Scaling for Mixtures of Hard Ions and Dipoles in the Mean Spherical Approximation

Using new scaling parameters ${\beta }_i$, we derive simple expressions for the excess thermodynamic properties of the Mean Spherical Approximation (MSA) for the ion-dipole mixture. For the MSA and its extensions we have shown that the thermodynamic excess functions are a function of a reduced set of scaling matrices ${\mathbf{\Gamma }}_{\chi }$. We show now that for factorizable interactions like the hard ion-dipole mixture there is a further reduction to a diagonal matrices ${\mathbf{\beta }}_{\chi }$. The excess thermodynamic properties are simple functions of these new parameters. For the entropy we get $$S=-\frac{kV}{3\pi }(\mathcal{F}[{\mathbf{\beta }}_{\alpha }]{)}_{\alpha \in \mathbf{\chi }}$$ where $\mathcal{F}$ is an algebraic functional of the scaling matrices of irreducible representations $\chi$ of the closure of the Ornstein-Zernike. The new scaling parameters ${\beta }_i$, are also simply related to the chemical potentials of the components. The analysis also provides a new definition of the Born solvation energy for arbitrary concentrations of electrolytes.