This paper is in a collection of - Electrochemical...

This paper is in a collection of

llHistoric Papers in Electrochemistry"

which is part of

Electrochemical Science and Technology InformationResource (ESTIR)

(http://electrochem.cwru.edu/estir/)

OF

BOARD OF EDITORS

J. W. STOUT. Editor

THE JOURNAL

Ter... •nding 31 Dec....ber 1967

MILTON BURTON

LEWIS FRIEDMAN

O. J. KLEPPA

KLAUS RUEDENBERG

J. S. WAUGH

ROBERT ZWANZIG

T.,....nding 31 D.c....ber 1966

A. D. BUCKINGHAM

H. L. FRISCH

JOHN L. MAGEE

R. A. MARCUS

E. A. MASON

W. E. WALLACE

Associate Editors

VOLUME 43

1 JULY-1S DECEMBER 1965

Published by the

AMERICAN INSTITUTE OF PHYSICS

Ter'" ending 31 Dec....ber 1965

L. S. BARTELL

RICHARD B. BERNSTEIN

H. P. BROIDA

]dARTIN ECARPLUS

J. A. POPLE

G. WILSE ROBINSON

CHEMICAL PHYSICS

Downloaded 18 Oct 2009 to 152.2.176.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

rHE JOURNAL OF CHEMICAL PHYSICS VOLUME 43, NUMBER 2 15 JULY 1965

On the Theory of Electron-Transfer Reactions. VI. Unified Treatment for Homogeneousand Electrode Reactions*

R. A. MARcust

Department of Chemistry, Brookhaven National Laboratory, Upton, New York, and

Noyes Chemical Laboratory, University of Illinois, Urbana, Illinois

(Received 8 March 1965)

A unified theory of homogeneous and electrochemical electron-transfer rates is developed using statisticalmechanics. The treatment is a generalization of earlier papers of this series and is concerned with seeking afairly broad basis for the quantitative correlations among chemical and electrochemical rate constantspredicted in these earlier papers. The atomic motions inside the inner coordination shell of each reactant aretreated as vibrations. The motions outside are treated by the "particle description," which emphasizesthe functional dependence of potential energy and free energy on molecular properties and which avoids,thereby, some unnecessary assumptions about the molecular interactions.

1. INTRODUCTION

ATHEORETICAL calculation of the rates ofhomogeneous electron-transfer reactions was de

scribed in Part I of this series1 and the method wassubsequently extended to electrochemical electrontransfer rates.2 The calculation was made for reactionsinvolving no rupture or formation of chemical bondsin the elementary electron-transfer step. In this sensethese electron transfers are quite different from othertypes of reactions in the literature. This property,together with the assumed weak electronic interactionof the reactants, introduced several unusual features:"nonequilibrium dielectric polarization" of the solventmedium,3 possible nonadiabaticity, unusual reactioncoordinate, and an approximate calculation of thereaction rate without use of arbitrary adjustableparameters.

Applications of the theoretical equations were madein several subsequent papers.2,4 The mechanism ofelectron transfer was later examined in more detailin Part IV using potential-energy surfaces and statistical mechanics.5 (In Part I the solvent medium outsidethe inner coordination shell of each reactant had beentreated as a dielectric continuum. The free energy ofreorganization of the medium, accompanying the formation of an activated complex having nonequilibrium

* This research was performed in part under the auspices of theU. S. Atomic Energy Commission while the author was a visitingSenior Scientist at Brookhaven National Laboratory. It was alsosupported by a fellowship from the Alfred P. Sloan Foundationand by a grant from the National Science Foundation. A portionof the work was performed while the author was a member of thefaculty of the Polytechnic Institute of Brooklyn, and was presented in part at the 146th Meeting of the American ChemicalSociety held in Denver in January 1964.

t Present address: Noyes Chemical Laboratory.1 R. A. Marcus, J. Chern. Phys. 24,966 (1956).2 R. A. Marcus, ONR Tech. Rept. No. 12, Project NR 051-331

(1957); cf Can. J. Chern. 37, 155 (1959) and Trans. Symp. Electrode Processes, Phila., Pa., 1959, 239-245 (1961).

3 R. A. Marcus, J. Chern. Phys. 24, 979 (1956).4 R. A. Marcus, J. Chern. Phys. 26, 867, 872 (1957); Trans.

N. Y. Acad. Sci. 19, 423 (1957).& R. A. Marcus, Discussions Faraday Soc. 29, 21 (1960).

dielectric polarization, was computed by a continuummethod.) In Part IV, changes in bond lengths in theinner coordination shell of each reactant were alsoincluded, and the statistical-mechanical term for thefree energy change in the medium outside was replacedonly in the final step by its dielectric continuumequivalent.

A number of predicted quantitative correlationsamong the data were made on the basis of Part IV.They have received some measure of experimentalsupport, described in Part V and in a recent reviewarticle.6 ,7 A more general basis for these correlations isdescribed in the present paper, which also presents aunified treatment of chemical and electrochemicaltransfers.

The form of the final equations for the rate constantsis comparatively simple, a circumstance which leadsalmost at once to the above correlations. (It permitsextensive cancellation in computed ratios of rateconstants.) This simplicity has resulted from severalfactors: (1) Some of the more complex aspects of therate problem are rephrased so that they affect only apre-exponential factor (p) appearing in the rate constant, a factor that appears to be close to unity.(2) Little error is found to be introduced when theforce constants of reactants and products are replacedby symmetrical reduced force constants. (3) Animportant term (X) in the free energy of activationis essentially an additive function of the properties ofthe two redox systems in the reaction.

The electron transfer rate constants can vary bymany orders of magnitude: For example, knownhomogeneous electron-exchange rate constants varyby factors of more than 1015 from system to system,and electrochemical rate constants derived from electrochemical exchange currents vary by about 108 atany given temperature.6 (An electron-exchange reaction is one between ions differing in their valence

6 R. A. Marcus, J. Phys. Chern. 67,853,2889 (1963).7 R. A. Marcus, Ann. Rev. Phys. Chern. 15, 155 (1964).

679


680 R. A. MARCUS

state but otherwise similar.) Thus, small factors of2 or 3 are of relatively minor importance in any theorywhich is intended to cover this wide range of values.Some approximations in this paper are made with thisviewpoint in mind.

In the present paper classical statistical mechanicsis employed for those coordinates which vary appreciably during the course of the reaction. This classicalapproximation is a reasonable one for orientational andtranslational coordinates at the usual reaction temperatures and, in virtue of the above remark, for the usuallow-frequency vibrations in inner coordination shells.Because of cancellations which occur in computationsof ratios of rate constants this approximation could beweakened for deriving the predicted correlations, evenwhen the quantum corrections would not be small.

In calculations of absolute values of the electrontransfer rate constants a classical approximation willintroduce some error when the necessary changes inbond lengths to effect electron transfer are so small asto be comparable with zero-point fluctuations. However, in this latter case, the vibrational contribution tothe free energy of activation is itself small and doesnot account for any large differences in reaction ratesin redox reactions which have been investigated experimentally. Hence, for our present purpose and, in theinterests of simplicity, this particular possible quantumeffect may be ignored.

2. ORGANIZATION OF THE PAPER

The paper is organized in the following way:

Individual and over-all rate constants are distinguished in Sec. 3, potential-energy surfaces for weakoverlap electron transfers are discussed in Sec. 4, andformal expressions for the rate constants are given inSec. 5. The latter expressions arise from a generalizationof activated complex theory.s The approximate relationof certain surface integrals appearing in Sec. 5 to morereadily evaluated volume integrals is described inSec. 6, where certain complicating features are rephrased so as to cast some of the difficulties into anevaluation of one of the pre-exponential factors p.In Sec. 6 a linear dependence of an effective potentialenergy function (governing the configurational distributions in the activated complex) on the potentialenergies of reactants and products is established[Eq. (13)]. The rate constants are expressed in Sec. 7in terms of the contribution of the coordinates of thesolvent molecules in the medium and of the vibrationsin the inner coordination shell of each reactant to thefree energy of formation of the activated complex.

To deduce from Eq. (13) a simple dependence ofthe free energy of activation on differences in molecular

8 R. A. Marcus, J. Chern. Phys. 41, 2624 (1964). The U in thepresent Eqs. (1) to (3) was denoted there by ut.

parameters, the contributions of the above two sets ofcoordinates are treated differently (Sec. 8), since oneset already has a desired property while the other doesnot. Changes in bond force constants accompanyingelectron transfer are responsible for this difference inbehavior. However, it is shown later in Appendix IVthat the introduction of certain "reduced force constants" circumvents the difficulty, with negligible errorin typical cases. The contributions of the two sets ofcoordinates are computed in Sees. 9 and 10. Themedium outside the inner coordination shell of eachreactant is treated by a "particle description."9,lo Thelatter is a considerable generalization over the customary permanent-dipole-induced-dipole treatment of polarmedia and serves to emphasize the functional dependence of the free energy of activation on variousproperties and to facilitate thereby the analysis leadingto the predicted correlations.

The standard free energy of reaction and the cellpotentials are introduced in Sees. 11 and 12, and areused in Sec. 13 to evaluate a quantity (m) closelyrelated to the electrochemical and chemical transfercoefficients. The final rate equations are summarizedin Sec. 14.

The additive property of X, mentioned in the previoussection, is discussed in Sec. 15 and further establishedin Sec. 16. The significance of the characteristic scalarquantity (m) appearing in the potential-energy function of the activated complex is deduced in Sec. 17.Deductions from the final equations are made in Sec.18.

In Sec. 19 the present paper is compared with earlierpapers of this series, and the specific generalizationsmade here are described. Detailed proofs are given invarious appendices. In Appendix VIII it is establishedthat under certain conditions the correlations derivedabove should apply not only for rate constants ofelementary steps but also for the over-all rate constantof a reaction occurring via number of complexes of thereactants with other ions in the electrolyte.

3. INDIVIDUAL AND OVER-ALL RATECONSTANTS

Many chemical and electrochemical redox reagentsare ions which possess inner coordination shells andwhich may form complexes with ions of opposite sign.Any such complex is "inner" or "outer" according asthe latter ions do or do not enter the inner coordinationor shell of the reactant. To a greater or lesser extent,all such complexes normally contribute to the measuredrate of the redox process. For this reason both a rate

9 R. A. Marcus, J. Chern. Phys. 38, 1335 (1963).10 R. A. Marcus, J. Chern. Phys. 39, 1734 (1963). The notation

differs somewhat from the rresent paper: 1', U, Uf, and p,o therebecome Vo', Uo, Ui , and Po here. A typographical error occurs inEq. (13): The fs should be deleted. No equations deduced from(13) need correction.


ELECTRON-TRANSFER REACTIONS. VI 681

constant for the over-all reaction, involving all complexes, and a rate constant for each individual step,involving a specific complex with a given inner coordination shell or involving a specific pair of complexes ina bimolecular step, have been defined in the literature.They equal the over-all reaction rate divided by thestoichiometric concentration (or product of such concentrations in the bimolecular case), in the case of anover-all rate constant, and the reaction rate dividedby the concentration of the particular complex (orproduct of such concentrations in the bimolecular case),in the case of an individual rate constant. Often theindividual rate constants are measured experimentally.Frequently, however, only the over-all rate constantis determined in the experiment.

The derivation up to and including Sec. 6 applies toover-all as well as to individual rate constants. TheSecs. 7 to 17 apply only to the individual rate constants.To calculate the over-all rate constant from the expression derived for the individual one in these lattersections, one must take cognizance of any reactionsleading to the formation and destruction of the complexes and must average over the behavior of allcomplexes, as in Appendix VIII.

