrumentation is required.

1.2. The structure of this book These considerations, along with those outlined in the foreword, are the main influences in the design of this book, which may be summarised as follows. Diffusion, control of reaction rates by diffusion, and the properties of encounter complexes are considered in Chapters 2 and 3; Chapter 2 is largely descriptive, while the more mathematical aspects are in Chapter 3, which may be judiciously skimmed by readers allergic to mathematical equations. It is advantageous to have in mind a preliminary pictorial sketch, and this is presented in the present chapter. Next follow Chapters 4,5,6 on the three strong-perturbation

Introduction:

Origins, Methods, Mechanisms,

Rate Constants

Table 1.1 Ranges of reaetion half-time (11/2 in s) aeeessible by various teehniques. Fulllines apparatus; dotted extensions refer to speeial equipment. All values are orders of magnitude Method

Ref.

Continuous flow Stopped flow Temperature-jump Pressure-jump Eleetric-field jump Ultrasonie relaxation Dieleetric relaxation Flash* Pulse radiolysis** Fluorescenee quenehing Eleetroehemieal E.p.r. N.m.r. (proton)

[a] lb] [e] [d] [e] [f] [g] [h] [j] [k] [1] [m] [n]

11 refer to standard

10-6 ..........

10-10

............

* Flash teehniques can be used for reaetions with tl/2 down to 2 x 10-13 s (see Seetions 4.3.6.2 and 7.3.4.3). ** Pulse radiolysis teehniques can be used for reaetions with tl/2 down to 3 x 10-11 s (see Seetion 5.2.2). [a] H. Strehlow, Ref. [12], Chapter 3; 1.F. Holzwarth, in Ref. [I0,a]. lb] B. Chanee, in Ref. [I,d], Chapter I1; B.H. Robinson, in Ref. [l,e], Chapter 1. [e] H. Strehlow, Ref. [12J, Chapter 4, p. 60 seq.; 0.0. Hammes, in Ref. [I,e], Chapter IV; D.H. Tumer, Ref. [I,e], Chapter III. [d] H. Strehlow, Ref. [12], Chapter 4, p. 64 seq.; W. Knoehe, in Ref. [I,e], Chapter V, and in Ref. [l,e], Chapter IV. [e] H. Strehlow, Ref. [12], Chapter 4, pp. 67-68; E.M. Eyring and P. Hemmes, in Ref. [I,e], Chapter V. [f] H. Strehlow, Ref. [12], Chapter 4, pp. 70-77; 1.E. Stuehr, in Ref. [l,e], Chapter VI. [g] H. Strehlow, Ref. [12], Chapter 4, pp. 77-80; 1. Everaert and A. Persoons, 1. Phys. Chem. 85 (1981) 3930; L. De Maeyer, R. Wolsehann and L. Hellemans, in Ref. [lO,a], p. 50\. [h] This book, Chapter 4 (esp. Tables 4.1 and 4.2) and Chapter 7; N. Hirota and H. Olya-Nishiguchi, in Ref. [\.e], Chapter Xl, Tables 6 and 7; and see footnote*. [j] This book, Chapter 5 (esp. Table 5.1). [k] This book, Chapter 6 (esp. Tables 6.1, 6.2, 6.3, 6.4). [1] H. Strehlow, Ref. [12], Chapter 8; H. Strehlow in Ref. [I,d], Chapter VIII; C.P. Andrieux and 1.M. Savéant in Ref. [1,eJ, Chapter VII. [m] N. Hirota and H. Olya-Nishiguru, in Ref. [I,e], Chapter XI, Tables 4 and 5. [n] K.R. lennings and R.B. Cundall, Prog. React. Kinet. 9 (1977) 2. Table 1.2 The time-spectrum

of moleeular proeesses

Rate scale (s-l)

Time seale (s)

t

t

1015 femto

Proeesses

10-15 (fs)

eleetronie motion eleetron orbital jumps

1012 pico

10-12 (ps)

vibrational motion bond cleavage (weak bonds) eleetron transfer proton transfer

109 nano

10-9 (ns)

rotational and translational motion (small molecules) bond cleavage (strong bonds) spin-orbit eoupling

106 miero

10-6 (¡.¡.s)

rotational and translational hyperfine eoupling

motion (large moleeules)

12

Chapter 1

techniques. Chapter 7 outlines the remarkable developments in studies of 'ultrafast' reactions due to the use of very fast detection devices and the combination of these with mass-spectrographic techniques. (We note the triple alliance of kinetic experiments with spectroscopy and molecular dynamics.) Chapters 8 and 9, which should be considered together, have a dual function. In the first place, they deal, respectively, with proton-transfer and electron-transfer reactions, in recognition of their important roles as elementary processes in numerous mechanisms. In the second place, they both deal with 'Marcus theory', which has been steadily developed from the 1950's onwards and is the leading general theory of reaction rates. (The Nobe1 prize was awarded in 1992 to R.A. Marcus, a leading participant in their development.) Marcus theory builds on transition-state theory, which relates rate constants and their temperature-dependence to quasi-thermodynamic activation parameters (activation free energy, entropy and enthalpy) in a formula to which experimental results on most reactions can be fitted,7 but it assumes a more specific model, in which effects of solvent reorganisation are made explicit as well as those of changes of electron distribution and of bond breaking and making.1t has extensive support on the experimental side from fast-reaction studies (cf. Tables 8.1 to 8.4 and 9.1 to 9.6), and on the theoretical side from quantum-mechanical calculations on potential-energy barriers. It is successful both with proton-transfer, where covalent bonds are made and broken, and with electron transfer, where (uniquely) they are noto Its greatest achievement is a successful calculation of rate constants for outer-sphere e1ectron-transfer reactions between transition-metal ions in aqueous solution, in terms of the fundamental wave-mechanical properties of reactant molecules and an e1ectrostatic model of the solvento The results are the closest approach yet made to the ultimate goal of a successful ab-initio calculation for a reaction in solution; the agreement between theory and experiment is within a factor of ~ 30, compared with a range of 15 powers of ten for the observed rate constant (cf. Section 9.1.3.4). The possibility of progress towards a universally-applicable theory can be envisaged. Our treatment begins (unconventionally) by outlining (in Chapter 8) the fundamental s of the theory as applied to proton transfer (and methyl-group transfer), and goes on to apply them to electron transfer (in Chapter 9).8

1.3. Theory of rate constants for diffusion-controlled

reactions

1.3.1. The energetics ofvery fast reactions There is evidently an upper limit to the rate at which a bimolecular reaction in solution can proceed, set by the rate at which reactant molecules encounter each other. Some reactions have rates close to this limit, but there are many which have much lower rates and yet are fast in the sense that special means are required to measure their rates. The temperaturevariation of the rates of these latter reactions is represented empirically by the Arrhenius 7 Transition-state theory was developed in lhe 1930's along with the wave-mechanical theory of covalent bonding (for an early survey, see: S. Glasstone, H. Eyring and K.J. Laidler, Theory 01 Rate Processes, McGraw-Hill, New York, 1941). 8 This order is lhe reverse of that of the historical development, which began with the problem of electron transfer; lhe lheory was later adapted to proton transfer and subsequent1y to group transfer (see Sections 8.1 seq., 8.3).

Introduction:

Origins, Methods, Mechanisms,

Table 1.3 Second-order rate constants at 298 K, calculated various values of EA, to nearest order of magnitude

50

60

40

Rate Constants from k = 1011 exp( -EA/

20

exp(-EA/RT)

2 x 10-11

3 x 10-4

k (M-1 s-l)

2

3 x 107

13 RT) for

o

equation k = A exp( - EA/ RT), where EA is taken to represent a critical energy without which a collision will not result in reaction, and exp( - EA/ RT) is the fraction of effective collisions. To illustrate the relationship between k and EA, Table 1.3 shows the secondorder rate constants at 298 K calculated from the Arrhenius equation for various values of EA, with A = 1011 M-1 s-l (a representative value for many reactions not involving two ions). A second-order rate constant higher than 102 M-1 s-l, for instance, may be attributable to an activation energy less than 50 kJ mol-l. For the fastest reactions, however, this interpretation of the Arrhenius equation breaks down. Many reactions are known with rate constants in the region of 1010 M-1 s-l (for some examples, see Table 2.1, in Chapter 2, Section 2.1). The rates of such reactions approximate to the rate of molecular encounter, which is dependent on the speeds at which the reactant molecules move about in the solution. A reaction taking place at practically every encounter is not subject to any appreciable energy barrier once the molecules have reached adjacent positions. The diffusion process itself, however, will require some activation energy, and indeed the rate is found to increase with temperature, varying inversely with the viscosity and giving an apparent activation energy of the order of 10 kJ mol-l. From the molecular point of view, diffusion is the random migration of molecules or small particles, arising from the motions which they continually undergo by virtue of their thermal energy. In the gas phase these motions lead to occasional collisions, one at a time; in solution, after a collision the molecules tend to stay close to each other and recollide several times before separating (see Sections 2.4, 2.5.2.2). If the concentration of the solution is not uniform, the molecules migrate along the concentration gradient at any point; this is the macroscopic change that can be directly observed experimentally. For example, if a drop of a coloured material is put into apure solvent, the colour spreads through the liquid as the molecules of the material, moving at random, gradually occupy the whole available volume uniformly. 1.3.2. Concentration-gradient

treatment

This treatment of the rate of diffusion-controlled encounters in solution was developed originally by Smoluchowski [13,a] for the rates of coagulation of colloidal solutions, and was later applied to reactions between molecules. It as sumes that the diffusive motions of molecules can be treated like those of macroscopic particles in a continuous viscous fluido A simplified version is as follows. Consider a solution containing two kinds of solute molecule, A and B (Figure 1.2), assumed to be spherical. Suppose that these can be treated as hard spheres in a continuous medium, and that intermolecular forces can be neglected. We wish to find the rate at which

Chapter 1

14

(B)r

o (a)

f

(b)

Figure 1.2 Diffusion-controlled encounter. (a) Schematic diagram showing solute B mo1ecu1es approaching an A mo1ecu1e; reaction occurs on contact, when the centre-to-centre A-B distance r becomes equal to rA + r¡¡ (= rAB). (b) Schematic p10t of (B)r, the local concentration of B mo1ecules, against r; at large distance (B)r will be the bulk concentration (B)o, but if r falls to rAB then reaction occurs and (B)r becomes zero. (After P.w. Atkins, Physical Chemistry, 4th edn. p. 847.)

A and B molecules will encounter each other by diffusion. The simplest assumption about the rate of diffusion of any species in solution is that the rate is proportional to the concentration gradient (Fick's law of diffusion). A local concentration gradient is set up when a pair of molecules A and B meet each other and react; the disappearance of the B molecule depletes the concentration of B in the immediate neighbourhood of the A molecule, thus producing a non-equilibrium spatial distribution of B molecules, Le., a concentration gradient, and similarly for the A molecules. This will result in a net flux of reactant molecules into the depleted region, tending to restore the equilibrium distribution. A steady state is reached when the average depletion is just enough to provide the concentration gradient needed to maintain a flux of molecules equal to the rate at which the molecules are lost by reaction. The mathematical theory of diffusion shows that the number of encounters between A and B molecules per second per unit volume is then (Section 2.2): (1.1)

Here nA and nB are the numbers of A and B molecules per unit volume, fA and fB are the radii of the molecules, DA and DB are their diffusion coefficients, and (DA + DB) represents approximately the relative diffusion coefficient. If reaction occurs at every encounter between A and B, the rate of encounter given by Equation (1.1) is also the rate of reaction. In terms of arate constant kD, this rate is given by the following expression (in which NA is the Avogadro number): 0.2)

Introduction:

Origins, Methods, Mechanisms,

Rate Constants

15

Equating the expressions (1.1) and (1.2), we find that the rate constant for a diffusioncontrolled reaction is given by the following, often called the Smoluchowski equation [13,a]: (1.3)

If we write r AB for the centre-to-centre encounter distance (r A + rB) and D AB for (DA + DB), we obtain the following compact expression for kD in terms of molecules per unit volume per unit time: (1.3a) or in terms of moles per unit volume per unit time: (1.3b) It must be remembered that in the derivation of these equations it was assumed that reaction occurs at every encounter. The calculated value of kD is thus an upper limit for reactions where reactant molecules meet by diffusion. Most reactions have smaller rate constants, often because they require activation energy, or because the reactant molecules have limited reactive sites which impose a need for correct orientation on collision. Such effects are considered in Chapter 2. 1.3.3. Random-walk treatment of dijfusion It is useful to complement this rather abstract treatment with one based on a more easily visualised model, which treats diffusion as due to random movements of the solute molecules, related to Brownian motion.9 The path of an individual molecule will be a three-dimensional zigzag, with abrupt changes of direction, similar to the two-dimensional zigzag shown in Figure 1.3 [14].1f the changes of direction are entirely uncorrelated, i.e., if each is completely independent of its predecessors, one can apply the mathematical theory of 'random walks'. A particular solute molecule may find itself, after a given time interval, anywhere within a wide range of distances from its initial position; but the average distance can be calculated. The result is that, in the simplest case, the mean-square displacement (r2) of the centre of mas s is related to the time (t) through the translational diffusion coefficient (D), by an equation derived by Einstein in 1905:10 r2

= 6Dt,

whencer~(r2)

-

1/2

~(6Dt)1/2.

