The Journal of
Physical Chemistry
0 Copyright, 1984, by the American Chemical Society
VOLUME 88, NUMBER 13
JUNE 21, 1984
LETTERS Diffusion-Controlled Reactions of Chemically Anisotropic Molecules S. I. Ternkin* Institute of Chemical Kinetics and Combustion, Novosibirsk 630090, USSR
and B. I. Yakobson Institute of Solid State Chemistry and Raw Materials, Novosibirsk 630091, USSR (Received: August 1 1 , 1983; In Final Form: January 25, 1984)
Rigorous expressions for orientationally constrained reaction rate constants in condensed media have been derived on the basis of the Wilemski-Fixman-Doi approach. The following assumptions have been made: the relative translational motion is diffusive, and the isotropic stochastic reorientation of each molecule is Markoffian and arbitrarily correlated. The general results have been reduced to simple formulae for small steric factors of either one or both partners.
Since Smoluchowski the theory of diffusion-controlled bimolecular reactions has been well developed for reagents with spherically symmetric reactivity.' The simplest and well-known formula for the rate constant of a purely diffusion-controlled reaction is
kDo= 47rRD
(1)
+
where R = a b is the sum of the reagent radii, and D = DA D B is the relative diffusion coefficient. However, the generalization of this result for chemically anisotropic molecules, based on the equation for diffusion in multidimensional space (including angular variables of both particles) with corresponding boundary conditions, remains a severe mathematical problem even for numerical methods.2 This fact has stimulated an active search for physical approximations that could simplify the p r ~ b l e m . ~ -The '~
+
(1) H. Eyring, S . H. Lin and S.M. Lin, "Basic Chemical Kinetics'', Wiley, New York, 1980. (2) K. Solc and W. H. Stockmayer, J. Chem. Phys., 54, 2981 (1971); In?. J . Chem. Kinet., 5, 733 (1973). (3) J. M. Schurr and K. S . Schmitz, J . Phys. Chem., 80, 1934 (1976). (4) K. M. Salikhov, Teor. Eksp. Khim., 13, 732 (1977).
0022-3654/84/2088-2679$01.50/0
most interesting approach is based on the "closure approximation"" which allows the rate constant to be calculated from the motion parameters in the configuration space of the nonreactive partners. A high accuracy for this approximation has been mathematically substantiated,12 giving the expression k = V/r (2) where Vis the reaction zone volume, and 7 is the total residence time of the system representation point inside the volume V. In the particular case of chemically isotropic reagents, eq 1 can be derived from eq 2 under the assumption that the reaction zone is a thin spherical layer with a width A
