= Sodium export cost from a capacitive vesicle
{title2=$\Delta G(n)=ne^2/C+k_BT\ln[n_0/(n_0-n)]$}
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 $n_0=c_oV$ for an ideal fixed external reservoir. Nonzero initial equilibrium potential permits unequal concentrations, so that latter equality is an additional assumption.
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