Charged systems obey exact asymptotic relations. Excluded volume for each ion is needed. These are included in the Mean Spherical Approximation, thus this theory is exact for dense systems. A method that interpolates between the MSA and the PY equation and is amazingly accurate for many systems (Blum Narten, J. Chem. Phys. 56, 5197 (1972)). The switching was done at a distance s such that all distribution functions were continuous everywhere. In the present work we extend this theory to the general case of an arbitrary mixture of charged hard spheres, and obtain an explicit analytical solution in terms of a few scaling parameters G : We use the new extended soft mean spherical approximation (ESMSA). (Blum Arias, J. Phys. Cond. Matt. 18, S2437,(2006), Mol. Phys. 104,3801,(2006) that also satisfies the low temperature and low density association limits.

In the ESMSA the closure of the Ornstein Zernike (OZ) equation near the excluded core region is divided in two regions: For the inner region we can use an exponential closure, for example the Percus-Yevick (PY) or the contact pair correlation function as derived in the BIMSA-EXP approximation ( Bernard and Blum,J. Chem. Phys. 104, 4746,(1996)). This leads to the correct exact limits of zero temperature and zero density. In the outer region far from the central ion we use the Mean Spherical Approximation (MSA) closure. An explicit solution is obtained from the continuity of the pair correlation function when the distances are additive. The thermodynamics and structure functions are also discussed.