Outer sphere reactions

Consider one of the most common types of outer-sphere reactions involving bivalent and trivalent metal complexes, ML and M,L + where L and L, represent the total ligand structure and M and M, or/and L and L, may be different, as in the reactant pairs, V(H20)g and Fe(H20)j+, FefHjO) and Peffopy) , and V(H20) and Ru(NH3). The outer-sphere pro- 

A 7j = change in equilibrium value of the jth normal coordinate, and when breathing vibrations are employed 

Ac/ = the difference of the metal-ligand distance between oxidized and reduced complex. 

Several workers, particularly Marcus and Hush, tackled the calculation of AG for an account and comparison of the various early attempts, the reader is referred to Refs. 19 and 20. This important area has been thoroughly reviewed and representative examples in Refs. 21-25 as well as in the Selected Bibliography give accounts of the theory in varying depths as well as an entry into the vast literature. 

The free energy barrier AG is considered to consist of various components  

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The observed rate law for inner-sphere, as for outer-sphere, reactions is commonly first order in each reactant but this does not indicate which step is ratc-dcteriiiining. Again, details should be obtained from more extensive accounts." ... 

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). 

In the case of other systems in which one or both of the reactants is labile, no such generalization can be made. The rates of these reactions are uninformative, and rate constants for outer-sphere reactions range from 10 to 10 sec b No information about mechanism is directly obtained from the rate constant or the rate equation. If the reaction involves two inert centers, and there is no evidence for the transfer of ligands in the redox reaction, it is probably an outer-sphere process. 

However, some quantitative interpretation of the rates of outer-sphere reactions may be made. It is possible to determine the rate constant, kn for the reaction of two complex ions and [M Lg] (Eq. 9.26). 

The nature of the group X determines the type of reaction which is the most important. For X = azide, thiocyanate, hydroxide, chloride bromide and iodide the inner-sphere bath operates while for X = ammonia or oxyanions (including carboxylates) the main pathway is the outer-sphere reaction. For X = fluoride or nitrite the concentration of the cyanide ion present determines which is the major reaction pathway. 

Reductions of various Co(ni) complexes by Fe(II) have been studied under high pressures . The motivation for performing such experiments resides in the possibility that the volume of activation (AF ), like the entropy of activation, might be a criterion for distinguishing between inner- and outer-sphere reactions. For reactions of the type... 

On this basis AF should be more positive for an inner-sphere than for an outer-sphere reaction since a water molecule occupies a greater volume in the liquid phase than if it is coordinated. Second-order rate coefficients were determined at various pressures in the range 0.001 to 3.5 kbars, the rate decreasing with increase in pressure. The apparatus used was a modification of that first described by Osborn and Whalley . Values of AF were calculated from the slopes of plots of log k versus pressure, since... 

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. 

A principal distinction is made between outer-sphere and inner-sphere mechanisms in ET reactions (Kochi, 1988). In the outer-sphere reactions the... 

N. Sutin, Brookhaven National Laboratory Strictly speaking, the outer-sphere and inner-sphere designations refer to limiting cases. In practice, reactions can have intermediate outer-sphere or inner-sphere character this occurs, for example, when there is extensive interpenetration of the inner-coordination shells of the two reactants. Treating this intermediate situation requires modification of the usual expressions for outer-sphere reactions — particularly those expressions that are based upon a hard-sphere model for the reactants. 

The aim in solution studies on metalloprotein is to be able to say more about intermolecular electron transfer processes, first of all by studying outer-sphere reactions with simple inorganic complexes as redox partners. With the information (and experience) gained it is then possible to turn to protein-protein reactions, where each reactant has its own complexities... 

Figure 5.1 Potential energy curves for an outer-sphere reaction the upper curve is for the standard equilibrium potential oo the lower curve for

Figure 5.3 Tafel plot for the anodic current density of an outer-sphere reaction.

On this basis, the outer-sphere reaction mechanism was concluded to be unlikely. 

Section II focuses on outer-sphere reactions of species that are stable in their adjacent oxidation states, which leads to a degree of confidence in the reaction mechanisms and the ability to define and measure the... 

Electrochemical reactions can be broken down into two groups outer-sphere electron-transfer reactions and inner-sphere electron transfer reactions. Outer-sphere reactions are reactions that only involve electron transfer. There is no adsorption and no breaking or forming of chemical bonds. Because of their simplicity, numerous studies have been performed, many entirely theoretical.18-25 By definition, though, electrode reactions are not outer-sphere reactions. However, if charge transfer is rate limiting for an electrode reaction, it typically takes a form similar to that of an outer-sphere reaction, which is described later in this section. 

In this notation, anodic current is positive, while cathodic current is negative. As the later section on oxygen reduction will show, the Tafel slope can change with overpotential. This is because the Butler-Volmer law only applies to outer-sphere reactions. Although it can describe electrode reactions, the equation does not account for repulsive interactions of the adsorbates or changes in the reaction mechanism as potential is changed. 

A recently proposed semiclassical model, in which an electronic transmission coefficient and a nuclear tunneling factor are introduced as corrections to the classical activated-complex expression, is described. The nuclear tunneling corrections are shown to be important only at low temperatures or when the electron transfer is very exothermic. By contrast, corrections for nonadiabaticity may be significant for most outer-sphere reactions of metal complexes. The rate constants for the Fe(H20)6 +-Fe(H20)6 +> Ru(NH3)62+-Ru(NH3)63+ and Ru(bpy)32+-Ru(bpy)33+ electron exchange reactions predicted by the semiclassical model are in very good agreement with the observed values. The implications of the model for optically-induced electron transfer in mixed-valence systems are noted. 

In the following sections, we shall explore the applicability of such relationships to experimental data for some simple outer-sphere reactions involving transition-metal complexes. In keeping with the distinction between intrinsic and thermodynamic barriers [eq 7], exchange reactions will be considered first, followed by a comparison of driving force effects for related electrochemical and homogeneous reactions. 

How important, though, is nuclear tunnelling for thermal outer-sphere reactions at ordinary temperature If we work in the Golden Rule formalism, an approximate answer was given some time ago. In harmonic approximation, one obtains from consideration of the Laplace transform of the transition probability (neglecting maximization of pre-exponential terms) the following expressions for free energy (AG ) and enthalpy (AH ) of... 

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