Stochastic theory for rate constants of chemical reactions in liquid solution

Abstract

The stochastic model of irreversible processes is developed in a fashion that yields expressions for the magnitudes and temperature dependences of chemical reaction rate constants. The model is sufficiently abstract to encompass reactions in liquids as well as those in gases. For liquid reactions both the general point of view and one feature of the results (the “frequency factor”) are apparently new. For gas reactions the viewpoints and the results are compatible with (though not as detailed as) those of well-established collision theory. An approximation (to the effect that nonreactive but energy-redistributing transitions are much more frequent than reactive ones) may limit the quantitative, though not the schematic, application of this development to reactions in the presence of an excess of inert diluent. The traditional assumption to the effect that reactants are in equilibrium with “activated complexes” (whether or not such exist in the sense of possessing well-defined microstates) is avoided. As in previous discussion of sufficient conditions for an Arrhenius rate law, in which certain complicating features (treated here) of the case of chemical reactions were ignored, the method used here involves taking explicit account of the role of those (rapidly equilibrating via frequent nonreactive transitions) degrees of freedom that serve as the activation-energy-supplying (and temperature defining) “reservoir” by use of an especially detailed form of stochastic master equation. A concise form for the master equation facilitates (1) the appropriate extension of the previously described steady state treatment of the case in which a “transition state” lasts sufficiently long to be internally equilibrated and (2) the treatment of the probably more realistic case in which reactants undergo transitions directly to products without a definable intermediate and in which “transition state” can only be defined in terms of an energy threshold in the reaction transition probability. The latter case is a generalization to arbitrary density of that originally treated by Ross and Mazur within the framework of collision theory for bimoleculear dilute gas reactions.