Inner-sphere reorganization

The elementary electrochemical reactions differ by the degree of their complexity. The simplest class of reactions is represented by the outer-sphere electron transfer reactions. An example of this type is the electron transfer reactions of complex ions. The electron transfer here does not result in a change of the composition of the reactants. Even a change in the intramolecular structure (inner-sphere reorganization) may be neglected in many cases. The only result of the electron transfer is then the change in the outer-sphere solvation of the reactants. The microscopic mechanism of this type of reaction is very close to that for the outer-sphere electron transfer in the bulk solution. Therefore, the latter is worth considering first. 

Let us consider the electron transfer between two rigid metal ions located some distance x from each other in the bulk of the solution. It is assumed that the inner-sphere reorganization of the donor D and acceptor A does not take place. The experiments show that the rate constants of these reactions differ by many orders of magnitude and the processes have an activated character even for identical ions D and A. The questions to be answered are Why does the electron exchange between identical ions in the solution require activation What is the reaction coordinate ... 

The change in the inner-sphere structure of the reacting partners usually leads to a decrease in the transition probability. If the intramolecular degrees of freedom behave classically, their reorganization results in an increase in the activation barrier. In the simplest case where the intramolecular vibrations are described as harmonic oscillators with unchanged frequencies, this leads to an increase in the reorganization energy ... 

Table 5-6. Comparison of ADEs and VDEs, and inner sphere reorganization energies (A.oxi) (eV) for iron-sulfur protein, Rd...

In contrast, with organometals there may be substantial reorganization changes in inner-sphere processes. With the tetraalkyl-tins, for example, it could involve a conversion of a tetrahedral structure to a trigonal bipyramidal structure, i.e.,... 

The model presented here is simplified in several ways (harmonic approximation, purely classical treatment of inner-sphere reorganization), and it says little about the pre-exponential factor A. But it does... 

There is a small complication in that the frequency to is different for the reduced and oxidized states so that one has to take an average frequency. Marcus has suggested taking u>av = 2woxu>red/(wox + a>reci)-When several inner-sphere modes are reorganized, one simply sums over the various contributions. The matter becomes complicated if the complex is severely distorted during the reaction, and the two states have different normal coordinates. While the theory can be suitably modified to account for this case, the mathematics are cumbersome. 

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. 

Let us consider the reorganization of an inner-sphere mode. Typically the modes have such high frequencies (hui > kT) that we can assume them to be in their ground state before the reaction.1 Therefore thermal averaging is not required, and Eq. (19.25) simplifies to ... 

If in addition one inner-sphere mode of frequency oj, with Tuo 3> kT, is reorganized, the total rate constant can be written as a sum over partial rates ... 

The Marcus classical free energy of activation is AG , the adiabatic preexponential factor A may be taken from Eyring s Transition State Theory as (kg T /h), and Kel is a dimensionless transmission coefficient (0 < k l < 1) which includes the entire efiFect of electronic interactions between the donor and acceptor, and which becomes crucial at long range. With Kel set to unity the rate expression has only nuclear factors and in particular the inner sphere and outer sphere reorganization energies mentioned in the introduction are dominant parameters controlling AG and hence the rate. It is assumed here that the rate constant may be taken as a unimolecular rate constant, and if needed the associated bimolecular rate constant may be constructed by incorporation of diffusional processes as ... 

The exothermicity dependence is in p, (p = — AG°/hco) whereas the reorganization energy is expressed in s, (s = X,/ho)). This limit can be appropriate for the inner sphere reorganization whereas for the outer sphere reorganization one can assume very low frequencies and take the high temperature (classical) limit, obtaining... 

A very practical comprehensive rate expression appropriate for long-range electron transfer in proteins and other large molecules yet which retains ease of computation by anyone and on the smallest computer is obtained by assuming one high frequency harmonic mode (inner sphere reorganization) and one very... 

The effect of ligands on the character and degree of the inner-sphere reorganization during electroreduction of aqua-, aquahydroxy-, hydroxy-, and ethylene-diamine tetraacetic acid (EDTA) complexes of Zn(II) [95] and electrochemical process of Zn(II) complexed by different ligands - glycinate [96], ethanol amine [97], azinyl methyl ketoximes [98], aspartame [99], glutathione [100, 101] and several cephalosporin antibiotics [102] -were studied at mercury electrodes in aqueous solutions. 

Single-electron transferring across electrified interfaces are traditionally characterized as either outer- or inner-sphere processes according to whether or not they are accompanied in a concerted manner with the formation or the breaking of chemical bonds. Outer-sphere reactions therefore solely involve the reorganization of the outer solvent sphere after the electron transfer has occurred. There are only very few truly outer-sphere reactions known to date such as [23]... 

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