4. POTENTIAL-ENERGY SURFACES

The potential energy of the system is a function ofthe translational, rotational, and vibrational coordinates of the reacting species and of the molecules inthe surrounding medium. A profile of the potentialenergy surface is given in Fig. 1 in the case of homogeneous reactions. (The related electrochemical plot isconsidered later.) The abscissa, a line drawn in theabove many-dimensional coordinate space, representsany concerted motion of the above types leading fromany spatial configuration (of all atoms) that is suitedto the electronic structure of the reactants to onesuited to that of the products. Surface R denotes thepotential-energy profile when the reacting species havethe electronic structure of the reactants, and SurfaceP corresponds to their having the electronic structureof the products. If the distance between the reactingspecies is sufficiently small there is the usual splittingof the two surfaces in the vicinity of this intersectionof Rand P. If the electronic interaction causing thesplitting is sufficient, the system will always remainon the lowest surface as it moves from left to right inFig. 1. Thus, the system has moved from surface R tosurface P adiabatically, in the usual sense that thecorresponding motion of the atoms in the system istreated by a quantum-mechanical adiabatic method.On the other hand, if the electronic interaction causingthe splitting is very weak, a system initially on Curve Rwill tend to stay on R as it passes to the right acrossthe intersection. The probability that as a result of

R

NUCLEAR CONFIGURATION

FIG. 1. Profile of potential-energy surface of reactants (R)and that of products (P) plotted versus configuration of all theatoms in the system. The dotted lines refer to a system havingzero electronic interaction of the reacting species. The adiabaticsurface is indicated by a solid line.

this nuclear motion the system ends up on CurveP is then calculated by treating this motion nonadiabatically.ll

It should be noted that the system can undergo thiselectron transfer either by surmounting the barrier ifit has enough energy or by tunneling of the atoms ofthe system through it if it has not. We confine ourattention to the case where the systems surmount thebarrier. Some atom tunneling calculations have beenmade, however,12

Since the abscissa in Fig. 1 is some combination oftranslational, rotational, and vibrational coordinates,this "reaction coordinate" is rather complex: The surfaces Rand P intersect, and the set of configurationsdescribing this intersection form a hypersurface inconfiguration space. The exact motion normal to thishypersurface depends on the part being crossed. Insome parts it involves changes in bond distances inthe inner coordination shells of the reactants, in otherparts it involves a change of separation distance of the

11 See, for example, L. Landau, Physik. Z. Sowjetunion 1, 88(1932); 2, 46 (1932) j C. Zener, Proc. Roy. Soc. (London) A137,696 (1932); A140, 660 (1933); C. A. Coulson and K. Zalewski,ibid. A268, 437 (1962). The present situation has been summarizedin Ref. 7, where the definition of nonadiabaticity was also discussed. Reference should also have been made there to the workof. E. C. G. Stueckelberg, Helv. Phys. Acta. 5, 369 (1932); d.,H. S. W. Massey, in Encyclopedia of Physics, edited by S. FHigge(Springer-Verlag, Berlin, 1956), Vol. 36, p. 297.

12 N. Sutin and M. Wolfsberg, quoted by N. Sutin, Ann. Rev.Nucl. Sci. 12, 285 (1962). These authors discussed the possibility of tunneling of the atoms in the inner coordination shell.Possible quantum effects which include atom tunneling in themedium outside this shell have been treated by V. G. Levich,and R. R. Dogonadze, Proc. Acad. Sci. USSR, Phys. Chern.Sec. [English transl. 133, 591 (1960)]; Collection Czechoslov.Chern. Comm. 26, 193 (1961) [transl., O. Boshko. Universityof Ottawa, Ontario.] Any conclusions concerning the contributionof atom tunneling depend in a sensitive way on the assumedvalues for the bond force constants and lengths in the inner coordination shell, properties on which data are now becoming available, and on the assumed value for a mean polarization frequencyfor the medium. [Atom tunneling is different from electrontunneling.>, the latter being a measure of the splitting in Fig. 1(Ref. 7).J


682 R. A. MARCUS

13 In these equations S is an abbreviation for Sint (made forbrevity of notation), since several integrations over "externalcoordinates" have been performed and there remains only theintegration over a hypersurface in internal coordinate space.8

Similarly, the symbols S', V, and V' discussed later should beara subscript int, which is omitted here for brevity.

tional coordinates of the entire system are such thatthe system is in the vicinity of the many-dimensionalintersection hypersurface in configuration space.

It is assumed below that the distribution of systemsin the vicinity of the intersection region of Figs. 1 or 2is an equilibrium one. The usual equilibrium-typederivation of the rate of a homogeneous or heterogeneous reaction in the literature employs a special formfor the kinetic energy, a form consistent with the setof configurations of the activated complex being describable by a hyperplane in configuration space. Amore general curvilinear formulation has been givenrecently.8 Upon integrating over a number of coordinates which leave the potential energy invariant oneobtains (1), (2), and (3) for homogeneous bimolecularreactions, homogeneous unimolecular reactions, andheterogeneous reactions, respectively8,13:

In these equations mt is the effective mass for motionnormal to the hypersurface S, R is the distance betweenthe two reactants (normally between their centers ofmass), Q is the configuration integral for the reactants,and dS is the area element in a many-dimensionalinternal coordinate space.l3 Both mt and R may varyover S. In (1) to (3) integration has already beenperformed over several coordinates, as follows: (i) in Q,the center of mass of each reactant; (ii) in the numerator of (1), the center of mass of one reactant and theorientation of the line of centers of the two reactants;(iii) in the numerator of (2), the center of mass of thereactant, and (iv) in the numerator of (3), the twocoordinates of this center parallel to the solution-solidinterface. Thus, these coordinates are to be held fixedin the internal coordinate space in (1) to (3).

In adapting these equations to electron-transferreactions one should consider the possibility of thereaction occurring nonadiabatically and, in the case ofelectrodes, should consider the existence of many levelswhich may accept or donate an electron to a reactantin solution. In the framework of a classical treatmentof the motion of the nuclei in (1) to (3), a factor K

(2)

(3)

(1)k (kT)'l exp( - U/kT) R2(mt)-!dS

bi= 8'1l" ' ,s Q

-(kT)!l exp( - U/kT) (mt)-!dSkhet

- 2'1l" sQ'

k .=(kT)!l exp( - U/kT) (mt)-!dSlim 2'1l" S Q '

5. EXPRESSION FOR THE RATE CONSTANT

reactants, and in still others it involves reorientationof polar molecules in the medium.

Analogous remarks apply to electrode reactionsexcept that the intersection region is more complexbecause of the presence of many electronic energy levelsin the metal. A blown-up portion of this region is indicated in Fig. 2. The diagram consists of many potentialenergy surfaces, each for a many-electron state of theentire macrosystem. All the surfaces are parallel sincethey differ only in the distribution of electrons among"single-electron quantum states" in the metal. (Onlyone distribution of the electrons among these singleelectron quantum states correspond to each surface inFig. 2 if the energy level of the entire macrosystem isnondegenerate. It corresponds to several distributionsin the case of degeneracy.) There is a probabilitydistribution of finding the macrosystem in any manyelectron energy level indicated in Fig. 2. As a consequence of a Fermi-Dirac distribution of the electronsin the metal, most electrons which are transferred to orfrom the many-electron energy levels in the metal willbehave as though they go into or from a level whichis within k T of some mean energy level, and hencepractically equal to it. Thus, except for the calculationof the transition probability associated with the transition from Surface R to Surface P in the intersectionregion, the situation is in effect very similar to that inFig. 1. We return to this point in the following section.

In the present paper we confine our attention inelectrode reactions, as in homogeneous reactions, toreaction paths involving a surmounting of the barrier.

NUCLEAR CONFIGURATION

FIG. 2. Same plot as Fig. 1 but for an electrode reaction. Thefinite spacing between the many-electron levels of a finite electrode is enormously magnified, and only three of them are indicated. The splitting differs from level to level.

We consider any particular pair of reactants (or areactant, in the case of intramolecular electron transfer). These "labeled" reactants may be any two givenmolecules in solution or one molecule and the electrode,and each may form complexes to various extents withother ions and molecules. In effect, we need to calculatethe probability that the vibrational-rotational-transla-



where €(0) is the value of €at cfi = 0 and p. is the chemicalpotential.16

The probability that electron transfer from theelectrode to the ion or molecule in solution will occurfrom a "one-electron model" level of energy € wouldb.e expected to depend on €by a factor roughly proportiOnal to

the third factor arising in the region where the "electrochemical transfer coefficient" is 0.5, a common value.6

Since n(€) is a weak function of € the last two factorsin (6) largely determine the most probable value of €.The maximum of (6) is then easily shown to occur at€= ,ue. Similarly, contribution to electron transfer froman ion in solution to a particular level € would beexpected to vary with € as in

n(€) [1-f(€)] exp( - €/2kT) , (7)

which also has a maximum at €= ,ue, of course.Because of this circumstance (that most contribu

tions arise from levels € near jIe), we approximate thesituation in Fig. 2 by replacing the set of Surfaces R byone surface and P by another surface, correspondingto an electronic energy in the electrode given by jIe asabove. IS If electron transfer accompanies each passage~hrough the intersection region in Fig. 2 the reactionIS referred. to as "adiabatic," purely by analogy withthe term m the homogeneous reaction. The reactionrate is given by (3), where the equation of S dependson the electrostatic potential. On the other hand whenthe transfer probability per passage is very ~eak aterm K. shoul? be introduced in the integral, K beinga velOCIty-weIghted transition probability appropriatelyst.Immed over all energy levels in the electrode (AppendiX I). A value for K in this weak interaction limithas ~een discussed elsewhere.7 When a complete calcula~iOn for the transfer probability from and to acontmuum of electrode levels becomes available it canbe used to estimate K. Normally, however, we assumethe electrode reaction to be "adiabatic" and so takeK"'l on the average.

ca~ be shown to appear in the integrand (Appendix I) ;K is a momentum-weighted average of the transitionprobability from the R to the P surface per passagethrough the intersection region. (It is momentumweighted since the transition probability depends onthe momentum.) K can vary over S. Normally, we takeK as approximately equal to unity when the reactantsare near each other, introducing thereby the assumption that the reaction is adiabatic.

In the case of (3) the situation is somewhat morecomplex because of the presence of the many electrodelevels. At present there is, in the literature, no theoretical calculation of the transfer probability from a levelR to a continuum (essentially) of levels P, per passagethrough the intersection range, for the entire range oftransfer probabilities from 0 to 1. Such a calculationwould take into account the fact that in an unsuccessfulpassage through the intersection region the system canalso revert to other R levels different from the originalone. At present only the limiting case of very smalltransfer probability has been considered in the literature.l4 In this case transfers to and from each of thelevels have been treated independently using perturbation theory; they do not interfere at this limit.

When the transfer probability in electrode reactionsis fairly large when ion and electrode are close adifferent approach must be employed. IS Here, we t~keadvantage of the fact that for a metal electrode mostof the electron transfers occur to and from levels nearthe Fermi leveps: In the terminology of a one-electronmodel, most of the levels several kT below the Fermilevel are fully occupied and cannot accept more electrons. The Boltzmann factor discourages transfer tothe rather unoccupied levels several k T above theFermi level. Conversely, transfers from the occupiedlevels below the Fermi level are discouraged by a higherover-all energy barrier to reaction while transfer froma higher level is discouraged by the fact that most ofthe higher levels are unoccupied. To illustrate this pointmore precisely, let n(€) be the density of the "oneelectron model levels" for the electrode and f( €) theFermi-Dirac distribution,

n(€)f(€) exp(€/2kT), (6)

where € is the energy of one of these levels and where,ue is the electrochemical potential of electrons in themetal. Both € and jIe depend on the electrostatic potential of the metal cfi:

14 R. R. Dogonadze and Y. A. Chizmadzhev, Proc. Acad. Sci.USSR, Phys. Chern. Sec., English Trans!. 144, 463 (1962) 145,563 (1962); V. G. Levich and R. R. Dogonadze, Intern. C~mm.Electrochem. Thermodyn. Kinet., 14th Meeting, Moscow (1963)preprints. This work is reviewed in Ref. 7. '

16 This approximation was used but not discussed in Ref. 2.

f(€) = lexp[(€-,ue)/kT]+1}-l,

,u= p.-ecfi,

(4)

(5)

6. RELATION OF THE SURFACE INTEGRALS(1) TO (3) TO VOLUME INTEGRALS

Although some deductions can be made from thesurf~ce integ:als in (1) to (3) when the equation ofthe mtersectiOn surface S is simple, we find it convenient to express the surface integral in terms ofvolume one. The same aim was followed in Part IVbut. in a less precise way. The principal equationdenved in this section is (26), which is later used inconjunction with Eqs. (1) to (3) to obtain an expression for kr • te•

16 For example, C. Herring, and M. H. Nichols Rev ModPhys.21, 185 (1949). ' . .


684 R. A. MARCUS

Let Ur be the potential-energy function for thereactant and UP be that for the products. As mentionedearlier the intersection of the Rand P surfaces in Fig.1 (and 2) forms a hypersurface in configuration space.This hypersurface is called the "reaction hypersurface."Its equation is given by (8). It is a hypersurface inthe entire coordinate space and also in the internalcoordinate space since (8) is independent of theexternal coordinates8

To establish (18), we first note from Appendix IIthat the distribution in volume which is centered on S'(but not confined to S', of course) isf*, given by (12)20

1*= exp( - ~;) / f exp( - ~;)dV" (12)

where(13)

Ur- UP=O (for points on reaction hypersurface). (8)

where U* is a function to be determined; RotdV' is anelement of volume of this internal coordinate spaceat fixed R.8

This surface is a member of a family of hypersurfacesin configuration space, represented by (9), where c is aconstant:

In deriving (16) we have also used the fact that U*equals Ur on S'.

Finally, the integral in (11) can be rewritten asf exp[-1(qN, R)]dqN, in virtue of (14) and (15). Onthe basis of a Gaussian expansion Eq. (17) can bederived (post).

where 1"(0, R) is d21(qN, R)/dqN2, evaluated on S'(and hence at qN=O). One then obtains, from (10),

20 If, for any R, a distribution functionj* is stated to be centeredon S', we mean that it is centered on the set of configurationswhich lie at the intersection of the hypersurface S and of thehypersurface R=constant. Occasionally, in some part of theinternal coordinate space the two hypersurfaces may be "cotangential," but this circumstance does not alter the argument. Atthese parts of space the value of Ur equals UP and (12) becomes"exact" for computing relative probabilities of various configurations, rather than approximate.

