The molecular origins of the common ion effect and the salting out of nonpolar molecules from aqueous solutions are investigated. Thermodynamic stability criteria for a common ion mixture in a polar solvent are derived. Kirkwood-Buff statistical thermodynamics is used to make the connection with the microscopic pair correlation functions. The observed sensitivity of the compositional stability with respect to ionic strength indicates that a demixing transition is the primary cause of the instability for the common ion effect for our model Lennard-Jones plus Coulomb Hamiltonian.