Individual ion species chemical potentials in the Mean Spherical Approximation

The Mean Spherical Approximation (MSA) is a commonly-used thermodynamic theory for computing the energetics of ions in the primitive model (i.e., charged hard-sphere ions in a background dielectric). For the excess chemical potential, however, the early MSA formulations (which were widely adopted) only included the terms needed to compute the mean excess chemical potential (or the mean activity coefficient). Other terms for the chemical potential $\mu_i$ of individual species $i$ were not included because they sum to $0$ in the mean chemical potential. Here, we derive these terms to give a complete MSA formulation of the chemical potential. The result is a simple additive term for $\mu_i$ that we show is a qualitative improvement over the previous MSA version. In addition, our derivation shows that the MSA's assumption of global charge neutrality is not strictly necessary, so that the MSA is also valid for systems close to neutrality.