J. Chern. SOC., Faraday Trans. I, 1989, 85(8), 1881-1895 Thermodynamics of the Transition State and the Application to Interfacial Reactions Denver G. Hall Department of Chemistry and Applied Chemistry, University of Salford, Salford A45 4WT According to transition-state theory, when the equilibrium hypothesis is valid, activated complexes formed from reactants may be treated thermo- dynamically in the same way as other species present in very small amounts. Inclusion of these complexes in the Gibbs-Duhem equation for the solution enables an expression to be derived for the dependence of the equilibrium concentration of activated complexes on the solution com- position. When the transmission coefficient of the reaction is insensitive to the reaction environment this expression leads directly to the effect of solution composition on the reaction rate.For bulk solutions, this approach is entirely equivalent to that published recently based on the Kirkwood-Buff theory of solutions. For reactions at interfaces the Gibbs adsorption equation plays the same role as the Gibbs-Duhem equation for reactions in bulk and similar arguments apply. These arguments are used to obtain a general expression for the transfer coefficient p in the Butler-Volmer equation of electrode kinetics and for the dependence of /3 on solution composition. A similar treatment can also be applied to the effects on reaction rates of macromolecules which interact with reactants. Finally, the situation in which adsorbed material is not in equilibrium with that in the bulk solution immediately adjacent to the interface is considered and general guidelines for dealing with this type of situation are forwarded.A key element of transition-state theory is the equilibrium hypothesis. In a recent paper' it was argued that when this hypothesis is valid the Kirkwood-Buff theory of s01utions~~~ can be used to provide an expression for the effect of solution composition on the concentration of activated complexes. This in turn leads to the corresponding change in reaction rate provided that any accompanying changes in the kinetic behaviour of the complexes are small. It is believed that this is usually so. Although it is general, the Kirkwood-Buff theory is also somewhat formal. For solutions in which molecules associate to form well defined aggregates alternative thermodynamic treatments which allow for aggregation explicitly are more informa- The application of these theories to calculate changes in rate via changes in the concentration of activated complexes has also been described recently.This new treatment appears to account better for the effects of micellar aggregates on reaction rates than its predecessors. It is fairly straightforward to show that this approach, and that based on the Kirkwood-Buff theory, are entirely consistent with each other and that the former is a special case of the latter. The philosophy underlying both is essentially the same as that used in the successful application of transition-state theory to calculate ionic strength effects on the rates of reactions involving ions.'0*'' In this paper an approach equivalent to that based on the Kirkwood-Buff theory but cast in more traditional thermodynamic terms is developed and applied to reactions at interfaces.The development of the argument provides a justification for some points of T Also at: Unilever Research Laboratory, Quarry Road East, Bebington, Wirral, Merseyside L63 3JW. 18811882 Transition-state Theory at Interfaces reasoning in the work on micellar catalysis cited above. A particular application is to eletron-transfer reactions at the metal/solution interface where an expression is derived for the transfer coefficient in the well known Butler-Volmer equation. Reactions in solution We consider the simple reaction A + B + products in a multicomponent solvent and suppose that the reaction proceeds via a transition state.According to transition-state theory" we may write for the rate per unit volume of the system rate = k*n* (1) where n* is the number density of activated complexes which have formed from reactants and k* is a term which involves quatitites such as (i) the rate at which complexes cross the top of the energy barrier whose height is the activation energy and (ii) the probability that a complex crossing in one direction recrosses in the other. The important features of eqn (1) are that n* is a number density not an activity and that k* is insensitive to the environment in which the reaction takes place so that changes in rate are mainly attributable to changes in n*. Let p denote chemical potential, let the subscripts ijk etc.denote solvent components and let += refer to the transition state. We state the equilibrium hypothesis in thermodynamic language by writing p* = pe * ( T, pi9 PA9 P B ) + RT1n * = + pB (2) and supposing that the activated complexes can be treated thermodynamically in the same way as any normal component which is present in very small amounts. Eqn (2) relates changes in n* to changes in pressure and solution