(l.4)

9 Brownian molion is lhe name given lo lhe small random molions of micron-sized particles in suspension in liquids, which can be observed by high-power microscopes. It is altribuled lo lhe random molions of!he molecules of lhe surrounding liquid, whose impacts on !he (much larger) colloidal particle do nol always cancel out. 10 For a derivalion of lhis relalion, based on a simplified one-dimensional model, see P.W. Atkins, Physical Chemistry, 5th edn., Oxford Universily Press, 1994, p. A39. Essentially one calculates the probability !hal a molecule will be found al a given dislance r from lhe origin at a time t. For references see Chapler 2, Seclion 2.3.

16

Chapter 1

Figure 1.3 A plOl of a compuler-simulaled lwo-dimensional random walk of n = 18.050 sleps. The walk slarts al lhe upper lefl-hand comer of lhe track and works its way lo lhe righl-hand edge. (Some regions are complelely black; this is due lo repeated traversals.) The straight-line distance, 'as lhe crow flies', is only 196 slep lenglhs. This is in agreement wilh lhe expected root-mean-square displacemenl which is (2n) I/2 = 190 slep lenglhs. (Diagram from H.C. Berg, Ref. [13J.)

This relation enables us to calculate the average volume swept out, or 'searched through', by a reactant molecule in unit time, and hence the average number of encounters per unit time with molecules of the other reactant. Assuming that reaction occurs at every encounter, we can then calculate the diffusion-controlled rate constant. On carrying through the algebra (see Chapter 2, Section 2.3), we obtain an equation of exactly the same form as the one derived from the concentration-gradient approach (Equation (1.3) or (1.3a)) and with much the same numerical coefficient (which in any event is only approximate). This is not in fact surprising, because Fick's law of diffusion can itself be deduced from the random-walk model, which indeed provides the simpJest molecular interpretation of that law. 1.3.4. Rate constant and viscosity Application of the Smoluchowski equation (1.3) requires a knowledge of the sizes and diffusion coefficients of molecules. An estimate of the effective values of r can usualIy be made from molecular volumes. Diffusion coefficients present more of a problem, since not very many have been experimentally determined over a range of temperature. They may, however, be eliminated from Equation (1.3) by using the Stokes-Einstein relation, which is theoreticalIy derived from the same molecular model and is often in fairly good accord with experimental data; it expresses D in terms of the viscosity of the solvent (r¡) and the

Introduction:

molecular radii:

Origins. Methods, Mechanisms.

Rate Constants

17

11

kT

DA=--, 4Jrr¡TA

kT DB=--.

(1.5)

4Jr1)fB

Gn substituting for DA and DB in Equation (1.3), one obtains for the rate constant calculated according to the Stokes-Einstein model: (1.6)

(1.6a)

This equation shows that ko at a given temperature depends primarily on the viscosity 1) of the solvento Variations in the sizes of the molecules do not affect the calculated rate unless the radius ratio is altered; even then the term in the second bracket is not much affected. The reason for this insensitivity of the rate to molecular size is that, on the model assumed, a larger molecule will move about more slowly than a smaller one in a solvent of given viscosity, but will present a bigger target for encounter with another solute molecule; these two effects of molecular size on the encounter rate willlargely compensate. If the molecular radii of A and B are assumed equal, we obtain from Equation (l.6a) simple expression for ko: 12 4RT ko=--.

(1.7)

1)

Since ko is not sensitive to molecular size, this is a useful though approximate relation. It predicts that the rate constant of such a diffusion-controlled reaction will be inversely proportional to the viscosity of the solvent; its temperature-coefficient will be comparable with that of the viscosity, and therefore small compared with that of most reactions. The numerical value calculated for ko in water at 25 °e is 1.1 x 1010 M-1 s-I ; the values in many common organic solvents are of the same order. All these conclusions agree semiquantitatively with the experimental results on very fast reactions, and in some cases the agreement is close (cf. Table 2.2)0 The expression for ko in Equation (106) or (lo7) cannot be more than an approximation, from the nature of its assumptions, which ignore the molecular structure of the liquid and the consequences of solute-solvent and solute-solute interactions, and take no account of the fact that reactant molecules are seldom spherical and commonly have localised reaction siteso None the less, it provides a general guide to the effects of various factorso An observed rate constant comparable with the calculated value of ko is a useful indication of diffusion control. II The factor 4 is related to the assumption that the diffusing molecule 'slips' rather than 'sticks', when passing a solvent molecule. Ifone as sumes that it 'sticks', the factor 6 is more appropriate. See Section 3.4.1 12 If the factor 6 is used instead of 4 in Equation (l.4), the factor 4 in Equation (1.7) must evidently be rep1aced by 8/3. This appears in various texts.

18

Chapter 1 Table 1.4 Diffusion-eontrolled rate eonstant for reaetion between ions, eompared . ions uncharged . wJth that for uneharged moleeules. Values of kD / kD = 8/(e ¡; - 1), for lons of eharges ZA and ZB, with distanee of closest approaeh a (in Á), for water at 25°C kions / k uncharged D D

lA lB

a (Á)

+2 +1 -1 -2

2.0

5.0

7.5

10.0

0.005 0.10 3.7 7.1

0.17 0045 1.9

0.34 0.60 1.6

0045

3.0

2.2

0.69 1.4 1.9

1.3.5. Reactions between ions The strongest molecular interactions are those due to charges on ions. If A and B are both ions, the Coulombic interaction energy is considerable and it is necessary to modify expressions for the rate constant such as that given in Equation (1.7). A simple electrostatic calculation (which will be outlined in Section 2.5.3 below) shows that they should be multiplied by a factor involving the charges on the ions, the distance of closest approach, and the relative permittivity of the solvent [l3,b]. Table 2.3 gives some values of this factor for ionic compounds in water. Unless the ions are exceptionally small, the value of kD is changed by les s than a power of ten. The brief outline given in this section will serve as an introduction to the next chapter, which deals with the phenomena of diffusion-controlled reaction rates.

References [1.11

[1.2] [1.3] [lA]

(a) H. Hartridge and F.J.W. Roughton, Proe. R. Soe. A 104 (1923) 376; (b) H. Hartridge and F.J.W. Roughton, Proe. R. Soe. B 94 (1923) 336; for reviews, see, (e) F.J.W. Roughton and B. Chanee, in: S.L. Friess, E.S. Lewis and A.Weissberger (Eds.), Investigation of Rates and Mechanisms of Reactions, Part /l, 2nd edn., Interseienee, New York, 1963, pp. 703-792; (d) B. Chanee, in: G.G. Hammes (Ed.), Investigation of Rates and Mechanisms of Reactions, Part 11, 3rd edn., Wiley-Interseienee, New York, 1974, pp. 5-62; (e) B.H. Robinson, in: C.F. Bemaseoni (Ed.), Investigation of Rates and Mechanisms of Reactions, Part /l, 4th edn., Wiley-Interseienee, New York, 1986, pp. 9-26. Diseuss. Faraday Soe. 17 (1954) 114-234; (a) M. Eigen, p. 194; (b) R.G.W. Norrish and G. Porter, p. 40; (e) G. Porter and M.W. Windsor, p. 178. Z. Elektroehem. 64 (1960) 1-204. This series is frequently mentioned, and for brevity's sake we shall designate it simply as Weissberger. Eaeh book in the series forrns Part 11of a two-volume set entitled lnvestigation of Rates and Mechanisms of Reaction, Wiley Interseienee, New York. Three editions have been published, with different editors: S.L. Friess, E.S. Lewis and A. Weissberger (1963), G.G. Hammes (1974) and c.F. Bemaseoni (1986). We shall distinguish these volumes by their dates, sinee (eonfusingly) the first of them belongs to the seeond edition of the complete work, the seeond volume belongs to the third edition, and the third belongs to the fourth edition. Thus we designate the three fast-reaetion volumes as Weissberger (1963), Weissberger (1974), and Weissberger (1986), respectively.

Introduction: [1.5] [1.6]

[1.7]

[1.8]

[1.9]

[1.10]

[1.11] [1.12] [1.13] [1.14)

Origins, Methods, Mechanisms,

Rate Constants

19

G. Porter (Ed.), Progress in Reaction Kinetics, Pergamon Press, Oxford, Vol. 1, 1961, Vol. 2, 1964, Vol. 3,1965, Vol. 4,1967, ete. (a) E.E Caldin, Fast Reactions in Solution, Blaekwell Seientifie Publieations, Oxford 1964; (b) Another, eovering reaetions in the gas phase as well as in solution, was published in 1971: D.N. Hague, Fast Reactions, Wiley-Interseienee, 1971. S. Claesson (Ed.), Fast Reactions and Primary Processes in Chemical Kinetics, Nobel Symposium 5, Almqvist and Wiksell, Stoekholm, Interseienee Publishers, New York, 1967; (a) R.G.W. Norrish, p. 33; (b) H.T. Witt, pp. 81, 261; (e) G. Porter, pp. 141,469; (d) ES. Dainton, pp. 185,485; (e) M. Eigen, pp. 245, 333, 477; (f) A. Weller, p. 413; (g) B. Chanee et al., p. 437. (a) E Basolo and R.G. Pearson, Mechanisms of Inorganic Reactions, Wiley, New York, 1958, Chapter 3. This was a pioneering monograph. In their prefaee, the authors refer to this 'renaissanee' of inorganie ehemistry in the preeeding two deeades, with sueeessive periods of advanee in studies of strueture (d. L. Pauling, Nature of the Chemical Bond), of thermodynamies, and last1y of kinetie and meehanistie studies. For instanee, (a) T.A.M. Doust and M.A. West (Eds.), Picosecond Chemistry and Biology, Seienee Reviews Ltd, 1983; (b) Symposium on Flash Photolysis and its Applieations, E 82 (1986) 2065-2451; (e) M. Ebert, J.P. Keene, A.J. Swallow and J.H. Baxendale (Eds.), Pulse Radiolysis, Aeademie Press, 1965; (d) J.H. Baxendale and E Busi (Eds.), The Study of Fast Processes and Transient Species by Electron Pulse Radiolysis, D. Reidel, Dordreeht, 1982; (e) P. Laszlo (Ed.), Protons and Ions in Fast Dynamic Phenomena, Elsevier, Amsterdam, 1978, mainly on ionie reaetions. For instanee, eonferenees whose proeeedings were published as (a) W.J. Gettins and E. Wyn-Jones (Eds.), Techniques and Applications of Fast Reactions in Solution, D. Reidel, Dordreeht, 1978; (b) A eonferenee open to all fast-reaetion teehniques has been held every year sinee 1977 by the Fast Reaetions in Solution group sponsored by the Royal Soeiety of Chemistry. A eonferenee on 'ultrafast' reaetions held at Berlin was published in J. Phys. Chem. 97 (1993) 1242312645, and an exeellent survey by A.H. Zewail appeared in J. Phys. Chem. 100 (1996) 12701. H. Strehlow, Rapid Reactions in Solution, VCH, Weinheim, 1992. (a) M. von Smoluehowski, Z. Phys. Chem. 92 (1917) 129; (b) P. Debye, Trans. Eleetroehem. Soe. 82 (1942) 265; (e) M. Eigen, Z. Phys. Chem. N.E 1 (1954) 176. H.C. Berg, Random Walks in Biology, Prineeton University Press, 1983, p. 12.

21

Chapter 2

The Rates of Diffusion-Controlled Reactions 2.1. Introduction The foregoing brief discussion on diffusion-controlled reactions serves to call attention to the importance of random translational diffusion as a means whereby reactant molecules can come into contact, and to the infiuence of intermolecular forces. In the present chapter, we continue to explore, with the aid of simple models, the motions of solute and solvent molecules that govern the rate at which reactive collisions occur. Throughout this book we shall come across rate constants in the region of 1010 M-1 s-l, often inversely related to the viscosity of the solvento There is compelling evidence that these are controlled by the rates at which the reactant molecules meet by random diffusive motions. Some examples of simple bimolecular reactions are collected in Table 2.1.1 A preliminary account of the theory of such very fast reactions was given in Chapter 1 (Section 1.3); in the present chapter we consider them in more detail, with the help of simple models. The basic assumptions are: (a) that reactant molecules meet in the course of their diffusive motions; (b) that they stay close together long enough for a number of collisions (an 'encounter') to occur; and (e) that during an encounter they may react, with or without an energy-barrier. This model may be treated by various methods. In this chapter we shall consider two approaches, one macroscopic and the other molecular, based, respectively, on the classical theory of diffusion and on the theory of random motions. We shall postpone much of the mathematics of diffusion to Chapter 3, in the interests of developing a working picture of diffusion-controlled reactions.

