Theory of Electron Transfer Reactions in Simple Fluids

Abstract

The purpose of this paper is to explore with as much detail and accuracy as possible the feasibility of formulating a theory of solution electron transfer reactions utilizing nonequilibrium statistical mechanics and the existing theory of simple fluids. Within the confines of a clearly stated, reasonably flexible set of approximations this has been accomplished. The theoretical analysis utilizes Yamamoto's generalization to chemical reactions of the Kubo nonequilibrium statistical mechanical response theory. We examine systems composed of electroactive solutes dissolved in an electrolytic solvent. The paper has two parts. The first part contains the treatment leading from a general Hamiltonian operator for a system of electron donors and acceptors dissolved in a solvent (with supporting electrolyte present) to an expression for the rate constant. The Hamiltonian operator adopted contains no physical solvent model assumptions such as, for example, that represented by the dielectric continuum approximation. The second part of the paper is devoted to the examination of a model, but representative, system: the exchange reaction ${\mathrm{H+H}}^{+}={\mathrm{H}}^{+}\mathrm{+H}.$ This is done to establish a correspondence between the theory and typical chemical interpretation, viz., the identification of activation energies and frequency factors in an Arrhenius type rate constant expression. While the H; H⁺ exchange has no physical reality, it is conceptually simple, and, apart from the hydrogenic mass, does correspond to systems such as Fe⁺²; Fe⁺³. The main reason for choosing this system is the clear simplicity of the Coulomb matrix elements and the hard sphere character of the reactants. Thus, this system typifies outer sphere transfer couples. In addition, the dielectric continuum model is used in the example calculations to compare this theory with that of Levich and Dogonadze. As a result of the study of the model system, we demonstrate a dependence on transport properties of the system. In particular, the electron transfer kinetics depend on the transport of reactants to a transfer state configuration. This transport differs from the familiar solution diffusion in that the simultaneous movement of the reactants and the distortion of the environment to a configuration favorable to electron transfer must occur. The general treatment of electron transfer reactions presented in this paper should prove valuable in a number of related areas, especially in assessing the validity of the use of the dielectric continuum approximation and the Born—Oppenheimer separation. Moreover, the approach provides the best route to the theoretical examination of more complicated electron transfer systems as typified by the inner sphere examples.