Marcus-Hush

In many cases, the values of A n and k2i may be directly or indirectly determined. We shall say no more about this relationship here, other than to indicate that it proves to be generally applicable, and is sufficiently accepted that the Marcus-Hush equation is now used to establish when an outer-sphere pathway is operative. In the context of this chapter, the involvement of the Kn term is interesting for it relates to the relative stabilization of various oxidation states by particular ligand sets. The factors which stabilize or destabilize particular oxidation states continue to play their roles in determining the value of Kn, and hence the rate of the electron transfer reaction. 

Reactions of practical interest involve the breaking or formation of chemical bonds, which require extra energy. The theory of Saveant and its subsequent developments are an ingenious extension of the Marcus-Hush type of theory to the breaking of a simple bond. The binding energy enters into the energy of activation, but the interaction with the metal electrode is still assumed to be weak, i.e., the reactants are not adsorbed. In this sense, the reaction is not catalyzed by the electronic interaction with the metal. 

Azo-bridged ferrocene oligomers also show a marked dependence on the redox potentials and IT-band characteristics of the solvent, as is usual for class II mixed valence complexes 21,22). As for the conjugated ferrocene dimers, 2 and 241 the effects of solvents on the electron-exchange rates were analyzed on the basis of the Marcus-Hush theory, in which the t/max of the IT band depends on (l/Dop — 1 /Ds), where Dop and Ds are the solvent s optical and static dielectric constants, respectively (155-157). However, a detailed analysis of the solvent effect on z/max of the IT band of the azo-bridged ferrocene oligomers, 252,64+, and 642+, indicates that the i/max shift is dependent not only on the parameters in the Marcus-Hush theory but also on the nature of the solvent as donor or acceptor (92). 

The first attempt to describe the dynamics of dissociative electron transfer started with the derivation from existing thermochemical data of the standard potential for the dissociative electron transfer reaction, rx r.+x-,12 14 with application of the Butler-Volmer law for electrochemical reactions12 and of the Marcus quadratic equation for a series of homogeneous reactions.1314 Application of the Marcus-Hush model to dissociative electron transfers had little basis in electron transfer theory (the same is true for applications to proton transfer or SN2 reactions). Thus, there was no real justification for the application of the Marcus equation and the contribution of bond breaking to the intrinsic barrier was not established. 

As with the Marcus-Hush model of outer-sphere electron transfers, the activation free energy, AG, is a quadratic function of the free energy of the reaction, AG°, as depicted by equation (7), where the intrinsic barrier free energy (equation 8) is the sum of two contributions. One involves the solvent reorganization free energy, 2q, as in the Marcus-Hush model of outer-sphere electron transfer. The other, which represents the contribution of bond breaking, is one-fourth of the bond dissociation energy (BDE). This approach is... 

Figure 5. Energy diagram for charge separation resolved into reactant-like and product-like diabatic surfaces. The two diabatic curves do not intersect, but interact, to give an avoided crossing, whose energy gap is twice the electronic coupling, Vei, for the interaction. Also depicted is the Marcus-Hush classical rate expression for nonadiabatic ET.

Marcus equation, 19 112 Marcus-Hush theory, 13 430 Marflex, 7 636... 

FIGURE 1.13. Free-energy profiles in outer-sphere electron transfer according to the Butler-Volmer approximation (a) and to the Marcus-Hush model (b). 

When electron transfer is forced to take place at a large distance from the electrode by means of an appropriate spacer, the reaction quickly falls within the nonadiabatic limit. H is then a strongly decreasing function of distance. Several models predict an exponential decrease of H with distance with a coefficient on the order of 1 A-1.39 The version of the Marcus-Hush model presented so far is simplified in the sense that it assumed that only the electronic states of the electrode of energy close or equal to the Fermi level are involved in the reaction.31 What are the changes in the model predictions brought about by taking into account that all electrode electronic states are actually involved is the question that is examined now. The kinetics... 

Coming back to solvent reorganization, the reduction of aromatic hydrocarbons in an aprotic solvent such as DMF provides a series of data that may be used for testing the Marcus-Hush model of solvent reorganization13,61-63... 

When conformational change and electron transfer are concerted, the structural change may be treated as an internal reorganization factor in the electron transfer dynamics. This is the A, term of the Marcus-Hush model (Section 1.4.2 see also Section 1.4.4 for experimental examples). The model is applicable as long as the conformational changes are not so strong as to invalidate the harmonic approximation. 

This favorable situation may not be encountered in every case. With radical reductions endowed with high intrinsic barriers, the half-wave potential reflects a combination between radical dimerization and forward electron transfer kinetics, from which the half-wave potential cannot be extracted. One may, however, have recourse to the same strategy as with the direct electrochemical approach (Section 2.6.1), deriving the standard potential from the half-wave potential location and the value of the transfer coefficient (itself obtained from the shape of the polarogram) under the assumption that Marcus-Hush quadratic law is applicable. 

In the case of stepwise electron-transfer bond-breaking processes, the kinetics of the electron transfer can be analysed according to the Marcus-Hush theory of outer sphere electron transfer. This is a first reason why we will start by recalling the bases and main outcomes of this theory. It will also serve as a starting point for attempting to analyse inner sphere processes. Alkyl and aryl halides will serve as the main experimental examples because they are common reactants in substitution reactions and because, at the same time, a large body of rate data, both electrochemical and chemical, are available. A few additional experimental examples will also be discussed. 

It is thus, in principle, possible to derive from the potential location and from its shape all the parameters contained in the Marcus-Hush model, namely, the standard potential, , and the intrinsic barrier, AGq (Klinger... 

Let us again emphasize the connection between the Marcus-Hush model. 

In other words, under these restrictive conditions, outer sphere electron-transfer reactions obeying the Marcus-Hush model are typical examples where the Hammond-Leffler postulate and the reactivity-selectivity principle (see, for example, Pross, 1977, and references cited therein, for the definition of these notions) are expected to apply. 

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