Activity coefficients in ionic solutions

The simplest model capable of reproducing the ion-ion interactions at high dilutions is that given by the Debye-Hiickel Theory. The model assumes that the solvent can be characterized as a structureless, isotropic medium of dielectric constant D which is identical with its macroscopic dielectric constant. The ions are regarded as point charges imbedded in hard spheres of fixed radius and of dielectric constant 1. 

With such a model, the ion-ion interactions are obtained by calculating the most probable distribution of ions around any central ion and then evaluating the energy of the configuration. If (r) is the spherically symmetrical potential in the solution at a distance r from a central ion i of charge z 8, then (r) will be made up of two parts ZxZlDv the coulombic field due to the central ion, and an additional part, ai(r), due to the distribution of the other ions in the solution around t. The potentials at(r) and must satisfy Poisson s equation = — 47rp/D at every point 

The charge density p(r) is determined by the Boltzmann equation, which says that the concentration of positive and negative charges in a spherical shell 4 dr at a distance r from the central ion is proportional 

In order to use this equation, we are forced to make the approximation that zfii/iIDkT 1 so that the exponential can be expanded in a power series  

Because of the condition of electric neutrality, the first term, vanishes and it is the next term which is used  

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