Marcus-Hush equation

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. 

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... 

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

Within exact C2 symmetry, the two states may cross. The system can, however, deviate from this ideal symmetry to allow coupling between the two states so that the crossing is avoided. If we equate the 81 -> 82 excitation energy in 1 to the reorganization energy. A, of Marcus theory [11], we can use the Marcus-Hush expression [Eq. (2)] to estimate the activation energy, AE, based on the calculated values for A and the heat of reaction, AE —12kcalmol" ) ... 

The famihar equihbrium form of the Marcus-Hush-Jortner equation is simply In... 

In the original Marcus-Hush model, the role of the solvent is described by the static equation of reorganization, Eq. 13 b. The time scale of the reorganization does not appear explicitly and this has been included in more recent treatments of e.t. processes. There are indeed several reports of e.t. reactions of which the rates seem to be controlled by solvent relaxation, related to the longitudinal relaxation time [72]. 

The classical (or semiclassical) equation for the rate constant of e.t. in the Marcus-Hush theory is fundamentally an Arrhenius-Eyring transition state equation, which leads to two quite different temperature effects. The preexponential factor implies only the usual square-root dependence related to the activation entropy so that the major temperature effect resides in the exponential term. The quadratic relationship of the activation energy and the reaction free energy then leads to the prediction that the influence of the temperature on the rate constant should go through a minimum when AG is zero, and then should increase as AG° becomes either more negative, or more positive (Fig. 12). In a quantitative formulation, the derivative dk/dT is expected to follow a bell-shaped function [83]. 

An alternative theory to the Butler-Volmer theory for electron transfer is provided by the Marcus-Hush theory (Marcus, 1968 Hush, 1968) which assumes a potential-dependent a. Since in most cases a is essentially independent of potential, use of the simpler Butler-Volmer equation is usually adequate. 

Fig. 4 Plot of RT In vj aGj3 in CH3CN (n = 0.1 M) at 22 2°C for the quenching of [Ru (bipy)J+] by viologens. The theoretical line is that obtained by using the Marcus-Hush equation having k o) = 8.4 x 10 M" s and X = 17 kcal mol . (From Bock et ai, 1979b)...

In the absence of ion pairing and rate limitation by solvent dynamics, the volume of activation for adiabatic outer-sphere electron transfer in couples of the type j (z+i)+/z ju principle, be calculated as in equation 2 from an adaptation of Marcus-Hush theory. In equation 2, the subscripts refer respectively to volume contributions from internal (primarily M-L bond length) and solvent reorganization that are prerequisites for electron transfer, medium (Debye-Huckel) effects, the Coulombic work of bringing the reactants together, and the formation of the precursor complex. 

The Marcus-Hush equation (which we shall not derive) is given by expression 25.58 and applies to outer-sphere mechanisms. 

Equation 25.59 gives a logarithmic form of equation 25.58 often, / 1 and so log / 0, allowing this term to be neglected in some cases. Thus, equation 25.60 is an approximate form of the Marcus-Hush equation. 

Manganese(IV) complexes magnetic behavior, 272 Manganese(V) complexes magnetic behavior, 272 Mannich reaction metal complexes, 422 Marcus cross-reaction equation electron transfer, 355 Marcus Hush theory electron transfer, 340 Masking... 

Wul-6 (In contrast to Marcus-Hush which refers to the theory of electron transfer activation, the Mulliken-Hush equation describes the preexponential factor of the rate constant. We spell out Mulliken-Hush each place it occurs in this chapter and use the acronym MH to refer to only Marcus-Hush.) In practice, however, FCWD(O) cannot be extracted from experimental spectra, and one needs a theoretical model to calculate FCWD(O) from experimental band shapes measured at the frequencies of the corresponding electronic transitions. This purpose is achieved by a band shape analysis of optical lines. 

The Marcus-Hush formula (11) is an integral of equation (9) assuming spherical ions of radii and 2 separated by distance R, but neglecting image... 

The classical, Marcus/Hush, limit corresponds to Equation (1) with /Cei= 1 and = (FC)r=o-This condition is achieved if either (i) the structural differences between the reactants and products do not implicate high-frequency vibrational modes or (ii) the exchange of energy (heat) between the high-frequency vibrational modes and the solvent is fast on the time scale for electron transfer. Statement (ii) is equivalent to the equilibrium assumption of transition-state theory. That this assumption is not always correct for reactions in solution has been demonstrated in ultrafast kinetic studies of reactions that vary ... 

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