Theory of Bimolecular Reaction Processes in Liquids

Abstract

A theoretical approach to the problem of diffusion controlled bimolecular reactions 1s presented. In order to take into account the ti:ne correlation of reaction process of our many-particle system, the probability of the fust reaction is introduced as a fundamental quantity. Time development of the ensemble of our system is formulated using the probability of the first reaction. An approximation which reduces the general formula to a problem of Markov process is adopted. Then it is shown that, if we assume stationary reaction rate, the usual phenomenological kinetic equation, i.e. the so called law of mass action can be derived as the first order approximation, and as the second order approximation the deviation from the law of mass action is examined. For the general case, in order to obtain the probability of the first reaction in an explicit form, it becomes necessary to solve. the multi-dimensional diffusion equation with pair absorbing interactions, which is calculated using the binary collision expansion method. § 1. Introduction 29 Diffusion contrqlled reaction processes have been widely known, for example, coagulation of colloidal particles, quenching of fluorescence, excitation transfers and usual bimolecular reactions of the type A+ B-> AB. Theoretical investigations have been developed by many authors,tl-lll since Smoluchowski.'s fundamental work, which was originally developed to explain the process of coagulation of colloids, appeared. Many of these works were devoted to deriving formulae for collision frequency or reaction rate as a function of time and friction constant, on the basis of Smoluchowski's equation. Assuming the independence of collisions , they solved essentially two-body problems under various boundary and initial conditions. If we consider the reaction process as a time dependent many-body problem, it becomes necessary to take into account complicated spatial and time correlations. The effects of these correlations have been also discussed by several authors. However, the ~ystematic formulation has seemingly not yet been established. The purpose of this paper is to derive a general formula which describes the time development of an ensemble of our many-particle system, and investigate the effects of these correlations on the reaction rates. In order to take into account the time correlations of reaction process, the probability of the first reaction is introduced as a fundamental quantity. By using this probability function, time development of the ensemble of our system is formulated in the most general form in § 2. On the assumption of the uniform