21 {(gNN)l) in (16) is defined as

f (gNN)l(aNN)-t exp( ~i)dS'/ f (aNN)-l exp( ~i)dS"

where aNN is conjugate to an element aNN in the lineelement of the many-dimensional configuration space.On recalling from Ref. 8 that mt equals aNN/gNN,where gNN is conjugate to an element gNN in the lineelement of the corresponding mass-weighted configuration space, the S' integral in (10) can then be rewrittenas in (16), where «gNN) i) is a suitable average over S'21

and m is a parameter which varies with the coordinateR and which is determined in Sec. 13. On S', one seesfrom (8), U* equals Ur for any given R.

We then recall from Ref. 8 that dV' and dS' arerelated by (14), and we introduce a quantity l(qN, R)defined by (15) :

dV'= (aNN)-!dS'dqN, (14)

exp[-1(qN, R)]=f exp( - ~;)(aNN)-idS" (15)

f exp( - k~)(mt)-!dS'= «gNN) i) exp[-1(0, R)].

(16)

f exp[-1(qN, R)]dqN

=[211"1"(0, R)]i exp[-1(0, R)], (17)

(9)

(11)i, exp(- ~;)dV"

17 This factoring of dS (or as it was called there dS int) wasdescribed in Ref. 8.

18 The K appearing in (10) is now a symbol representing

r(mt)-t exp( ~f)dS' / f (mt)-l exp( ~f)dS"where K is the original kappa.

19 These "internal coordinates" were defined8 as those coordinates for which integration was not performed in obtaining(1) to (3).

where a is 2, 0, or °according to whether (1), (2), or(3) is the equation involved.

We wish to relate the above integral over S' to avolume integral (11) over the internal coordinatespace at fixed R, as in (18) and finally as in (26)19:

The surface (8) can be obtained from the surface (9)by lowering the P surface in Figs. 1 or 2 by an amount c.

We employ a coordinate system q1 to qn used in thederivation of (1) to (3) and recall that one coordinate,qN, in the internal coordinate space was chosen to be acoordinate constant on the hypersurface (8). Let qNbe zero there. In fact, each member of the family ofhypersurfaces (9) is made a coordinate hypersurfacefor qN.

We consider any of the integrals in (1) to (3),include the factor K in the integrand, and write dS asdS'dR.I7 The factor Kdepends primarily on R. In thefollowing expression the same symbol K is used todenote this K, averaged over S',18 Each of the integralsin (1) to (3) can be rewritten as

LKR{l, exp( - k~) (mt)-idS'}R, (10)



(18)

(16), and (17),

LKR{L exp( -k~)(mt)-idS'}R

=/. KRa«gNN)i) exp[-F*(R)/kTJdR

R [21TI"(0, R)Ji '

where F*(R) is the configurational free energy of asystem having the potential-energy function U* forthis separation distance R

for small s's:

«(OS)2)= «(aNN )-1(oqN)2)= «(aNN)-l )«(oqN)2), (25)

where «aNN)-l) is a suitable average of (aNN )-1.23We make use of the fact that «gNN)i)«aNN)-l)i

has units of (mass)-i, and denote it by (m*)-i, andthat the integrand in (18) has a maximum at somevalue of R, denoted in (26) by R. [When R becomeslarge K tends to zero and when R is small the van derWaals' repulsion makes F*(R) large.J On treating theintegrand as a Gaussian function of R, (18) becomes

( F*(R)) f (U*)exp - -----:;;r- = exp - k T dV' .

where «oR) 2) is the mean square deviation in thevalue of R; p and K can be calculated from more specificmodels when the various integrals defining them canbe evaluated.

= KpRa(m*)-i exp[- F*(R) /kTJ, (26)

where K is evaluated at this value of R and where p isa ratio (27) whose value should be of the order ofmagnitude of unity:

To complete the proof of (18) we must verify (17).We recall from the definition of qN that Ur- UP dependsonly on qN on any hypersurface (9). To ensure centeringof the system on 5', i.e., at qN = 0, meR) is to beselected so that (20) is satisfied:

(20)

where ( ) denotes average with respect to the distribution function 1*. On using (12), (14), and (15),Eq. (20) becomes

p= [( (oR)2)/ «( OS)2)Ji, (27)

7. RATE CONSTANT IN TERMS OF !:IF*

23 This average is defined here as

f (alVN

)-1(oqN)2 exp( ~~)dV' / f (oqN)2 exp( ~~*)dV"For the proof that ds2equals (aNN)-1(dqN)2, see Ref. 32 Appen-dix III. '

Let Fr be the configurational free energy associated~ith the Q of Eqs. (1) to (3) as in (28). Thereby, itIS the free-energy contribution for an equilibrium distribution of "V' coordinates" when the reactants arevery far apart but fixed in position,

(28)Fr=kT lnQ.

Let peR) be the corresponding quantity when thereactants are a distance R apart. We then have

wr= Fr(R) - Fr, (29)

where wr can be called the reversible work to bringthe reactants from fixed positions infinitely far apartto the cited separation distance.

We also introduce !:IF*(R) :

!:IF*(R)=F*(R)-Fr(R). (30)

Equations (1) to (3) for krate now yield (31) to (33),when (26) and (28) to (30) are used,

kbi= KpZbi exp( -wr/kT) exp[-!:IF*(R) /kTJ, (31)

kuni= Kp(kT/21Tm*)i exp( -!:IF*/kT) , (32)

khet= KpZhet exp(-wr/kT) exp[-!:lF*(R)/kT], (33)

(24)

22 This average, «(oqlV)2), is defined as f (oqlV) 2 f*dV'. It isreadily shown that (qN) vanishes.

Because of the centering of 1*, expansion of I(qN, R)about qN=°is permissible, as is one of Ur- UP

f exp[-I(qN, R)J(ur-Up)dqN=O. (21)

where «qN)2) is the mean-square deviation of qN.22The mean-square deviation of the perpendicular distance s from the reaction hypersurface is given by (25)

I(qN, R) =1(0, R)+qNI'(O, R)

+[(qN)2/2 IJI"(O, R)+···, (22)

Ur_Up=O+qN(Ur_Up)'+.··, (23)

where ' indicates a derivative with respect to qN,evaluated at qN = 0. We retain only leading terms ineach case. Insertion of (22) and (23) into (21) followedby integration reveals that 1'(0, R) vanishes. Introduction of (22) into the left-hand side of (17) thenestablishes (17).

Some of the terms in (18) can be expressed in termsof quantities of more immediate physical significance.It may be shown from (12), (14), (15), (22) and thevanishing of I' (0, R) that


686 R. A. MARCUS

[In Eq. (32) fiF* is simply F*-Fr, there being onlyone reactant.] Zbi is in fact the collision number oftwo uncharged species in solution when they have unitconcentration, when their reduced mass is m*, and whentheir collision diameter is R. Zhet is the collision numberof an uncharged species with unit area of an interface(here, the electrode), when it has unit concentrationand when its mass is m*.

F* and Fr in (31) to (33) involve an integrationover the orientation of each reactant. The integrandin Fr is independent of these coordinates and, in thecase of the "outer-sphere electron-transfer mechanism"discussed here, the integrand in F* is assumed to beindependent of them also. (For purposes of derivingmany of the correlations in Sec. 16, this assumptioncould be weakened because of cancellations.) Integration over these coordinates is regarded as having beenperformed in (31) to (34), since the orientationalfactors now cancel in fiF*(R). Thus, in the subsequentcalculation of F* and pr each reactant may be regardedas fixed not only in position, as before, but in orientation also.

those of the inner coordination shells. Thus, V; dependsonly on the V'; coordinates; V o depends primarily onthe V'o coordinates, but also depends on the V'; ones,

(37)

(38)dVI=dVI;dV'O'

The quantities V;* and Vo* are defined in terms ofVi', etc., to be given by (13), with i and 0 subscripts,respectively. Then, V* is the sum of V;* and Vo*.

The volume element dV' is written as

where dV' i is defined as the product of the differentials(llidqi) of the V'; coordinates. Thereby, dV'o contains the Jacobian appearing in dV'. It may vary,therefore, with the V'; coordinates.

In calculating F* and Fr we may evaluate theintegrals appearing in them by first integrating overthe V'o coordinates and then over the V'; ones. Thisprocedure is convenient since the V'i ones performsmall oscillations while the others can undergo considerable fluctuations. With this procedure in mind,we define new quantities f;* and fo*, the former depending only on the V'i coordinates, the latter depending on the V'o coordinates and parametrically on theV'; ones:

(34)

where Zbi and Zhet are given by

8. DISTRIBUTION FUNCTION AND THEFREE ENERGY

The main purpose of this section is the derivationof Eqs. (47) to (49).

Equation (19) for F*(R) can be rewritten as in (35),with the aid of (12), (13), and (20):

Since - k (lnf*) is the configurational entropy of asystem having the distribution function f* and since(Vr) is the mean potential energy of a nonequilibriumsystem having a potential-energy function Vr but adistribution function of f* inappropriate to this Vr, wesee that F*(R) is also equal to the configurational freeenergy of this nonequilibrium system.

In obtaining an expression for F*(R) it is convenientto divide, as one usually does in related problems, theinternal coordinates at the given R into two groups:V'; coordinates describing the positions of the atoms inthe inner coordination shells of the reactants, andV'o coordinates describing the positions of the atoms ofthe medium relative to each other and to those in theinner coordination shells. It is also convenient to writeVas the sum of two terms, Vi and V o, one describingthe intramolecular interactions of the atoms in eachcoordination shell, the other describing the interactionsof the atoms of the medium with each other and with

(43)f*=fo*f;*·

One then obtains

fo*= exp[(xo*- Vo*)/kT], (39)

f;*= exp( - ~;) / f exp( - ~;)dVli' (40)

where

0;*= V;*+xo*, (41)

exp( -~~)=f exp(- ~~)dVlo. (42)

Quantities fo', fi', ai', and xo' can be defined, byreplacing the * by an r superscript in (39) to (42).However, xo' is simply Fo', the V'o contribution to theconfigurational free energy of the reactants for thegiven value of the V'i coordinates

We also introduce Fo*, defined by

Fo*= (Vo')'o+kT(lnfo*)'o, (45)

where the average ( )'0 is computed with respect to fo*.Fo* is the V'o contribution to the free energy of thenonequilibrium system having the potential energyfunction Vr and the distribution function fo *. Thefirst and second terms in (45) are the energy andentropy contributions, respectively.

To obtain an expression for 0;*, the function largelycontrolling the V'i coordinate distribution, we first

(35)

(lnf*)=f (lnf*)f*dV' . (36)

F*(R) = (Vr)+kT(lnf*),

(Ur)=fV1*dV' ,

where



obtain (46) by introduction of (39) into (45) and byuse of (13) with subscript o's added. Equations (41),(46), (37) and, with subscript i's added, (13) thenyield (47), since Ut and UiP are independent of theV'0 coordinates:

On multiplying numerator and denominator of (40)by exp[Ui*(Q.) /kTJ, introducing this expression forfi* into (48b) , then using (50), integrating,24 andfinally introducing an expression for Ui(Q*) [Eq. (47)evaluated at Q= Q*J, Eq. (51) follows:

Xo*= Fo*+m(Uo'- Uop).o,

Ui*= Ut+Fo*+m(ur- Up).o.

(46) F*(R)=Ut(Q.)+Fo*(Q.)-m(Ur(Q.)-Up(Q.) ).0

(47) -!kT In[(27l"kT) ni/I J;k* IJ, (51)

(48a)

(48b)

Equations (35), (37), (43), and (45) yield (48a) ,when one notes that Ut and fi* are independent ofthe value of the V'0 coordinates. Equation (48b) thenfollows from (20), (47), and (48a):

F*(R) = (Ut+Fo*).i+kT(lnf;*).i,

= (Ui*+kTlnfi*).i,

where the average( ).i is computed with respect tothe distribution function f;*.

The free energy pr(R), given by

-kT Inf exp(-k~)dVI

evaluated at R, can also be shown to be given byexpressions similar to (48) but with the asterisksreplaced by r's

To evaluate krate, we compute!:J.F* from (30), (48),and (49), and use (47).

The similarity of (48) and (49) and later of (51)and (52) is an example of the fact that properties of the[rJ system can be obtained from those of the [*Jsystem by setting m= o. The origin of this behavior isseen in the original Eq. (13) defining the [*J system.

9. VIBRATIONAL CONTRmUTION TO !:J.F*(R)

While it is not necessary to introduce the harmonicapproximation, the expressions are appreciably simplified by it. There is evidence that the approximation isadequate for many reactions of interest.

