DYNAMICS OF BROWNIAN-MOTION UNDER A POTENTIAL AND THEORY ON THE RATE OF CHEMICAL-REACTIONS

We have introduced a two square-well potential system, solved the Smoluchowski equation, calculated exactly the concentration of the reactant, compared it to that resulting from the phenomenological rate equations and finally obtained expressions for rate constants for a first-order reversible reaction. It is found that the pre-exponential factors for both constants are just the diffusion coefficients divided by the square of the widths of potential wells. Our expressions for rate constants satisfy a relation between standard free energy and the equilibrium constant arising from chemical thermodynamics. We have also developed a general theory on the relaxation processes of approachment of a system to the equilibrium Boltzmann for an arbitrary potential with sufficiently deep well and the escape rate from the well for an asymptotically long time limit of the reaction, which enables us to calculate the rate constants for given potentials. Furthermore, we obtained the rate constant for a dissociation reaction of an ion-pair interacting through the Coulomb potential and found a maximum as the sum of the radii of the ion-pair increases.