2.2. Application of the diffusion law to rates of encounter and of chemical reaction: the Smoluchowski equation 2.2.1. 1ntroduction In the 'classical' approach to the rate of encounter of particles in liquid solutions, initiated by Smoluchowski [2], the random diffusive motions of particles are treated in terms of Fick's law of diffusion, which (summarising experimental evidence) states that 'rates of diffusion are proportional to concentration gradients'. Consider a bimolecular reaction between reactants A and B. When a pair of A and B molecules meet and react, the local 1

Reactions between ions have been excluded fram this table; they are briefly treated in Section 2.5.3.

22

Chapter 2

Table 2.1 Some reaetions with very high rale eonslanls. Values of k (M-1 s-l) al or near 298 K, eompared with ko == 4RT Ir¡. Values of kl ko are rounded. For referenees [l,a], [l,b] ete. for the rate eonstants, see Ref. [1]. These reaetions do not inelude any between two ions; sueh reaetions are treated separately in Seetion 2.5.3 Seetion

Solvent

10-9 k klko

Ref.

H+ + NMe3 ---*HNMej OH- + PhOH ---*PhO- + HZO e- + PhCHZCI---* PhCHi + CI-

8.1.2.1 8.1.2.1 5.3.1.1

water water ethanol

25 e.14 5.1

[1, al [1, b] [1, e]

Os(bipy)~+ + Fe(bipy) (CN)z 2PhCOzH ---*(PhCOZHlz 3-hydroxypyrenea + pyridine 2CMej ---*Me3CCMe3

9.1.1

water

6.2.4 4.6.2

CCl4 ete. benzene oetane ete.

3.2 5 9 3

Hm+CO---* HmCO IZ + (MeZNlzCS ---*(MeZNlzCS ... Iz Aeetylcholinesteraseb + HNMe-aer+

2.2.2 2.5.2.2 2.5.2.4

Reaetion type Proton transfer: Electron transfer: Hydrogen bonding: Radical recombn: Molecular combination: Haemoglobin Charge-transfer Enzyme

glyeerol n-BuCI water

0.002 10 1.2

2 1 0.6 0.3 0.5 0.5 0.7

[1, [1, [1, [1,

d] e] f] g]

0.3 0.3 0.1

[1, h] [l,j] [1,1]

a Exeited moleeules. b Miehaelis-Menten

kineties assumed; see, Ref. [31,a,b].

concentrations of both A and B are depleted, giving rise to concentration gradients, under which a net migration of A and B molecules to the region occurs by diffusion; a steady state is reached when the average rate of reaction is equal to that of migration. The reactant molecules are treated as hard spheres in a continuous medium. Fick's law is applied to derive from this model a differential equation which when sol ved leads to an expression for the encounter rate constant kD in terms of the translational diffusion coefficients (DA, DB) and the radii (rA, TB) ofthe molecules. This expression is mentioned in Chapter 1 (Section 1.2.1) and derived in Chapter 3 (Section 3.2.1). If we denote the centre-to-centre distance of closest approach of the reactants by r AB = r A + rB, and as sume that the relative diffusion coefficient DAB can be approximated as DAB = DA + DB the expression in terms of molecular concentrations is: (2.1)

or in terms of molar concentrations: (2.2)

If chemical reaction occurs at every encounter, kD is al so the rate constant for reaction. Equation (2.1) is often known as the Smoluchowski equation. Since experimental values of diffusion coefficients are known in relatively few instances, it is often convenient to express them in terms of the viscosity (r¡) of the medium, which is known for many solvents and is easy to measure. Application of classical hydrodynamic theory to the hard-sphere modelleads, as was shown in Section 1.3.2, to the Stokes-Einstein equation (1.3) for the relative diffusion coefficient DAB, which may be

The Rates of Diffusion-Controlled

Reactions

23

written DAB = kT /4lf17(rA + 113). On substituting this expression into Equation (2.1), we obtain (cf. Equation (1.6a» for the rate constant of a diffusion-controlled reaction: (2.3) For equal-sized molecules A and B, this takes the simple form: 4RT kD=--. r¡

(2.3a)

This Equation (2.3a) is also a good approximation when r A and 113 are different but comparable; for instance, even if r A is doubled, the expression for the rate constant will be identical in form to Equation (2.3a) and the numerical factor will change only from 4 to 4.5. These are theoretical equations, derived from a simplified model. They provide a useful starting-point; we shall examine, in the present chapter and the next, how far they agree with observation. At present we note that Equation (2.3a) not only predicts rate constants of the right order of magnitude for the fastest reactions in water and ordinary organic solvents; it predicts also that these high rate constants will ron parallel with the viscosity of the solvent, and will exhibit relatively low temperature-coefficients determined by that of the viscosity.2 Quantitatively the theory predicts that if the viscosity is varied, by changing the solvent (or the pressure), then the observed rate constant k will vary as l/r¡, while if the temperature is varied with a given solvent then k will vary as T Ir¡. There are a considerable number of reactions where these predictions are at least qualitatively fulfilled (see Table 2.1). Although many diffusion-controlled reactions are extremely fast, some are not; they may have rate constants down to 107 M-1 s-I in common solvents, yet show the viscositydependence characteristic of diffusion control (Section 2.5.2 below). This is because orientational constraints can in principIe decrease the rate constant by several orders of magnitude, by reducing the chance of reaction at a particular collision. Such orientational effects are important in the general theory of diffusion control, and are especially prominent for reactions of macromolecules (Section 2.5.2.4). The fundamental rate expression to be considered is the Smoluchowski relation k = 4rr N DABrAB (Equation (2.1». The derived expression 4RT Ir¡ (Equation (2.3a», is a useful approximation, but deviations from it are observed, because the Stokes-Einstein equation which is involved is derived by hydrodynamic theory for spherical particles moving in a continuous ftuid, and does not accurately represent the measured values of translational diffusion coefficients in real systems. Although the proportionality D ex l/TI is indeed a reasonable approximation for many solutes in common solvents, the numeral coefficient 1/4 is subject to uncertainty. In the first place, this theoretical value derives from the assumption that in translational motion there is no friction between a solute molecule and the first layer of solvent molecules surrounding it, i.e., that 'slip' conditions hold. If, however, one assumes instead that there is no slipping ('stick' conditions), so that momentum is 2 Many common organic solvents have viscosities in the region of 10-3 kgm-1 s-1 (l centipoise) at room temperature and 1 atm pressure. Their temperature-variation is often represented fairly well by an equation of the form r¡ = A exp( B / R T), where A and B are constants; B commonly lies in the range 4 to 12 kJ mol-1 .

24

Chapter 2

transferred between solute and solvent molecules, the coefficient becomes 1/6. Secondly, the experimental values of diffusion coefficients for many reactant molecules in ordinary organic solvents are 10-30% higher than the Stokes-Einstein 'slip' values, implying that the coefficient could be around 1/3, so that in place of Equation (2.3a) we should find kD ~ 5RT Ir¡. It is often convenient to write the numerical coefficient simply as n, so that: kT

(2.4)

DAB=---

nnr¡rAB

and kD =

(~6)(

Rr¡T).

(2.5)

Further, even the proportionality D ex 1/ r¡ breaks down when the solute molecules are small compared with the solvent molecules. These matters are considered further in Chapter 3 (Section 3.4.3). Measurements of diffusion coefficients, for which there is now a range of methods (Section 3.4.2), will permit the wider use of the Smoluchowski equation (2.1) in place of (2.3a), thus avoiding the assumption of Stokes-Einstein behaviour. Historical note The original treatment by Smoluchowski [2] was designed to account for the rates of coagulation of colloids, and was applied by him to experimental data on copper and gold soIs. An early application of the theory to homogeneous chemical processes was to the quenching of the fiuorescence of dye solutions, by Sveshnikoff in 1935 [3]. Debye [4] in 1942 replaced diffusion coefficients by the solvent viscosity, and introducted a correction term for interionic forces. The mathematical theory was improved by Collins and others (1949 seq.), who also considered random-walk theory. In the 1950's, the rapid development of experimental methods for the study of very fast reactions [5] was accompanied by theoretical advances; the position in 1961 was summed up in a seminal review by Noyes [6]. Since then there has been much interest in developing theoretical approaches [7] to the many problems thrown up by the increasing mass of experimental results, including those on macromolecular systems such as enzymes. The following account of the concentrationgradient approach draws largely on the treatment ofNoyes {6]. 2.2.2. Extension of diffusion theory to include activation requirements The assumption made in the simple version of diffusion theory outlined above, that every encounter results in reaction, refers to a limiting case; in general, there will be encounters which do not result in reaction before the molecules separate. This will be the case if either activation energy or a particular geometrical orientation is required for reaction. Moreover, Equation (2.2) leads to the anomalous result that as the viscosity approaches zero the calculated rate constant tends to infinity, whereas its maximum value must in fact be the collision number corresponding to absence of solvent, Le., the gas-phase value. An improved calcu-

The Rates of Dijfusion-Controlled

Reactions

25

lation removes this anomaly and allows us to refine the model so as to include activation requirements [8].3 Suppose we consider any bimolecular reaction in solution, whatever its rate, and apply Fick's laws to the general case where activation energy and orientational factors are important as well as diffusion. Carrying through the calculation (see below, Chapter 3, Section 3.3.2), one finds that in the steady state the observed second-order rate constant k will be given by (in place of Equation (2.1)) the following equation:

(2.6)

where the newly-introduced quantity ko is the value of the rate constant that would be observed if there were no diffusion effects and the rate were limited only by a collision number equal to that in the gas phase, along with any chemical interactions between molecules in collision, including both activation energy and orientational requirements. (In that imaginary situation, the equilibrium spatial distribution of molecules would be maintained; however fast the reaction, it would not lead to local depletion of reactant molecules.) The Equation (2.6) may be rewritten as: (2.7) With the express ion for kD in Equation (2.1), this becomes: (2.8)

The significance of this useful relation may be seen if we replace each rate constant by a reciprocal relaxation time «). Then Equation (2.8) becomes: <

=
(2.9)

This indicates that the average time required for reaction is simply the sum of the average times required for the molecules to diffuse together «D) and then to react «o) after overcoming the activation and orientation constraints. This is the physical interpretation of Equations (2.7) and (2.8). 2.2.2.1. Viscosity-dependence

01 rate

constants

The dependence of reaction rate constants on the viscosity of the solvent predicted by this model can be derived fram Equation (2.8) along with the assumption that D oc TIr¡ as in the Stokes-Einstein theory. For simplicity we take as an example the case of a reaction between equal-sized molecules, for which kD = 4RT Ir¡ (Equation (2.2)). Inserting this in Equation (2.8), we obtain: (2.10)

3 Mathematically, the difference from the Smoluchowski solution of the differential equation.

treatment is in the boundary

condition used in the

Chapter 2

26

10.0

10.0

Diffusion control

Activation control 1.0

--------+-1.0

5.0

/ /

o

5

10

r¡/T

(a)

O

5

10

r¡/T

(b)

Figure 2.1 Effect of viscosity on rate constant in solution (1). Schematic plots of rate constant k calculated from Equation (2.8) against viscosity, for (a) a diffusion-controlled reaction with k = 1010 M-1 s-l at 25 °e in water; (b) an activation-controlled reaction with k = 1.0 M-1 s-1 at 25 °e in water.