composition. The quantity pe * is a standard chemical potential and, as has been supposed previously,' changes in p** are given by the Kirkwood-Buff theory of solutions. At constant T we write the Gibbs-Duhem equation of our reacting system as dp = ~nidpi+nAdpA+nBdpB+n*dp* i where the n, are number densities. Subtracting d(p*n*) from both sides of eqn (3) we obtain d [p - n*p*] = C ni dpi nA dpA 4- nB dpu, - p*dn* i (3) (4) which gives on cross differentiation ( a ~ * /api)pj, pA, pB, n+ = - (ani/an* ) p i , PA, pB Ian * >pi, pA, pB ( 5 a) (5 b) By writing eqn (5a) and (5b) we are in effect supposing that n* can be varied at constant ninA and nB.This appears to contradict eqn (2). However, when dealing with variations at equilibrium it is not uncommon to use quantities that are defined in terms of variations which do not maintain equilibrium. For example, partial molar enthalpies and volumes of species which participate in equilibria are defined in terms of adding a small amount of the species concerned with the amounts of all other species held constant. ('p * / ' p A ) p i , pB, n = -D.G. Hall 1883 Changes in rate are governed by changes in n*. In particular A In (rate) = A Inn* (C71nn*/~,ui)iij.p~,pB. eq and (2 In n* / Z P . A ) i i i . / i B . eq (6) Hence the quantities we wish to evaluate are derivatives such as where eq denotes that the derivatives are evaluated under the condition that eqn (2) is valid. The first derivative reflects solvent composition effects on the rate, whereas the second derivative gives the influence of changes in activity of reactants. We proceed by regarding n* as a function of pi,,uA,,uR and ,u*. This entitles us to write We note that and that according to eqn (2) We now use eqn (5a) and (5b) to substitute for derivatives of p* in eqn (8a) and (8b), feed the results into eqn (7) and then multiply both sides by (8,u*/&z*)pi,pA,pB, we obtain Since, under the conditions of interest eqn (2) and (6) apply, we find that eqn (10) gives (1 1) Eqn (1 1) is a general result expressing the dependence of the reaction rate of pressure and solution composition at constant T.The derivatives on the right-hand side are easily written in terms of integrals of pair distribution functions. For example, (3) = N? * = ni J: (g'(r) - 1) 4 nr2 dr (12) an+ PtVpAvpB where g'(r) is the pair distribution of i molecules around an activated complex. It is apparent from eqn (12) and (13) that the above treatment is entirely equivalent to that given in ref. (1). Indeed the final expressions differ only in notation. The application to electrolyte solutions and to reactions involving ions is straightforward and can be done in such a way that all thermodynamic quantities refer1884 Transition-state Theory at Interfaces to electrically neutral combinations of ions.All that is required is to replace the pi by the quantities gi and proceed as above. The 9, are defined by where v denotes ionic valency including the sign and c is some ionic species which can be chosen in any way that is convenient. It is usually best not to choose species c as a reactant or a product. Evidently 9, = 0. Also, for uncharged species it is obvious from eqn (13) that Bi = pi. When i and c have opposite signs, gi refers to the chemical potential of the neutral salt formed from i and c where the unit amount is that containing one mole of i. When i and c have the same sign, gi can be expressed as the chemical potential difference between a neutral salt of i and a neutral salt of c, where the units whose chemical potentials are compared contain the same amount of a common ion and again refer to one mole of i.When one or both reactants are ionic we write instead of eqn (2) 9' = Se*(T, gi, 9,, 9,) + RTln n* = 9, + 9,. (14) An equation of this form can be written for any normal species present in small quantities. The set of independent variables we have used so far, namely T and the pi or gi, have the advantage that they enable the relationship between the present approach and that based on the Kirkwood-Buff theory of solutions to be displayed in the simplest way. For many purposes, however, it is more convenient to work with the set of variables T, p and the solute chemical potentials.Thus at constant T and p we may write instead of eqn (3) dpo = - C Ci d9, - C* d9* i + O where Ci = ni/no, C* = n*/no and subscript 0 denotes solvent. We may now proceed exactly as above to obtain the expression where in this case the rate refers to that amount of solution containing one mole of solvent. The various derivatives with respect to C* are at constant T and p , and are related to the molecular distribution functions by In a recent treatment of micellar effects on reaction ratesg similar reasoning to that given above was used to discuss change in the micellar concentration of activated complexes. The treatment given here is more detailed and more thorough and provides the justification for some of the procedures used in the earlier work.Reactions at Interfaces The importance of reactions at interfaces can hardly be