Improved rate theories of chemical reactions

Abstract

A mathematical difficulty in Kramers’ theory on reaction rate is pointed out and his treatment is improved by considering the Brownian motion near the bottom of potential minimum explicitly. We used the Fokker–Planck–Kramers equation (FPK) and derived a modified Smoluchowski’s equation by eliminating the velocity dependence. Moreover, we assumed the stationary state and defined the reaction coefficient by the flux divided by the concentration of the reactant. Finally, we solved the modified Smoluchowski by using computer assisted symbolic manipulation programs and obtained the expression for the rate coefficient by a single theory. It is found that there exists a lower limit for the friction value which is corresponding to the critical damping motion for an ordinary harmonic damped oscillator. It is also shown that we can have Kramers’ turnover for the case of the steep decline of the potential after passing the maximum. The Boltzmann transport equation for the strong collision case with the collision term by Bhatnager, Gross, and Krook (BBGK) is also solved exactly by the symbolic manipulation to find the rate coefficient which does not lead to the Kramers turnover.