It is recalled that the generalized coordinates weredenoted by qi. Let the first ni of these be vibrationalcoordinates of the reacting species, i.e., the V/ coordinates, and let q.i denote the value of the jth vibrational qi occurring at the minimum of Ui*. We have

U.*= U,*(Q.) +! fJik*(qi_ q .f) (qk_ q .k)i,k=l

where Q-Q. denotes a column vector whose elementsare qi_q.i. F* denotes a square matrix whose elementsare /i/. The superscript T denotes a transpose (a rowvector here), and the dot indicates the scalar productof this row vector with the column vector F*(Q-Q.).

where IJ;k* I is the determinant of the J;k*'s.If qri denotes the value of a vibrational qi occurring

at the minimum of Ut, it can be shown that pr isgiven by (52) after a quadratic expansion of Ut(Q)about Qr,

pr(R) = Ut(Qr) +Fo'(Qr) -!kT In[(27l"kT)n i/1 J;kr IJ,

(52)where

J;kr= (a2Ut/aqiaqk) at Q=Qr. (53)

Equation (54) is then obtained from (51) and (52)by noting that (Ur(Q.)-UP(Q.) ).0 vanishes (Appendix V), that Ut equals Ut-For at any Q, and thatUt(Q.) can be expanded about the value of Ut at Qr:

F*(R) -pr(R) =!(Q.T_QrT). Fr(Q.-Qr)

+!:J.Fo*(Q.) +!kT In(j J;k* I/IJ;kr\), (54)

where

!:J.Fo*(Q)=Fo*(Q)-Fo'(Q) (atanygivenR). (55)

It is shown later that at any given Rand Q !:J.F0 *(Q)equals m2x,,(Q), where Ao(Q) is given by (69), andthat !:J.Fo*p(Q) , which is Fo*p(Q)-For(Q) , equals(m+1)2x,,(Q). We then obtain (56) from (47)26

Ui*= Ut+m(Ut- UiP) -m(m+ 1)x,,(Q). (56)

Since U;* is a minimum at Q= Q., the first variationin U i* vanishes for any arbitrary infinitesimal oQ. InAppendix VI it is found that Ao may be neglected inobtaining

Since the oq' are selected to be independent, thecoefficient of OQT vanishes. Hence,

Q.= [(m+ 1) Fr - mFpJ-l[ (m+ 1) FrQr-mFpQpJ,

(58)and the first term in (54) becomes

HQ.T_Ql) ·Fr(Q.-Qr) =!m2!:J.QT.F!:J.Q, (59)

24 We use Eq. (2) in R. Bellman, Introduction to Matrix Analysis (McGraw-Hill Book Company, Inc., New York, 1960),p. 96, to obtain the last term of (51).

25 On recalling the definition of Uir and flip, and adding andsubtracting mkT(lnfo).o it follows that Ui· in (47) can bewritten as (m+1)U;,-mUip plus /1Fo·+m(/1Fo·-/1Fo·p).Equation (56) then follows.


688 R. A. MARCUS

in powers of the permanent charge density Pa° of thereactants. The usual approximations in the literaturecorrespond to neglect of powers higher than the second,together with the assumption of specific forms for theseterms.9

In terms of the symbols U; and Uo introduced earlier,we have

In (37) Ui is the intraparticle term for the reactantsand Uo is the sum of the intraparticle term for themedium and of the interparticle term. U (0), U (1) ,and U(2) depend functionally on zeroth, first, andsecond powers of Pa° and, respectively, on second,first, and zeroth powers of PMo, the permanent chargedensity of the medium.9 U(O) also contains the intraparticle term for the medium and the electron correlation interparticle term. U; and pa° depend only on theintraparticle coordinates, VI;, of the reactants, and PModepends only on those of the medium, Vl

o.9

The distribution function fo* defined in (39) can beshown to be similar to that which occurs when thepermanent charge distribution on a reactant A is Po0*,given by (66) for all A:

In (67) pop and F denote the polar contributions to thefree energies of two hypothetical equilibrium and dielectrically unsaturated systems, each having a pa

0

equal to m (Par0- Pap0) on each reactant. The first

system is an "optical polarization" system,9 i.e., asystem whose medium responds to these Pao's only viaan electronic polarization. The second system respondsvia all polarization terms. Both Fop and F are quadraticfunctions of the m (Par

0- Pap0) 'So

It can be shown26 that Fop- F depends on the square

26 According to Eqs. (10) and (11) of Ref. 10 Fop-F equals[(U(1)2)- (U(1) )2J/2kT. The latter depends only on the secondpower of the charge distribution, since U (1) is a linear functionalof the first power.

where Par0 is the permanent charge distribution of

Molecule A when it is actually a reactant and Pap0 is

that when it is a product. The proof is given in AppendixIII and utilizes the facts that U (1) is a linear functionalof po 0 and that U (2) is insensitive to the usual translational-rotational fluctuations in condensed media, forreasons noted there, unlike the U(O) and U(1).Normally, as will be seen later, m will be close to -i.

The Vlo contribution to the free energy of formationof a system with a nonequilibrium Vlo distribution,tlFo*, at any given R and at any given Q, has beenevaluated elsewhere on the basis of the particle description described above and of an assumption of (at most)partial electric saturation1o :

where

~=~-~ (~

F= Fp[ (m+ 1) Fr-mFpjlFr[ (m+ 1) Fr-mFpjlFp,

(61)

and the equality of Fr, Fp, and [(m+l)Fr-mFpJ-lwith their transposes have been used.

On differentiating (56) twice and noting that ana posteriori calculation shows that the last term in (56)may be ignored in the differentiation we find (62), foruse in the In term in (51)

!ik*=a2U;*/aq jaqk= (m+l)!ikr-m!ikP. (62)

Later it is shown that Eqs. (54) and (59) can besimplified considerably to a good approximation byintroduction of symmetrical and antisymmetrical functions of the force constants and then neglecting termsinvolving the antisymmetrical ones

kjk = 2!ik1jkP/(jjkr+!ikP) , (63)

ljk= (jjkr-!ikP)/(jjkr+!ikP). (64)

The first of these quantities was chosen so as to havedimensions of a force constant and the second of theseso as to be dimensionless.

10. ORIENTATION AND OTHERCONTRIBUTIONS TO tlF*

For purposes of generality we employ the particledescription of the potential energy in a macrosystem.9.10

It introduces fewer assumptions than those normallyused in condensed polar media. Because of its comparative generality it also permits a simultaneous formulation of the theory of homogeneous intermolecularelectron transfers, electron transfers at electrodes, andintramolecular electron transfers. In this descriptionthe system consists of particles each of which is areacting molecule or any electrode present, the latterincluding as part of it any strongly bound layer of ionsor solvent. The remainder of the system, the medium,can then be regarded as one giant particle.

The potential energy is the sum of an intraparticleterm (the energy when the particles are isolated, eachhaving the given intraparticle coordinates) and aninterparticle term (the energy change when the particlesare brought together for the given values of the intraparticle coordinates). The solvent particle possesses a"cavity" for each reactant particle, which the latterfills when they are brought together.

The intraparticle terms below contain the electronicand potential energy of the reactants and of the medium.The interparticle term is, in the first approximation,the sum of interparticle polar terms and of interparticleelectron correlation (i.e., exchange repulsion andLondon dispersion) energies.9It can then be expanded

whereUo= U(O)+U(I)+U(2).

0*_ 0+ (0 0)Po -par m par -Pap ,

tlF0 *= Fopm(r-p) - Fm(r-p).

(37)

(65)

(66)

(67)



of the permanent charge distribution on the reactants, temperature, and pressure. Hence,in this case m(Pa,o-Pap0). We may then describe thedependence of AFo* by Fp- FT= AFo,. (72)

(68)

where

It equals -kT In K, where K is the equilibrium"constant" measured under these conditions. Both AFo,and K can vary with electrolyte concentration, withtemperature, and with pressure.

12. ACTIVATION OVERPOTENTIAL ANDELECTRODE-SOLUTION POTENTIAL

DIFFERENCE

For electrode systems, the counterpart of (72) isobtained by considering the free energy of Reaction(73) for a labeled molecule at any fixed position in thebody of the solution, but far from the electrode, M

and the averaging function is27 :

ex (- (UoT+UoP))av' flex (- (u/+uoP))av'p 2kTOP 2k T o·

(70)

To use Eq. (68) and those derived earlier, an expression is needed for m. It is derived below after somepreliminary analysis involving the standard free energyof reaction, the electrochemical cell potential, and theactivation overpotential.

Red+M= Ox+M(ne), (73)

11. STANDARD FREE ENERGY OF REACTION

where Lp and LT denote summation over productsand reactants. There are one or two terms in each sum,according as the reaction is unimolecular or bimolecular.

The right-hand side of (71) is AFo" the "standard"free energy of reaction for the prevailing medium,

27 If the dielectric unsaturation approximation is used, one canshow10 that (UOT+Uov) /2 would be replaced by U (0) in Eq. (70).Within the range of validity of the partial dielectric saturationapproximation, the average of the fluctuation term (69) wouldbe the same if (UoT+Uov)/2 were replaced by Uo*, by UOT or byU.v. We have simply selected some mean value for the exponent,symmetrical in T and p.

The configurational free energy of the system whenthe reacting species are labeled reactant molecules,fixed in position but far apart, was denoted by FT. Thecorresponding quantity when the pair refers to labeledproduct molecules was denoted by Fp. The momentumand translational contributions of each member of thereacting pair to the free energy of the initial statecancels that in the final state in these reactions involving no change in total number of moles of redoxspecies. Thus, the difference Fp- FT is equal to the freeenergy of reaction when a pair of labeled reactantsform a pair of labeled products in the prevailingmedium.

This free energy of reaction in the prevailing mediumcan be expressed in terms of "standard" chemicalpotentials. The chemical potential J.li can be written asJ.lio'+kT In Gi, where J.lio, is the "standard" chemicalpotential, defined here as the value of J.li at Gi= 1.Because of the labeling, Fp- pr does not contain a contribution from entropy of mixing of the reactants.Since it is these mixing terms which contribute thek T In Gi to J.li, we therefore have

(74)

where A is independent of E, and where we have useda standard convention regarding the sign of E. [Thisconvention is one which makes Reaction (73) increasingly spontaneous with increasing positive Eo', aquantity defined later.]

Because of the labeling the entropy-of-mixing termof the oxidized molecules and that of the reducedmolecules are again absent in FT and Fp. When thesystem is at electrochemical equilibrium and when theprobability of finding the labeled species as a reactantis the same as that for finding it as a product, Fp- FTmust vanish. Also, E then has its equilibrium value,which is Eo' for the case of equal concentrations of thelabeled species. [Eo' is the "standard" oxidation potential or, as it is sometimes called, the formal oxidationpotential of the half-cell; Eo' is defined in terms of the

where Ox and Red denote the oxidized and reducedforms of the labeled molecule in the body of the solution.This free energy change, which accompanies the transferof n electrons from the ion or molecule to the electrodeat a mean energy level discussed in Sec. 5, has a numberof contributions, such as one from the change in electronic energy, one from the change in ion-solventinteractions in the vicinity of the ion, and one fromthe change in vibrational energy. Let FT now denotethe configurational free energy of the system containingthe electrode and a labeled reactant, the latter fixedin a position far outside the electrode double layer.Let Fp denote the corresponding quantity when labeledmolecule is a product, the electrode having gained nelectrodes as in (73).

The term Fp- FT is linear in the metal-solutionpotential difference, as may be seen from the discussionin Sec. 5, and thereby in the half-cell potential E. (Eis defined to be the half-cell potential corrected for anyohmic drop and concentration polarization.) We havethen

(71)Fp-FT= LJ.lpo,- LJ.l/',P


690 R. A. MARCUS

equilibrium half-cell potential E e by (75) for anyratio of concentrations (Red)/(Ox)]

Ee=Eo'+(kT/ne) In[(Red)/(Ox)]. (75)

One then obtains, from (74),

0= !::l.+neEo'. (76)

counterpart of this equation for !::l.F*p(R), the equationfor m is obtained. The final equations for the reactionrate become quite simple when one notes that to anexcellent approximation terms involving the ljk's defined in (64) can be neglected. The proof is given inAppendix IV.

Hence,Fp- pr= ne(E- Eo'). (77) 14. SUMMARY OF FINAL EQUATIONS

On using the results of Appendix IV and referringto Eqs. (31) to (33), it is found that the rate constantfor a bimolecular homogeneous reaction or a unimolecular electrochemical reaction is given by

where Z is given by (34), !::l.F* by (81) and (82), andP by (27).

The rate constant of an intramolecular electrontransfer reaction, on the other hand, is given by Eq.(32), with !::l.F* given by (81) but with the work termswr and wP omitted:

We observe from (77) that E- Eo', rather than theactivation overpotential E- E e, plays the role of the"driving force" in these reactions. The same role isplayed by !::l.Fo1 in the homogeneous reaction.

