The Theory of Marcus

In this chapter, we wiU review electrochemical electron transfer theory on metal electrodes, starting from the theories of Marcus [1956] and Hush [1958] and ending with the catalysis of bond-breaking reactions. On this route, we will explore the relation to ion transfer reactions, and also cover the earlier models for noncatalytic bond breaking. Obviously, this will be a tour de force, and many interesting side-issues win be left unexplored. However, we hope that the unifying view that we present, based on a framework of model Hamiltonians, will clarify the various aspects of this most important class of electrochemical reactions. 

For outer sphere electron transfer reactions the Butler-Volmer law rests on solid experimental and theoretical evidence. An outer sphere electron transfer reaction is the simplest possible case of an electron transfer reaction, a reaction where only an electron is exchanged, no bonds are broken, the reactants are not specifically adsorbed, and catalysts play no role (see, e.g.. Ref. 2). Experimental investigations such as those by Curtiss et al. [206] have shown that the transfer coefficient of simple electron transfer reactions is independent of temperature. The theoretical basis is given by the theories of Marcus [207] and of Levich and Dogonadze [208] these theories also predict deviations at high overpotentials which were experimentally confirmed [209, 210]. 

A study of the irreversible reduction of several Co ", Rh" and Ir" complexes revealed no correlation between the polarographic Ey and several spectroscopic parameters but, interestingly, it was found that a linear correlation existed for several of the Co " complexes between the y, and In where was the rate constant for homogeneous electron transfer, when [Ru(NH3)6] was used as reductant. The theoretical foundation for this relationship is that Ey is linearly related to In (the heterogeneous rate constant for electrochemical reduction) and, from the theories of Marcus and Hush, the ratio of k for a series of compounds is the same as the ratio of the rate constants k for a constant reductant provided both pathways are outer sphere. The mechanistic implication of the relationship is not clear it may simply mean that both pathways proceed via an outer sphere mechanism as no correlation was found between y, and the values of kgx for reduction by which can undergo homogeneous electron transfer by an inner sphere mechanism. 

ISM encompasses, as particular cases, several current models of chemical reactivity such as the theory of Marcus [18], the BEBO [4a], Agmon and Levine [5a], Koeppl and Kresge [ 19] models and two dimensional models such as the ones of Kreevoy and Lee [20] and of Grunwald [21]. It is also in general accord with qualitative electronic theories of chemical... 

Two important modifications to the theory of Marcus, included in ISM, are known to account for some "disparity progress" in chemical reactions one is the asymmetry of the potential energy curves, fr fp [19,24,28] and the other is the linear dependence of d on (AG)2 [19]. Here we will consider another important factor, the siphoning of electtonic density at the transition state which makes nt>l/2. 

If one considers the Brpnsted coefficient, a =9AGV9AG, given by the theory of Marcus (eq(45)), one gets... 

To account for the disparity, Kreevoy and Lee consider -1< 6 1. Even within the limiting conditions of the theory of Marcus, the one-dimensional Intersecting-state Model can accommodate the disparity progress of the Kreevoy-Lee model negative values for the parameter 8 correspond to ni[Pg.176]

Hydride transfers are a very important category of reactions in organic and biological chemistry, and can be a model for electrophilic reactions, because they can be viewed as transfer of a proton with a pair of electrons between electron deficient sites [31], A + BH AH + B+. A large number of theoretical studies have addressed this topic, and the theory of Marcus gives a reasonable representation of the relation between rate and... 

Proton transfers in electronically excited states have not been amenable to any reasonable interpretation in terms of the theory of Marcus, in part due to the implicit assumption of the symmetry of the potential energy curves of reactant and product [24,38]. In contrast, ISM provides a simple interpretation of this kind of reactions [39]. The excited-state reactions appear to follow the same basic principles of their ground-state analogues the transition state bond order does not change appreciably from the ground to the excited state. However, the mixing entropy parameter X decreases an enhancement of the dipole moment upon eletronic excitation can increase the suddenness of the repulsive wall of the reaction and decreases X. 

In contrast with this view, in the theory of Marcus [18] one simply minimizes the potential energy of two harmonic oscillators... 

Equation (69) shows that within the theory of Marcus lc=N < 1 < lc=N. and when one substitutes eq(69) into eq(68) obtains an energy barrier... 

With the relevant force constant and bond length data [33], fc=N=l-lxlO and fc=N=6-3xl03 kJ mol l A 2, 1c=n=1-157 and 1c=n=1-34 A, eq(70) leads to a barrier AG = 67 kJ mol l which leads to a rate constant at room temperature of 20 s ca. 8 orders of magnitude slower than experiment. As Table 9 illustrates, in general, the theory of Marcus overestimates AGq when lred lox 0 and underestimates it when lred ox-... 

ISM can also be applied to the calculation of electron exchange reaction rates of coordination compounds and aquo-metal ions [60,62]. Table 10 presents some of the calculated data with the force constants estimated through eqs(38) and (41), with a coordination number of 6. The n values are identical to the order of the reactive bonds in reactants and products. In contrast with the theory of Marcus, our model provides calculated rates within an order of magnitude of experiment [60]. 

Equation (29) is also verified with several electron transfer reactions between coordinated metal ions [60,69]. The consideration of the role of the mixing entropy parameter can even explain anomalous "cross-reaction" estimates given by the theory of Marcus [60,70] and shines light on the controversy of the "inverted region" at low AG. 

ABSTRACT. A theoretical study of the electron self-exchange in porphyrins and in cytochrome c, and of the free-energy dependence of poq)hyrin-cytochrome c systems ows that these reactions are not easily amenable to an explanation in the framework of the theory of Marcus. On the other hand the intersecting-state model can be used to calculate the self-exchange rates and ] ovide an useful rationalization of the free-energy relationsh obs ed in these systems. 

The theoretical formalism proposed to estimate the rates of ET reactions is known as the theory of Marcus (TM) [3,5,6]. However it is relevant to make a distinction between two components of the theory. One component is concerned with the estimation of the intrinsic barrier, AG(0), for homonuclear reactions in terms of molecular parameters of the reactants, which we will call TM-1. The other component of the theory addresses the effect of the reaction energy, AG°, on the reaction rates, presented in Chapter 7 in terms of the quadratic expression of Marcus (eq. (7.6)) this will be called TM-2. It is currently employed to estimate the rates of cross-reactions, when the reaction energies of the heteronuclear reactions are known together with the rates of the corresponding homonuclear reactions. 

In general terms, the theory of Marcus involves a model for ET reactions based on the approximation that the inner-coordination sphere energy is independent of the outer-sphere reorganisation. In its classical formulation, TM provides the rate for a self-exchange reaction such as reaction (16.1)... 

The theory of Marcus was not only able to account for the kinetics of self-exchange reactions (TM-1), but could also estimate rates for moderately exothermic processes where AG° < 0. Marcus further develop the theory, TM-2, to relate the rates for which AG° = 0, to cross reactions... 

TM-1 and TM-2 are collectively known as the theory of Marcus. However, the approximations involved in the derivation of the two formalisms are different. The approximations involved in TM-2 have already been discussed with respect to Figure 15.16. 

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