Diffusion-limited reaction rate theory for two-dimensional systems

The Noyes and Smoluehowski diffusion-limited reaction rate theories are proved to be equivalent on a lattice. The Noyes theory is analysed and used to predict the kinetics of the two-dimensional irreversible reaction A + B → P. Only condensed phase reactions with molecules of A and B undergoing Brownian motion (diffusion) are discussed. For comparison, all calculations are done in both two and three dimensions. The two-dimensional rate function k N ( t ) in the equation d[A]/d t = d[B]/d t = - k N ( t ) [A] [B] asymptotically goes to zero as (In t ) -1 as t increases; the asymptotic expansion of k N ( t ) is derived from the expansion for the first-return probability in a random walk on a square lattice. The theoretical rate function is determined as a function of the probability α of reaction given an encounter. Although k N ( t ) is not significantly different from ‘empirical’ rate functions in a Monte Carlo simulation of a two-dimensional chemical reaction, it does differ from the rate function in a two-dimensional fluorescence quenching experiment.