Thus if the viscosity is varied (by varying the temperature, pressure, or solvent), a plot of k-1 against 1]/ T should, according to the model, show a linear correlation, with an intercept representing k 1 (Figure 2.1 (a)). All activation-energy and orientational limitations will decrease ko and so in crease the intercept. Orientational effects will diminish ko and so will also increase the slope as well. In principIe it should be possible to leam much from such plots; in practice there are only a few reactions for which it has be en possible to determine the intercept with any accuracy (see below, Section 2.2.3), or even to distinguish it from zero. The rates of very fast reactions are found to be viscosity-dependent, but those of slower reactions show no close correlation. Inspection of Figure 2.1 shows why this is so. A diffusion-controlled reaction in water at 25 °e might have ko ;::,;1010 and ko ;::,; 1011M-I s-l, giving (by Equation(2.8)) k;::,; 1010 M-I s-1; a tenfold increase ofviscosity would decrease both ko and k about tenfold (Figure 2.1 (a)). An activation-controlled reaction, with a much smaller value of ko, say ko ;::,;1 M-I s-l, would have the same decrease in ko, but this would decrease k only negligibly (Figure 2.1 (b)). Two particular cases of Equation (2.8) are of special interest; we will consider them in tum.

a

2.2.2.2. Rate constant lor activation control The condition for the observed rate constant k to approximate to the value for purely activation control (ko) is, from Equation (2.7): (2.11)

This condition will be satisfied within 1% for small molecules in ordinary solvents at room temperature (rAB ;::,;10-8 cm and DAB ;::,;10-5 cm2 s-1) if ko is less than about 10-14 cm3 molecule-1 s-l, i.e., 107 M-1 s-l. Reactions with rate constants lower than this will not be appreciably affected by diffusion rates, unless they occur in viscous solvents or involve very large reactant molecules. This implies that diffusion effects can be

The Rates 01 Dijfusion-Controlled

Reactions

ignored when the activation energy is greater than about 25 kJ mol-l Arrhenius pre-exponential factor is around 1011 M-l s-l. 2.2.2.3. Rate constantfor

27

(6 kcalmol-l),

if the

reaction at every encounter

It may be asked whether Equation (2.7) becomes identical with the Smoluchowski equation (2.1) in the limit. The condition for the two equations to become effectively identical is, from Equation (2.7): (2.12) For small molecules in ordinary solvents, this implies that k» 109 M-l s-l. The fastest reactions not involving ions commonly have rate constants of about 1010 M-l s-l, so the condition is not rigorously fulfilled. The Smoluchowski equation must therefore be regarded as only an approximation for these reactions. The difference between Equations (2.7) and (2.1) is, however, not much greater than the experimental errors. For the recombination of iodine atoms, for example, it amounts to a factor of 1.35, which may be experimentally significant. For this reaction, in n-heptane at 300 K, the rate constant varies nearly inversely with the viscosity, in agreement with Equation (2.10), with nearly the expected slope [9,a]; this is a particularly favourable case, however, since the reactant particles are indeed spherical and of equal size. For the addition of ea to haem in glycerol [9,b], the intercept of the plot of k-l against r¡/ T (shown in Figure 2.2) is close to zero (i.e., k ex T /r¡), but the slope is about four times greater than is predicted by Equation (2.10), possibly because of an orientation requirement or because the Stokes-Einstein value for D is too small. 0

2.2.2.4. Reactions showing both diffusion and activation effects Rate constants in the intermediate region where k¡;l is comparable with r¡/4RT will be infiuenced both by diffusion and by chemical interactions. There are a few instances where the two-term Equation (2.7) or (2.10) can be tested, by varying the viscosity of the solvent. For reactions in hydroxylic solvents, a wide range of viscosity can be obtained by using a series of mixtures of water with glycerol, or ethylene glycol, in which the viscosity is sensitive both to temperature and to composition. The binding of ea to the haem group in ferrous microperoxidase (MP) [1O,a] in aqueous ethylene glycol gives a nearly linear plot (Figure 2.3) of k-l against r¡ / T, but with a finite intercept corresponding to ko ~ 1 X 107 M-l s-l, much smaller than ko, implying a contribution of activation (or configuration) factors; also the slope is about ten times the value predicted by Equation (2.7), presumably because k is reduced by the limited size of the reaction site. Similarly the binding of Hea3 to carbonic anhydrase in aqueous glycerol [1O,b] gives a good linear plot of k-l against r¡/T, with an intercept corresponding to ko ~ 5 X 107 M-l s-l (Figure 2.3). In such cases the rate constant may be controlled either by diffusion or by chemical factors depending on the conditions. Evidence for the simultaneous roles of diffusion and activation control has also been found for some other reactions in highly viscous solvents, in that they show curved Arrhenius plots, indicating that the rate is controlled by diffusion at low temperatures where the viscosity is highest, but by chemical activation at ordinary temperatures. Examples include the reactions of ea with haemoglobin and myoglobin in aqueous glyerol [1O,c,d]. Reactions of simple radical s such as H· and Ha· with solvated electrons

28

Chapter 2

35 30

~

t"-

o ,....,

~

l>

25 20

~

:lO

~

o ,....,

15

D O

10

'-"

5

0.0

0.5

1.0

(O D) 103(r¡/T)

1.5

2.0

2.5

3.0

(\1) 102( r¡/T)

Figure 2.2 Effect of viscosity on rate constant in solution (2). Experimental plot of observed k-1 against r¡/ T for lhe reaction ofhaem with ea in glyceral. 108k-! against 103 (r¡/ T), D: at 72.5 ce, 108k-1 against 103 (r¡/T), O: at 48.5 ce, 107 k-1 against 102 (r¡/T), \1: at 20.1 ce. Data fram Ref. [9,b], replotted.

in water and the recombination of these radicals to form H20 [11] have been studied at temperatures from 20°C to 200°C; the results show agreement with Equation (2.8), the term kD I dominating at the lower temperatures and kü I at higher. 2.2.3. Transient effects in the early stages of reaction [12] A consequence of the disturbance of the initial equilibrium distribution of molecules by a fast reaction is that, after the initiation (assumed instantaneous) of reaction, a finite time will be required before the value of the observed rate constant k becomes steady. The theoretical equations for the time-variation of k have been solved; Noyes' treatment sums the matter up [12,a]. The result is that at a time t the rate constant k will be given by the expression in Equation (2.6) multiplied by a factor which depends upon l/tl/2. The new general equation is

k

=

[ 1 + rAB(JT DABt)-

1/2( 1 + 4JTrAB DAB)-I][ ko

k

ü 1 + (4JTrABDAB)-

1]-1

.

(2.13) As time increases, the expression in the first square bracket tends to unity and the equation becomes identical with Equation (2.6), as would be expected. For short times, we can simplify Equation (2.13) as follows: (a) for activation-controlled reactions, ko «kD and so

The Rates 01 Dijfusion-Controlled

29

Reactions

100

80

MP +

eo_--

MP -

'"::E

eo

" 1

Ethylene glyen!: water

2

60

1Glycerol: water

40

11 20 Polyethylene glycol: water

o

2

4 ryIT(mPasK1)

(a)

6

o

2

4

6

ryIT(mPasK-1)

(b)

Figure 2.3 Effect ofviscosity on rate constant in solution (3). Experimental plots of observed k-! against r¡ or 'lIT for: (a) binding of eo to the haem group in ferrous microperoxidases (MP) in aqueous etbylene glycol; (b) binding of He0 to carbonic anhydrase in aqueous glycerol.

3

k ::::;ko; (b) for diffusion-controlled reactions, ko is considerably larger than kD and we can approximate Equation (2.13) (using kD = 4n DABrAB, Equation (2.1» as: (2.14) So the theory predicts that k is initially larger than the steady-state rate constant kD, and varies linearly with 1/ tI 12. For large molecules in high-viscosity solvents, the second term in the bracket in Equation (2.14) dominates, and the equation may be further approximated to: (2.15) In favourable cases it is possible, by fitting experimental data to Equation (2.13), to estimate both ko and the distance of closest approach r AB. The rate constant predicted by Equation (2.13) for reactant molecules of ordinary dimensions in ftuid solvents at room temperature may differ from the steady-state value (Equation (2.6» by 1% at times of the order of 10-7 s, and by 10% at 10-9 s. These deviations may become significant when half-lives of the order of 10-8 sor less are concerned. Deviations up to 20-30% have been observed in ftuorescence quenching [13] and in picosecond-ftash phenomena [14].

30

Chapter 2

300 Hb + CO in Glycerol- Water, -81 °

Mb+ O2 in Glycerol-Water, -79"

~

o

2

Figure 2.4 Time-dependence of rate constant in the initial stage of a fast reaction. Plots of reciprocal change of absorbance (proportional to ~c-l) against square root oftime, for the recombination after flash photolysis of ea and 02 with myoglobin and haemoglobin in glycerol-water mixtures at "" -80 De. From Ref. [15,b].

The prediction that the initial stage of reaction can be dominated by the term in t-I/2 (Equation (2.14)) is strikingly fulfilled by the reactions of haemoglobin and myoglobin with oxygen and carbon monoxide in viscous solvents, studied by monitoring the absorbance changes during recombination after flash photolysis of the corresponding complexes [15].4 The rates were determined in glycerol-water mixtures at temperatures around -SO °C, where the viscosity is extremely high and one can make use ofthe limiting equation (2.15). Combining this with the usual second-order rate equation for equal concentrations c of two reactants (el = cOl + kt), one obtains:

(2.16)

We therefore expect a linear plot of 1/ C against Figure 2.4; they are satisfactorily linear [15,b].

ti /2.

Experimental

plots are shown in

A time-dependent rate 'constant' is observed also for the recombination of iodine atoms produced by flash photolysis, but this is attributable in par! to peculiarities of geminate recombination as well as to diffusion conditions (see Section 7.3.4.1).

4

The Rates of DijJUsion-Controlled

Reactions

31

2.3. A molecular model for translational diffusion and diffusion-inftuenced rates of reaction: random-walk theory So far our theoretical discussion has be en in terms of concentration gradients and Fick's law of diffusion. It is useful to complement this with a picture of diffusion in which the motions of the reactant molecules are more directly visualised. An illuminating model is provided by the theory of 'random walks'. We shall outline a simple version of this.5 Consider a spherical molecule dissolved in a liquid and undergoing diffusive motions. These motions can be treated as a series of discrete displacements, the direction of each being random. We as sume that they are not infiuenced by intermolecular attractions, nor by any external force such as that of an electric field on an ion. We as sume also that theyare 'uncorrelated', in the sense that such a displacement does not depend on the previous history of the molecule; metaphorically, the molecule 'has no memory' of its previous displacements. Many of these motions will be no more than oscillations, but occasionally the solute molecule will undergo a finite displacement. These movements will result in a zigzag path with abrupt changes of direction - a three-dimensional random walk. Computer simulation gives results illustrated in Figure lA, Section 1.2.2. How far does the molecule move in a given time? A particular solute molecule may find itself, after a given time interval, anywhere within a wide range of distances from its initial position; but the average distance can be calculated. A detailed model is required. Let us suppose: (a) that the solvent can be treated as a structureless continuum (Le., that its molecules are sufficiently smaller than those of the solute); (b) that solute-solvent forces can be neglected; and (c) that the motions of a solute mo1ecule are small equal-sized steps which occur very rapidly in random directions. Then we can use the random-walk relation between the time-interval (t) and the mean-square displacement (r2) of the centre of mass, derived for such a model of Brownian motion by Einstein in 1905:6 (2.17) Here D is the translational diffusion coefficient. Assuming that this relation holds even at molecular distances, the average time (t) required for the displacement of a solute molecule A over a distance of one molecular diameter (2rA) in any direction will be 4rl/6DA. In each such displacement the molecule will sweep out a cylindrical volume of approximately (rrrl)(2rA), ~.e., 2rrrÁ. This is effected in time t, so the mean volume swept out in unit time is 2rrrÁ/(4r1l6DA), i.e., 3rr DArA. (Some of the regions swept out in successive displacements will overlap, so the factor 3 will not be exact.) 5 A pre1iminary aeeount is given in Chapter 1 (Section 1.2.2). The present section owes much to the treatment by O.G. Berg and P.H. von Hippe1, Annu. Rev. Biophys. Biophys. Chem. 14 (1985) 131.

6 (a) For a derivation of this re1ation based on a simp1ified mode1 (equal-sized steps, in one dimension), see, P.w. Atkins, Physical Chemistry, 5th edn., Oxford University Press, 1994, pp. 855, A39. The more realistic the model, the more reliab1e is the result. An essential assumption in any model is that each step is independent of its predeeessor as regards direetion; this can be justified if the time between steps is 10nger than a few pieoseeonds. (b) For a standard treatment, see, S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1. (e) For the original papers, see, A. Einstein, Investigations on the Theory ofthe Brownian Movement, A.D. Cowper (Trans.), R. Fürth (Ed.), Dover Publieations, New York, 1956. (d) For the eorresponding re1ation for rotational molíon, see, Seelíons 3.2.3,3.4.1. (e) For a review ofmore reeent work on Brownian motion, see, J.M. Deuteh and 1. Oppenheim, Faraday Discuss. Chem. Soco 83 (1987) 1. (l) For advaneed treatments, see, S.A. Rice [7, and referenees thereinl.

