Equations, mathematical Coulomb potential

To solve the diffusion equation (141) for motion of an ion pair in their mutual coulomb potential and an applied electric field [the potential energy of eqn. (145)] is quite difficult [321]. Rather than discuss the mathematics in detail, the reader is referred to articles by Hong and Noolandi [72, 323—326] which amplify the mathematical details of Onsager s analyses [321], Here, the results alone are mentioned. The escape probability p(r0, d0) depends on both the initial separation of distance between the ions and the initial orientation of the ion-pair with respect to the applied electric field, d0. 

Schrodinger s equation for a single electron and a nucleus with Z protons is an extension into three dimensions (x,y,z) of Equation 6.8, with the potential replaced by the Coulomb potential U(r) = Ze2/Ajt8or. This problem is exactly solvable, but it requires multivariate calculus and some very subtle mathematical manipulations which are beyond the scope of this book. 

Aqueous solutions can be modeled by writing a virial equation such as (17.37) in which osmotic pressure replaces pressure. Friedman (1962) describes applications of cluster expansion theory, which include long-range Coulombic potentials as well as short-range square-well potentials that operate when unlike ions approach within the diameter of a water molecule. These models are mathematically quite cumbersome and are not easily used for routine calculations. They do predict the non-ideal behavior of simple electrolytes such as NaCl quite admirably at moderate concentrations however, they use the square-well potential as an adjustable parameter and so retain some of the properties of the D-H equation with an added adjustable term. For this reason these are not truly a priori models. 

Method (a), the use of the position of the coulombic attraction theory minimum with the Od = 0 value for g, leads to the same mathematical formula for s as that expressing the Donnan equilibrium. However, we cannot say that this constitutes a derivation of the Donnan equilibrium from the coulombic attraction theory because it does not correspond to a physical limit. If Od = 0 really were the case, there would be no reason for the macroions to remain at the minimum position of the interaction potential. Nevertheless, the identity of the two expressions is an interesting result. Because Equation 4.20 is derived in the case in which there is no double layer overlap and Equation 4.1 (the Donnan equilibrium) is likewise derived without reference to the overlap of the double layers, it is precisely in this limit that the calculation should reproduce the Donnan equilibrium. The fact that it does gives us some confidence that our approximations are not too drastic and should lead to physically significant results when applied to overlapping double layers. 

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