The following derivation is adapted from Foundations of Chemical Kinetics.
This derivation assumes the reaction . Consider a sphere of radius , centered at a spherical molecule A, with reactant B flowing in and out of it. A reaction is considered to occur if molecules A and B touch, that is, when the distance between the two molecules is  apart. 
If we assume a local steady state, then the rate at which B reaches  is the limiting factor and balances the reaction. 
Therefore, the steady state condition becomes 
where 
 is the flux of B, as given by Fick's law of diffusion, 
where  is the diffusion coefficient and can be obtained by the Stokes-Einstein equation, and the second term is the gradient of the chemical potential with respect to position. Note that [B] refers to the average concentration of B in the solution, while [B](r) is the "local concentration" of B at position r.
Inserting 2  into 1 results in 
It is convenient at this point to use the identity 
 allowing us to rewrite 3 as 
Rearranging 4 allows us to write 
Using the boundary conditions that , ie the local concentration of B approaches that of the solution at large distances, and consequently , as , we can solve 5 by separation of variables, we get
or
7. <math> \frac{k[B]}{4\pi D_{AB}\beta }= 
 [B]-[B](R_{AB})\exp(U(R_{AB})/k_{B}T ) )
For the reaction between A and B, there is an inherent reaction constant , so . Substituting this into 7 and rearranging yields