32

Chapter 2

Consider now a solution containing two reactants A and B, at concentrations CA and molecules per unit volume, and let us calculate the collision rateo The volume swept out in unit time by a single molecule of reactant A will, as has just been shown, be :::::: 3rr DATA; but since we are interested in collisions, which will occur whenever the centre of an A molecule comes within a distance rAB (:::::: rA +~) of the centre of a B molecule, the quantity of interest is the volume which is swept out (searched through) in unit time, Le., the volume of the 'collision tube' of radius r AB. This, by reasoning similar to the preceding, will be approximately 3rr N D ABr AB. Here D AB is a mutual diffusion coefficient; it may be approximated as DA + DB.7 The number of B molecule centres in this volume will be 3rrNDABrABcB, and this will be the number of collisions made by one A molecule in unit time. The number made by all the A molecules in unit volume will therefore be 3rr N DABrABcAcB. Hence, if reaction occurs at every encounter, the predicted theoretical diffusion-controlled rate constant will be: CB

(2.18)

This expression has exactly the same form as the corresponding approximate form of Smoluchowski's expression 4rr N DABrAB (Equation (2.1» although derived in such a very different way. The approximate numerical coefficient also is in reasonable agreement, despite the simplifying assumptions. The random-walk treatment thus provides a molecularscale model embodying the results of the successful concentration-gradient treatment. The agreement between the results of the concentration-gradient treatment and the randomwalk treatment reflects the fact that Fick's law can itself be deduced from the random-walk model, which is the simplest interpretation of it. 2.3.1. Hydrodynamic and molecular models models

01 the

solvento Assumptions

01 hydrodynamic

Let us recall the main assumptions of the simple model presented so faro (a) Whether we use the concentration-gradient or the random-walk approach, the reactant molecules are 7 This assumption is commonly used, but the author has not come across a thorough treatment of it. Support for it appears to depend on the successes of theoretical arguments in which it has been assumed. A general treatment would no doubt be a complex exercise, but a beginning might be made by considering reactions in the gas phase, where simple kinetic theory can be applied; for the simplest systems, Le., assemblies of hard-sphere molecules in thermal equilibrium interacting only on contact, rigorous derivations are available (see, S.W. Benson, The Foundations 01 Chemical Kinetics, McGraw-Hill, New York, 1960, p. 148. seq.). These provide exact expressions for the number of collisions per unit time per unit volume (a) in a homogeneous system composed of molecules of a single gas, A or B, and (b) in a mixture of the two gases A and B. The molecules are assumed to be all of uniform size and mass and in thermal equilibrium with a Maxwell distribution of velocities. On comparing the expression for the collision numberfor single gases (ZAA or ZBB), Le., for collisions between two like molecules, with the expression for the mixture of gases (Z AB), it turns out (1) that the two expressions have the same form, i.e., the same dependence of Z on the mass (m) and diameter (d) of the molecules, but differ in that m in the expression for ZAA or ZBB is replaced in the expression for ZAB by the reduced mas s ¡.L, and (2) that (for uniform mass and size) ZAB = ZAA + ZBB. Since it is to be expected that in each case the collision number will be equal to the diffusion-controlled rate constant kD if each collision results in reaction, and since according to the Smoluchowski relation (Equation (2.2), in Section 2.2) this rate constant is proportional to the diffusion coefficient D, it follows that D AB = D AA + DBB. This result is thus justified for diffusion in the gas phase. In solution, however, the matter will be greatly complicated by the solvent-solute interactions, which may welllead to appreciable divergences (cf. D.F. Calef and J.M. Deutch, Annu. Rev. Phys. Chem. 34 (1988) 493).

The Rates of DljJusion-Controlled

Reactions

33

treated as spheres, with uniformly reactive surfaces, reacting at every impact, and negligible intermolecular forces. (b) The mutual diffusion coefficient DAB is assumed to be constant, at all A-B distances, right down to the collision distance, and to be approximated by D AB = DA + DB. (c) The solvent is assumed to be a continuous isotropic fluido (d) The movements of the reactant molecules are assumed to be small uncorrelated motions; proximity effects are not considered, and oscillating motions are ignored. All these assumptions involve approximations of various degrees. The resulting limitations can be reduced, however, by refinement of the model, with the aid of more advanced mathematical methods. Effects due to restricted reaction sites, molecular reorientation, activation requirements, short-range interactions, longer-range ionic interactions, and solvent structure have all been examined (cf. below, Section 2.5.2.2 seq.). 2.3.2. Averaging An essential feature of any such theory is that it must average the behaviour of a large number of molecules. In concentration-gradient theory the averaging is implicit in the use of an average diffusion coefficient; in random-walk theory it is implicit in the use of basic equations such as Equation (2.17). Averaging can alternatively be achieved by starting from standard treatments of ensembles of molecules. One such type of treatment is nonequilibrium statistical thermodynamics [16]. For the Smoluchowski model the results agree with those given above, but the method also permits the use of flexible mathematical techniques which can handle the refinements as well. It also makes possible the extension of calculations to diffusion in one or two dimensions, such as occurs in membranes. 2.3.3. Molecular-pair theories In order to take account of the fact that the solvent is made up of discrete molecules, one must abandon the simple hydrodynamically-based model and treat the solvent as a manybody system. The simplest theoretical approach is to focus on the encounters of a specific pair of molecules. Their interactions may be handled by calculating the radial distribution function, whose variations with time and distance describe the behaviour of a pair of molecules which are initially separated but eventually collide. Such a treatment leads (as has long been known) to the same limiting equations for the rate constant as the hydrodynamically based treatments, including the term ko through which an activation requirement can be expressed, and the time-dependent term in t-1/2 (Equation (2.13» [17]. The procedure can be developed, but the mathematics is somewhat complex. Non-equilibrium statistical thermodynamics provides an alternative approach [16]. The kinetic theory ofliquids provides another model that readily permits the inclusion of a variety of interactions; the mathematics is again fairly complex [37,a]. In the computer age, however, mathematical complexity is no bar to progress. Refinement of the model is considered further below (Section (2.6».

2.4. Encounter in solution: the solvent 'cage' [18] The theoretical models of molecular diffusion in solution presented above have allowed us to calculate the frequency with which solute molecules of two different types, A and

34

Chapter 2

(a) Figure 2.5 Encounter in a solvent cage. Model assumed for a solution of A and B molecules in solvent S. (a) Spatially uniform distribution of A and B in S. (b) A and B in the same cage. From Ref. [Z3,dl.

B, come into contact by random diffusion motions. Having reached adjacent positions, they will remain close until either they drift apart by diffusion or reaction occurs. The two molecules are then said to have undergone an 'encounter', and the pair, though very short-lived, may be caBed an 'encounter pair' or an 'encounter complex'.8 The number of encounters per second in unit volume, when A and B are at unit concentration, may be caBed the 'encounter number'. For the simplest case, where the molecules of A and B are supposed to be spheres of equal size, moving in a continuous ftuid of viscosity r¡, we have seen that the theoretical value of the encounter number comes out as ~ 4RT Ir¡, regardless of size (Equation (2.2)). In water and solvents of comparable viscosity at room temperature, it is ofthe order of 1010 M-1 s-l. This wiB also be the rate constant for reaction between A and B if the molecules always react immediately on coming into contact without needing activation energy or reorientation. The highest observed rate constants are indeed in this range (see Table 2.1), so it seems that even the simplest theory can be a useful guide. The model of the solution as composed of hard spheres without intermolecular forces in a continuous medium is, however, clearly inadequate. The solution is held together in some kind of structure, by the intermolecular forces (dispersion, Coulombic, etc.) between solvent molecules; it is recognised that within the solution there is 'free space', with occasional cavities of molecular size, as sketched in Figure 2.5. When two solute molecules undergoing random diffusion approach to within one or two molecular diamcters of each other, having recently displaced solvent molecules to get there, they are hemmed in by a surrounding waB of solvent molecules. Random-walk calculations suggest that the solute molecules undergoing diffusive motions within this solvent 'cage' will collide with one another a good many times before one of them changes places with a solvent molecule 8 The term 'encounter' refers to lhe set of repetitive collisions that occur before the two solute molecules drift aparto 'Encounter' may be said to begin when the solute molecules are at a distance such that the solvent 'cage' begins to form, perhaps at about 1.5(r A + r A). 'Collision' refers to the shorter distance (r A + 'll) where 'contact' occurs; this is easy to define for hard spheres, but otherwise depends on the potential-energy surface. On these definitions, see, R.M. Noyes [Z]; S.A. Rice [4, p. 131]; and S.H. Northrup and J.T. Hynes, J. Phys. Chem. 71 (1979) 871. (Note that in some more recent publications these usages of the words 'encounter' and 'collision' have unfortunately been interchanged.)

The Rates of Diffusion-Controlled

Reactions

35

and so breaks up the encounter. Separation by one molecular diameter, say ~ 10 Á, by translational diffusion would require 10-10 s, whereas oscillations would occur at intervals of ~ 10-12 s on average, so there would be enough time for ~ 100 recollisions. This calculation is obviously very approximate (cf. Section 2.5.2.2), but it serves to show the importance of such cages. The cage is not envisaged as a permanent cavity in the liquid; it is simply the fluctuating set of molecules that surround a solute molecule at a given moment.9 Experimental evidence bearing on the cage effect includes the following. (a) Photolysis of a solution containing azomethane (CH3N = NCH3) and perdeuteroazomethane in iso-octane at temperatures between O °C and 100°C gives as products C2H6 and C2D6 but not CH3CD3; this suggests that CH3 or CD3 radicals formed from a given molecule recombine before they have time to separate by diffusion [19]. (b) In photochemical polymerization, the quantum yield is often less than unity, and varies with the initiator, monomer and medium [20]; these effects may likewise be attributed to recombination of the fragments formed by cleavage of the initiator molecules before they have time to escape from each other. (c) Direct evidence comes from studies on the recombination of radical ions produced by photo-dissociation or by photo-induced electron-transfer (Section 4.6.4). The spins of the radicals in the geminate pair so produced will initially be correlated, and must remain so if they are to recombine. A magnetic field will promote spin change and will therefore decrease the rate of geminate recombination, but will not affect the rate for radicals which have drifted aparto This difference will appear both in rate measurements and in the intensity distribution of the nmr spectrum. Striking effects of magnetic fields have in fact been observed [21]. (d) The recombination of iodine atoms produced by flash photolysis of iodine (1 + 1 ~ Iz) has been much studied (Section 7.3.4.1); it is easily monitored by fast opticalabsorbance experiments. Rate measurements on the microsecond time-scale showed simple bimolecular kinetics, with k ~ 1010 M-1 s-l, indicating that the recombination is diffusion-controlled; this interpretation is confirmed by the fact that k is inversely proportional to the viscosity of the solvent (cf. Equation (2.3a)). The quantum yield is much less than unity, indicating that most of the iodine atoms recombine with their original partner, without permanently separating (this is called 'geminate recombination'). Investigations on the picosecond and sub-picosecond time-scales have provided much striking information. The reaction during the first 20 ps after the dissociating flash shows arate constant much higher than that observed in later stages. This suggests that the atoms generated from many of the iodine molecules have separated far enough for the molecule to have lost its identity, but not far enough for a solvent molecule to come between them, while at longer times the recombination occurs between geminate iodine atoms that initially became separated by one (or more) solvent molecules but soon re-collided and recombined. (e) Investigations in which ultrafast flash methods have been combined with fast molecular-beam techniques have yielded remarkable results. Iodine can be vaporised into argon and other inert gases, and the resulting gaseous mixtures have been examined, by The eagepieturewas introdueedin papersby (a) E. Rabinowiteh[18,e]and (b) E. Rabinowitehand W.c. Wood[18,d].In the seeondof thesepapersit was shownthat the situationeouldbe simulatedby a meehanieal model(seeRef.[18,e]andbelow).[Author:I haveregularlyshownthebehaviourof sueha modelwhenleeturing to undergraduates.]

9

36

Chapter 2

conventional mass-spectroscopy and other methods. These show that in argon gas there are many clusters of argon molecules, about 10 to 40 in each. By adjusting the pressure, up to supercritical values, the state of the argon can be continuously varied from one close to an ordinary liquid to the gaseous state. When iodine is introduced into argon, each iodine molecule is surrounded by such a cluster. With the aid of molecular-dynamics calculations, models of the iodine-argon clusters can be constructed (see Figure 7.10); these give a clear picture of the role of the solvent cage in geminate recombination. Variation of the temperature and pressure of the gas gave results for the rate experimental constant in reasonable agreement with the calculated values. When the carrier gas is varied (helium, neon, argon, krypton), the effects on the experimental rate constants and on the quantum yield are in line with the predictions of the model (Figure 7.13). The cage effect may be convincingly simulated with the aid of a mechanical device in which steel balls are agitated on a fiat surface and their contacts recorded electrically, thus providing a crude model of equal-sized spherical particles in a confined space [ 18,d,e]. When the balls are few and occupy little of the available area, contacts occur singly, at random intervals. When the number of balls is increased so that little vacant space remains, the contacts occur in groups but at longer intervals on average; visual inspection shows that once two balls have made contact they often bounce back from their neighbours in a series of collisions, and even after becoming separated they often come into contact again after a short excursion. Eventually the pair of balls separate permanentIy. All this simulates, or at least illustrates, the behaviour expected of an encounter cage (cf. the next section). Quantitatively, the total number of collisions in a given time is found to remain constant within a few per cent over a considerable range of the fraction of vacant area, a result that encourages the view that collision numbers in gas and in solution can reasonably be assumed to be approximately equal (cf. Section 2.5.1.1).

