# Nernst–Planck equation

The time dependent form of the Nernst–Planck equation is a conservation of mass equation used to describe the motion of a charged chemical species in a fluid medium. It extends Fick's law of diffusion for the case where the diffusing particles are also moved with respect to the fluid by electrostatic forces:[1][2] It is named after Walther Nernst and Max Planck.

## Equation

It describes the flux of ions under the influence of both an ionic concentration gradientc and an electric field E = −∇${\displaystyle \phi }$ A/t.

${\displaystyle {\frac {\partial c}{\partial t}}=-\nabla \cdot J\quad |\quad J=-\left[D\nabla c-uc+{\frac {Dze}{k_{\mathrm {B} }T}}c\left(\nabla \phi +{\frac {\partial \mathbf {A} }{\partial t}}\right)\right]}$
${\displaystyle \iff {\frac {\partial c}{\partial t}}=\nabla \cdot \left[D\nabla c-uc+{\frac {Dze}{k_{\mathrm {B} }T}}c\left(\nabla \phi +{\frac {\partial \mathbf {A} }{\partial t}}\right)\right]}$

Where J is the diffusion flux density, t is time, D is the diffusivity of the chemical species, c is the concentration of the species, z is the valence of ionic species, e is the elementary charge, kB is the Boltzmann constant, T is the temperature, ${\displaystyle u}$  is velocity of fluid, ${\displaystyle \phi }$  is the electric potential, ${\displaystyle \mathbf {A} }$  is the magnetic vector potential.

If the diffusing particles are themselves charged they are influenced by the electric field. Hence the Nernst–Planck equation is applied in describing the ion-exchange kinetics in soils.[3]

Setting time derivatives to zero, and the fluid velocity to zero (only the ion species moves),

${\displaystyle J=-\left[D\nabla c+{\frac {Dze}{k_{\mathrm {B} }T}}c\left(\nabla \phi +{\frac {\partial \mathbf {A} }{\partial t}}\right)\right]}$

In the static electromagnetic conditions, one obtains the steady state Nernst–Planck equation

${\displaystyle J=-\left[D\nabla c+{\frac {Dze}{k_{\rm {B}}T}}c(\nabla \phi )\right]}$

Finally, in units of mol/(m2·s) and the gas constant R, one obtains the more familiar form:[4][5]

${\displaystyle J=-D\left[\nabla c+{\frac {zF}{RT}}c(\nabla \phi )\right]}$

where F is the Faraday constant equal to NAe.