Non-Equilibrium Thermodynamics Formalism for Marcus cross-exchange electron transfer reaction rates Richa Sethi and M.V.Sangaranarayanan * Department

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1 Non-Equilibium Themodynamics Fomalism fo Macus coss-exchange electon tansfe eaction ates Richa Sethi and M.V.Sangaanaayanan * Depatment of hemisty Indian Institute of Technology, Madas hennai India sangaa@iitm.ac.in FAX : ABSTRAT It is shown that Macus coss-exchange electon tansfe ate constant expession can be deived fom Onsage s linea Flux-Foce fomalism of Non- Equilibium themodynamics. The elationship among the Onsage s phenomenological coefficients fo coss-exchange electon tansfe pocesses is deduced and the significance of the methodology is pointed out.. Intoduction The estimation of coss-exchange electon tansfe ate constants using the constituent self-exchange ate constants occupies a pivotal ole in the theoy of eaction ates in view of its extensive validity,.the cossexchange ate constant k fo a edox eaction is elated to the self-exchange ate constants k and k as k (k k K f ) / W () whee K denotes the uilibium constant of the coss-exchange electon tansfe pocess while f and W consist of vaious wok tems involving the eactants and poducts. If it is assumed that f and W as is customay, a simplified n k (k k K ) / () is obtained, enabling the estimation of the coss-exchange ate constants without any adjustable paametes. The validity of n() has been extensively investigated fo divese types of eactions 3 and is paticulaly valuable when one of the two self-exchange ate constants is difficult to measue 4

2 In geneal, n () is consideed as a linea fee enegy elation on account of the linea dependence of the activation fee enegy upon the standad Gibbs fee enegy change. The oiginal fomalism of Macus leading to n () is based upon statistical mechanical consideations in conjunction with classical electostatics; howeve, seveal subsuent attempts have been made to analyze the functional dependence of the activation enegy on the intinsic baie and eaction coodinate vis a vis pogess vaiable. Notable among them ae the investigations of Rehm and Welle 5 Agmon and Levine 6 Thonton 7 and Mudoch 8 It is of inteest to note that the exponent to which the uilibium constant in n is aised may need not always be ual to 0.5 9,0.A notewothy featue of the Macus fomalism undelying n ()is that its basic pemise holds good not only fo electon tansfe, but also fo methyl tansfe,hydide tansfe, poton tansfe 3 etc. In this Lette, we demonstate that the coss-exchange elation () has a non-uilibium themodynamics pespective when viewed using Onsage s flux-foce fomalism 4. Futhe, the methodology popounded hee, indicates that thee exists a deepe theoetical basis undelying n ().. Non-uilibium Themodynamics fomalism fo coss-exchange electon tansfe eactions The desciption of chemical kinetic schemes using non-uilibium themodynamics concepts has pofound significance in so fa as it povides a geneal famewok in an unified manne. Fo example, the impotance of fluctuations fom uilibium states and the concept of coupled and noncoupled bio-chemical eactions ae elegantly bought about solely fom the magnitude of Onsage s phenomenological coefficients 4,5.. hemical kinetics desciption of coss-exchange eaction onside the coss-exchange electon tansfe eaction epesented as k p p (3) A B A B

3 whee A and B epesent the two edox couples while and p denote the eactant and poduct states. Analogously, the constituent self-exchange eactions ae as follows: A A A A a p k p (3 ) B B B B b p k p (3 ) The uilibium constant fo the coss-exchange eaction is given by K p p A B A B The velocity of the eaction (3) is v k (5) A B whee k is the coss-exchange electon tansfe ate constant. In nonuilibium themodynamics fomalism fo chemical kinetics, it is customay to intoduce the depatue fom uilibium concentations of the species involved in the eaction. onsuently, n(4) becomes (4) ( )( v k ) A A B B k A B [ ( ) ( )] A B A B k A B A B ( ] A B A B A B Since v k A B, we may wite v v ; k (5a) A B A B ( ) A B A B A B Fo small depatues fom uilibium viz. A A << and B B <<, the above uation can be appoximated as A B ; ( ) (6) A B A B v k

4 This uation is analogous to the velocity expession fo a fist ode eaction 4. Onsage s flux-foce fomalism fo coss-exchange electon tansfe eactions In ode to obtain insights povided by non-uilibium themodynamics, it is customay (efeence) to conside the same eaction using the flux-foce fomalism of Onsage as has been discussed fo a fist ode eaction 5.. Identification of Onsage s phenomenological coefficients A ( µ µ ) ( µ µ ) The Affinity of the eaction ( 3 ) is defined as 6 p p A B A B whee µ A, µ B etc. denote the chemical potentials of the indicated species. Since the liquid phase eactions ae consideed heein, we may employ the concentations of the species. onsuently, we wite the chemical potential fo A as µ µ RTln 0 A A A if we neglect the activity coefficient coections. Analogous uations hold good fo µ, and µ p µ p B A B Thus, the Affinity of the eaction becomes A [ µ RTln µ RTln ] [ µ RTln µ RTln ] (7) o o o o p p p p A A B B A A B B Intoducing the depatue fom the uilibium concentations A A etc. we obtain A RT RT A B [ µ ln( ) µ ln( )] A B A B p p A B [ µ p RTln( ) µ RTln( )] (8) A B p p p A B Howeve, at uilibium,