overstated. The simplest case to consider is that in which the reaction is sufficiently slow compared with mass transportD. G. Hall 1885 effects that the chemical potentials of all components in the system may be regarded as uniform throughout the system. Let N* be the total number of activated complexes in a system consisting of two bulk phases a and B separated by an interface of area A . For the overall rate in the system we write We note that N * , like the amount of any other species in the system, can be written as rate = k*N*. (18) where Va and V, are the volumes of the two phases, T* is the surface excess of activated complexes and n z and n$ are the bulk number densities.When k* is uniform throughout the system we may also think of an excess surface rate given by surface rate = k*T*A. (20) To formulate expressions for changes in r* at constant T we base our discussion on the Gibbs adsorption equation which we write in the form do =-CTid8i-rAd8A-rBd8B-T'd8* a where ci denotes surface tension and the Ti are chosen in accordance with an appropriate convention such as that used by Parsons12 and Hansen13 which at constant T and p eliminates the dp terms of the principal components of the two phases in contact. In analogy with eqn (2), (8) and (15) we suppose that This expression is entirely in accord with the assumption stated above that the activated complexes can be treated thermodynamically like any other normal species present in very small amounts.For such species the two-dimensional analogue of Henry's law is and it is readily shown that eqn (22) follows from this expression. is quite straightforward. The results are The derivation of expressions analogous to eqn (9) and (1 1) from eqn (22) and (23) and Eqn (24) shows that increasing Oi at constant 8,,8, and 8, increases the surface rate if the adsorption of activated complexes leads to an increase in Ti. Eqn (25) shows how increasing 8, affects the surface rate. In analogy with eqn (12) we have RTd In (surface rate) For reactions which proceed via a transition state, and which conform to the equilibrium hypotheses, eqn (24H26) provide a general, but not exact, treatment of how1886 Transition-state Theory at Interfaces bulk phase composition affects their rates at surfaces.These equations can be used in conjunction with any equilibrium theory or model which enables one to estimate the effect of adsorbing an activated complex on the amounts adsorbed of the other species present. So far we have assumed implicitly that I?* is positive. This covers most situations of practical interest. However, the case where r* is negative may also be treated. For a negatively adsorbed species a present throughout the system in very small amounts we expect k / h , to be independent of Inn, in the limit that n, + 0. Hence if activated complexes can be treated in the same way as normal species eqn (22) should still apply. However, since In T* is not now defined we must replace it instead by In (- r*).The left- hand side of eqn (26) now becomes RTdln(-surface rate) but the right-hand side remains unaltered. When r* is positive, the derivative (Xt/X*)T,ei,e ,e, may be regarded as the amount of i associated with an 'adsorbed' activated compqex in much the same way as the corresponding bulk quantity. In this context association may be used in a positive or negative sense. Thus when represents the amount of adsorbed i displaced by an activated complex. When I'* is negative the interpretation of (Wt/X*) in terms of association or displacement of i is less easily visualised. Comparison with Brensted-Bjerrum Theory According to the Brarnsted-Bjerrum theory,'O the rate of the reaction A + B + products, occurring in solution is given by 'A YA 'B YB rate = k Y* where yA, yB and y* denote activity coefficients defined with respect to an infinite dilution standard state and where k is the rate constant at infinite dilution.It follows from eqn (27) that However, by application of the Kirkwood-Buff theory to a mixture of uncharged species it is readily shown that RTdln(rate) = dpA+dpB-RTd1ny*. (28) RTdlny* = dp*-RTcilnC* When eqn (29) is substituted into eqn (28) we recover the analogue of eqn (16) for a mixture of uncharged species. When one or both reactants are charged, the y terms in eqn (27) refer to individual ionic species but the combination refers to a set of species which is electrically netural. However, eqn (29) no longer applies in a simple way. To recover eqn (16) in this case we proceed instead as follows.We define the quantities f A and J y by writing V* V C 8* = p*( T, p ) + RT In C* - - RTln Cc + RTlnfYwhere D. G. Hall lnfA = In yA --In V A Y c V , 1887 (31 4 (31 b) V* VC l n p = In?