Sodium export cost from a capacitive vesicle ($\Delta G(n)=n{e}^2/C+k_{B}T\ln [n_0/(n_0-n)]$)

Start at sodium electrochemical equilibrium with $n_0$ internal ions. Exporting $n$ ions changes the membrane potential by $-ne/C$ and the internal number density to $(n_0-n)/V$. Subtracting the initially balanced chemical and electrical contributions gives the displayed incremental cost. An initial zero potential further implies $n_0=c_{o}V$ for an ideal fixed external reservoir. Nonzero initial equilibrium potential permits unequal concentrations, so that latter equality is an additional assumption.