5 µ µ µ µ p A A B p B (9) Fo nea-uilibium conditions, we may expand the logaithmic tems and neglect tems othe than linea as is customay in the non-uilibium themodynamics desciption of chemical kinetics 5. onsuently, n (9) becomes A RT RT p A B A p B ; ( ) ( ) (0) p p A B A B Since the eaction (3) is consideed to be an ievesible pocess (cf. n A9 of Macus, the above uation becomes A RT A B ; ( ) A B Since the velocity is linealy elated to the Affinity in the linea flux-foce fomalism, we may wite v L A Howeve at uilibium, A 0; hence ( v v ) L A.Hence v v L RT A B ( ) () A B ompaing ns (6) and () we obtain L k A B () RT This uation is eminiscent of the Onsage s coefficient fo evesible fist ode eaction X Y with k and k - denoting the fowad and evese ate constants; in this case 7, L k x /RT. In an analogous manne, Onsage s coefficients fo the two self-exchange eactions may be witten as L k p B B ( 3 ) and RT

6 L k p A A (4) RT. Relation among Onsage s phenomenological coefficients fo cossexchange electon tansfe eactions Since the coss-exchange eaction is composed of the two self exchange eactions and if the pinciple of micoscopic evesibility is valid in this context, (5) v v v which implies that k k k RT RT RT A p p B A A B B Thus L L L viz. L (6) L L (7) Substituting the appopiate expessions fo L s in tems of ate constants, we obtain k k k K (8) which is identical with uation () aising fom Macus theoy. 3. Results and Discussion The foegoing analysis demonstates that it is possible to deduce Macus coss-exchange elation solely fom Onsage s Flux- Foce fomalism when cetain appoximations ae intoduced. Inteestingly, expeimental tests of n() do indicate satisfactoy validity in geneal 3 and the assumptions made in the pesent methodology seems easonable. It is of inteest to enquie whethe any new insights have emeged fom the appoach suggested hee. Fistly, in the pesent vesion, the depatue fom uilibium concentations of the eactants wee explicitly intoduced and these wee assumed small enabling us to neglect highe ode tems in the expansion of the Affinity tem in tems of the uilibium concentations. A possibility that emains un-clea is whethe incopoation of these will yield the complete Macus expession () consisting of the wok

7 tems too. Secondly, it is customay 7 to deive and estimate the Onsage s phenomenological coefficient fo evesible fist ode eaction X Y wheein the phenomenological coefficient L k c x /RT whee k is the fist ode ate constant in the fowad diection, x being the uilibium concentation of X. The fact that an analogous execise is feasible fo a cossexchange eaction (composed of the two self-exchange pocesses)involving two diffeent eactants may indicate that non-uilibium themodynamics fomalism is capable of yielding entiely new insights undelying moe complicated chemical kinetic schemes. One of the methods of veifying Onsage s Recipocity Relation (ORR) consists in analyzing a tiangula cyclic eaction scheme of coupled chemical eactions. On the othe hand, the phenomenological coefficients ae elated in the pesent context via n(7). The ate of entopy poduction of coss-exchange electon tansfe pocesses can be estimated using the above pesciption fo the Affinity and Onsage s coefficient; this may povide new insights in cetain cases. Since thee is a coelation between the homogeneous and heteogeneous ate constants, it appeas that the pesent appoach can be employed mutatis mutandis to heteogeneous chemical kinetics too. In this context, it is woth noting that the Flux-Foce fomalism of Onsage has been employed to deive Butle-Volme uation aising in electode kinetics as shown by Keize elsewhee 8. Analogously, the pesent fomalism has led to the analysis of a hieachy of diffusion-migation uations fo electon hopping in edox polyme electodes 9,0. This implies that incopoation of the electochemical potentials in the Affinity expessions would yield a coelation between homogeneous and electochemical ate constants. It is woth emphasizing that the non-uilibium themodynamics fomalism applied hee povides a pointe to the geneal validity of Macus appoach fo diffeent classes of eactions such as electon tansfe, poton tansfe, methyl tansfe, hydide tansfe etc. 4. onclusions The linea Flux-Foce fomalism of Onsage is shown to yield the coss-exchange electon tansfe ate constant expession of Macus.

8 The phenomenological coefficients ae identified and the ole played by the (depatue fom) uilibium concentations of the species is indicated. REFERENES..Macus R.A, J.hem.Phys 965, 43, 679..Macus R.A and Sutin N Biochim.Biophys.Acta 985, 8, See fo example Bochat D and Wheland S. Inoganic hem., 986, 5, 90 and efeences theein. 4. See fo example Simmoneaux Gead and Bondon Anaud hem.rev., 005, 05, Rehm D and Welle A. Is.J.hem 970, 8, Agmon N and Levine R.D, hem.phys.lett., 977, 5, Thonton E.R, J.Ame.hem.Soc., 967, 89, Mudoch J.R, J.Ame.hem.Soc., 984, 06, Rossinsky D.R,hem.ommun., 97, Phani K.L.N and Sangaanaayanan M.V.J.Electoanal.hem 99,38, 3...Lewis Edwad S J.Phys.hem 986, 90, Han Lee In Sook, how Kim Hung and Keevoy Mauice J.Ame.hem.Soc., 00,4, Suhnel Jugen J.Phys.Og.hem., 993, 6,8. 4.Onsage L, Phys.Rev., 93,37, 405; ibid, 93, 38, Katchalsky A and.uan Pete F Non-uilibium Themodynamics in Biophysics Havad Univesity Pess ambidge, Massachusetts 965 hapte Pigogine I. and Defay R,hemical Themodynamics Wiley-New Yok McQuaie Donald.A and.simon John D Molecula Themodynamics- Viva Books Pivate Limited, 004 hapte Keize Joel Statistical Themodynamics of Non-Equilibium Pocesses- Spinge Velag Belin 987 p Aldin Denny R and Sangaanaayanan M.V.J.Phys.hem 998,0,3.

9 0 Uma Maheswai, J and Sangaanaayanan M.V.J.Phys.hem 00,05, 465..