In terms of formal electrochemical potentials of theproduct and reactant ions and in terms of the electrochemical potential of the electrons in the electrode wehave, incidentally, for Reaction (73),

(78)

13. EQUATION FOR m

krate = KPZ exp( -!::l.F*/kT) , (31), (33)

We first note that !::l.Fo 1 can be written as the algebraic sum of the following terms: The free energychange when the reactants are brought together to theseparation distance R, wr; the free energy of reorganization of the reacting system at this R, !::l.F*; the freeenergy difference of reactants and products in thisreorganized state, which equals

(Up+kT ln1*)- (Ur+kT lnf*»)

Homogeneous:

wr+wp A !::l.Fo1 (!::l.F0/+wp-wr) 2

!::l.F*=-2-+4+-2-+ 4A ' (81)

Electrochemical:

wr+wp A ne(E-E ')!::l.F*=--+-+ 0

2 4 2

(neE- neEo'+ wP- Wr) 2

+ 4A (82)

In (81) and (82) Ais given by

where Ai is given by (84) and Ao is given by (69) atQ= Q•. On introducing the symmetrical force constantsone finds Q.=Qr+m(Qr-Qp). Since Ao depends butweakly on Q. and since m is usually close to -!, itsuffices to evaluate Aoat Q.=!(Qr+Qp) in the typicalcase. This result is used in deriving Eq. (88a),

The reduced force constants kjk are defined in (63)and the !::l.q/s are differences in equilibrium values ofbond coordinates (e.g., independent bond lengths andangles), q/-qp.

It is expected that typically p should be about unity.As noted earlier, Z is essentially the collision number,being about 1011 liter mole-1'sec1 and 104 cm sec1forhomogeneous and electrochemical reactions, respectively.

In Ref. 6 the above equations were written in an

(84)

(83)

because of cancellation of momentum and of translational contributions; minus the free energy of reorganization of the product system at this R to the abovestate, -!::l.F*p; and minus the free energy change whenthe products are brought together to the separationdistance R, -wp • Thus, (79) is obtained when (20) isused,

(Homogeneous)

!::l.F0/=wr+!::l.F*(R) -!::l.F*p(R) -wp. (79)

The electrochemical equation corresponding to (79)is (80), as one may show from (77),

(Electrochemical)

ne(E-Eo') =wr+!::l.F*(R) -wP-!::l.F*p(R). (80)

Here, !::l.F*p is obtained from !::l.F*, and wP from wr byinterchanging rand p superscripts and, at the sametime, interchanging - m and m+ 1. To establish thisresult it suffices to note from (13) that U* and all itsassociated properties are unaffected by such a transformation, but the properties of the reactants becomethose of the products.

Upon introducing Eqs. (54) and (60) for !::l.F*(R),using (68) for !::l.F0 *(Q .), and upon introducing the



equivalent formAF*=wT+m2X, (85)

Homogeneous:- (2m+l)X=AFo'+wp-w2, l

Electrochemical: - (2m+1)X =ne(E-Eo') +wP-wT.

(86)

The value of m defined by (86) can be shown todiffer very slightly from that in the preceding sections,due to the approximation of neglect of the 1jk's, butthe final equations obtained when (86) is introducedinto (85) are identical with (81) and (82).

According to Eqs. (81) and (82) AF* depends onAFo, or on neE according to (87a) and (87b) whenwT and wP are held constant.

case, and the deviation from the 0.5 value arising fromthis source correspondingly smaller.

In summary, the transfer coefficient at constant w'sis expected to be close to !, reflecting a type of symmetry of the Rand P surfaces in the vicinity of thereaction hypersurface (compare also Sec. 17). A sourceof deviation from this symmetry arises from a differencein corresponding force constants in reactants andproducts. It appears as an (1.) term in (87) and hasbeen shown to be small. A second source of deviationarises when the R or P surface is appreciably lowerthan the other, and is reflected in the presence of theAFo, and ne(E- Eo') terms in (87). This source ofdeviation, too, is normally small. The leading term in(87), !, arises from the quadratic nature of both theV'0 and the V'i contributions to AF*.

15. PROPERTIES OF THE REORGANIZATIONTERM A

Furthermore, in the electrochemical case there is onlya contribution from one ion (assuming that any distortion of atomic structure of the electrode yields onlya relatively minor contribution to AFo*). Denoting thevalues of Xo for the electrochemical redox system Aand for the homogeneous redox system A by Ao

o! andXoeJ< respectively, we have

For use in subsequent correlations, we examine anadditivity property of A and the relation between thevalues of Ain related homogeneous and electrochemicalsystems. We consider first the (hypothetical) situationwhen R is very large, so large that the force field fromone reactant does not influence the other. On notingthat Ao is given by (69) and that the fluctuationsaround each reactant are now independent (large R), Ao

can be written as the sum of two independent terms,one per reactant. It then follows that when R is largethe value of Ao for a reaction between reactants fromtwo different redox systems A and B, Ao

ab, is the arithmetic mean of the values Ao

aa and Aobb of the respective

systems:(88a)

(88b)

(R large).

(R large).

(iJAF*/aAFO')w=!(1/2X) (AFo'+wP-wT) , (87a)

(aAF*/aneE)w=!+ (1/2X) (neE-neEo'+wP-wT).

(87b)

We refer to these slopes as "transfer coefficients atconstant work terms." The second term in (87a) and(87b) can be calculated when X is known, and this inturn can be estimated from the experimental value ofAF* at AFo,=O, or at E=Eo' using (81) or (82),when the work terms can be estimated or are negligible.Typically, this second term is found to be small, sothat these "transfer coefficients" are then 0.5.

Equations (87a) and (87b) are based on the neglectof the antisymmetrical functions 1jk defined in (64).When these functions are not neglected, the transfercoefficient is not exactly 0.5 for zero (AFo'+wP-wT)/Xor zero (neE-neEo'+wP-wT)/A, but is given insteadby Eq. (A14) in Appendix IV. When these two sourcesof deviation from a 0.5 value are small, we may addthem and so obtain (87c) and (87d) instead of (87a)and (87b) :

(aAF*/aAFO')w=!+ (1/2A) (AFo'+wP-wT+!Xi(l.» ,

(87c)(aAF*/aneE)w=!+ (1/2A)

X (neE-neEo'+wP-wT+!Xi(l.». (87d)

As noted in Appendix IV the (t.) term could causea deviation from the 0.5 value by 0.04 when the forceconstants in the products are all twice as large (or alltwice as small) as the corresponding ones in the reactants and when Xi/X is about !. Smaller differencesin force constants would lead to even smaller deviationsthan 0.04. This source of deviations would be difficultto detect experimentally, since there are other sourcesof deviation as well. In the case of homogeneousreactions, force constants on one reactant may stiffenand those in the other weaken, so that the averagevalue of (1.) may be even less than that for the above

Relations similar to (88a) and (88b) also hold for Ai,independent of R, as may be seen from (84): Part ofthe sum for Ai is over the bonds of the first reactantand the remainder is over those of the second one.While the kjk's of one reactant in the activated complexdepends slightly on the fact that there is a neighboringreactant, this influence is taken to be weak.

In the absence of specific interactions, Eqs. (88a)and (88b) would also hold for smaller R, since in theequation for Ao each ion would merely see anothercharge, -mAe, and the surrounding medium, in boththe homogeneous and electrode cases. In the homo-


692 R. A. MARCUS

geneous case, the -m!:J.e is centered on the other ion.In the electrode case it is an image charge on theelectrode.28 To obtain some estimate of deviations from(88a) due to differences in ion size (one type of "specificeffects") we examine in the next section the evaluationof Ao in the dielectric continuum approximation.

16. DIELECTRIC CONTINUM ESTIMATE AT !:J.Fo*

The present section on a continuum estimate of !:J.Fo*is included partly for what it can reveal approximatelyabout certain aspects of the statistical mechanical valuefor !:J.Fo* and partly for making some approximatenumerical calculations. It does not form a necessarypart of the present electron-transfer theory itself, ofcourse, for the latter rests on statistical mechanics.

We note that !:J.Fo* can be regarded as the sum oftwo contributions, !:J.F*sol and !:J.F*atm· !:J.F*sol is definedas the contribution if the atmospheric ions have notadapted themselves to the change m(Pa.c - Pap0), and!:J.F*atm is defined as the contribution due to theiradaptation ("reorganization"). !:J.F*sol in an electrolytemedium will not have exactly the same value it has atinfinite solution, since the local dielectric propertiesnear the reactants will be altered somewhat by thepresence of salt.

These two contributions are estimated in AppendixVII by treating the medium as a dielectric continuum,the ion atmosphere as a continuum, and the reactantsas spheres, and by neglecting dielectric image effects.29

We obtain (89) and (90) for the value of !:J.F*sol for amedium treated as dielectrically unsaturated continuumoutside the inner coordination shell of each reactant.If partial saturation occurs, Eq. (67) still applies.9 Ifone then introduces "differential" rather than "integral"dielectric constants, as defined in the Appendix, andtreats them approximately as constants Eqs. (89) and(90) again apply but now Dop and D. are mean valuesof these differential constants

where ne is !:J.e, the charge transferred from one reactantto the other; al and a2 are the radii of the two reactantscomputed at intramolecular coordinates qi= q.i (theradii are of spheres each of which includes any innercoordination shell) ; R is taken to be the sum al+a2;Dop is the square of the refractive index of the medium;and D. is the static dielectric constant of the medium

Electrochemical:

!:J.F*sol=m2(ne)2(~_~)(_1 -~), (90)2 al R Dop D.

where R is twice the distance from the center of theion to the electrode surface and al is the radius of thereactant (and hence of the product) computed at qi= q.i.

The value obtained in Appendix VII for !:J.F*atm inthe electrically unsaturated region (i.e., in the DebyeHuckel region for the atmosphere around the ion and,in electrode systems, for the diffuse part of the doublelayer) is given by Eq. (91) for homogeneous systemsfor the case of al = a2 ( = a), and by (92) for electrodesystems. The value for !:J.F*atm for partially electricallysaturated systems can also be obtained from (67).Once again, one introduces "differential" quantities. Ifthe latter are replaced by "mean" values near thecentral species Eqs. (91) and (92) are again obtained,but with D. and te reinterpreted; te is given by (A23)in Appendix VII. A more reliable procedure, however,would be to use the position-dependent value of te insolving this particular linearized Poisson-Boltzmannequation, since the electric fields in electrolyte mediadie out fairly rapidly, namely as r-1 exp( -ter). Equations (91) and (92) are based on the solution of alinearized Poisson-Boltzmann equation with a localmean te

Homogeneous:

m2(ne)2!:J.F*atm= D.R

Homogeneous:

( 1 1 1)( 1 1 )!:J.F*sol=m2(ne)2 -+--- ---,2al 2a2 R Dop D.

(89)

X[teR+ exp[-te(R-a) J(1+te2a2/2)

1+tea+ exp[-te(R-a) Jte2a3/3R

Electrode:

1J (91)

Calculated as above, !:J.F*atm is much smaller than!:J.F*sol and is also expected to be less than the salteffects on wr and wp • Even at high te it is only m2(ne)2(R-a)/D.aR. Since R"'"'2a, its value there is aboutm2(ne)2/D.R, which is only about 2% of !:J.F*sol'Parenthetically, we note that this term arising from(91) and (92) just cancels the D. term in (89) and (90),respectively.

Added electrolyte can influence the rate constant, weconclude, principally by affecting wr, wP, and (by affecting dielectric properties) !:J.F*sol.

Comparison of (89) with (90) reveals that Ao for an

28 Quantum-mechanical calculations in support of the classicalimage law are given by R. G. Sachs and D. L. Dexter, J. App!.Phys.21, 1304 (1950). At a distance of 5 A from the electrode thecomputed energy of an ion in vacuum may be estimated from theirresults to be about 8% higher than that estimated from the imagelaw. Experimental evidence for the validity of the image law atdistances of 5 A has been offered by L. W. Swanson and R. Gomer,J. Chern. Phys. 39,2813 (1963) (d. p. 2835).

29 The dielectric image contribution to tJ.F001* is estimated to benegligible: It makes essentially no contribution to the value ofFm(r_p)Ol' since this hypothetical system has a low diectric constant equal to the optical dielectric constant throughout. Itscontribution to Fm(r-p) is only about 8% of the value of theterm containing liD. in (90). Since this term is only a negligiblefraction of the 1/Dol' term in (90), the dielectric image contribution to tJ.Fool* can be neglected. We note later that tJ.Fatm* isapparently much smaller than tJ.Foo1*. Dielectric image effectsmay be estimated from electrostatic calculations to contributeabout 8% to w' when two charges of equal magnitude meet.

!:J.F*atm=![rhs of (91)]. (92)


ELECTRON.TRANSFER REACTIONS. VI 693

(97)

isotopic exchange reaction has twice the value of Xo foran electrode reaction involving this redox couple whenthe value of R is the same in each case. It is recalledthat R is the value for which Kp exp( -f1F*/kT) had amaximum. If one presumes this R to be the distance ofclosest approach of the "hard spheres" and assumesthe reactant to just touch the electrode, then R is thesame in each case. In Eq. (89) al= a2 for an isotopicexchange reaction since these are the radii evaluatedfor q= q., it is recalled, and since typically the transition state should be symmetrical in this respect. (Fromthe equation cited the actual q.i's can be computedand the presumed symmetry verified for typicalconditions.)