* --ln y,. WhenvA/vc is negative, lnfA is (1 - v,/v,) times the mean ionic activity coefficient of the neutral salt formed by A and C. Since v* = V, + v,, it is clear from eqn (31) that we may replace the y terms in eqn (27) by the corresponding f terms. Thus at constant T and p we may write RTd In (rate) = RTd In CAY, + RTd In CBfB -RTd lnf+ (32) but with eqn (30) and (31) this equation becomes V* VC RTd In (rate) = do, +do, +- RTd In C, - RTd lnf* = doA +do, -(do* - RTd In C*). (33) On substituting for the final term of this expression, which according to the Kirkwood-Buff theory is given by dB*-RTdInC* =-F(s) doi-(%) doA-(%) do, (34) ac* ei,eA,eB ac* e,,e,,e, ac* e,,o,,e, we recover eqn (1 6).The analogous expression to eqh (27) at surfaces is where yA is defined by writing and stipulating that yA --+ 1 as rA and all Ti + 0. However, eqn (35) is of limited value unless we have a way of handling changes in the y terms. For bulk solutions this is provided formally by Kirkwood-Buff solution theory. There is, however, no equally convenient formalism for surfaces. To provide such a formalism we may proceed as follows. Suppose we have a species a present which is identical to A in all of its interactions (e.g. a may be a radiolabelled form of A). For such a species we may write (2) dpa = dpA + RTd In (3 7) but pa may be regarded as a function of the variables pipA and r,.Hence we may write Now when a is present in very small amounts1888 Transition-state Theory at Interfaces also we have in general Eqn (40a) and (406). follow straightforwardly from the Gibbs adsorption isotherm. Eqn (37)-(40) enable us to write dpA-RTdInrA = dp,-RTdInr, which after some rearrangement gives The derivatives with respect to r, in this expression have exactly the same significance as the derivatives with respect to T* in eqn (24) and (26). Thus (WA/aTn)pA,p may be particular A molecule in the surface. [Note that the summations in eqn (38)-(42) include B.] For bulk solutions a derivation parallel to that of eqn (42) has been given previously.2 It is apparent from eqn (42) that Similarly for the transition state For charged species we may switch variables from the pi to the Oi in exactly the same way as outlined above for bulk solutions.When this is done eqn (26) is recovered from eqn (35) in exactly the same way as eqn (16) was recovered from eqn (27). A difficulty with eqn (35) which does not arise with eqn (27) is that some of the r terms may be negative. Examples of such species include co-ions adjacent to a charged surface, potential determining ions such as I- at the AgI/solution interface when the crystal is positively charged and electrons at the metal/solution interface when the metal is positively charged. A further difficulty is that activity coefficients for the latter two species cannot be defined with respect to an infinite dilution standard state.In contrast the more general approach described in the preceding section copes readily with such species and circumvents the above problems. When discussing reactions in bulk solution, one is usually interested in the dependence of rates on reactant concentrations. In this context eqn (27) is particularly useful and it is arguable that all the present treatment adds is a formal route to the various bulk activity coefficients. In contrast for reactions at surfaces, variables other than amounts adsorbed are often of interest. Hence, even when all terms in it are well defined and positive, eqn (35) may be less useful than its bulk solution counterpart. The approach developed in the preceding section has greater flexibility. It is not confined to particular sets of variables because these can easily be altered by standard thermodynamicD. G.Hall 1889 manipulations Clearly it amounts to much more than a formal prescription for calculating surface activity coefficients and in some cases it may actually be more convenient to use than more conventional approaches. Electrode Kinetics Typically the reactions involved in kinetics are redox reactions such as which involves the transfer of a single electron. 0 and R, respectively, denote the oxidised and reduced forms of a redox couple such as Fe3+ and Fe2+. Since reactions which involve the transfer of more than one electron usually occur as a sequence of elementary steps involving one or no electrons, and transition-state theory applies to elementary reactions only, the single-electron case is sufficiently general for our present purposes. It is sometimes found in practice that the net rate of reactions such as the above can be described by the well known Butler-Volmer equation.l4 (45) where the current density i is a measure of the net rate per unit area, q is the overpotential, io is the exchange current density which is in effect the equilibrium forward or backward rate when q = 0 and /? is the so-called symmetry coefficient which is expected to be between 0 and 1 and is typically of the order of 0.5. Eqn (45) is the difference between forward and backward rates where the forward rate is given by In pracitice it is sometimes found that p does not depend significantly on q, solution composition or temperature.However, there is no obvious reason why this must be the case. For the reaction 0 + e + R it is straightforward to relate eqn (46) and (26). We identify if with the rate, species A with 0 and species B with electrons. Thus 8, is given by However at