Equation used to calculate the electromigration of ions in a fluid
The Nernst–Planck equation is a
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 of a chemical species:
where is the
concentration gradient
,
flow velocity , and an
electric field :
where is the diffusivity of the chemical species, is the valence of ionic species, is the elementary charge, is the Boltzmann constant, and is the absolute temperature. The electric field may be further decomposed as:
where is the electric potential and is the magnetic vector potential. Therefore, the Nernst–Planck equation is given by:
Simplifications
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.
Applications
The Nernst–Planck equation is applied in describing the
See also
References