A solution of the Debye–Smoluchowski equation for the rate of reaction of ions in dilute solution

Reactions of isolated ion pairs in solution have been modelled using the Debye–Smoluchowski equation for diffusion and conduction. An activation step was incorporated using a partially reflecting boundary condition. The method of matched expansions and the Abelian theorem of Laplace transforms was used to give an approximate solution of the Debye–Smoluchowski equation. Numerical integrations based on the finite-difference method confirmed these approximate analytic formulas.