constant T and p we may write do, = - F d E where E is the e.m.f. of a cell in which one electrode is the metal of interest and the other is an electrode reversible to species c. Also, since free electrons are not present in significant amounts in the bulk solution, the bulk composition remains constant when 6, is varied. In this case (49) where (ym- vS) is the inner potential difference between the bulk metal and the bulk solution. Eqn (49) relates changes in the overpotential q to changes in 8, at constant T, p and solution composition. do, = - FdE = - Fd(vrn - vS) = - Fdq The surface excess of electrons re is given by re = -q/F where q is the charge per unit area on the metal.1890 Transition-state Theory at Interfaces From eqn (26), (48) and (49) it is apparent that Likewise from eqn (46) we find that When p is independent of q we obtain on comparing eqn (51) and (52) Eqn (35) shows that surface excess of electrons. On electrostatic grounds one might expect is simply related to the effect of the activated complexes on the to be positive for a positively charged transition state in which case we should have p > 1.However, for reactions such as Fe3+ + e -, Fe2+ this turns out not to be the case in practice.14 We conclude therefore that if the assumptions underlying transition-state theory apply to reactions of this kind, then the activated complexes tend to repel electrons in the metal.A possible explanation is that the transition state has an asymmetric charge distribution with the negative part closer to the metal and that this negative part repels electrons to a greater extent than the more highly charged positive part attracts them with its greater net positive charge. The effect of the concentration of reactant A on the rate is given by R T ( Y ) , , = [ 1 + (s) 1. ar* eA,ei,E (54) In the common case where the contribution of A to the overall ionic strength of the system is small, we will have ( 5 5 ) do, = RTd In C, also will usually be very small unless there is either strong specific adsorption of A or, alternatively, association between A and the transition state.The reason for this is that (X,/X*) may be expressed as a volume integral which includes the potential of mean force between the transition state and A together with the number density of A throughout the region of influence of the transition state. If this number density is small, as is the case in the absence of specific adsorption, then the integral is also small unless the potential of mean force is strongly attractive, but this corresponds to association between A and the transition state. Such association should perhaps be included explicitly in the mechanistic equation for the reaction. When (X,/i3T*) is small, eqn (54) predicts that at constant T,p, Oi and E the rate will be proportional to C, as expected.However, eqn (54) also caters for situations in which this simple expectation is not realised.D. G. Hall I891 The effects of solution composition on the rate at constant T , p , E and 8, are described by the expression RT(T) 2 In rate =(s) O j , E Bi, E which follows straightforwardly from eqn (26). In some cases it may be possible to estimate the right-hand side of eqn (56) using double-layer theory. As an illustrative example we consider the case where none of the species i = I -+ c or A are specifically adsorbed and their Ti are given by Gouy-Chapman theory. In this case, for a given bulk composition, the Ti are determined by the primary surface charge q* given by 4* - F = v*r* -re. (57) The activated complexes are included in this equation because they are confined to the interface and count as a specifically adsorbed species.Their contribution to q* is, of course, very small indeed but it is necessary to include them if we wish to form derivatives with respect to r*. If follows that However from eqn (57) it is immediately apparent that hence By calculating (i3ri/c3q*)oi we may obtain an estimate of the left-hand side of eqn (60) which in turn enables us to estimate the effect of changing Oi on the rate. Alternatively, if it is assumed that (c3r,/i3q*),i in the reacting system is the same as (X&) in a solution without reactants, and the latter quantity can be estimated from studies of the double layer, then the left-hand side of eqn (56) may be estimated empirically. However, if i is specifically adsorbed it may interact specifically with the activated complexes in which case the above identification is likely to be invalid.The effects of solution composition on p can also be determined from eqn (26). This expression gives (g) - _ - - ' [ a - ( a ' i ) - 1 a8i ojv 0,. E F aE ar' o,,e,,E B,.e, We note that the derivatives in eqn (61) at const T, p and Bi refer to the situation where activated complexes are ' in equilibrium with reactants '. Under these circumstances any change in r* due to the change in E are entirely neglibible and the quantities can be found simply from the effect of surface charge on the right-hand side of eqn (60). In a recent paper various model-based predictions of (ap/aT) have been discussed and1892 Transition-state Theory at Interfaces compared with experiment.15 An expression for this quantity can be obtained via the approach developed above.The expression concerned relates (t$/aT) to @AS* / d E ) where AS* is the entropy of activation. By so doing it provides a means of estimating this latter derivative but is somewhat sterile as a basis for predicting (8p/i3T). Consequently we do not pursue this issue further. Non-uniform Chemical Potentials So far we have discussed the case of reactions which are sufficiently slow that the reactant chemical potential may be regarded as uniform throughout the system. However there are many instances of reactions at interfaces where transport of one or both reactants to the surface is rate-determining. Conceptually this situation can be dealt with by regarding the overall rate as the volume integral of the local rate and arguing that the equilibrium hypothesis applies locally.This viewpoint is satisfactory provided that the surface may be regarded as in equilibrium with the solution immediately adjacent to it even though there may be diffusive transport of reactants from the bulk. For systems in which there is an energy barrier to adsorption the above situation may not apply. To deal with this case we proceed by regarding adsorbed molecules as separate species from their bulk counterparts which just happen to have equal chemical potentials at equilibrium. Similarly, activated complexes formed from adsorbed reactants may be regarded as separate species from bulk complexes. Thus if the reactive events can take place only between reactants both of which are adsorbed or both of which are in the bulk the overall rate is the sum of two terms involving the two types of complex.For this type of situation, in analogy with eqn (2) and (14), we write the equilibrium hypothesis as ~JA+G = ezu (62 a) 0; + 0; = 6Zb (62 b) where superscripts cz and b refer to adsorbed and bulk material, respectively. The reason for the dual subscripts 00 and bb will emerge as the discussion proceeds. If eqn (22) applies to activated complexes formed from adsorbed reactants and also to activated complexes formed from bulk reactants it follows that we may write where r'20 is the amount per unit area of complexes formed from adsorbed reactants and In general we expect @: to be dependent on T,p, the composition of the bulk solution immediately adjacent to the interface, on f$ and on the chemcial potentials (or surface excesses) of any other adsorbed species which are not in equilibrium with the adjacent bulk solution.Similarly 6; in eqn (63b) can be expected to depend on < and etc. as well as on the intensive properties of the solution concerned. Suppose now that we may also have reaction between adsorbed A and bulk B and between adsorbed B and bulk A. These reaction pathways are not allowed for in the above discussion. To take them into account we write, in anology with eqn (62) and (63), is the surface excess of complexes formed from bulk reactants. where routes concerned. and r'Cu are amounts per unit area of activated complexes formed via theD. G.Hail 1893 Now in analogy with eqn (20) we write for the overall excess surface rate per unit area (65) surface rate = k"[Tzo + I-& + rzb + If we so wish a similar subdivision of complexes may be made for a reaction which proceeds at equilibrium. For this case we may write 0: = e* +RTlnr: (66) where e* depends on T,p, O,, 0, and the other Oi and where r: is given by with the subscript e denoting that surface and bulk are in complete equilibrium. Now in this case we also have (68) 0: == g* = g* - g* = g* It follows therefore that uo bb -- uu bu' which shows how the various t P f are related when surface and bulk reactant chemical potentials are equal. The assumptions underlying this treatment are firstly that it makes sense to talk about an adsorbed state as distinct from a bulk state, and secondly that the chemical potential gradients in the solution immediately adjacent to the interface are such that changes in chemical potential with distance are negligible compared with the range of molecular interactions. However eqn (69a)-(69d) do not apply when surface and bulk reactant chemical potentials are unequal and the various standard chemical potentials depend on 6, and O R ' Application to Solutions of Macromolecules For solutions of non-interacting macromolecules which may bind or otherwise interact with reactants, a combination of multiple equilibrium theory and the Kirkwood-Buff theory of solutions leads at constant T and p to the expression d(0,- RTln C,) = - C ( N i + ri) dgi - N* dO* 1 where Ni denotes the average amount of i specifically bound per macromolecule and Ti denotes the average relative 'adsorbtion ' of i per macromolecule from bulk solution defined according to the convention that the adsorption of solvent is zero.Eqn (70) is in fact almost identical to eqn (34) of ref. (9) which in turn is based on the thermodynamic theory developed in ref. (7) and (8). Apart from the explicit