A formula relating the potential difference between two planes, to the ratio in concentration of Ions at the two planes, when ignoring the interionic interactions (ok at low ionic concentrations), and at equilibrium. For out-of equilibrium, on gets a Fokker-Planck equation (advection-diffusion equation).

For derivation in electrochemical terms, see wikipedia.

For a derivation in terms of Kinetic theory, see the Nelson's biological physics book (sec 4.6.3 page 125)

At equilibrium, the flux $j=0$, giving (after integrating over $x$):

–>>This is also the potential that forms across a membrane which is permeable to one ion species, but not another! See here