Derivation of the Debye—Smoluchowski equation

Attention has so far been focused on reaction between spherical particles diffusing in a hydrodynamic continuum with no forces acting between reactants. In this chapter, the most important force, the coulomb interaction, between ions in solution is included. The potential energy, 

Now the probability of reactants 1 and 2 simultaneously being at ri and r2 is n(ri, r2 ). The two-body density or concentration changes with time due to a net flux into the small volume dridr2, that is the law of conservation of matter or the equation of continuity of mass 

Ji and J2 are the fluxes of density n into the hypervolume dr 1 dr2 due to the entrance of reactant 1 or 2, respectively, into that volume. These 

This is the diffusion equation for simultaneous motion of two particles in the field of force of each other. In Chap. 9, Sect. 2, the equation is further reduced to two uncoupled diffusion equations, which is valid providing the potential energy, U, is dependent only on the relative separation of particles, —12. In this case, n can be shown to be the product of the density of finding the pair of reactants with their centre of diffusion coefficient coordinate, x = (D2ri + i r2)/(Di + 2)) M(x,t), and the density of finding the pair of reactants separated by r = rj — r2, p(r,t), i.e. 

The equation for p is the diffusion equation for motion of one reactant about the other stationary reactant with a relative diffusion Z) = + D2 

---