The Nernst-Planck Equation
Cody Zundel and Jim Hutchins
[latex]J = J_{drift} + J_{diff} \\ \mkern12mu = -\mu z [C] \frac{\partial V}{\partial x} - D \frac{\partial [C]}{\partial x}[/latex]
[latex]J = -(\mu z [C] \frac{\partial V}{\partial x} + \frac{\mu k T}{q} \frac{\partial [C]}{\partial x})[/latex]
[latex]J =J/N_{A} = \frac{-\mu z [C]}{N_{A}} \frac{\partial V}{\partial x} - \frac{\mu k T}{N_{A}q} \frac{\partial [C]}{\partial x} \\ \mkern12mu = -(\mu z [C] \frac{\partial V}{\partial x} + u \frac{RT}{F} \frac{\partial [C]}{\partial x })[/latex]