Free energy electron transfer reactions

Within this framework, by considering the physical situation of the electrode double layer, the free energy of activation of an electron transfer reaction can be identified with the reorganization energy of the solvation sheath around the ion. This idea will be carried through in detail for the simple case of the strongly solvated... 

Sekiguchi S, Kobori Y, Akiyama K and Tero-Kubota S 1998 Marcus free energy dependence of the sign of exchange interactions in radical ion pairs generated by photoinduced electron transfer reactions J. Am. Chem. Soc. 120 1325-6... 

An electron transfer reaction may be separated into two half-reactions or redox couples so that the free energy, AG°, can be separated into AGa and AGg the free energies of reduction of the donor (D) and the acceptor (A), respectively, by... 

The environmental (i.e., solvent and/or protein) free energy curves for electron transfer reactions can be generated from histograms of the polarization energies, as in the works of Warshel and coworkers [79,80]. 

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... 

G King, A Warshel. Investigation of the free energy functions for electron transfer reactions. J Chem Phys 93 8682-8692, 1990. 

M Tachiya. Relation between the electron-transfer rate and the free energy change of reaction. J Phys Chem 93 7050-7052, 1989. 

Use die activated complex theory for explaining clearly how the applied potential affects the rate constant of an electron-transfer reaction. Draw free energy curves and use proper equations for your explanation. 

The reactivities of potassium and silver with water represent extremes in the spontaneity of electron-transfer reactions. The redox reaction between two other metals illustrates less drastic differences in reactivity. Figure 19-5 shows the reaction that occurs between zinc metal and an aqueous solution of copper(II) sulfate zinc slowly dissolves, and copper metal precipitates. This spontaneous reaction has a negative standard free energy change, as does the reaction of potassium with water ... 

Equation (34.10) describes the dependence of the activation free energy on the free energy of transition AF for electron transfer between two discrete energy levels (one in the donor, Eq, and one in the acceptor, e ). The quantity AF involves the difference of these electron energies, the solvation free energies of the reaction products, wfi and the initial reactants, wf and the works required to bring the reaction products, w, and the reactants, w,., from infinity to a given interreactant distance 34. 

FIGURE 34.8 Free-energy surfaces for the dissociative electron transfer reaction (a) for the solvent polarization (b) along the coordinate r of the molecnlar chemical bond. corresponds to stable molecule in oxidized form. U" is the decay potential for the rednced foim. AFj and AF are the partial free energies of the transition determining mntnal arrangement of the two sets of the free-energy surfaces. 

FIGURE 35.3 Free-energy functions for reactant (AE) and product Ag (AE) of an electron transfer reaction as calculated using umbrella sampling within a simple dipolar diatomic solvent. AG° is the reaction free energy. Solid lines are polynomial fittings to the simulated points. Dashed lines are parabolic extrapolations from the minimum of the curves. (From King and Warshel, 1990, with permission from the American Institute of Physics.)... 

The expression in Eq. (10) for the exponent in Eq. (9) is quite similar to that for the activation free energy in electron transfer reactions derived by Marcus using the methods of nonequilibrium classical thermodynamics8 ... 

Instead of the quantity given by Eq. (15), the quantity given by Eq. (10) was treated as the activation energy of the process in the earlier papers on the quantum mechanical theory of electron transfer reactions. This difference between the results of the quantum mechanical theory of radiationless transitions and those obtained by the methods of nonequilibrium thermodynamics has also been noted in Ref. 9. The results of the quantum mechanical theory were obtained in the harmonic oscillator model, and Eqs. (9) and (10) are valid only if the vibrations of the oscillators are classical and their frequencies are unchanged in the course of the electron transition (i.e., (o k = w[). It might seem that, in this case, the energy of the transition and the free energy of the transition are equal to each other. However, we have to remember that for the solvent, the oscillators are the effective ones and the parameters of the system Hamiltonian related to the dielectric properties of the medium depend on the temperature. Therefore, the problem of the relationship between the results obtained by the two methods mentioned above deserves to be discussed. 

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... 

Force-field methods, calculation of molecular structure and energy by, 13,1 Free radical chain processes in aliphatic systems involving an electron-transfer reaction, 23, 271 Free radicals, and their reactions at low temperature using a rotating cryostat, study of, 8. I Free radicals, identification by electron spin resonance, 1, 284... 

In our description of the Marcus theory of electron-transfer reactions we have found it helpful to plot the free energy change in the three dimensional picture shown in Fig. 10 (Albery, 1975c, 1980). This picture emphasizes that... 

Fig. 10 Free energy plot for an electron-transfer reaction.

Our problem now is to determine the functional form of this experimental free energy curve for the intrinsic rate constant ki for electron transfer. In addition to the Marcus eq 4, two other relationships are currently in use to relate the activation free energy to the free energy change in electron transfer reactions (15, JL6). 

The forward and reverse rate constants are thus equal at zero standard free energy. However, this will be difficult to check in practice, for both reactions are very slow, since a bond-breaking/bond-forming process endowed with a quite large internal reorganization is involved. The result is that dissociative electron transfer reactions are usually carried out with electron donors that have a standard potential largely negative to the dissociative standard potential. The reoxidation of the R, X- system is thus possible only with electron acceptors, D +, that are different from the D,+ produced in the reduction process (they are more powerful oxidants). There is no reason then that the oxidation mechanism be the reverse of the... 

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