Diffusion and Conductivity: the Nernst-Einstein Equation
The equation fo2r the flux of a species in a concentration gradient and a potential gradient is
where ji is the flux, ci is the concentration of species i, R is the gas constant, T is the absolute temperature, x is the distance in the x direction, zi is the charge on ion i in coul, F is the Faraday constant, and f is the potential in volts.
In the absence of an electric field, i.e., if 
, then the equation becomes
or
or
which expresses flux due to a concentration gradient only. This is a form of Fick’s first law. Di is the the diffusion coefficient of i and has units cm2 s-1. Under conditions of zero concentration gradient, the flux is given as
Here flux is expressed only as migration, the movement of charged species through the solution under the influence of an electric field (potential gradient). The units: c is in mol m-3. F is in coul mol-1. The potential gradient is in V m-1. The product of these gives mol m-3 coul mol-1 V m-1 or joule m-2. Since the flux is in mol m-2 s-1 the units of the proportionality constant k are (mol m-2 s-1)/( joule m-2) or mol m2 s-1 joule-1. This can be arranged to give (m2 s-1)/(joule mol-1) or Di/RT.
To convert flux ji to current density i we multiply both sides by ziF. Flux is in mol m-2 s-1. F is in coul mol-1. Mol m-2 s-1 coul mol-1 gives coul m-2 s-1, the units of current density.
Conductivity can be expressed in the form k=l/RA. By Ohm’s law R=E/I. Substitution gives
where I is the current and A is the area. Thus conductivity can be described as the ratio of current density to potential gradient. Substituting for i gives
Since the molar conductivity of ion i is k/c we have
which is a form of the Nernst-Einstein equation. From this a relationship between the diffusion coefficient and molar conductivity of an electrolyte can be derived.
This is another form of the Nernst-Einstein equation. D0 represents the diffusion coefficient at infinite dilution.