Outer-sphere theory

Fig. 11. Computer-generated data for spheres with a volumic fraction v = 5 x 10 and Ad) = 2.64 x 10 rad/s, plotted vs. TD.The open symbols represent I/T2 value, while the filled symbols represent rates obtained, respectively, with tcp = 0.1 ms ( , and line a), 0.2 ms ( , and line b), 0.5 ms ( , and line c), 2 ms (T, and line d), 5 ms (A, and line e), 10 ms ( , and line f), 20 ms ( , and line g). The short dashed line is the rate predicted by outer sphere theory and the long dashed line is the Static Dephasing Regime model from 22). Lines (a-g) are the rates given by Eq. (11).

The outer-sphere theory has been developed using an electrostatic approach to calculate the energy necessary to bring reactants together, to reorganize the solvent around the transition state and to prepare the metal centers for election transfer. 

A powerful application of outer-sphere electron transfer theory relates the ET rate between D and A to the rates of self exchange for the individual species. Self-exchange rates correspond to electron transfer in D/D (/cjj) and A/A (/c22)- These rates are related through the cross-relation to the D/A electron transfer reaction by the expression... 

According to the Marcus theory [64] for outer-sphere reactions, there is good correlation between the heterogeneous (electrode) and homogeneous (solution) rate constants. This is the theoretical basis for the proposed use of hydrated-electron rate constants (ke) as a criterion for the reactivity of an electrolyte component towards lithium or any electrode at lithium potential. Table 1 shows rate-constant values for selected materials that are relevant to SE1 formation and to lithium batteries. Although many important materials are missing (such as PC, EC, diethyl carbonate (DEC), LiPF6, etc.), much can be learned from a careful study of this table (and its sources). 

Comparison of equations (2.11) and (2.15) reveals q and r to be kikilk i and A 2//r i, respectively. This enables k to be calculated from qjr. In its simplest forms the structure of the reactive intermediate can be viewed as V(OH)Cr " (when n is 1) or as VOCr (when n is 2). Similar species which have been characterized or implied kinetically are CrOCr (ref. 33), Np02Cr (ref. 37), U02Cr (ref. 31), VOV " (ref. 34), U0Pu02 + (ref. 41), Pu02pe + (ref. 42) and FeOFe + (ref. 38). Predictions on the rate of the V(III)- -Cr(lI) system, based upon Marcus theory", have been made by Dulz and Sutin on the assumption that an outer-sphere process applies. The value arrived at by these authors is 60 times lower than the experimental value. 

Here we mention as an example that in the coordination-chemistry field optical MMCT transitions between weakly coupled species are usually evaluated using the Hush theory [10,11]. The energy of the MMCT transition is given by = AE + x- Here AE is the difference between the potentials of both redox couples involved in the CT process. The reorganizational energy x is the sum of inner-sphere and outer-sphere contributions. The former depends on structural changes after the MMCT excitation transition, the latter depends on solvent polarity and the distance between the redox centres. However, similar approaches are also known in the solid state field since long [12]. 

Figure 2.1 (Plate 2.1) shows a classification of the processes that we consider they aU involve interaction of the reactants both with the solvent and with the metal electrode. In simple outer sphere electron transfer, the reactant is separated from the electrode by at least one layer of solvent hence, the interaction with the metal is comparatively weak. This is the realm of the classical theories of Marcus [1956], Hush [1958], Levich [1970], and German and Dogonadze [1974]. Outer sphere transfer can also involve the breaking of a bond (Fig. 2. lb), although the reactant is not in direct contact with the metal. In inner sphere processes (Fig. 2. Ic, d) the reactant is in contact with the electrode depending on the electronic structure of the system, the electronic interaction can be weak or strong. Naturally, catalysis involves a strong...

Here, n denotes a number operator, a creation operator, c an annihilation operator, and 8 an energy. The first term with the label a describes the reactant, the second term describes the metal electrons, which are labeled by their quasi-momentum k, and the last term accounts for electron exchange between the reactant and the metal Vk is the corresponding matrix element. This part of the Hamiltonian is similar to that of the Anderson-Newns model [Anderson, 1961 Newns, 1969], but without spin. The neglect of spin is common in theories of outer sphere reactions, and is justified by the comparatively weak electronic interaction, which ensures that only one electron is transferred at a time. We shall consider spin when we treat catalytic reactions. 

In order to account for the foregoing kinetic behavior, we rely on the Marcus theory for outer-sphere electron transfer to provide the quantitative basis for establishing the free energy relationship (8), i.e.,... 

Electron Transfer Far From Equilibrium. We have shown how the Marcus Theory of electron transfer provides a quantitative means of analysis of outer-sphere mechanisms in both homogeneous and heterogeneous systems. It is particularly useful for predicting electron transfer rates near the equilibrium potential,... 

The value of E° was hence determined by the reaction of R4M with Fe3+ complexes as outer-sphere SET oxidizers. Using five complexes with a range of different E° values, from 1.15 to 1.42 V, the rate constants were determined193. This was followed up by Eberson who, by application of the Marcus theory, was able to determine from the E° values (shown in Table 18) standard potentials and reorganization energies. Most compounds... 

Chemical and electrochemical reactions in condensed phases are generally quite complex processes only outer-sphere electron-transfer reactions are sufficiently simple that we have reached a fair understanding of them in terms of microscopic concepts. In this chapter we give a simple derivation of a semiclassical theory of outer-sphere electron-transfer reactions, which was first systematically developed by Marcus [1] and Hush [2] in a series of papers. A more advanced treatment will be presented in Chapter 19. 

To develop these ideas into a quantitative theory, we require models for the inner and outer sphere and their reorganization. The problem is similar to that encountered in infrared and Raman spectroscopy, where... 

However, a closer inspection of the experimental data reveals several differences. For ion-transfer reactions the transfer coefficient a can take on any value between zero and one, and varies with temperature in many cases. For outer-sphere electron-transfer reactions the transfer coefficient is always close to 1/2, and is independent of temperature. The behavior of electron-transfer reactions could be explained by the theory presented in Chapter 6, but this theory - at least in the form we have presented it - does not apply to ion transfer. It can, in fact, be extended into a model that encompasses both types of reactions [7], though not proton-transfer reactions, which are special because of the strong interaction of the proton with water and because of its small mass. 

The theory of electron-transfer reactions presented in Chapter 6 was mainly based on classical statistical mechanics. While this treatment is reasonable for the reorganization of the outer sphere, the inner-sphere modes must strictly be treated by quantum mechanics. It is well known from infrared spectroscopy that molecular vibrational modes possess a discrete energy spectrum, and that at room temperature the spacing of these levels is usually larger than the thermal energy kT. Therefore we will reconsider electron-transfer reactions from a quantum-mechanical viewpoint that was first advanced by Levich and Dogonadze [1]. In this course we will rederive several of, the results of Chapter 6, show under which conditions they are valid, and obtain generalizations that account for the quantum nature of the inner-sphere modes. By necessity this chapter contains more mathematics than the others, but the calculations axe not particularly difficult. Readers who are not interested in the mathematical details can turn to the summary presented in Section 6. 

There are three points of significance of this result. One is that it provides strong support for the 10-step mechanism originally proposed for reaction 1. Another is that it facilitates a more robust fitting of the mechanism to the kinetic data obtained for that reaction. Thirdly, it confirms that reaction 2 has a rate constant that is four orders of magnitude greater than predicted by Marcus theory. It is concluded that reaction 2 is poorly modeled as an outer-sphere process and is better described as... 

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