Variational derivation of the MSA

These results reveal the physical meaning of the MSA solution as an interpolation between two exact lower bounds, the Debye lower bound, effective when 

We can conceive the MSA as a variational problem [5], in which the free energy AA is minimal 

The generalization to a more general case by minimization of a functional which is identical to the MSA excess free energy requires that the excess energy be of the form [6, 7]  

Since this is a function of the single parameter F, the derivative must be zero 

In the last decades the progress of statistical mechanics has opened the possibility of treating quantitatively the effect of ionic interactions at the Mc-Millan Mayer level for clusters [8] [9] [10]. It is possible to include the non ideal contribution in the statistical formulation of the thermodynamic properties of ionic solutions [11] [12] [13]. This can be done combining the concept of ionic association to the evaluation of excess thermodynamic properties. 

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