It may be seen from (89) that Xo is essentially equalto the sum of two terms, one per reactant, and thatfor the same R the value of Xo for the homogeneousreaction in any redox system A equals twice its valuefor the electrochemical case. While the presence of theR term makes Xo not quite additive, the deviation fromadditivity can be shown to be small: On denoting theradii for ions of the two systems by a and b we obtain(93) .

aLl aa bb _ [1-(b/a)J2 2(~_~)Xo 2(XO +Xo )- 4b[1+ (b/a) J(ne) D

opD.' (93)

Even if b/a is !, a fairly extreme case, the ratio of theabove difference to Xobb is

(1-bja)2

2(1+bja) ,

i.e., -h. In virtue of this result, Xo has been treated asan additive function in applications6.7 of the equationsof this paper.

17. SIGNIFICANCE OF m

The parameter m was chosen in Sec. 13 so as tosatisfy the centering condition (20), a condition whichled to the vanishing of 1'(0). On differentiatingI(qN, R) given in (15) and setting qN=O one finds:

m=_<aur>/<~(ur_uP» (94)aqN aqN '

where the average ( ) is over the distribution functionon the reaction hypersurface 5' at the given R,

From (94), -m is seen to be the mean slope at thereaction hypersurface 5', (aUrjaqN), of the R surface,for the given separation distance R, divided by the sumof the mean slopes, (aUrjaqN) and (-auP/aqN) of theRand P surfaces at 5'. If the intersection surface 5' atthis R passed through the stable configurations of thereactants, on the average, then (aUrjaqN) would bezero. If it passed through those of the products instead

(auPjaqN) would be zero. In these two cases one seesfrom Eq. (94) that m would be 0 and -1, respectively.When in the vicinity of 5' the Rand P surfaces are,on the average, mirror images of each other about 5',(aUrjaqN) equals (-aUPjaqN) and one sees thatm= -!. Values of m close to -! are typical6•7 and areto be expected, one sees from (86), when f1Fo, is nearzero or when E is close to Eo' (typically of the orderof or less than 10 kcal mole-lor 0.25 V, respectively).

18. DEDUCTIONS FROM THE FINAL EQUATIONS

Equations (31) to (33), together with the additivityproperty of X (Sec. 15), and the relation betweenthe electrochemical and chemical X's described earlierlead to the following deductions, if K and p are aboutunity, or if they satisfy milder conditions in somecases.30

(a) The rate constant of a homogeneous "crossreaction," k12, is related to that of the two electronexchange reactions, kn and k12, and to the equilibriumconstant K 12, in the prevailing medium by Eq. (96),when the work terms are small or cancel,

k 12

OXl+ Red2~ Red1+ OX2, (95)

k12= (knk22K 1d)i, (96)where

lnf= (lnK12)

2

4ln(knk22jZ2)

Frequently, f is close to unity.(b) The electrochemical transfer coefficient at metal

electrodes is 0.5 for small activation overpotentials318

(Le., if InFTJ I < If1Fo* I, where f1Fo* is the free energyof activation for the exchange current) ,31b when thework terms are negligible.

(c) When a substituent in the coordination shell ofa reactant is remote from the central metal atom and isvaried in a series, a plot of the free energy of activationf1F* versus the "standard" free energy of reaction inthe prevailing medium f1Fo, will have a slope of 0.5, iff1PO' is not too large (i.e., if If1Fo, I is less than theintercept in this plot at f1Fo, = 0). In this series, for asufficiently remote substituent, X and the work termsare constant but f1Fo, varies, as in (87a). The slope ofthe f1F*-versus-f1Fo, plot has been termed the chemicaltransfer coefficient,6 by analogy with the electrochemicalterminology.

(d) When a series of reactants is oxidized (reduced)by two different reagents the ratio of the two rateconstants is the same for all members of the series in

30 For example, it suffices for some of the deductions that Kp beconstant in a given series of reaction or that it have a geometricmean property.

31 (a) We have phrased this condition for the case that (Ox) =(Red). For any other case, 7J should be replaced by E-Eo'. (b)The exchange current cited refers to the value observed when(Ox) = (Red).


694 R. A. MARCUS

the region of chemical transfer coefficients equal to 0.5[i.e., in the region where jl1FO f I < (I1F*) AFo°'=0 in eachcase].

(e) When the series of reactants in (d) is oxidized(reduced) electrochemically at a given metal-solutionpotential difference the ratio of the electrochemical rateconstant to either of the chemical rate constants in (d)is the same for all members of the series, in the regionwhere the chemical and (work-corrected) electrochemical transfer coefficient is 0.5.

(f) The rate constant of a (chemical) electronexchange reaction, kex, is related to the electrochemicalrate constant at zero activation overpotential,3la kel, forthis redox system, according to Eq. (98) when the workterms are negligible:

(98)

where Zsoln and Zel are collision frequencies, namelyabout 1011 mole-I. secl and 104 cm seci •

When the work terms are not negligible, or do notcancel in the comparison, the deductions which dependon this condition refer to rate constants, to K I2 and toan electrochemical transfer coefficient corrected forthese terms. Again, a minor modification of the transfercoefficients from the value of t in (b) or (c) can alsoarise from the antisymmetrieal force constant term (18 >in Eqs. (87) and (AI4).

It is shown in Appendix VII that under certainconditions these expected correlations apply to over-allrate constants as well as to those involving only onepair of reactants.

19. GENERALIZATION AND OTHERIMPROVEMENTS

Some of the extensions or improvements in thepresent paper, compared with the earlier ones in thisseries, are the following:

(1) Use is made of a more general expression for thereaction rate as the starting point.

(2) A more detailed picture of the mechanism ofelectrode transfer is given for the electrochemical case.

(3) The derivation is now given for both electrodeand homogeneous reactions, and in a single treatment.

(4) The statistical-mechanical treatment of polarinteractions, based in Part IV on the interactions ofpermanent and induced dipolar molecules in themedium, was replaced by a more general particledescription of polar interactions, through the use ofthe potential-energy function (37) and (65).

(5) The equivalent equilibrium distribution madeplausible in Part IV was proved more rigorously here.

(6) The functional form (68) for I1Fo*, obtained inPart IV only by subsequently treating the medium asa dielectric continuum, was derived here using astatistical-mechanical treatment of nonequilibriumpolarization systems.

(7) The basic equation for krau. has been convertedto a simple form [e.g., (31) and (81)], a form used inPart V, by neglecting the antisymmetrical function ofthe force constants, a neglect which has only a minoreffect numerically.

(8) The symmetry arguments used in Part IV toconvert the kT/h and a portion of a I1F* term to theZ factor in (31) have now been given more rigorously.

(9) The ion atmospheric reorganization term wasbut mentioned in Part IV. It is now incorporated intoI1Fo*. The nonpolar contribution to w" and wP is alsoformally included.

(10) The contribution of a range of separationdistances to the rate constant is included.

The results in the present paper may be comparedwith earlier papers in this series. In Part I, I1Fo* wascomputed for homogeneous reactions at zero ionicstrength, and dielectric continuum theory was used.Equation (89) was obtained. The actual mechanismof electron transfer was discussed there, but withoutthe detailed description which the use of many-dimensional potential energy surfaces provides. The latterwas used in later papers of this series, a use whichadded to the physical picture. The counterpart ofPart I for electrode systems was also derived andapplied to the data in a subsequent paper.2

In the earliest papers, the dielectric continuumequivalent of the equivalent equilibrium distributionwas derived by a method apparently different fromthat used in the present paper. The distributionselected was the one which minimized the free energysubject to the constraint embodied in Eq. (20), orreally embodied in the dielectric continuum counterpart of (20). In Appendix IX this method is in factshown to yield the same result for the equivalentequilibrium distribution as the functional analytic oneused in Appendix II. It is entirely equivalent.

APPENDIX I. NONADIABATIC ELECTRONTRANSFERS

Several estimates are available for the probability ofnonadiabatic reactions, per passage through the intersection region of two potential energy surfaces, andhave been referred to and discussed in Ref. 7. In eachcase the motion along the reaction coordinate wasassumed to be dynamically separable from the remaining motions. (For conditions on separability see, forexample, Ref. 32 and references cited therein.) Theprobability of electron transfer per passage throughthe intersection region in Fig. 1 will depend in the firstapproximation on the momentum PN conjugate to thereaction coordinate qN, as, for example, in the LandauZenerll equation. While the value of K is not so simplyrepresented in more rigorous treatments, we simplywrite it as K(PN). In the above treatments the reactioncoordinate was assumed to be orthogonal to the others

;12 R. A. Marcus, J. Chern. Phys. 41,603 (1964).



APPENDIX II. PROOF OF EQ. (13) FORTHE CENTERED DISTRIBUTION

f'" ( gNNPN2)gNNPN exp - -- dpN

o 2kT

Suppose, for possibly more general applications, thatthere are n linear equations of constraint of the typerepresented by (A3). Here, we are especially interested

f'" ( gNNPN2)K(PN)gNNPN exp --- dpN

PN~O 2kTK=-------------

(A4)

(A3)j=l,·",n.

x=w[J(x)/ J(w)]+y,

!f*YidV'= 0,

where Y belongs to M. In the present instance w= 1 issuch a function. On applying (A4) to the functionx= U*- Ur and using (A2) one sees that x=y, i.e.,that x belongs to M and can so be written as a linearcombination of the functions Yi. In the present case,M is one dimensional, the only Yi being Ur- UP, since(A1) is the only equation of constraint. Thus, x, i.e.,U*- Ur, equals Ur- UP multiplied by a real scalar,and Eq. (13) is established.

APPENDIX III. DISTRIBUTION OF Yo' COORDINATES IN THE ACTIVATED COMPLEX

Since U(l) is linearly dependent on the Pao of eachreactant, ur(l) +m[ur(l) - Up(l)] equals the U(l)for a system in which each reactant has a Pa 0, Pa0*,

33 A. E. Taylor, Introduction to Functional Analysis (JohnWiley & Sons, Inc. New York, 1958), p. 138.

For any temperature and U*, this integral is a linearfunctional of Yi. Although one can find functions, U,

other than Yi (and other than linear combinations of Yi)for which f1*udV' vanishes at some temperature T,the y/s are the only ones for which this integral isspecified to vanish for all T. That is, there are onlythe n equations of constraint (A3) on the U* in 1*.The space functions Y for which ff* Y dV' is real andfinite form a linear vector space over the field of realscalars. Moreover, the integral, denoted by J ( Y), is alinear functional on this space. For some subspace Mof it, the integral vanishes. The functions Yi( j = 1 to n)form a basis for M. If there exists some function w forwhich J (w) does not vanish, then an elementarytheorem33 of functional analysis shows that any function x can be written as

in the case n= 1,

We first note that U(2) in Eq. (65) does not dependon PMo, the pO of the "medium," and so is insensitiveto the usual rotational and translational fluctuations ofthe solvent molecules, unlike U(O) and U(l). SinceUo* is given by (13), with 0 subscripts added, one termin Uo* is ur(2) +m[ur(2) - Up(2)]. Since this can beextracted from the integral in the denominator of theabove distribution function because of this insensitivityto the V'o coordinates, it cancels a corresponding termextracted from the numerator. The distribution function fo* then becomes (AS) :

exp(- {U(O) +ur(l) +m[Ur(l) - UP(l) ]}/kT)

! exp(- {u(o)+ur(l)+m[ur(l)- UP(l)]}/kT)dV'o

(AS)

(Al)

(A2)!f*U*dV'= !f*urdV'.

The centering is of both a horizontal type (horizontalin terms of Fig. 1 or 2) and of a vertical type, represented by Eqs. (A1) and (A2), respectively:

!f*urdV'= !f*UpdV',

This K can depend on all the other coordinates, qi(i~N)at the given value of qN characterizing the intersectionsurface S. The denominator in the above equation iseasily shown to equal kT, and so to be independent ofthe qi. In the discussions of K(PN) in the literature, thederivation of the Landau-Zener equation, for example,the reaction coordinate has been assumed to be rectilinear; gNN is then a constant and the integral in thenumerator then becomes independent of the qi and maybe removed from the integral in Eqs. (1) and (2).

There appears to be no treatment in the literature fornonadiabatic reactions involving many closely spacedenergy surfaces as in Fig. 2, covering the range of K(PN)from 0 to 1. If K (PN) is sufficiently small, the transitionto each P surface from the initial R surface may beassumed to be independent, as mentioned earlier, andthe reverse transition to the initial R surface duringthis passage may also be neglected. In this case onlydoes the method of Levich and co-workers in thisconnection become appropriate. (For references seeRef. 7.) In this case the above K appears in the integrand of Eq. (3) and care is taken to sum over all levelsin an appropriate fashion, as done by Levich et al.(see Ref. 7 for bibliography). One can then evaluatethe K appearing in Eq. (33) and defined earlier. Usually,however, we assume that K(PN) is close to unity (withinsome small numerical factor, say) for the PN'S ofinterest.

in mass-weighted configuration space, so that gNi vanishes for i~N (and so, therefore, does gNi) in thekinetic energy. On recalling the derivation of Equations(1) and (2)8 and on introducing the above assumptions,the rate constant is given, one can show, by Eqs. (1)or (2), but with the integrand multiplied by K:


696 R. A. MARCUS

given by (66). Next, on multiplying the numeratorand denominator of (A5) by the exponential of the-U(2)/kT corresponding to these Pao*,S and placingit under the integral sign, we see that the distributionfunction jo* is the same as that corresponding to thePa o*'s given by (66).

