A CONTRIBUTION TO THE FORMULATION OF DIFFUSION-LIMITED CHEMICAL REACTION

The previous theoretical work on diffusion-limited reactions is analyzed. All theories are approximate and are proven to be equivalent (Smoluchowski-Noyes theory). For the reaction A+B(--->)P in two and in three dimensions (2 D and 3 D) the rate function k(t) appearing in the rate equation dA/dt = -k(t)AB is given as a function of (alpha), the probability a pair reacts given an encounter (1 (GREATERTHEQ) (alpha) (GREATERTHEQ) 0). For a reaction in 2 D the rate function goes to zero as 1/ln t whereas the 3 D rate function decays rapidly to a constant value. With the connection between a random walk and diffusion, the probability of not returning to the origin in the first n steps is proportional to the rate function; the first return probability is evaluated asymptotically in 2 D. Computer simulations of 2 D reactions yield a rate function with a time dependence indistinguishable from the theoretical rate function. The irreversible diffusion-limited reaction A+A(--->)P taking place on a ring is formulated exactly, as a stochastic process. Initially 2N particles are placed at random on a ring of length L. Each diffuses with coefficient D until it collides with another resulting in the removal of the pair. The expected fraction of the initial number of particles remaining at the dimensionless time (zeta) = Dt/L('2) in an ensemble of rings is given for all values of N. In the limit of an infinitely long ring with particles put on at the initial density A(,0), the survival fraction S((zeta)') is e('8(zeta)')erfc(8(zeta)')(' 1/2). ((zeta)' = A(,0)('2)Dt). The fluctuations about the mean number are of the order N(' 1/2). The exact solution for an infinite system is compared with the Smoluchowski-Noyes theory. It is found that the exact rate function is always larger than the theoretical rate function with the ratio increasing from one at the beginning of the reaction to (pi)/2 at its completion.