Exact Derivation of the Onsager Limiting Law

Abstract

The limiting law for the conductance of ionic solutions proposed by Onsager in 1927¹ was a major result in the early statistical-mechanical theory of Brownian motion. The model used in the derivation of the limiting law represented the ions as charged point particles moving in a viscous incompressible fluid medium. The velocity with which an ion moves is taken to be the velocity of the surrounding fluid plus the velocity of the ion relative to the surrounding fluid; these two contributions are calculated independently. If interactions among the particles were to be ignored, the velocity of an ion of species i relative to the fluid would be given by

$$ \overrightarrow V _i^{rel} = {D_i}\left( {{{\overrightarrow F }_i}/kT - {\nabla _i}\ln \,{n_i}} \right) $$

where ${\overrightarrow F _i}$ is the average electrostatic force on the ion and n_i is the density of the ions of species i. The diffusion constant D_i is given by the Einstein diffusion relation

$$ {D_i} = kT\,{\omega _i} $$

where k is Boltzmann’s constant, T the temperature, and ωi is the mobility in the fluid of ions species i.