An accurate understanding of the dynamics of membrane potential formation underpins modern electrophysiology and much of cell biochemistry. Computer simulations using a Nernst-Planck-Poisson (NPP) finite difference method are used to model the dynamic evolution of a series of membrane systems in which two reservoirs of electrolyte solution are separated by a thin membrane which is impermeable to selected species. Two specific examples are considered in detail. The first ("type 1") is the case in which the solutions are monophasic but of unequal concentration, and the second ("type 2") is the case in which the solutions are of equal concentrations but different phase, with a common impermeant ion (a bi-ionic membrane). The validity of the Goldman equation for membrane potential, as applied to each case, is investigated. The type 1 case is shown to reach a steady state, and strong agreement with the Donnan equation for potential difference is observed. For the type 2 case, it is shown that the potential difference consists of two separable components: a localized, Donnan-type potential that reaches a pseudosteady state and a dynamically expanding diffuse component, with properties similar to those of a liquid junction potential, that does not reach a steady state but rather discharges at constant potential difference. This is contrary to the classical interpretation of a static diffuse layer, due to Planck, Henderson, and Goldman.