2.5. The course of an encounter 2.5.1. Encounter without reaction The encounter pair and the surrounding solvent cage constitute a many-body problem of great complexity, and theoretical treatments necessarily have recourse to correspondingly complex mathematics. Rigorous treatments have been developed [22], but the results are not easy to apply to experimental data. We can, however, gain some insight by the application of simple collision and random-walk theories to a hard-sphere model [23]. 2.5.1.1. Duration of encounter and number of collisions [23,a,b,cJ We make the supposition that the total collision rate between A and B solute molecules in a solution (including the solvent-cage region) is the same as in the gas phase.1O The 10 The equality of the collision number in solution to that in lhe gas phase cannot be derived from simple kinetic theory. It does, however, lead to reasonable conclusions (see below). Equality within an order of magnitude for hard-sphere non-interacting equal-sized molecules is indicated by the following argument based on consideration offree volumes [18,a]. The collision number Z in an assembly of such molecules depends on (a) their translational energies (b) their diameters and (c) lhe space in which lhey are free to move. Both (a) and (b) are lhe same in

The Rates 01 Diffusion-Controlled

Reactions

37

corresponding second-order rate constant Zeoll is therefore given by Maxwell's kinetictheory expression nrÁB (SkT/nlJ.,) 1/2, where rAB = rA + 'B and IJ., is the reduced mass. The rate constant for encounter (Zene), may be expressed by Equation (2.6), or approximately by Equation (2.1) as 4n N r AB D AB. The ratio of these two expressions gives the average number of collisions per encounter. Simple algebra shows that the average number of recollisions per encounter (Nene) is:

Nene- _ (~)(SkT)1/2 4DAB

n IJ.,

-_ (~)

4DAB

e,

(2.19)

where

DAB

= DA + DB,

and

(~/~)1/2

e __ "r

For hard-sphere molecules with radius in the range 2-5 Á (which includes many ordinary organic reactants), in solvents such as water or cyclohexane with viscosity about 1 cp, this expression gives the average number of recollisions per encounter as 25 to 150, increasing approximately with rÁB [23,c]. Extending the argument, it is possible, without a specific model of the encounter cage, to deduce the average frequency of recollisions in an encounter [23,a] and hence its reciprocal, which is the average time between such recollisions (treeol¡). This comes out to be: (2.20) The calculated values of treeoll, for the same range of molecular radii as before, are in the region 0.5 to 1.5 ps. The average duration of an encounter (tene) may now be deduced. It is the product of the number of recollisions and the average time between them, i.e., Nenetreeoll. Inserting the expressions above, one obtains tene ~ rÁB/ DAB. Since DAB ()( l/rAB according to the Stokes-Einstein relation (if r A = 'B), this result implies that tene should vary approximately as rÁB' The calculated values [23,c] are in accordance with this expectation, and are in the region of 5 to 200 ps. 2.5.1.2. Random-walk

treatment

01 encounter11

The encounter may be visualised in terms of the random molecular motions, both translational and rotational, of two adjacent solute molecules A and B. The average encounter duration is the average time required for one of the molecules (A) to leave its partner (B) solution as in the gas phase, but the free volume will be smaller in solution, thus inereasing Z. The maximum effeet will be for close-paeked spheres, giving an inerease of 4. Thus in general the ealeulated differenee between gas and solution will be a factor between l and 4. 11 The following aeeount is based on the treatment of A.M. Lopez-Quintela et al. [23,d].

Chapter 2

38

and penetrate into the solvent far enough to allow one or more solvent molecules to be interposed between A and B, thus making a recollision unlikely and usually bringing the encounter to an end. This average time can be calculated by a simple random-walk treatment. (The assumption that the motions are random is not strictly accurate at such short separations (cf. Ref. [7, p. 215]), but as an approximation its use may be tolerated.) As before, we consider the simplest case, where the solute and solvent molecules are hard spheres of equal size (rA = TE = rs). We use the random-walkexpression relating the mean square displace2 ment in three dimensions (r ) to the time-interval l, namely r2 = 6Dl (Equation (2.17)). For l we substitute the average encounter duration, which we shall write as lene; and for r we must then substitute some value for the critical distance beyond which A is very unlikely to return to B. The minimum distance is evidently one molecular diameter (2r A), but a more realistic value is 1.5 diameters (3rA) [23,a]. For the translational diffusion coefficient D, we can use the Stokes-Einstein equation with 'slip' conditions (DA = DB = kT /4JT1]TA), and take the relative diffusion coefficient as DA + DB = kT /2JT1]TA. The random-walk expression then becomes:

l

r2 (3r A)2 ------------ene - 6D - 6(kT /2JT1]TA)

VAr¡) lene ~ 2 ( kT

'

3JT r¡rl

-

kT

'

(2.21 )

(2.22)

where VA is the molecular volume (4/3JTri). For a typical organic solute molecule, with radius 3 Á, in a solvent with viscosity ~ 1 cp (such as water or cyclohexane) at 25 °e, this expression gives lene ~ 240 ps. The Stokes-Einstein equation, however, often underestimates D (Section 3.4.3); the experimental values for many organic molecules are better represented by D = kT /3JT1]TA, which gives lene ~ 180 ps. For molecules of other sizes, the calculated values of lene will be proportional to and hence to the molecular volume VA. Many common organic reactant molecules of moderate size have volumesl2 which in spherical molecules would correspond to r A = 2-5 Á. The corresponding calculated values of the average encounter duration are in the range 50 to 900 ps. The values differ from the results of the preceding calculation by about an order of magnitude. This discrepancy could be reduced to a factor of about 3 if we adopted a critical distance close to one molecular diameter, instead of 1.5. This remaining discrepancy illustrates the shortcomings of simplified hard-sphere models.

ri,

2.5.1.3.

Rolalion between collisions

We saw above (Section 2.5.l.l, Equation (2.20)) that according to the hard-sphere model the average time between A-B collisions in an encounter, for common bimolecular organic reactions between neutral molecules in ordinary solvents, is in the region of 1 ps. How much 12 Moleeular volurnes can be estirnated by surnrning atornie inerernents based on van der Waals volumes: (a) J.T. Edward, J. Chem. Educ. 47 (1970) 261; (b) A. Bondi, 1. Phys. Chem. 68 (1964) 441; or by more 80phistieated rnethods: (e) A.Y. Meyer, Chem. Soc. Rev. 15 (1986) 449; (d) A. Gavezzotti, J. Amer. Chem. Soco 105 (1983) 5220; 107 (1985) 962. The rnoleeular volurne range eorresponding to rA = 2-5 A is about 50-500 A3

The Rates of Diffusion-Controlled

Reactions

39

will the molecules rotate in this time-interval? (The extent of reorientation will evidently affect the chance of a reactive collision when the reactant molecules have limited reaction sites.) Rotational motion, like translational motion, can be treated as random motion due to the impacts of solvent molecules. The solute molecule may be visualised as oscillating about a particular orientation in space for much of the time, occasionally receiving an impulse which makes it rotate through a small but appreciable angle into a new orientation before being brought to rest by the viscous resistance of the surrounding solvento Each change of orientation is assumed to be independent and uncorrelated with its predecessors. The average angle (8) through which a spherical molecule (or more strictly some axis in the molecule, which is assumed to be rigid) rotates in a time-interval ( can be found by a random-walk calculation [24] analogous to that for translational motion (Section 2.3). The result is: (2.23) (2.24) Here DR is the rotational diffusion coefficient. It has the dimensions (angle)2 (time)-I and is commonly expressed in the units (radian)2 s-l. We note that the average time for rotation through unit angle, which we may write rR, is given (according to Equation (2.23)) by: R

=

r

1 6DR.

(2.25)

Hydrodynamic calculation [25] relates D~ for a particular spherical molecule A to the radius of the molecule (rA) and the viscosity of the solvent (1)) by the expression: D

R

---

A -

kT mJr1]r3 ,

(2.26)

where m is a numerical coefficient which is calculated as 8 if one as sumes that there is no slip between the solvent and the solute molecule, but experimentally is found to be in the region of 4 for common reactant molecules (Section 3.4.3 below). For the relative rotational motion of equal-sized molecules of two types, A and B, we shall use the approximation D~B ~ D~ + D~ ~ 2D~, analogous to the expression for relative translational motion. We then obtain from Equations (2.23) and (2.26) (with m = 4):

(2.27) For typical solute molecules with rA = ~ = 3 Á, in a solvent with viscosity 1 cp at 25 °e, this gives (j ~ (4 x ]05)( l/2. We have already noted that according to the hard-sphere model the time between recollisions for such molecules in an encounter is around 1 ps. Hence we find that the average relative rotation (j of the two molecules between collisions in an encounter is in the region of 0.4 radian s (~20 degrees). If the molecules are reactive only

40

Chapter 2

at certain sites, such changes of orientation will evidently increase the change of these sites coming into contact, and so will have an important infiuence on the rate reaction (cf. Section 2.5.2.2). 2.5.2. Encounter leading to reaction 2.5.2.1. Introduction: diffusion-controlled

reaction rates

In this discussion of molecular motions during encounter, we have so far considered only non-reacting solutes. We now turn to the consequences for diffusion-controlled reactions. For reactions that occur at every collision, the diffusion-limited rate is simply the encounter rateo The above calculations, whether based on the concentration-gradient approach or on random-walk theory, agree in predicting arate constant given by ko = 4JrrABDAB (Equation (2.1». But diffusion coefficients over a range of temperature are seldom available, and there are uncertainties about the dimensions of molecules and the effects of their irregular shapes [23,c,d]. For a broad survey one can fall back on the equation ko = 4RT Ir¡ (Equation (2.2», which was derived for equal-sized spherical reactant molecules; this is more widely applicable, because the viscosities of many common solvents are known over ranges of temperature. Since the theory has omitted consideration of intermolecular forces, we must expect agreement with this equation to be best for reactions of neutral reactants in non-polar solvents. The data in Table 2.1, which could be greatly augmented, show that these reactions have rate constants agreeing with the simple theory within a factor of ten. This agreement is remarkable, given the approximations implicit in the theory. It seems clear that the rate is dominated by that of diffusional encounter. We must, however, consider some possible complications. 2.5.2.2. Diffusion-controlled

reactions at restricted reaction sites

The reactions with the highest values of k in Table 2.1 are those where orientational requirements would be expected to be unimportant. Reaction usually occurs at specific sites on molecules; a collision will not result in reaction unless the re active sites come into contact. Supposing the relative orientations of the reactant molecules A and B to be random as they approach each other, only a fraction of the colliding pairs will be in the correct orientation for reaction. The chance that reaction will occur immediately on collision is therefore less than unity; it will depend on the fraction of the 'surface' of each reactant molecule that is reactive.13 But once the reactant pair have met, they will undergo a series of recollisions (cf. above) between which there will be time for some reorientation, which may result in the next collision being correctly oriented for reaction. The restricted reaction sites will reduce the rate below that of a diffusion-controlled reaction occurring at every collision, but the reduction will be less than it would have been if there were no repeated collisions. Some attempts have been made to formulate theories for this situation, as follows. We as sume the reactant molecules A and B to be hard spheres, each with a small re active region. Let
The Rates of Diffusion-Controlled

Reactions

41

reactive. When a pair of molecules (A and B) meet by diffusion through the solvent, the chance that in the first collision the reactive sites will come into contact is 1. Calculations have beencarried out by a statistical method [23,a] and also by a random-walk method [23,d] (Section 2.5.1.2). They indicate that, for values of
t:j:

t

t t# :j:# t:j:

t:j: t:j:

CC14 e-hexane CH3CN

CC4 e-hexane n-heptane I-Cl-butane e-hexane

Solvent

"" 295

295 293 298 298 298 298 298

Temp.

Solvent varied. # Substituent varied. * Exeited state.

"" 0.1 "" 0.1

""0.1 ""0.02 ""0.02

1.0 1.0 1.0


""28

10 10 25 23 II 10 II

1O~9ko (K)

9.36 20

2.1 4.7

6.9 6.7 20.5 11.7

1O-9k

(M-1 s-l)

0.85 ""0.7

0.7 0.7 0.8 0.5 0.2 0.4

klko

""5 ""10 ""20 ""8 ""7

0.7 0.7 0.8

kl koI
6.2.2.1 6.3.2.2 2.5.2.2 4.6.2 6.2.2.2 6.2.2.2

7.3.4.1 lb] [e] [d] [e] [f] [g] [h]

Seetion [a]

Ref.

References for data: [a] See J.E. Willard et al., Ref. [7.13]; lb] J.B. Birks and L.H. Munro, Prog. React. Kinet. 4 (1967) 238, see table on p. 292, also p. 255; [e] w.R. Ware and J.S. Novros, Ref. [13,b]; [d] Ref. [I,h]; [e] Ref. [I,g]; [f] Ref. [I,e]; [g] M.-H. Hui and W.R. Ware, J. Amer. Chem. Soco 98 (1976) 4718; [h] A. Weller, H. Staerk and R. Treiehe, Faraday Discuss. Chem. Soco 78 (1984) 271.

