Theoretical Treatment of the Kinetics of Diffusion-Limited Reactions

TLDR

The kinetics of diffusion‑limited reactions A+B→AB are formulated using pair probability densities for every A–B pair. The study aims to derive general equations that enable treatment of non‑random initial distributions, such as those occurring during radiation‑damage annealing. The authors model the diffusion‑reaction dynamics by tracking pair probability densities, accounting for competition and particle removal, and formulate coupled differential equations solvable under various boundary conditions, including non‑random initial distributions. For random initial distributions the model predicts a time‑dependent rate equal to the diffusion‑controlled encounter rate times the numbers of A and B, yielding second‑order kinetics for A+B→AB and first‑order for A+B→B after a transient, while irregular transients dominate the reaction in non‑random cases.

Abstract

The problem of the kinetics of the diffusion-limited reaction $A+B\ensuremath{\rightarrow}\mathrm{AB}$ has been formulated in terms of the pair probability densities of the reacting particles (every $A$ taken with every $B$). The alteration of these pair densities due to diffusion and reaction have been considered. The competition of every $A$ for every $B$ and the removal of particles from the system upon reaction have been appropriately accounted for. The formulation leads to a set of coupled differential equations that can be solved for a variety of boundary conditions. The problem has been solved in detail for a random initial distribution. The rate of reaction at any time is just the probable rate at which a single $A$ and a single $B$ diffuse together (with an appropriate boundary condition for reaction on close approach) multiplied by the product of the number of $A'\mathrm{s}$ and the number of $B'\mathrm{s}$ present at that particular time. The rate of the reaction $A+B\ensuremath{\rightarrow}\mathrm{AB}$ will be second order and the reaction $A+B\ensuremath{\rightarrow}B$ will be first order after times long compared to a transient whose form is given explicitly. More general equations are obtained to permit the treatment of non-random initial distributions, as occur, for example, in the annealing of radiation damage. In such cases the irregular transients may account for a major portion of the reaction.