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.
It is assumed that the total flux is composed of three elements: diffusion, advection, and electromigration.
is the diffusivity of the chemical species,
is the valence of ionic species,
The electric field may be further decomposed as:
is the magnetic vector potential.
Assuming that the concentration is at equilibrium
and the flow velocity is zero, meaning that only the ion species moves, the Nernst–Planck equation takes the form: 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: Finally, in units of mol/(m2·s) and the gas constant
, one obtains the more familiar form:[3][4] where
is the Faraday constant equal to
; the product of Avogadro constant and the elementary charge.
The Nernst–Planck equation is applied in describing the ion-exchange kinetics in soils.
[5] It has also been applied to membrane electrochemistry.