:j:

12 + (Me2N12CS Me3C . + . CMe3 PhC02H + H02HPh antbraeene* + PhNMe2 pyrene* + PhNM2

1+1 pyrene* + pyrene anthraeene* + CBr4

t TemperalUre varied.

l 2 3 4 5 6 7 8

Reaetants

Table 2.2 Rate eonstant for some very fast assoeiation reaetions, eompared with ko = 4RT Ir¡. k is observed forward rate constan!. Values given for
Q N

...•

"

{l

The Rates of Dijfusion-Controlled

Reactions

43

[28] involves a pairing of electron-spins, and is therefore retarded by an external magnetic field (Section 4.6.4). The related phenomena of chemically-induced dynamic electron or nuclear polarisation (CIDEP or CIDNP) give information on the recombination and escape probabilities [29]. These techniques open possibilities for the study of the structures and kinetics of radical pairs. Flash photolysis must also be expected to produce radical pairs initially at various distances aparto Much theoretical work has been done on such reactions [30]. 2.5.2.4. Macromolecules and biological systems Many reactions of macromolecules involve binding of a relatively small molecule, with severe orientational constraints. An example is the association of an enzyme and its substrate; another is the binding of ligands to receptor sites on biological membranes [31]. The rate constant for association of an enzyme with a small ligand is commonly in the region of 107 to 108 M~l s~l, and for association with another protein it may be up to 106 M-1 s-1 [3l,a,b]. Such values may be attributed to the fact that active sites are restricted in area, sometime even buried in cavities, and to activation energies; various models have been considered [3l,c]. These reactions cannot be represented by the simplified model of an encounter that we have used in Sections 1.2 and 2.2, which assumes that the surfaces of both the reactant molecules are uniformly reactive and that every encounter leads to reaction. In enzyme molecules, on the contrary, most of the surface is unreactive; only a few limited areas are active sites. Moreover, access to these sites is often hindered by severe orientational constraints; in some enzyme reactions the active site is buried in a cavity (as in haemoglobin). Various detailed models have been considered [3l,c]. Another consideration is that molecules are much more likely to meet if they are constrained to move on a surface; diffusional 'search' is considerably more efficient in two dimensions than in three. This applies to the probability of a molecule finding an acceptor site (a small target) on a membrane; it appears that ligands can slide on the surface of membranes, thus accelerating bonding and reaction. One-dimensional diffusion is even more efficient; the repressor protein of E.coZ¡ lac locates its specific binding site on a long DNA chain considerably faster than would be expected from three-dimensional diffusion [31 ,d]. 2.5.2.5. Effects of solvent structure When the discrete molecular structure of the solvent is taken into account, the assumption that the distribution and motions of solute molecules are always random (implicit in the classical theories of diffusion in a continuous fiuid) must be expected to break down at short distances. 'Proximity effects' of two kinds can be distinguished. (a) Liquids have a short-range structure, which may be studied by neutron-scattering experiments, or by subjecting simple models to computer simulation. Analysis of radial distribution functions shows that, at an intermolecular distance of two molecular diameters, the solvent molecules are relatively tightly packed and will tend to exclude solute molecules, while at 1.5 diameters apart the packing is less dense and solute molecules will be drawn in. Calculations suggest that the latter process will often dominate, and will contribute to the cage effect by prolonging the life of the encounter complex [32,a,c]. (b) The fact that the solvent is made up of discrete molecules also has a dynamic effect. As the reactants approach each other the intervening solvent molecules have to be squeezed out of the way, thus reducing the diffusion coefficient at short inter-reactant distances and hindering the formation of encounter

44

Chapter 2

complexes. This effect is known as 'hydrodynamicrepulsion' [32,b,c]. Calculations on theoretical models indicate that by itself this might reduce the calculated rate constant by as much as one-half. However, it will usually act in the opposite direction to the effect of short-range structure just described; the slower formation of encounter complexes will then be partly compensated by their longer life. It appears that the net effect on the calculated rate constant may be ~ 10-30%. These effects would be predicted even for hard-sphere molecules, but they will no doubt be increased by any solvent-solvent attractions. 2.5.2.6. Effects of solvent-solvent

intermolecular forces

The simplest behaviour is of course expected for reactions between neutral molecules in non-polar solvents, which will correspond most closely to hard-sphere theoretical models. Complications may be expected when the reactants are polar or ionic, and when the solvent molecules interact strongly with them or with each other. Solvent-solvent interactions have been introduced into Smoluchowski-type diffusion theory [32,c], by assuming that the solvent molecules interact according to a Lennard-Jones potential. In general, however, models that treat the solvent as a continuum must be abandoned when short-range interactions are considered. Kinetic theory and computer simulation are more flexible means of taking account of these and other additional factors (Section 2.6 below). 2.5.3. Effects of intermolecular forces between reactants: reactions betwcen ions 2.5.3.1. Effects of size and charge ofreactants When both reactants are ions, there will be a long-range Coulombic interaction which affects rates of diffusion and so necessitates modification of Equation (2.1). Analysis shows that the form of the expression is unchanged but that it must be multiplied by a factor which depends on the charges on the ions and the distance of closest approach [33]. In Debye's treatment, Fick's law is modified by including a term representing the potential energy (U) due to the Coulombic forces. This term is U = ZAZBe2 / Er, where ZAe and ZBe are the charges on the ions, r is their distance apart, and e is the relative permittivity of the solvent. Solution of the modified differential equation leads to an expression for the encounter rate constant for reactions between ions (k~ns) as a modification of the expression for ko in Equation (2.1). The full equation including ion-atmosphere effects is complex, but in the limit at low concentrations it takes the simple form: koions

= ko

[8]

-8--

e-l

= 4JT DABrAB

[ 8 ] -8--

e-l

,

(2.28)

where 8 = ZAZBe2/ekTrAB and rAB is the distance of closest approach. Equation (2.28) is known as the Debye-Smoluchowski equation. Some values of the factor 8/ (e8 - 1) for ions of various charges with different values of a are given in Table 2.3. The factor is not very sensitive to the size of the ions unless this is very small. The resulting value of k~ns for reactions between oppositely-charged univalent ions in water at 25°C is between 1010 and 1011 M-I s-l. A selection of values of rate constants for fast ionic reactions of various types is given in Table 2.4. They are in the expected range. Despite the uncertainties in the theoretical calculation of ko, it seems

The Rates oi Dijfusion-Controlled

Table 2.3 Calculated diffusion-controlled for uncharged molecules. Values of a

Reactions

45

rate constant for reaction between ions, compared with that 1), for ions of charges ZA and ZB,

kD'ns / koolecules = 8/(é -

with distance of closest approach rAB (Á), for water at 25°C Charge product

rAB (Á)

ZAZB

2.0

5.0

7.5

10.0

0.17 0.45 1.9

0.34 0.60 1.6

3.0

2.2

0.45 0.69 1.4 1.9

kt'ns / koolecules

+2 +1 -1

-2

0.005 0.10 3.7 7.1

clear that for these reactions the rate is dominated by the rate of diffusion, just as it is for non-ionic reactions, and that the various proximity and other effects do not alter it by an order of magnitude. 2.5.3.2. Effect 01 added non-reacting ions In the presence of non-reacting ions, the rate constant of a reaction between ions (at low concentration) would be expected to follow the Debye-Hückellaw, which takes account of the non-uniform distribution of positive and negative ion s in the ionic atmospheres. Experimental resuIts in accordance with this expectation have been obtained for the ftuorescencequenching of some ionic ftuorophores by added ion s [34], and in continuous-ftow experiments on some inorganic electron-transfer reactions [35]. The results plotted in Figure 2.6 on ftuorescence quenching of acridinium ion s (AH+) by positive and negative ion s show good agreement with the limiting law [36]. Taking a ~ 5 Á, they can be checked against the values shown in Table 2.4. The plot also shows that at the higher ionic concentrations the changes of viscosity should be taken into account when comparing these very high rate constants with theoretically-calculated values.

2.6. Alternative theoretical approaches: refinement of theory We have seen that to make progress in theoretical investigations of diffusion-limited reactions it is necessary to take into account the discrete molecular structure of the solvent, and therefore to discard the continuum model on which the concentration-gradient approach is based. This implies discarding also the hydrodynamic models by which the required diffusion coefficients have been calculated, and the simplest random-walk assumption (Section 2.3.1) that the reactant molecules undergo diffusive motions in which each jump is in a random direction, uncorrelated with the previous ones. Although hydrodynamic theory is extremely useful in providing approximations for systems in which diffusion coefficients have not been experimentally deterrnined, and is satisfactory when the solute molecules are large compared with those of the solvent, it is not well suited to dealing with systems where the solute and solvent molecules are of comparable size and are).A:Wac.tj.ng

~

.,

. ,

,.

-6 -9 -12 -12

+8 +8 +8

MnO~Os(bipy)~+ IrCI~Mo(CN)¿W(CN)¿Os(bipy)~+ Fe(bipy)z(CN)g Os(bipy)~+ Fe(CN)¿W(CN)¿~ fAg TPPTS4-

Fe(phen)~+

Mo(CN)~-

Fe(DMP)~+

Os(bipy)~+

Os(bipy)~+

IrCI~-

Os(bipy)~+

Fe(bipy)~+ IrC12-

IrCI~-

IrCI~0.1

0.1 1.0

0.1 1.0

O

O

O

O

0.5

0.1

0.5

0.1 0.45

?

varo

Ionic strength 1 (M)

25

25

25

22

22

22

22

10

10

10

25

10

25

25

25

Temp. (OC)

2

1.6

2.3

2.0 1.7

3.2

X

1.2

X

X

X

X

109

108

108 109

X 109

X

2.5

1010 1010

109

1.0 X

2

X 109

4.0

I

109

7

X

X

1.2

1011

109

2.0

X

X

X

X

X

0.2 0.05 0.2

1010 1010

0.2

2.7 1.0

kobs/ kcalc

1010

109

1.5 x 109 4.8 x 109

4.0 x 109 4.6 x 109 2 X 1010 1.1

kcalc

kobs

Calculated

Rate constant (M-1 s-l) Observed

r

a Data fram R.D. Cannon, Electron-Transfer Reactions, Butterworths, 1980, p. 105. bData fram J.F. Holzwarth, in: Techniques and Applications of Fast Reactions in Solution, Reidel, Dordrecht, 1979, p. 515, Figure 2, appraximate values taken fram graphs in figure. ('Data fram H. Bruhn, S. Nigain and J.F. Holzwarth, Faraday Discuss. Chem. Soco 74 (1982) 129, see Figure 2.6 and Table 2.3. dThe figure shows that above I = 0.5 the plots level out, indicating that the electrastatic interactions become smal!. eln KCI aq. at pH 5.1. Ag TPPTS4- is silver mesotetraphenylporphyrin tetrasulphate.

6

6

O

-4

-6

-4

Fe(DMP)~+

6

MnO'¡ IrC12-

O

ZAZB

Charge praduct

O -1

H

Reactants B

H Cd+

Fe(CN)~FeCI2+

A

Table 2.4 Rate constants for some fast reactions in aqueous solution

c.e

c.e

c.e

b.d

b.d

b.d

a

a

a

a

a

a

a

Note

The Rates of Diffusion-Controlled

••• ".

Reactions

47

'.~.

10·0

'.

, ,'"

• -------

"O

::

.

- -.- - •

•.............•

'- .....•

.••........•.

o

9·5

o ::0 ..... 0

o

o

o

o o

"- ...• o

. 'O,

o

P

ti

9

9·0 O

Figure 2.6 Rate constants kq at 25°C for quenching of acridinium ion f1uorescence by Br- (.) and Ca2+ (o) in aqueous NaN03 solution as a function of ionic strength, l. The dotted lines represent the limiting gradients expected in dilute solution. The dashed line represents the theoretically-calculated diffusion-controlled rate constant calculated for neutral reagents in the same solution. From Ref. [36J.

at short distances inside an encounter cage. It takes no account of the coupling between solvent and solute motions, or between translational and rotational diffusion, which must then occur. Simple random-walk theory has similar limitations. More flexible approaches are required. Molecular-pair theories offer the simplest ways to handle the system. The use of statistical non-equilibrium thermodynamics has already been mentioned [16]. Kinetic theory provides another approach [37]. Beginning with the theory of dilute gases, it has been extended to dense gases and to liquids [37,a). The application to diffusion-limited reactions gives as a first approximation the Smoluchowski Equation (2.1), but it is a fairly straightforward matter to include more details of molecular motions. Solute and solvent molecules are treated alike from the start. The coupling of reaction to diffusion is explicit. Cage effects can be modelled. If analytical solutions cannot be obtained, fast computers can provide numerical solutions. Computer simulation [38], in which fast computers are used to solve numerically the simultaneous equations of motions of the many-body systems, has been rapidly developed. It eliminates the complexities of mathematical analysis, and is the most flexible theoretical approach. For instance, simulations of Brownian-motion dynamics have been made to describe the simultaneous effects of orientational constraints, long-range attractions, and hydrodynamic interactions [38,c,d,e]. The recombination of iodine atoms (Section 7.3.4.1) has been the subject of several such investigations [38,c,d,e] as have been the diffusion-controlled reactions ofbiological macromolecules [38,f] and the formation of colloidal aggregates [38,g]. It has been used with great effect as an integral part of studies of ultrafast reactions [38,h].