APPENDIX IV. SIMPLIFICATION OF EQ. (54) ANDTHE EQUATION FOR krate

We introduce the quantities kjk and ljk defined in(63) and (64). The first was chosen so as to havedimensions of a force constant, and the second so as tobe dimensionless.

Principally, it is the diagonal stretching contributionswhich are usually important. Purely for simplicity ofargument we confine our attention to the diagonalterms. We denote the new force constants by j:, j.p,and their symmetric and antisymmetric combinationscited above by k. and 1•. In terms of k8 and 1., j: equalsk./ (1-1.) and j.p equals k8/ (1 + 18), To make use ofthe symmetry of the resulting equations we use theparameter e, equal to (m+!).

We obtain (A6) from (54) and (60):

.1F.*= H e-!)2l:). (.1q80) 2(1-1.) (1 +2el.)-2

+!kT2:: In[(1+2e18 )/(1+1.)J, (A6)•

where .1F;* is defined as .1F*(R) - .1Fo*(R). Similarly,we find (A7) by noting that it is obtained from (A6)by replacing -m by m+l and interchanging rand psubscripts (see Sec. 13)

.1F/P= He+!) 22::k8(.1q.0)2(1+1.) (1 +2el.)-2•

+!kT2:: In[(1+2e18 )/(1-1.)J, (A7)

where .1Fi*p is .1F*p(R) - .1Fo*p(R).In terms of e, Eq. (79) can be written as (AS), upon

introducing Eq. (68) for .1Fo* and its counterpart for.1Fo*p[= (m+l)2A"J

where

whereXi =!2::k8(.1q8 0) 2,

•

Furthermore, according to (79) .1F* equals .1F*P+.1FRo,. Hence,

.1F*= H.1F*+.1F*) = H.1F*+.1F*p) +!.1FR0'.

On introducing (A6) and (A7) one finds

.1F*= !.1FRo'+X(e2+t) +X i O(1]4) +!kT2:: (4e18+1.2).•

(A12)

On introducing (AlO) for e one finds that (A12)becomes

.1F*= !.1FRo'+tX+ (1/4X) (.1FRo'+!Xi (18»2

+tkT[2::18L (kT/X) (2::18)2J+0(1]4). (A13)

The same expression obtains for electrode processes,with the .1Fo, in .1FRo, replaced by ne(E-Eo').

In an isotopic exchange reaction which involves mereinterchange of charges in the electron transfer step, theterm (1.) vanishes by symmetry. In other reactionsthere will be some tendency for it to vanish, for while 1.increases on one reactant on going from State R toState P (due to an increase of charge), it will tend todecrease on the other. As a somewhat extreme caseinvolving no compensation, consider two reactantsone of which has vanishing 1. and also vanishing contribution to Xi. (Hence, we include the possibility thatthis "reactant" is an electrode.) For the other moleculelet the force constants k: and k.p differ by as much asa factor of 2. Then one finds (1. )"'t. If X/X",! thenXl(18)2/16X is only about 1% of A/4, the main term at.1FRo,=O. When A/4 has its usual value of 10 to 20kcal mole-I, say 10, and when the reactant has acoordination number of six, then the kT term in (A13),is estimated to be about 4% of the X/4 term at roomtemperature.

We consider next the effect of nonvanishing (1.) onthe derivative (iMF"/a.1Fo'ho,x, at .1FR0'= 0, theregion of greatest interest. This derivative equals

(A9) (A14)

Most of the data are obtained in the vicinity of.1FRo'=0.6,7 We consider this region first. Near thepoint (l.=0, .1FRo,=O) one readily verifies from theequations below that e is close to zero and that itvanishes at that point. We let 1] denote e or 18 , and 0denote the "order of." (1] is a small quantity in thevicinity of this point.) One then finds from (A6) to (A8)

-2eX-!Xi(1.)+0(1]3) = .1FRo'+kT2::1., (AlO)•

In the case cited above the Xi (1.)/4X term is about+0.04. Thus, the derivative differs by only 8% forthis case. Hence, the (1.) term may be neglected whene (and hence I.1FRo,/X I) is small. When I.1FRo,/X 1isnot small, one finds that (A13) should be replaced by(A14a), to terms correct to first order in the 1.

.1F*= !.1FRo'+tX+t(.1FR°')2



The term containing (1.) is still small: A fairly extremecase is one where the activated complex resembles thereactants (m=O) or the products (m=1). At eachextreme I t!.Fnol/A I is about unity, since the expressionfor E( = - t!.Fn01/2A) is but slightly affected and sinceIE I equals! when m is 0 or 1. In the interval

O~ I t!.Fnol/A I~1the last term in (A14a) has a maximum at

I t!.Fnol/A I = (!)!.

At this point it equals about 1 kcal mole-I for the valuesof (1.), Ai/A, and A/4 cited above. When one does notneglect second and higher-order terms in I., and solves(A6) to (A8) numerically in this region one obtainsthe same result: The 1. terms may be neglected.

APPENDIX V. SMALLNESS OF (Ur(Q*) - Up(Q*) )*0

If (ur-up).o at any Q is expanded about its valueat Q. and if it can be shown that the linear termsuffices, it follows that (Ur-up).o averaged over fi*equals the value at Q. plus the average of the linearterm. In virtue of (SO) the averaged linear term vanishesand, in virtue of (20), the average of (Ur-Up).oover f;* vanishes. Hence, (ur (Q.) - UP (Q.) ).0 alsovanishes.

To show by a posteriori calculation that the linearterm in the expansion suffices we make use of somenotation introduced after (54). After use of (37), ofthe equality of (U/-Uop).o with Fo*-Fo*P, of (68),of the definition of Ot and OiP, and of their quadraticexpansions about Qr and Qp, respectively, of the essential equality of the vibrational entropy of reactantsand products, and of the justifiable neglect of theantisymmetrical functions (64) (Appendix IV) onefinds (A15) for any given Q.

(Ur- Up).o= - (2m+1)Ao(Q) -t!.F01

+!L:)ii(qLqri) (qi_qri)i,i

-!L:)ii(qLqpi) (qi_ q/). (A15)i,i

The quadratic term, kiiqiqi, is seen to cancel. A linearexpansion of Ao(Q) about Xo(Q*) is sufficient, for eventhe linear term is small (compare Appendix VI) . Hence,he linear term in an expansion of (ur (Q) - UP (Q) ).0

suffices. The vanishing of (ur(Q.) - Up(Q.) ).0 thenfollows.

APPENDIX VI. JUSTIFICATION OF NEGLECT OFaxo/aqi IN THE DERIVATION OF EQ. (58)

It is shown here that the error in neglecting thedependence of Ao on Q in deducing (58) from (56) isminor.

Since the arguments in Appendix IV reveal that theerror in neglecting the antisymmetrical functions (64)is minor, we may simplify the present analysis by

neglecting them. To this purpose all force constantsmay be replaced by the symmetrical ones, kik , definedby Eq. (63).

Let Ao be a column matrix whose components areaXo/aqi:

Xo(Q) =>'o(Q.) + L(aAo/aqi) (q'-q.i) +...i

=Ao(Q.)+AoT. (Q-Q.)+.... (A16)

The first variation in an expansion of Oi*(Q) aboutQ. is found from (56)

00.*= oQT{(m+1)K(Q.-Qr)

-mK(Q.-Qp) -m(m+1)Ao], (A17)

where the elements of K are the kik'S.On setting 00.* equal to zero, one obtains, instead

of (58);

Q.=m(m+1)K-IAo+ (m+1) Qr-mQp. (A18)

Equation (54) for t!.F* then becomes

t!.F*= (m2/2) [t!.QT+ (m+1) (K-lAo)T]

·K[t!.QT+(m+ 1)K-lAo]

+m2>'0(Q.)+!kTln Ihk* III kik I. (A19)

For present purposes it suffices to consider the casewhere t!.Fo1 is small. An expression for t!.F*p can beobtained from (A19) by replacing m by - (m+1)and t!.Q by -t!.Q. On letting t!.F*-t!.F*p equal zero(since t!.Fo1 is zero) the resulting equation is solvedfor m, which is thereby found to be -!. A simplenumerical estimate then shows that the presence of theK-IAo terms have negligible effect: Other than the Interm the rhs of (A19) is given by

i>'(Q.)+!t!.>'o+~AoT(KT)-I·Ao, (A20)

where t!.Aois the total change in Aowhen Qr is changedto Qp. Typically Ao/4 is of the order of 5 kcal mole-I

and is inversely proportional to ion size. When themean bond length changes by as much as 0.15 A(compare the probable Fe-O bond length difference inFe2+ and Fe3+ hydrates) and when the radius of thereactant including inner coordination shell is 3 A, t!.>'0/4is about !(0.15/3), i.e., about 0.25 kcal mole-I. Thethird term in (A20) is even less. For example, if oneconsiders the stretching of bonds only, and if thestretching ki/s for metal-oxygen bonds in a hydratedcation are taken to be the same one finds

i7%AoT(KT)-IAo= (t!.Ao/Ai)2iA.. (A21)

(Similar remarks apply to other coordination complexes.) Since Ai/4 is of the order of 10 kcal mole-I forthe cited case (A21) is about 0.006 kcal mole-I.

APPENDIX VII. CALCULATION OF t!.Fo* INCONTINUUM APPROXIMATION

When dielectric unsaturation and electric unsaturation prevail there is, respectively, a linear response of


698 R. A. MARCUS

where ei is the charge of Species i in this atmosphere,

34 R. A. Marcus, J. Chern. Phys. 38, 1858 (1963).

where aP and aE are the change in polarization and inelectric field at r. The effective dielectric constantdescribing the response to this aE is V s (r) equal to1+41lx(r). The quantities x(r) and Vs(r) can betensors.

Then, again, if P(r) is the charge distribution in theion atmosphere and, if one wishes, in the electricaldouble layer at the electrode-solution interface, and ifp(r) is approximated by a continuum expression

p(r) = Lc;'°ei exp (- eiif;jkT) ,i

the solvent polarization and of the charge density ofions in an ion atmosphere to the charging of the centralion (or ions), and not merely to a small change in itscharge. In real systems, some partial dielectric saturation outside of the coordination shells may occur and,at appreciable concentration of added electrolyte, theresponse of the atmospheric ions is certainly nonlinear.(The region of linear response of an ion atmosphere toa charging of the "central ion" is confined to theDebye-Htickel region.)

We introduce the partial saturation approximation,wherein only a linear response to a small change incharge of the central ion (ions) is assumed. The specialcase of unsaturation is automatically included, therefore. We are interested, typically, in changes of magnitude, mfie, i.e., about t an electronic charge unit.Equation (67) was derived for both the partiallysaturated and for unsaturated systems, but in theformer case the definition of FOPm(r_p) and Fm(r-p) hasto be interpreted carefully.

To calculate Fm(r-p) appearing in (67) and to takepartial saturation into account, one considers twocharge distributions: (i) The original charge distribution of the reactants and the medium for the cited R.(ii) A hypothetical charge distribution in which thereactants' charge distribution is altered from (i) byan amount m (Par 0- pap 0), in a hypothetical systemwhich has responded linearly to this change. To obtainthe properties of the hypothetical system in Fm(r-p) onesubstracts the above two charge distributions on thereactants and also substracts the portions of the remaining charge distributions, induced or otherwise,which did not respond. One now has in this hypothetical[m(r-p)] system reactants which have permanentcharges given by the distribution m (Par 0- pap0) andare imbedded in a medium of solvent and atmosphericions which has linear "response functions" describingthe above response. For example, if we use a continuummodel, then the effective dielectric susceptibility of thesolvent is the proportionality constant x( r) in34

(A26)

(A23)x'(r) = [Lcicoei2 exp( -eiy;/kT) ]!.i

c/-O is its concentration at infinity, and y; is the potentialat r relative to the value at infinity, then

ap(r) = - (LciCOei2e-ei'i/kT/kT)01/;(r).i

when al=a2.The difference of (A24) and (A25) is the value of

Fop- F when the atmosphere does not respond, and

Fopm(r-p) = - [(mfie) 2(1_~)+ (mfie) 2(1_~)J2al V op 2a2 V op

On recalling that the Debye kappa is defined as thesquare root of the proportionality constant of p(r)and -y;(r) in linear systems, the quantity which playsthe same role in this hypothetical system is x' (r) .

