Outer-sphere reaction rates

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). 

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... 

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. 

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. 

The reaction of Fe + with Fe(phen)3+ is considered to be an outer-sphere reaction because of the inertness of the Fe(III) complex. Marked effects of anions on the rate have been interpreted however... 

The rates of very sluggish outer-sphere reactions of (also V(II) and V(III)) with Co(NHj)5+ and Co(NH3)5py + are increased markedly by addition of small (mM)... 

The rates of electron-transfer reactions can be well predicted provided that the electron transfer is a type of adiabatic outer-sphere reaction and the free-energy change of electron transfer and the reorganization energy (X) associated with the electron transfer are known [1-7]. This means that electron-transfer reactions can be designed quantitatively based on the redox potentials and the reorganization energies of molecules involved in the electron-transfer reactions. 

More extensive data in other systems concerning the dependence of rate on ligand is being accumulated, and as yet we are far from understanding these, particularly in the bridged reactions. The subject of outer sphere reactions and the factors affecting the rate are in a somewhat better state. 

There has been some exploration of the mechanism of reduction of d transition metal complexes by M2+(aq) (M = Eu, Yb, Sm). Both inner- and outer-sphere mechanisms are believed to operate. Thus the ready reduction of [Co(en)3]3+ by Eu2+(aq) is necessarily outer-sphere. 2 However, the strong rate dependence on the nature of X when [Co(NH3)5X]2+ or [Cr(H20)5X]2+ (X = F, Cl, Br or I) are reduced by Eu2+(aq) possibly suggests an inner-sphere mechanism.653 The more vigorous reducing agent Yb2+ reacts with [Co(NH3)6]3+ and [Co(en)3]3+ by an outer-sphere route but with [Cr(H20)5X]2+ (X = halide) by the inner-sphere mechanism.654 Outer-sphere redox reactions are catalyzed by electron-transfer catalysts such as derivatives of isonicotinic acid, one of the most efficient of which is iV-phenyl-methylisonicotinate, as the free radical intermediate does not suffer attenuation through disproportionation. Using this catalyst, the outer-sphere reaction between Eu2+(aq) and [Co(py)(NH3)5]3+ proceeds as in reactions (18) and (19). Values found were ki = 5.8 x KFM-1 s 1 and k kx = 16.655... 

Marcus has derived a relationship from first principles that enables one to calculate rate constants for outer sphere reactions 43... 

A point of note in the data in Table 1 is the extraordinary range in electron-transfer reactivity that can exist even for outer-sphere reactions among what appear to be closely related reactions. For example, the self-exchange rate constants for Co(NH3)63+/2+ and Ru(bipy)33+/2+ differ in magnitude by 1014. 

For an outer-sphere reaction there are three factors which play a role in determining the rate of electron transfer. The first is the approach of the reactants to be in sufficiently close proximity to create an electronic interaction which provides a basis for the delocalization of the exchanging electron. The second is a barrier to electron transfer that is created by the equilibrium structural differences between reactants and products. The third is an additional barrier that is created in the surrounding solvent by the change in charge distribution associated with the electron transfer act. 

For an outer-sphere reaction, given the translation mobility of the reactants, electron transfer may occur over a range of distances. The problem can be treated in a general way since from statistical mechanics the equilibrium distribution of intemuclear separations can be calculated based on pairwise distribution functions. Integration of the product of the distribution function and ket(r) over all space gives the total rate constant et-32b 48... 

An important feature to emerge from the comparisons in Table 2 is that variations in the electronic coupling term play a relatively small role in dictating the magnitudes of self-exchange rate constants for outer-sphere reactions, at least for transition metal complexes. Even for reactions... 

In contrast to outer-sphere reactions, the simple observation that a reaction occurs by an inner-sphere mechanism necessarily introduces an element of structural definition. The relative dispositions of the oxidizing and reducing agents are immediately established and, except for structurally flexible bridging ligands such as NC5H4(CH2) C5H4N, the internuclear separation between redox sites can be inferred from known bond distances. Even so, bimolecular inner-sphere reactions necessarily occur by a sequence of elementary steps (Scheme 2) and the observed rate constant may include contributions from any of the series of steps. 

Two separate but somewhat interwoven themes have emerged from the study of inner-sphere reactions. The first is the use of product and rate studies to establish the existence of inner-sphere pathways and then the exploitation of appropriate systems to demonstrate such special features as remote attack . In the second theme the goal has been to assemble the reactants through a chemical bridge and then to study intramolecular electron transfer directly following oxidation or reduction of the resulting dimer (note equation 7). It is convenient to turn first to chemically prepared, intramolecular systems since many of the theoretical ideas and experimental results for outer-sphere reactions can be carried over directly as an initial basis for understanding the experimental observations. 

Relatively little attention has been given in the literature to the electronic transmission coefficient for electrochemical reactions. On the basis of the conventional collisional treatment of the pre-exponential factor for outer-sphere reactions, Kel has commonly been assumed to equal unity, i.e. adiabatic reaction pathways are followed. Nevertheless, as noted above, the dependence of xei upon the spatial position of the transition state is of key significance in the "encounter pre-equilibrium treatment embodied in eqns. (13) and (14). Thus, the manner in which Kel varies with the reactant-electrode separation for outer-sphere reactions will influence the integral of reaction sites that effectively contribute to the overall measured rate constant and hence the effective electron-tunneling distance, Srx, in eqn. (14). 

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