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 describes the flux of ions under the influence of both an ionic concentration gradient ∇c and an electric field E = −∇φ −∂A/∂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:

Where

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.

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

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

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

where F is the Faraday constant equal to N_Ae.

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