# Nernst–Planck equation

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The time dependent form of the NernstPlanck equation is a conservation of mass equation used to describe the motion of a charged chemical species in a fluid medium. It describes the flux of ions under the influence of both an ionic concentration gradientc and an electric field E = −∇φA/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]

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

Where

• t is time,
• D is the diffusivity of the chemical species,
• c is the concentration of the species
• z is the valence of ionic species,
• e is the elementary charge,
• kB is the Boltzmann constant,
• T is the temperature,
• 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),

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

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

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

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

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

where F is the Faraday constant equal to NAe.