Free energy-reaction coordinate relations

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... 

In the introductory chapter we stated that the formation of chemical compounds with the metal ion in a variety of formal oxidation states is a characteristic of transition metals. We also saw in Chapter 8 how we may quantify the thermodynamic stability of a coordination compound in terms of the stability constant K. It is convenient to be able to assess the relative ease by which a metal is transformed from one oxidation state to another, and you will recall that the standard electrode potential, E , is a convenient measure of this. Remember that the standard free energy change for a reaction, AG , is related both to the equilibrium constant (Eq. 9.1)... 

Recently, much attention has been paid to the investigation of the role of this interaction in relation to the calculations for adiabatic reactions. For steady-state nonadiabatic reactions where the initial thermal equilibrium is not disturbed by the reaction, the coupling constants describing the interaction with the thermal bath do not enter explicitly into the expressions for the transition probabilities. The role of the thermal bath in this case is reduced to that the activation factor is determined by the free energy in the transitional configuration, and for the calculation of the transition probabilities, it is sufficient to know the free energy surfaces of the system as functions of the coordinates of the reactive modes. 

The y-axis is the free energy. In general the free energy is related to the probability density function P( ) of the reaction coordinate through... 

Fig. 9.24. The free energy versus solvent coordinate curve showing the relation between reorganization energy Es, free energy of reaction AE°, and free energy of activation AF .

Much information can be gained by examining trends in redox potentials within series of compounds. Let us consider such a series of coordination compounds, M0, Mj, M2. .. M , which undergo reversible, one-electron oxidation reactions at potentials E°0, E°lf E°2,... E° respectively, with respect to the same reference electrode. If we define the oxidation of M0 as our standard reaction (equation 8), then we can examine the variation of the free energy difference, F(E°0— E° ), in terms of the structural difference between M0 and each other member M . Such an analysis is directly comparable to the classic approach of Hammett17 which relates a free energy difference term, log(AH/Ax), for equilibrium reactions such as (9) and (10), to the nature of the aryl substituent, X. 

Using the reaction free energy we are now in the position of calculating the reaction (classical) kinetics by means of the equations outlined in Section 5.2. In particular, for solving DE using Eq. 8-49, it is first necessary to evaluate the related diffusion coefficient D of the reaction coordinate. 

But the symmetry factor p has been defined strictly for a single step and is related to the shape of the free-energy barrier and to the position of the activated complex along the reaction coordinate. To describe a multi step process, p must be replaced by an experimental parameter, which we call the cathodic transfer coefficient a. Instead of Eq. 41E we then write ... 

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