Theoretical Treatment of the Kinetics of Diffusion-Limited Reactions

The problem of the kinetics of the diffusion-limited reaction $A+B\to \mathrm{AB}$ 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 $A\text{'}\mathrm{s}$ and the number of $B\text{'}\mathrm{s}$ present at that particular time. The rate of the reaction $A+B\to \mathrm{AB}$ will be second order and the reaction $A+B\to B$ 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.