Marcus theory classical equation

Volmerian electrode reaction — This term has been used for electrode reactions in which the - charge transfer coefficient is constant. Reactions for which the latter is potential dependent were called non-volmerian. According to the - Marcus theory there is generally a potential dependence of the charge transfer coefficient, however that is usually very small. The terms Volmerian and non-volmerian refer to the classic Butler-Volmer theory (-> Butler-Volmer equation) where no potential dependence was assumed. See also -> Volmer. 

V. Like the classical Marcus theory, equation 1 predicts an inverted region, although the decrease of rates for highly exoergic reactions may be less pronounced than in the Marcus theory. The quantum mechanical theory also predicts modifications of the effects of temperature and polarity. Some principal features of these predictions have been verified by experiments using both pulse radiolysis and laser photoexcitation. 

The semi-classical Marcus equation derives from quantum-mechanical treatments of the Marcus model, which consider in wave-mechanical terms the overlap of electronic wave-functions in the donor-acceptor system, and the effects of this overlap on electronic and nuclear motions (see Section 9.1.2.8 above). Such treatments are essential for a satisfactory theory of D-A systems in which the interaction between the reactant and product free-energy profiles is relatively weak, such as non-adiabatic reactions. A full quantum-mechanical treatment, unfortunately, is cumbrous and (since the wave-functions are not accurately known) difficult to relate to experimental measurements but one can usefully test equations based on simplified versions. In a well-known treatment of this type, leading to the semi-classical Marcus equation introduced in Section 9.1.2.8, the vibrational motions of the atomic nuclei in the reactant molecule (as well as the motions of the transferring electron) are treated wave-mechanically, while the solvent vibrations (usually of low frequency) are treated classically. The resulting equation, already quoted (Equation (9.25)), is identical in form with the classical equation (9.16) (Section 9.1.2.5), except that the factor... 

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]. 

We have thus found that in this high temperature, strong electron-phonon coupling limit the electronic transition is dominated by the lowest crossing point of the two potential surfaces, that is, the system chooses this pathway for the electronic transition. It is remarkable that this result, with a strong characteristic of classical rate theory, was obtained as a limiting fonn of the quantum golden-rule expression for the transition rate. Equation (12.69) was first derived by Marcus in the context of electron transfer theory (Chapter 16). 

The recent theoretical approaches include a theory of barrierless electronic relaxation which draws on the model of nonradiative excited state decay, and a general treatment of the effect of solvent dielectric relaxation based on the theory of optical line shapes, as well as treatments based on classical and quantum rate theories. Equation(5) does not hold for all solvents and, more generally, may be frequency-dependent. Papers by Hynes, Rips and Jortner, Sumi and Marcus, and Warshel and Hwang " contain good overviews of the theoretical developments. 

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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