Smoluchowski equation distance probability distribution

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

Noyes [269, 270] and, more recently, Northrup and Hynes [103] have endeavoured to incorporate some aspects of the caging process into the Smoluchowski random flight or diffusion equation approach. Both authors develop essentially phenomenological analyses, which introduce further parameters into an expression for escape probabilities for reaction, that are of imprecisely known magnitude and are probably not discrete values but distributed about some mean. Since these theories expose further aspects of diffusion-controlled processes over short distances near encounter, they will be discussed briefly (see also Chap. 8, Sect. 2.6). 

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