Outer rate constant

Smoluchowski theory [29, 30] and its modifications fonu the basis of most approaches used to interpret bimolecular rate constants obtained from chemical kinetics experiments in tenus of difhision effects [31]. The Smoluchowski model is based on Brownian motion theory underlying the phenomenological difhision equation in the absence of external forces. In the standard picture, one considers a dilute fluid solution of reactants A and B with [A] [B] and asks for the time evolution of [B] in the vicinity of A, i.e. of the density distribution p(r,t) = [B](rl)/[B] 2i ] r(t))l ] Q ([B] is assumed not to change appreciably during the reaction). The initial distribution and the outer and inner boundary conditions are chosen, respectively, as... 

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). 

The Marcus treatment applies to both inorganic and organic reactions, and has been particularly useful for ET reactions between metal complexes that adopt the outer-sphere mechanism. Because the coordination spheres of both participants remain intact in the transition state and products, the assumptions of the model are most often satisfied. To illustrate the treatment we shall consider a family of reactions involving partners with known EE rate constants. 

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). 

FIGURE 34.4 Dependence of electrochemical rate constant on the electrode potential for outer-sphere electron transfer. An exponential increase in the normal region changes for the plateau in the activationless region. 

The first estimations of for photoinduced processes were reported by Dvorak et al. for the photoreaction in Eq. (40) [157,158]. In this work, the authors proposed that the impedance under illumination could be estimated from the ratio between the AC photopotential under chopped illumination and the AC photocurrent responses. Subsequently, the faradaic impedance was calculated following a treatment similar to that described in Eqs. (22) to (26), i.e., subtracting the impedance under illumination and in the dark. From this analysis, a pseudo-first-order photoinduced ET rate constant of the order of 10 to 10 ms was estimated, corresponding to a rather unrealistic ket > 10 M cms . Considering the nonactivated limit for adiabatic outer sphere heterogeneous ET at liquid-liquid interfaces given by Eq. (17) [5], the maximum bimolecular rate constant is approximately 1000 smaller than the values reported by these authors. 

The aquation of [IrCl6]2- to [Er( E120)C1S] and Ir(H20)2Cl4 has been found to activate the complex toward the oxidation of insulin in acidic solutions, with measured rate constant of 25,900 and 8,400 Lmol-1 s 1, respectively.50 The oxidation reaction proceeds via an outer-sphere mechanism. 

For the series of -branched alkyl radicals, the second-order rate constant in eq 3 is relatively unaffected by steric effects [compare Figure 2 (right)] as expected for an outer-sphere process. In strong contrast, the rate constant kL for ligand substitution in eq 21 is adversely affected by increasing steric effects, as shown in Figure 17. 

Where solvent exchange controls the formation kinetics, substitution of a ligand for a solvent molecule in a solvated metal ion has commonly been considered to reflect the two-step process illustrated by [7.1]. A mechanism of this type has been termed a dissociative interchange or 7d process. Initially, complexation involves rapid formation of an outer-sphere complex (of ion-ion or ion-dipole nature) which is characterized by the equilibrium constant Kos. In some cases, the value of Kos may be determined experimentally alternatively, it may be estimated from first principles (Margerum, Cayley, Weatherburn Pagenkopf, 1978). The second step is then the conversion of the outer-sphere complex to an inner-sphere one, the formation of which is controlled by the natural rate of solvent exchange on the metal. Solvent exchange may be defined in terms of its characteristic first-order rate constant, kex, whose value varies widely from one metal to the next. 

Similar reactions are catalyzed by Mn and Fe centers of MnSOD and FeSOD. It is obvious that before participation in Reaction (2), superoxide must be protonized to form hydroper-oxyl radical HOO by an outer-sphere or an intra-sphere mechanisms. All stages of dismuting mechanism, including the measurement of elementary rate constants, have been thoroughly studied earlier (see, for example, Ref. [2]). 

Reaction (2) is an outer-sphere exothermic process (AE° is about —0.4 V) and therefore, the equilibrium of this reaction is completely shifted to the right, i.e., the reoxidation of reduced cytochrome c by dioxygen is impossible. However, the rate constant for Reaction (2) (2.6 + O.lxlO5 1 moR1 s 1) is unexpectedly low for the exothermic one-electron transfer... 

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... 

This limited amount of kinetic evidence suggests that the kinetic models developed for reactivity in aqueous micelles are directly applicable to reactions in vesicles, and that the rate enchancements have similar origins. There is uncertainty as to the appropriate volume element of reaction, especially if the vesicular wall is sufficiently permeable for reaction to occur on both the inner and outer surfaces, because these surfaces will have different radii of curvature and one will be concave and the other convex. Thus binding, exchange and rate constants may be different at the two surfaces. 

A significant technical development is the pulsed-accelerated-flow (PAF) method, which is similar to the stopped-flow method but allows much more rapid reactions to be observed (1). Margerum s group has been the principal exponent of the method, and they have recently refined the technique to enable temperature-dependent studies. They have reported on the use of the method to obtain activation parameters for the outer-sphere electron transfer reaction between [Ti Clf ] and [W(CN)8]4. This reaction has a rate constant of 1x108M 1s 1 at 25°C, which is too fast for conventional stopped-flow methods. Since the reaction has a large driving force it is also unsuitable for observation by rapid relaxation methods. 

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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