  We study fluid-fluid equilibrium in the simplest model of ionic solutions
where the solvent is explicitly included, i.e., a binary mixture consisting of
a restricted primitive model (RPM) and neutral hard-spheres (RPM-HS mixture).
First, using the collective variable method we find free energy, pressure and
partial chemical potentials in the random phase approximation (RPA) for a
rather general model that takes into consideration solvent-solvent and
solvent-ion interactions beyond the hard core. In the special case of a RPM-HS
mixture, we consider two regularizations of the Coulomb potential inside the
hard core, i.e., the Weeks-Chandler-Andersen (WCA) regularization leading to
the WCA approximation and the optimized regularization giving the optimized RPA
(ORPA) or the mean spherical approximation (MSA). Furthermore, we calculate the
phase coexistence using the associative mean spherical approximation (AMSA). In
general, the three approximations produce qualitatively similar phase diagrams
of the RPM-HS mixture, i.e., a fluid-fluid coexistence envelope with an upper
critical solution point, a shift of the coexistence region towards higher total
number densities and higher solvent concentrations with increasing pressure,
and a small increase of the critical temperature with an increase of pressure.
As for a pure RPM, the AMSA leads to the best agreement with the available
simulation data when the association constant proposed by Olaussen and Stell is
used. We also discuss the peculiarities of the phase diagrams in the WCA
approximation.
