Some Mathematical Formulas and Identities

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. 

The preceding treatment considered separation in a time-varying output from a separator of the chromatographic type. Similar methods of describing separation, namely Rs, Pi, 72, ate., may also be adopted for describing the separation between two concentration profiles in Figure 2.5.1 if the abscissa is a distance coordinate z instead of a time coordinate t. One then replaces tg, by z,-, t tt by (7zj, Cj(t) by Cj(z), tc by z etc. The mathematical formulae will be identical with the proper substitutions made, and we may conclude that no special treatment is needed to describe the separation between two concentration profiles spatially displaced but having some overlap. 

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