Activation free energy theory

This section contains a brief review of the molecular version of Marcus theory, as developed by Warshel [81]. The free energy surface for an electron transfer reaction is shown schematically in Eigure 1, where R represents the reactants and A, P represents the products D and A , and the reaction coordinate X is the degree of polarization of the solvent. The subscript o for R and P denotes the equilibrium values of R and P, while P is the Eranck-Condon state on the P-surface. The activation free energy, AG, can be calculated from Marcus theory by Eq. (4). This relation is based on the assumption that the free energy is a parabolic function of the polarization coordinate. Eor self-exchange transfer reactions, we need only X to calculate AG, because AG° = 0. Moreover, we can write... 

The vertical axis is free energy, showing AGO for the net conversion of A to P, and AG, the activation free energy for each of the kinetic steps. According to Eyring s transition state theory (Chapter 7), AG is given by... 

If the interaction between the donor and acceptor in the encounter pair (D. .. A) is weak (Scheme 4.2), the rate constant kET can be estimated by the Marcus theory. This theory predicts a quadratic dependence of the activation free energy AG versus AG° (standard free energy of the reaction). 

The reorganization energy of a self-exchange reaction is denoted A(0) (from the fact that AG° = 0) and is an important quantity in the Marcus theory, where it can be shown that the activation free energy of a self-exchange reaction, AG(0), is equal to X.(0)/4. It is also possible to measure rate constants of self-exchange processes experimentally and thus get access to (0) via this relationship. 

My conclusion (based on double-layer theory) was that, under the usual experimental conditions, the image term is almost entirely screened off. When this is so, the dielectric contribution to the activation free energy is essentially the same for the homogeneous and the heterogeneous process. 

Early theories of electron transfer attempted to give a theoretical foundation to the phenomenological rate expression (Eq. 15) and the constants A, a, and o. A brief summary of the history and references to recent reviews have recently been given by Marcus. " Marcus theory has been able to derive an expression similar to Eq. (15), in which the relationship between the activation free energy and the reaction free energy is given in terms of a new quantity called the reorganization free... 

In Marcus original formulation of ET theory, the free energy curves Gj and Gj are assumed to be quadratic in x (linear response approximation). Using this assumption, Marcus derives the relationship between the activation free energy and the reaction free energy... 

The activation free energy of desorption may be computed from the rate of desorption as determined experimentally from the change in the surface potential with time. The theory of absolute rates has been applied to desorption by Eley (120) and Higuchi et cd. (107) to obtain energies and entropies of activation as a function of coverage. The rate of desorption is given by,... 

Our calculations of the activation free energy barrier for the cuprous-cupric electron transfer were not precise enough to permit a very accurate estimate of the absolute value of the exchange current for comparison with experiment. In principle, a determination of the absolute rate from the activation energy requires a calculation of the relevant correlation function [82] when the ion is in the transition region within the molecular dynamics model. We have not carried out such a calculation, but can obtain some information about the amplitude by comparing experiments with the transition state theory expression [84]... 

In terms of traditional Transition State Theory (TST) for solution reactions [40,41], in which e.g. the activation free energy AG can be estimated with equilibrium solvation dielectric continuum theories [42-46], the nonequilibrium or dynamical solvation aspects enter the prefactor of the rate constant k, or in terms of the ratio of k to its TST approximation kTST, k, the transmission coefficient, k and kTST are related by [41]... 

One of the key outcomes of this theory is the concept of critical size which must be achieved by an assembly of molecules in order to be stabilized by further growth. The higher the operating level of supersaturation the smaller is this size (typically a few tens of molecules). Now, in Fig. 2.9 the supersaturation with respect to II is simply Go - Gn and is lower than Go - Gj for structure I. However it can now be seen that if for a particular solution composition the critical size is lower for II than for I then the activation free energy for nucleation is lower and kinetics will favour form II. Ultimately form II will have to transform to form I, a process that we discuss later. Overall we can say that the probability that a particular form i will appear is given by... 

The activation free energy AG is dehned by two terms. One term is the reaction free energy [AG ], which is derived from the redox potential difference of the donor and acceptor. The second term is the medium reorganization energy (k), which is the energy stored in the solvent inertial degrees of freedom when the electron is shifted suddenly from donor to acceptor (1-5). Marcus theory predicts the activation free energy ... 

According to Marcus theory [7, 8], the activation free energy (AG ) of outer-sphere electron transfers depends on the free energy (AGet) and the reorganization energy (A) as follows (Eq. 90). 

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