= Solution
The lipid <cell membrane> separates ionic solutions and has selective <ion channels>, ion pumps, and a finite <capacitance>. Ionic gradients and unequal permeability allow a small charge separation to generate a <membrane potential>. The bulk compartments can remain almost electrically neutral even when the narrow interfacial layers carry opposite <electric charges>.
For one permeant ionic species with valence $z$, its <electrochemical potential> is $\widetilde\mu=\mu^0+k_BT\log a+ze\phi$, where $a$ is its activity. With no applied drive and no net ionic current, electrochemical equilibrium across a permeable membrane gives
$$
k_BT\log a_{\rm in}+ze\phi_{\rm in}
=k_BT\log a_{\rm out}+ze\phi_{\rm out}.
$$
Hence, with the inside-minus-outside convention,
$$
\boxed{\Psi_m=\phi_{\rm in}-\phi_{\rm out}=\frac{k_BT}{ze}\log\frac{a_{\rm out}}{a_{\rm in}}.}
$$
For dilute ideal solutions the activities may be replaced by <concentrations>. At $25^\circ\mathrm C$ this is $(59.16\,\mathrm{mV}/z)\log_{10}(c_{\rm out}/c_{\rm in})$. The resulting <Nernst equilibrium potential> balances diffusive and electrical driving, not necessarily equal <concentrations>. Physically there must be countercharge, such as impermeant counterions; the single-species idealization concerns the permeant current. With multiple permeant species, zero total current need not mean zero current of each species, and the resting <membrane potential> is not generally any single species' <Nernst equilibrium potential>.
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