## 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 j_{i} is the flux, c_{i} is the
concentration of species *i*, R is the gas constant, T is the absolute temperature,
x is the distance in the x direction, z_{i} 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. D_{i} is the the diffusion coefficient of *i*
and has units cm^{2} 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 m^{2} s^{-1} joule^{-1}. This can be
arranged to give (m^{2} s^{-1})/(joule mol^{-1}) or D_{i}/RT.

To convert flux j_{i} to current density *i* we
multiply both sides by z_{i}F. 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. D^{0}
represents the diffusion coefficient at infinite dilution.