# Nernst–Planck equation

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

The Nernst–Planck equation is a continuity equation for the time-dependent concentration ${\displaystyle c(t,{\bf {x}})}$  of a chemical species:

${\displaystyle {\partial c \over {\partial t}}+\nabla \cdot {\bf {J}}=0}$

where ${\displaystyle {\bf {J}}}$  is the flux. It is assumed that the total flux is composed of three elements: diffusion, advection, and electromigration. This implies that the concentration is affected by an ionic concentration gradient ${\displaystyle \nabla c}$ , flow velocity ${\displaystyle {\bf {v}}}$ , and an electric field ${\displaystyle {\bf {E}}}$ :

${\displaystyle {\bf {J}}=-\underbrace {D\nabla c} _{\text{Diffusion}}+\underbrace {c{\bf {v}}} _{\text{Advection}}+\underbrace {{Dze \over {k_{\text{B}}T}}c{\bf {E}}} _{\text{Electromigration}}}$

where ${\displaystyle D}$  is the diffusivity of the chemical species, ${\displaystyle z}$  is the valence of ionic species, ${\displaystyle e}$  is the elementary charge, ${\displaystyle k_{\text{B}}}$  is the Boltzmann constant, and ${\displaystyle T}$  is the absolute temperature. The electric field may be further decomposed as:

${\displaystyle {\bf {E}}=-\nabla \phi -{\partial {\bf {A}} \over {\partial t}}}$

where ${\displaystyle \phi }$  is the electric potential and ${\displaystyle {\bf {A}}}$  is the magnetic vector potential. Therefore, the Nernst–Planck equation is given by:

${\displaystyle {\frac {\partial c}{\partial t}}=\nabla \cdot \left[D\nabla c-c\mathbf {v} +{\frac {Dze}{k_{\text{B}}T}}c\left(\nabla \phi +{\partial {\bf {A}} \over {\partial t}}\right)\right]}$

### Simplifications

Assuming that the concentration is at equilibrium ${\displaystyle (\partial c/\partial t=0)}$  and the flow velocity is zero, meaning that only the ion species moves, the Nernst–Planck equation takes the form:

${\displaystyle \nabla \cdot \left\{D\left[\nabla c+{ze \over {k_{\text{B}}T}}c\left(\nabla \phi +{\partial {\bf {A}} \over {\partial t}}\right)\right]\right\}=0}$

Rather than a general electric field, if we assume that only the electrostatic component is significant, the equation is further simplified by removing the time derivative of the magnetic vector potential:

${\displaystyle \nabla \cdot \left[D\left(\nabla c+{ze \over {k_{\text{B}}T}}c\nabla \phi \right)\right]=0}$

Finally, in units of mol/(m2·s) and the gas constant ${\displaystyle R}$ , one obtains the more familiar form:[3][4]

${\displaystyle \nabla \cdot \left[D\left(\nabla c+{zF \over {RT}}c\nabla \phi \right)\right]=0}$

where ${\displaystyle F}$  is the Faraday constant equal to ${\displaystyle N_{\text{A}}e}$ ; the product of Avogadro constant and the elementary charge.

## Applications

The Nernst–Planck equation is applied in describing the ion-exchange kinetics in soils.[5] It has also been applied to membrane electrochemistry.[6]