Activation free energy Marcus equation

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, Relationship of the activation free energy for electron transfer with the electrode potentials of various FeL33 according to Equation 6 (left), and the driving force according to the Marcus Equation 4 (right).

It has in fact been anticipated for many years that the CT free energy surfaces may deviate from parabolas. A part of this interest is provoked by experimental evidence from kinetics and spectroscopy. Eirst, the dependence of the activation free energy, Ff , for the forward (/ = 1 ) and backward i = 2) reactions on the equilibrium free energy gap AFq (ET energy gap law) is rarely a symmetric parabola as is suggested by the Marcus equation,Eq. [9]. Second, optical spectra are asymmetric in most cases and in some cases do not show the mirror symmetry between absorption and emission.In both types of experiments, however, the observed effect is an ill-defined mixture of the intramolecular vibrational excitations of the solute and thermal fluctuations of the solvent. The band shape analysis of optical lines does not currently allow an unambiguous separation of these two effects, and there is insufficient information about the solvent-induced free energy profiles of ET. 

Equation 5.7 Classical Marcus relationship for the activation free energy of electron transfer... 

The results of kinetic studies on the reduction of the siderophores ferrichrome and ferrichrome A by Eu(II) were presented by Kazmi et al. (1984). The comparison of the kinetic data for the reduction of the same siderophores with V(II) and Cr(II) led the authors to postulate an outer-sphere electron transfer mechanism providing an adequate interpretation of the data. Apparent values of the activation free energy for the self-exchange reaction of Eu(II)/Eu(III) were calculated from the reduced form of the Marcus equation. 

Of more direct interest to kineticists is a reconsideration by Marcus and Sutin of the basis for the well known equation (1) relating activation free energies AG to... 

The classical Marcus equation, derived from the basic model by the simplest treatment, in which all motions are treated by classical mechanics, is Equation (9.46), in which AG is the theoretical activation free energy and AG is the free energy of reaction ... 

In Eq. (7-21), AGo is the intrinsic barrier, the free energy of activation of the (hypothetical) member of the reaction series having AG" = 0. It is evident that the Marcus equation predicts a nonlinear free energy relationship, although if a limited... 

Equation (3.44) (in the Arrhenius form) is usually called the Marcus equation [74,75]. A special feature of the Marcus equation is that it predicts the parabolic dependence of the activation energy AEa on the free energy change AG/, that is, AEa is related to the free energy change AG/ in a parabolic form. 

Note that calculated energies have been substituted for the more normal free energies of Marcus theory. Equation (2) predicts a low 7.2 kcalmoPM activation energy for the 1 -+ 2. The experimental work in Freon glasses did not reveal the existence of 1, although recent work has shown that it does have a finite lifetime in solution and that the estimated rearrangement barrier is 4.8 kcalmo] [12],... 

Arnett and coworkers later examined the reaction of lithium pinacolone enoiate with substituted benzaldehydes in THE at 25 °C. The determination of the heat of reaction indicated that the Hammett p value for the process is 331. Although the aldol reaction was instantaneous in THF at 25 °C, the reaction with o- or p-methylbenzaldehyde could be followed using a rapid injection NMR method in methylcyclohexane solvent at —80 °C. Application of Eberson s criterion based on the Marcus equation, which relates the free energy of ET determined electrochemically and the free energy of activation determined by kinetics, revealed that the barriers for the ET mechanism should be unacceptably high. They concluded that the reaction proceeds via the polar mechanism . Consistent with the polar mechanism, cyclizable probe experiments were negative . The mechanistic discrepancy between the reactions of benzaldehyde and benzophenone was later solved by carbon kinetic isotope effect study vide infraf. ... 

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]. 

Modern theoretical developments in the theory of proton-transfer reactions suggest that such linear Bronsted plots are only a first approximation when the range of the P-KHA-values is narrow. When a wider range of bases is used, the curve obtained should be such that the Gibbs free energy of activation AG fits the Marcus eqn (6). This equation was derived (Cohen and Marcus, 1968 ... 

The simple geometric properties of parabolas enable the free energy of activation to be expressed as A Gf = (A G° + A)2/4A, in which case, equation (1) may be rewritten as equation (2), generally called the classical Marcus equation ... 

Rate constants for electron transfer may be related to the free energy AG° of the reaction through the classical Marcus equation Eq. (5), where AGq is the intrinsic activation barrier of the reaction process [90, 91]. 

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