There are three primary differences with this result versus the original Larson-Qin theory, 100 but which we attribute to the differences in the original interpretation of the model as due to actual chemical binding versus the charge localization that we consider here. The first difference is that we do not explicitly include the fraction of un-paired charges into the RPA portion of the free energy. This is expected in the Larson-Qin theory because they are explicitly removing charges upon reacting, so the fraction reacted would both change -and be affected by -the RPA fluctuations. 100 In our case, however, the charges remain present and contributing to the RPA. The second difference is similarly due to this difference in interpretation, which is that there is a subtle difference in how the translational entropy of the ions are treated. In the original theory, their mixing entropy is based on the number of 'free' ions due to the binding reactions. In this version, there is a similar correction, but this only affects the prefactor of the natural log term and not the quantity in the natural log.