Extend the analysis of a system for transport in a charged membrane to the case of a mixture of two electrolyte salts with a common anion (a mixture of \(\mathrm{NaCl}\) and \(\mathrm{KCl}\), for example).

Derive a formula for the membrane potential when one side of the membrane is exposed to concentrations of \(C_{1}(0)\) for \(\mathrm{KCl}\), and \(C_{2}(0)\) for \(\mathrm{NaCl}\) while the other side at \(L\) is exposed to \(C_{1}(L)\) and \(C_{2}(L)\).

Calculate the potential for a membrane that is \(100 \mu \mathrm{m}\) thick with concentrations of \(0.1 \mathrm{M}\) and \(0.01 \mathrm{M}\) on either side.