For outer-sphere electron transfer reactions

An expression of the type in Eq. (29) has been rederived recently in Ref. 13 for outer-sphere electron transfer reactions with unchanged intramolecular structure of the complexes where essentially the following expression for the effective outer-sphere reorganization energy Ers was used ... 

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. 

In the general case R denotes a set of coordinates, and Ui(R) and Uf (R) are potential energy surfaces with a high dimension. However, the essential features can be understood from the simplest case, which is that of a diatomic molecule that loses one electron. Then Ui(R) is the potential energy curve for the ground state of the molecule, and Uf(R) that of the ion (see Fig. 19.2). If the ion is stable, which will be true for outer-sphere electron-transfer reactions, Uf(R) has a stable minimum, and its general shape will be similar to that of Ui(R). We can then apply the harmonic approximation to both states, so that the nuclear Hamiltonians Hi and Hf that correspond to Ui and Uf are sums of harmonic oscillator terms. To simplify the mathematics further, we make two additional assumptions ... 

An ideal photosensitizer must satisfy several stringent requirements (Balzani et. al., 1986) 1) stability towards thermal and photochemical decomposition reactions 2) sufficiently intense absorption bands in a suitable spectral region 3) high efficiency of population of the reactive excited state 4) long lifetime in the reactive excited state 5) suitable ground state and excited state potentials 6) reversible redox behavior 7) good kinetic factors for outer sphere electron transfer reactions. 

Several studies of bimetallic complexes in which the donor and acceptor are linked across aliphatic chains have demonstrated that these are generally weakly coupled systems. " Studies of complexes linked by l,2-bis(2,2 bipyridyl-4-yl)ethane (bb see Figure 5), indicate that these are good models of the precursor complexes for outer-sphere electron-transfer reactions of tris-bipyridyl complexes. A careful comparison of kinetic and spectroscopic data with computational studies has led to an estimate of //rp = 20cm for the [Fe(bb)3pe] + self-exchange electron transfer. In a related cross-reaction, the Ru/bpy MLCT excited state of [(bpy)2Ru(bb)Co(bpy)2] + is efficiently quenched by electron transfer to the cobalt center in several resolved steps, equations (57) and (58). ... 

Rate Constants for Outer-Sphere Electron Transfer Reactions ... 

Rate constants for outer-sphere electron transfer reactions that involve net changes in Gibbs free energy can be calculated using the Marcus cross-relation (Equations 1.24—1.26). It is referred to as a cross-relation because it is derived from expressions for two different self-exchange reactions. 

For outer sphere electron transfer reactions the Butler-Volmer law rests on solid experimental and theoretical evidence. An outer sphere electron transfer reaction is the simplest possible case of an electron transfer reaction, a reaction where only an electron is exchanged, no bonds are broken, the reactants are not specifically adsorbed, and catalysts play no role (see, e.g.. Ref. 2). Experimental investigations such as those by Curtiss et al. [206] have shown that the transfer coefficient of simple electron transfer reactions is independent of temperature. The theoretical basis is given by the theories of Marcus [207] and of Levich and Dogonadze [208] these theories also predict deviations at high overpotentials which were experimentally confirmed [209, 210]. 

The situation is fundamentally different from that for outer sphere electron transfer reactions where, according to Marcus theory, the solvent reorganization determines the reaction. In contrast, the model calculations discussed in this section indicate that the energy of activation for the ion transfer step is not related to the electron exchange with the electrode, since the crossing between the two diabatic energy states of 1 and 1° occurs only at such short distances where the ion has already surpassed the solvent barrier. Contrary to the approach discussed here, Xia and Berkowitz [235] assumed the validity of the outer sphere mechanism from the outset. The analysis of the dependence of the solvent barrier on external electric field and temperature indicates that a in Eq. (17) is indeed not constant but depends on temperature. 

The importance of a nonadiabatic path for outer-sphere electron transfer reactions of Eu(III)/Eu(Il) was again examined by Yee et al. (1983) via a study of a series of reactions with Eu(III)/(II) cryptates (table 12). The cryptate (polyoxadiazamacrobi-cyclic) ligands form thermodynamically stable and substitution inert complexes with both Eu(lll) and Eu(ll), markedly changing the primary coordination spheres of the Eu ions. The dramatic variation in the values for the Eu exchange reactions with such a change is demonstrated by the respective calculated values for EUavalues calculated from the cross reactions are consistent with the values of the Franck-Condon barriers estimated from structural data. 

Figure 93 Three-step mechanism for outer-sphere electron-transfer reactions. Line 1. Chemical species and processes. Line 2. R are the reactants I is the precursor intermediate complex I is the successor intermediate complex P are the products j is the activated complex. Line 3. a is the association to I et is the electron-transfer step d is the dissociation of I to products b, c is the ligand and solvent reorganisation. Line 4. Free-energy changes.

Derivation of the Marcus relation for outer-sphere electron-transfer reactions between metal ions in solution... 

The electron-transfer reactivity of a species is related to the driving force for the reaction, which is essentially the difference in reduction potentials of the two redox couples to the intrinsic reactivity of the couples and, for charged species, to electrostatic terms. For outer sphere electron transfer reactions, the intrinsic reactivity of a couple can be expressed in terms of the self-exchange rate constant for the exchange of an electron between the two halves of the couple, for example, reaction (20) ... 

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