# Nernst–Planck equation

The NernstPlanck equation is a conservation of mass equation used to describe the motion of chemical species in a fluid medium. It describes the flux of ions under the influence of both an ionic concentration gradient $\nabla c$ and an electric field $E=-\nabla \phi - \frac{\partial \mathbf A}{\partial t}$. 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]

$\frac{\partial c}{\partial t} = \nabla \cdot \left[ D \nabla c - u c + \frac{Dze}{k_B T}c(\nabla \phi+\frac{\partial \mathbf A}{\partial t}) \right]$

Where

• t is time,
• D is the diffusivity of the chemical species,
• c is the concentration of the species, and u is the velocity of the fluid,
• z is the valence of ionic species,
• e is the elementary charge,
• $k_B$ is the Boltzmann constant
• T is the temperature.

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