inclusion of the term for the activated complexes, the only other difference arises from the factthat each aggregate contains one, and only one, macromolcule so that C, = C, and Pp =1894 Transition-state Theory at Interfaces NP = 1. Evidently eqn (70) is isomorphic with eqn (21) and can be handled in the same way. Solutions of interacting macromolecules pose additional problems.In principle these can be handled by making use of the general thermodynamic formalism developed in ref. (16) and (8), with the latter adapted as suggested therein for the case under consideration here. Alternatively, fairly dilute solutions of highly charged macro-ions of moderate ionic strengths can be dealt with using the thermodynamic treatment developed in ref. (17) together with the corrections for non-ideality developed in ref. (8). Although as it stands this treatment is only strictly applicable to solutions in which there is only a single counter-ion species, it can be extended to deal with counter-ion mixtures in the same way as the corresponding treatment for micelles has been extended. To deal with particular systems further specific details can be introduced as required.The above discussion provides a tractable framework on which such detail can be built. For example, eqn (1 9) may be applicable to enzyme-catalysed reactions but is unlikely to be useful unless explicit information is available concerning the environment of the active sites of the enzyme molecules. Conclusion The above treatment is neither exact nor in itself predictive. However, it is rigorous within the confines of transition-state theory provided that the effects of environment on the transition coefficient K make only a small contribution to changes in rates. Consequently, any model based on transition-state theory should be consistent with the above treatment unless large changes in IC are involved. The theory thus provides a framework in which models can be developed.It also acts as a brake on plausible but ill founded speculation. The approach can also be used to help interpret data. A good example is (53) which leads to the conclusion that for the reaction considered a positively charged transition state in effect repels electrons. Such a conclusion requires some explanation. However, in the present context the correctness or otherwise of the explanation proposed above is less important than the conclusion itself and the way in which it was obtained. This conclusion can be avoided only by arguing either that transition-state theory does not apply to the type of redox reaction considered or that K is very sensitive to changes in the electrode potential. For reactions in solution the approach developed above is entirely equivalent to the alternative formulation of the same issues based on the Kirkwood-Buff theory of solutions.' The latter approach can also be extended to reactions at interfaces and again the results should be entirely equivalent to those given in this paper. However, the Kirkwood-Buff theory as applied to interfaces, is couched in terms of local number densities which must be integrated across the interface to obtain the overall interfacial rate.The approach based on the Gibbs adsorption equation given above has the advantage that it avoids this complication. Consequently it is much more concise. It is also better suited to deal with discontinuities in chemical potentials arising from activation energies of adsorption. Finally we note that any thermodynamic theory which enables the effects of composition on p** to be calculated can be used in conjunction with the approach developed above to predict the effects of composition on reaction rates. References 1 D. G. Hall, J . Chem. SOC., Faraday Trans. 2, 1986, 82, 1297. 2 J. G. Kirkwood and F. P. Buff, J . Chern. Phys., 1951, 19, 774. 3 D. G. Hall, Trans. Faraday SOC., 1971, 67, 2516.D. G. Hall 1895 4 T. L. Hill, Thermodynamics of Small Sj-stems (Benjamin, New York 1963, 1964), vol. 1 and 2. 5 D. G. Hall and B. A. Pethica, Nonionic Surfhctants, ed. M Schick (Marcel Dekker, New York, 1967), 6 D. G. Hall, Trans. Faraday Soc., 1970, 66, 1351 ; 1359. 7 D. G. Hall, J . Chem. Soc., Faruday Trans. 2, 1972, 68, 1439; 1977, 63, 897; 1981, 77, 1121. 8 D. G. Hall, Aggregation Processes in Solution, ed. E. Wyn-Jones and J. Gormally (Elsevier, Amsterdam, 1983), chap. 2. 9 D. G. Hall, J . Phys. Chem., 1987, 91, 4287. chap. 16. 10 J. N. Brcansted, Z . Phys. Chem., 1922, 102, 109; 1925, 115,537; N. Bjerrum, 2. Phys. Chem., 1924,82, I 1 S . Glasstone, K. J., Laidler and H. Eyring, The Theory of Rate Processes (McGraw-Hill, New York, 12 R. Parsons, Can. J . Chem., 1959, 37, 308. 13 R. S. Hansen, J . Phys. Chem., 1962, 66, 410. 14 J. O’M. Bockris and A. K. N. Reddy, Modem Electrochemistry (McDonald, London, 1970), vol. 2. 15 J. O’M. Bockris and A. Gochev, J. Phys. Chem., 1986, 90, 9232. 16 D. G . Hall, J . Chem. SOC., Faraday Trans. 2, 1974, 70, 1526. 17 D. G. Hall, J . Chem. Soc., Faraday Trans. I, 1985, 81, 885. 108. 1941). Paper 7/00136C; Received 18th September, 1987