Distribution of charge and potential in an electrolyte bounded by two plane infinite parallel plates

Abstract

The density of electricity, ρ , in a positively charged binary electrolyte associated with the distribution of ions in it is connected with the electrostatic potential, ψ , by the equations ∇ 2 ψ = — 4 πρ/D , (1.1) ρ = vFC [exp( — vFψ/RT ) — exp( vFψ/RT )] (1.2) (See § 2 for the meaning of the various symbols.) If the electrolyte is contained between two plane infinite parallel plates which cut the x -axis at right angles at the points x = 0 and x = l the equation (1.1) assumes the form d 2 ψ / dx 2 = -4 πρ / D , (1.1 a ) while equation (1.2) remains unchanged. The basis of the following theory is due to Gouy (1910), and the above equations can be found, with a somewhat different notation, in a paper due to Chapman (1913). Gouy calculated the distribution of charge in a semi-infinite electrolyte bounded by a plane plate. Rosenhead and Miller (1937) applied his results to the case of an electrolyte between two plane plates by assuming that the charges associated with the two plates could be superposed. This assumption can only be justified when vFψ/RT is so small that powers of it higher than the second can be neglected so that the differential equation for ψ and hence that for ρ , becomes linear. In this paper an exact solution of the problem of the distribution of charge between two plane plates is obtained.