48

Chapter 2

References [2.1]

[2.2] [2.3] [2.4] [2.5] [2.6] [2.7] [2.8]

[2.9] [2.10]

[2.11] [2.12] [2.13]

[2.14] [2.15] [2.16]

[2.17]

[2.18]

[2.19]

Referenees for the values of the rate eonstant k in Table 2.1 are as follows: (a) M.T Emerson, E. Grunwald and R.A. Kromhout, J. Chem. Phys. 33 (1960) 547; (b) M.L. Ahrens and G. Maass, Angew. Chem. (lnt. Ed. Engl.) 76 (1968) 88; (e) M. Anbar and EJ. Hart, J. Amer. Chem. Soe. 86 (1964) 5633; LA Taub, O.A Harter, M.e. Sauer and L.M. Oorfman, J. Chem. Phys. 41 (1964) 979; (d) J.F. Holzwarth, in Ref. [l.IO,a, p. 509, Fig. 2]; (e) W. Maier, Z. Elektroehem. 64 (1960) 132; (f) A. Weller, Ref. [6.I,e, p. 199]; (g) KV. lngold, in J.K. Koehi (Ed.), Free Radicals, Vol. 1, Wiley, 1973, p. 43, Table 1; (h) E.F. Caldin and B.B. Hasinoff, J. Chem. Soe. Faraday Trans. I 71 (1975) 515; (j) E.F. Caldin, L. de Forest and A. Queen, Ref. [26]; (k) K Hiromi, Kinetics of Fast Enzyme Reactions, Wiley, 1978, p. 268; (1) A. Fersht, Enzyme Structure and Mechanism, 2nd edn., Freeman, New York, 1985, Table 4.4. M. von Smoluchowski, Z. Phys. Chem. 92 (1917) 129. B. Sveshnikoff, Acta Physieoehimiea URSS 3 (1935) 257; B. Sveshnikoff, Acta Physieoehimiea URSS 7 (1937) 755. P. Oebye, Trans. Eleetroehem. Soe. 82 (1942) 265. See, e.g., The study offast reaetions, Oiseuss. Faraday Soe. 17 (1954); S. Claesson (Ed.), Fast Reactions and Primary Processes in Chemical Kinetics, Nobel Symposium 5, Wiley-Interseienee, 1967. RM. Noyes, in: G. Porter (Ed.), Progress in Reaction Kinetics, Vol. 1, Pergamon Press, 1961, pp. 131160. For summaries, see, e.g., S.A. Rice, Diffusion-limited Reactions, Elsevier, 1985. See also Chapter 3 below. For reviews, see, (a) Ref. [6]; (b) Ref. [7, Chapters 2, 3 and 8]; The improved ealculation is due to (e) F.C. Collins and G.E. Kimball, J. Colloid Sei. 4 (1949) 425; (d) F.C. Collins, J. Colloid Sei. 5 (1950) 499. (a) K Luther, J. Sehroeder, J. Troe and U. Unterberg, J. Phys. Chem. 84 (1980) 3072; (b) E.F. Caldin and B.B. Hasinoff, 1. Chem. Soe. Faraday Trans. 1 71 (1975) 515. (a) B.B. Hasinoff, Areh. Bioehem. Biophys. 211 (1981) 396; (b) B.B. Hasinoff, Areh. Bioehem. Biophys. 233 (1984) 676; (e) B.B. Hasinoff, Areh. Bioehem. Biophys. 191 (1978) 110; (d) B.B. Hasinoff and S.B. Chisti, Bioehem. J. 21 (1982) 4275. (a) AJ. Elliot, O.R. MeCraeken, G.v. Buxton and N.O. Wood, J. Chem. Soe. Faraday Trans. 86 (1990) 1539; (b) G.V. Buxton and A.J. Eliot, J. Chem. Soe. Faraday Trans. 84 (1993) 485. (a) RM. Noyes, Ref. [6]; (b) F.C. Collins and G.E. Kimball, J. Colloid Sei. 4 (1949) 425; (e) S.A. Rice, Ref. [7, p. 31, ete.]. (a) A. Weller, in: G. Porter (Ed.), Progress in Reaction Kinetics, Vol. 1, Pergamon Press, 1961, Chapter 7 and referenees therein; (b) W.R Ware and J.S. Novros, J. Phys. Chem. 70 (1966) 3246, on anthraeene + earbon tetrabromide in heptane ete.; (e) T.L. Nemzek and W.R. Ware, J. Chem. Phys. 62 (1975) 477, on 1,2-benzanthraeene + earbon tetrabromide in viseous solvents; (d) M.H. Hui and W.R. Ware, J. Amer. Chem. Soe. 98 (1976) 4718, on anthraeene + dimethylaniline in eyclohexane; (e) G.S. Beddard, S. Carlin, L. Harris, G. Porter and CJ. Tredwell, Photoehem. Photobiol. 27 (1978) 443, on ehlorophyll a+ nitrobenzene in ethanol. TJ. Chuang and KB. Eisenthal, J. Chem. Phys. 62 (1975) 2213. (a) B.B. Hasinoff, J. Phys. Chem. 82 (1978) 2630; (b) B.B. Hasinoff, J. Phys. Chem. 85 (1981) 526. For clear summary expositions ofthis approaeh, see, (a) J. Keizer, Aee. Chem. Res. 18 (1985) 235; (b) J. Keizer, Chem. Rev. 87 (1987) 167; (e) J. Keizer, J. Phys. Chem. 86 (1982) 5052; (d) J. Keizer, Statistical Thermodynamics of Non-equilibrium Processes, Springer, New York, 1987, see also Seetion 2.6 below. (a) R.M. Noyes, Ref. [6] above, and earlier papers eited therein, including J. Amer. Chem. Soe. 77 (1955) 2042; J. Chem. Phys. 22 (1954) 1349; (b) TR Waite, Phys. Rev. 107 (1957) 462; TR. Waite, J. Chem. Phys. 28 (1958) 103; T.R. Waite, J. Chem. Phys. 32 (1960) 21; (e) G. Wilemski and M. Fixman, J. Chem. Phys. 58 (1973) 4009; G. Wilemski and M. Fixman, J. Chem. Phys. 60 (1974) 866, 878; (d) J. Keizer, Ref. [l6,b, Seetion IIIA]; S.A. Rice, Ref. [7, esp., Chapter 8, Seetion 3]. (a) RP. Bell, Trans. Faraday Soe. 35 (1939) 324; (b) RH. Fowler and E.A. Guggenheim, Statistical Thermodynamics, Cambridge, 1939, p. 530 seq.; (e) RA. Fairclough and C.N. Hinshelwood, J. Chem. Soe. (1939) 594; (d) S.A. Rice, Ref. [7, esp., Chapters 6, 7 and 8]. RK. Lyon and O.H. Levy, J. Amer. Chem. Soe. 83 (1961) 4290; RK Lyon, J. Amer. Chem. Soe. 86 (1964) 1907.

The Rates of Diffusion-Controlled [2.20J [2.2IJ

[2.22J

[2.23J

[2.24] [2.25] [2.26] [2.27]

[2.28J [2.29J [2.30J

[2.3IJ

[2.32J

[2.33J

[2.34] [2.35J [2.36] [2.37J

[2.38J

Reactions

49

C. Walling, Free Radicals in Solution, Wiley, 1957, pp. 76-79. (a) A Weller, H. Staerk and R. Treiehel, Faraday Dise. Chem. Soe. 78 (1984) 271; (b) L.T Muus, P.W. Atkins, K.A. MeLauehlan and J.B. Pedersen (Eds.), Chemically Induced Magnetic Polarization, Reidel, Dordreeht, 1977, see esp., EJ. Adrian, pp. 57 seq.; (e) S.A Rice, Ref. [7, pp. 147-149J. (a) K. Sole and WH. Stoekmayer, J. Chem. Phys. 54 (1971) 2981; (b) K. Sole and WH. Stoekmayer, Int. J. Chem. Kinet. 5 (1973) 733; (e) K.S. Sehmitz and J.M. Sehurr, J. Phys. Chem. 76 (1972) 534; (d) J.M. Sehurr and K.S. Sehmmitz, J. Phys. Chem. 80 (1976) 1934; (e) For a review, see, O.G. Berg and P.H. von Hippel, Annu. Rev. Biophys. Biophys. Chem. 14 (1985) 131. (a) AJ. Benesi, J. Phys. Chem. 86 (1982) 4926; (b) AJ. Benesi, J. Phys. Chem. 88 (1984) 4729; (e) A.J. Benesi, J. Phys. Chem. 93 (1989) 5745; (d) M.A López-Quintela, J. Samios and W Knoehe, J. Mol. Liq. 29 (1984) 243. A. Einstein, see footnote in Seetion 2.3, pp. 31-33, 102, 112. P. Debye, Polar Molecules, Dover Publieations, New York, 1929, Chapter 5, p. 83. E.F. Caldin, L. de Forest and A. Queen, J. Chem. Soe. Faraday Trans. 86 (1990) 1549. (a) A.L Burshtein, LV. Khudyakov and BJ. Yakobson, Prog. Reaet. Kinet. 13 (1984) 222; (b) EH. Westheimer, in: M.S. Newman (Ed.), Steric Effects in Organic Chemistry, John Wiley and Sons, New York, 1963, Chapter 12, p. 524. S.A. Rice, Ref. [7, Chapters 6 and 7, esp., pp. 119-121, 132-137, 140-146], see also Ref. [21,aJ. See, e.g., Ref. [21 ,b,eJ. (a) S.A. Rice, Ref. [7, esp., Chapters 11 and 12]; for a kinetie-theory treatment, see, (b) R. Kapral, J. Chem. Phys. 68 (1978) 1903; for eomputer-simulation treatments, see, (e) S.A. Rice, Ref. [7, pp. 336337 and (d) Ref. [38J below. (a) For tables of rate constants, see, A. Fersht, Enzyme Structure and Mechanism, 2nd edn., Freeman, New York, 1985, pp. 150 seq.; (b) K. Hiromi, Kinetics of Fast Enzyme Reactions, Chapter 5, pp. 262 seq.; (e) K.-C. Chou and G.-P. Zhou, J. Amer. Chem. Soe. 104 (1982) 1409; (d) O.G. Berg and P.H.von Hippel, Ref. [22,e, Seetion 4J; (e) H.C. Berg, Random Walks in Biology, Prineeton Univ. Press, 1983, Chapter 3. (a) S.A. Rice, Ref. [7, Chapters 2,6 and 8, esp., pp. 42-43, 125-132, 215-221, 235-237], on the 'potential of mean force'; (b) S.A. Rice, Ref. [7, Chapters 2, 7, 8, and 9, esp., pp. 43, 162-164,232-235, 261, seq.]; (e) S.H, Northrup and J.T Hynes, J. Chem. Phys. 71 (1979) 871. (a) P. Debye, Ref. [4]; ef. also (b) M. Eigen, Z. Phys. Chem. N.E 1 (1954) 176; (e) R.M. Noyes, Ref. [6J; (d) H. Strehlow and W. Knoehe, Fundamentals ofChemical Relaxation, Verlag Chemie, Weinheim, 1977, p. 101. R.W Stoughton and G.K. Rollefson, J. Amer. Chem. Soe. 61 (1939) 2634; R.W Stoughton and G.K. Rollefson, J. Amer. Chem. Soe. 62 (1940) 2264. H. Bruhn, S. Nigain and J.E Holzworth, Faraday Dise. Chem. Soe. 74 (1982) 129, Figure 6. NJ. Bridge and P.D.L Fleteher, J. Chem. Soe. Faraday Trans. 179 (1983) 2161; see also L. Bass and WJ. Greenhalfh, Trans. Faraday Soe. 62 (1966) 715. (a) P.M. Résibois and M. De Leener, Classical Kinetic Theory of Fluids, Wiley, New York, 1977; (b) R. Kapral, Ref. [30,bJ; (e) R. Kapral, Adv. Chem. Phys. 48 (1981) 71-181; (d) M.W Evans, Aee. Chem. Res. 14 (1981) 253; (e) M.W Evans and GJ. Evans, Adv. Chem. Phys. 63 (1985) 377; (f) S.A. Rice, Ref. [7, Chapters 5, 8,12, esp., pp. 218, 247, 344J. (a) B.J. Alder and T.E. Wainwright, J. Chem. Phys. 31 (1959) 459, (Brownian motion); B.J. A1der, D.M. Gass and TE. Wainwright, J. Chem. Phys. 53 (1970) 3813; (b) C.A. Emeis and P.L. Fehder, J. Amer. Chem. Soe. 92 (1970) 2246 (diffusion in two dimensions); (e) J.T Hynes, R. Kapral and G.M. Tome, J. Chem. Phys. 72 (1980) 177, (Morse potentia1); (d) S.H. Northrup, S.A Allison and J.A MeCammon, J. Chem. Phys. 80 (1984) 1517; (e) P. Bado, P.H. Behrens, J.P. Bergsma, S.B. Wilson and K.R. Wilson, in: Picosecond Phenomena nr, 1982, (iodine atom reeombination); (f) J.A. MeCammon, R.T. Baequet, S.A Allinse and S.H. Northrup, Faraday Diseuss. Chem. Soe. 83 (1987) 213, (biologieal maeromoleeules); (g) R. Jullian, R. Botet and P.M. More, Faraday Diseuss. Chem. Soe. 83 (1987) 125, (formation of eolloidal aggregates); (h) See, Chapter 7, passim, e.g., remarks in Seetions 7.2.3 and 7.3.5.