_ (mAe)2[K'R+ exp[-x'(R-a)](1+K'2a2/2) JD'sR l+K'a+ exp[-x'(R-a)]x'2a3/3R 1,

To calculate FOPm(r_p) we recall that this systemresponds only via the electronic polarization of themedium, and so K' vanishes for this system and x'(r)becomes X'e(r), the proportionality constant replacingx(r) in (A22). The medium in this hypotheticalsystem behaves as though it had a dielectric constantV'op(r) equal to 1+41Txe.

If we take V'op to be approximately a constant, forsimplicity, then FOPm(r_p) is easily calculated. Weneglect dielectric image effects.29 POPm(r-p) is the sum ofthe free energy of solvation of the central species whenthey are far apart, plus the free energy change whenthey are brought together in this "op" medium. Theformer is given by the Born formula (it is not the freeenergy of solvation of the bare ion, but of the coordinated ion) and the latter by the Coulombic term.Hence,

(mfie) 2

- V'op R ' (A24)

The Fm(r-p) term is the sum of its value when the ionatmosphere does not respond

[(mfie)2( 1) mfie( l)J (mfie) 2

- ~ 1-V's + 2a2 1-V's - V'sR ' (A25)

and the contribution due to their response via K' (r) ,fiF*atm. On taking K' to be approximately a constantnear the central series the leading terms of the secondcontribution are36

35 Since dielectric image effects are being neglected one maym~rely use the ~xpressions obtained by G. Scatchard and J. G.Kirkwood, PhYSIk. Z. 33, 297 (1932), for the contribution to thefree energy of interaction of a pair of ions with their atmospheredue to a response described by >e. We may merely replace >e by>e' and D. by D,' under the approximations stated.

(A22)aP(r) = -x(r) aE(r),



In this way (90) and (92) of the text were obtained.

(A34)

(A33)

(A31)

(A30)

n

nm

n

m,n

m

On neglecting I1Fmno'2/4Amn in (81) as discussedearlier one obtains (A3S) , using (A30) to (A34) :

kab=ZKab!L exp{ -[wmnr+WmnP+!(Am+An) J/2kTIm,n

X (7rmP7rmr7rnP7rnr)!, (A3S)

where K ab is given by (A36) and is, in fact, easilydemonstrated to be the formal equilibrium constant ofthe reaction in the given medium, expressed in termsof the stoichiometric concentrations

Each kmn is given by a pair of equations of the type(31), (81), where for A we write Amn and recall theadditivity of A

On using (A32) the I1Fo, for Step (A29) is seen to be

mn contributions, per unit stoichiometric concentrations of Ar and of Br:

kab= L)mnr(Amr) (Bnr)/L)Amr) L(Bnr) ,

In virtue of their definition these F's depend on theconcentration of X/so The free energy of any reaction(A32) in the prevailing medium is in fact Fm,r-Fmr:

AmT+L(m'i-mi)X,~Am,r. (A32)i

m

then (A23) becomes

kab= Lkmnr7rmr7rnr.m,n

where ( ) denotes concentration. If 7rmr and 7rnr denotethe probabilities that an Ar species exists as Amr andthat a Br one exists as Bnr, respectively, i.e., if

Let Fmr+FnT denote the free energy of the systemcontaining a labeled Amr and a labeled Bnr moleculefar from each other, fixed in the medium, under theprevailing conditions. Let the corresponding propertybe FmP+FnP when the two labeled molecules are AmPand BnP. We subdivide Fmr+Fnr such that Fmr dependson the properties of Amr and its environment alone. Itis therefore independent of the nature of Bnr. We notethat the 7r'S can be expressed in terms of these F's, ifwe assume, as we do, that the complexes Amr and Bnrhave an equilibrium population,

exp( - Fmr/kT)7rm

r= L exp( _ Fmr/kT) ' etc.

(A29)

(A28)

kmnr

Amr+Bnr~AmP+BnP.

_ (ml1e)2(1_~)_ (ml1e) 2

2a D'. 2D'.R·

Let n play the same role for the B system

was called I1F*sol. In (89) to (92) we have omittedthe prime superscripts for brevity.

In the case of electrode systems, there is only one ion,but there is also the image charge of opposite sign inthe electrode.28

Instead of (A24) to (A26) one finds,

(electrode) Fop (_ )= _ (ml1e) 2(1-_1_)_(ml1e) 2mT P 2a D'op 2D'opR

(A27)

and that Fm(T-P) is the sum of (A28) and of one-half(A26) ,

Let the reactants and products be denoted by rand psuperscripts, respectively. A typical contribution tothe over-all redox reaction is (A29). Let it have abimolecular rate constant kmn for the forward step

n= (nl, n2, •", ni, ••• ).

The over-all second-order rate constant kab then involves a weighted sum over the rates of all bimolecular

APPENDIX VIII. CORRELATIONS OF OVER-ALLRATE CONSTANTS

Equations (31), (33), (81), and (82) describe therate constant for any reactants with intact, specifiedinner coordination shells. I1Fo, there refers to the changefor those species. Consider now the rate constants expressed in terms of the stoichiometric concentration ofeach redox reagent. The region of (81) linear in I1Fo,is the most important one in terms of the correlationsmade in Part V, and we restrict our attention here tosuch cases for each elementary redox step (A29) below.We consider only the case where the dissociation orformation of any important complex does not contributeappreciably to the reaction coordinate near the intersection surface: We make use of (81) and note that itsderivation was based on intact coordination shells in asystem near the intersection surface; the properties ofthe "reactants" or "products" appearing in Eq. (81)refer to those with such shells, even though they mightbe unstable.

We consider the homogeneous case first. Let mdenote the totality of any ligands Xl, X 2, ••• in areacting member of the A redox system having miligands of Type Xi,

m= (ml' m2, ••• , mi, ••• ).


700 R. A. MARCUS

This equilibrium constant is, by definition,

L(AmP) L(BnP)/L(Am') L(Bn').m n m 110

linearly on E, as in (74). FM P and FM' are independentof the properties of A. They depend only on those ofthe electrode and the electrical double-layer region

n

(A44)

(A43)

m

FmP+FMP- Fm'- FM'=ne(E- E.)

-kTln(AmP)/(Am'). (A45)

where Am is independent of E.When electrochemical equilibrium exists (E equals E s

then), it does so for each m. Adding to the free energydifference (A43) the mixing term, kT In(AmP)/(Am') ,the result must equal zero at equilibrium. We therebyobtain from (A43) the value of each Am,

Utilizing the fact that E. is related to Eo' accordingto (75), where (Ox) now equals Lm(AmP) and (Red)equals Lm(Am'), (A45) can be rewritten as

FmP+FMP- Fm'-FM'=ne(E-Eo') -kT In7rmP/7rm'.

(A46)From (A31) and (A46) one obtains:

exp[-ne(E-Eo')/kT) = exp[-(FMP-FM')/kT]

Lm exp( - FmP/kT)X Lexp( _ Fm' /kT) (A47)

Equation (A45) is finally obtained for the free-energydifference

(A41)

(A37)

(A38)kaa= L kmm,'7rm'7rm'p.m,ml

kmm,r

Am'+Am,p~AmP+Am'"

m

X (7rmP7rm'7rm,p7rm,')i. (A39)

When the work terms can be neglected one finds

kab=ZKah!L exp( -Am/4kT) (7rmP7rm')i

kaa is obtained from (A35) by noting that Kaa is unity

m

kaa=Z L exp{ -[Wmm"+Wmm,P+HAm+Am,)]/kT}mImI

From (A35) one can at once derive an expressionfor the isotopic exchange rate constant. On consideringthe A redox system a typical contribution to theexchange will be (A37) when m and m' describe anytwo complexes. The over-all rate constant, kaa, is thenobtained by multiplying kmm,' by 7rm'7rm'P and summingover all m and m'. The result is given by (A38), andis then counterpart of (A35) :

From (A40) and (A41) one then obtains

kab= (kaakbbKah )!.For the over-all electrochemical rate constant of the

(A42) forward reaction in (73), ke1 , we have

On considering next the electrochemical case, let Mdenote the electrode, M' describing its state beforeelectron transfer and Mp after. As in the text we assumethat the acquisition or loss of an electron by the electrode has essentially no effect on the force constantsor equilibrium bond distances in any adsorbed layer ofions or molecules. (To be sure, one or more electronson the electrode may be fairly localized when thereacting species is near it, and this number changeswhen the species gains or loses electrons.) We regarddifferent compositions of the adsorbed layer as corresponding to different domains of the coordinates inmany-dimensional space.

The free energy of a system having a labeled Am'molecule far from the electrode and fixed in position iswritten as Fm'+FM', the corresponding term when themolecule is AmP (and the electrode has lost n electronsthereby) is FmP+FMP. The free energy of Reaction (73)for the case where the reactant is Am' is then given by(A43) , since the translational contribution for Amcancels in computing FmP- Fm'. The change depends

(A48)

where km' is the rate constant for (Am') going to (AmP)at the given E. For each m, the km' is given by anequation analogous to (82), with ne(E- Eo') replacedby ne(E-Eo') -kT In7rm

P/7rm' [compare Eqs. (77) and(A46)]. One then obtains

ke1=Zel exp[-ne(E-Eo')/2kT]

XL exp[- (Wm'+WmP+!Am)/2kT] (7rmP7rm') i.m

(A49)

The work terms naturally depend on E. When theycan be neglected one has

ke1 = Zel exp[-ne(E- Eo') /2kT]

XL exp( -Am/4kT) (7rmP7rm')!. (A50)m

In the light of Eqs. (A40) to (A42) , (A49), and(A50) , we see that the correlations (a) to (f) in Sec. 18



1 (Ur+m(Ur- Up) +kT lnj*+a) of*dV'=O. (AS4)

we obtain (AS4) , where a and m are LagrangianmUltipliers:

Setting the coefficient of 01* equal to zero, andevaluating a from (AS2) we find

f*= exp(- ~;) / 1 exp( - ~;)dV"

(AS3)

(AS1)

(AS2)

!1*dV*=1,

where U* equals Ur+m(Ur-Up). This equation wasalso obtained by the method in Appendix II. Onceagain, m is determined by the energy condition (AS2).

Minimizing (AS1) subject to the energy equation ofconstraint (AS2) and to (AS3) ,

I(Ur-UP )1*dV*=O,

determined, is given by (AS1) to an additive constant

Fnon= If*UrdV'+kT11* Inf*dV'.

As we have seen in the text, the configurationaldistribution of the V'i and V'o coordinates in theactivated complex is not one which is appropriate toSurface R nor one appropriate to Surface P. That is,it is appropriate to neither electronic structure (theinitial or final one) of a reacting species. Cognizanceof this nonequilibrium distribution of solvent moleculeswas taken in Part I, using a dielectric continuumtreatment of systems possessing nonequilibrium dielectric polarization. An expression for the free energy ofa system with arbitrary polarization was minimized,subject to an energy equation of constraint, the dielectric continuum counterpart of (20). In this Appendix we show that this method, formulated now in termsof statistical mechanics yields the same result as themethod used in Appendix II.

The configurational contribution to free energy of anonequilibrium system described by a potential energyur and a distribution function 1*, where 1* is to be

still hold, even when applied to over-all rate constantsbut, as one sees from (A42) , (a) is now restricted tothe region of chemical transfer coefficient equal to ![i.e., to!'"......1 in (96)].

APPENDIX IX. ALTERNATIVE DERIVATIONOF (13)

THE JOURNAL OF CHEMICAL PHYSICS VOLUME 43, NUMBER 2 15 JULY 1965

Current Oscillations in Solid Polystyrene and Polystyrene Solutions*

A. WEINREB, N. ORANA, AND A. A. BRANER

Department of Physics, The Hebrew University of Jerusalem, Israel

(Received 19 March 1965)

Application of a dc voltage across a plate of polystyrene gives rise to oscillatory currents which reachconsiderably high negative values. The dependence of current intensity (maximum, minimum, and plateauvalues) on various parameters (voltage, dimensions of samples and electrodes, nature of dissolved solute,etc., as well as repetitive use) is treated. The pattern of oscillation is found to depend on all these parameters,too. The length of the oscillation period decreases very quickly with increasing voltage. It depends also verystrongly on the nature and pressure of the surrounding gas.

1. INTRODUCTION

I N trying to measure the extremely low dark conductivity of polystyrene we found a prohibitively

strong influence of air on the measured intensities of thecurrents. The variation in current intensity whichusually follows any mechanical handling of a plasticwas also found to be strongly influenced by the presenceof air. In order to avoid the effect of air the "chamber"

* Performed under the auspices of the U. S. Atomic EnergyCommission, Contract NYO-2949-6.

which houses the investigated specimen was evacuated.Upon evacuation the following effect was observed:The current oscillates with a definite pattern reachinghigh negative values although a dc voltage is applied. I

The period of oscillation as well as its pattern dependsstrongly on the voltage. It depends also strongly onthe nature and pressure of the surrounding gas. Theseoscillations present a serious obstacle in measuring

1 A. Weinreb, N. Ohana, and A. A. Braner, Phys. Letters 10,278 (1964).

This paper is in a collection of - Electrochemical...

Documents

Transcript of This paper is in a collection of - Electrochemical...