Nernst–Planck equation

The Nernst–Planck 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 $E=-\nabla \phi$. 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]

The Nernst–Planck equation is given by:

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

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 and 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].

Notes

1. ^ Kirby BJ. (2010). Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices: Chapter 11: Species and Charge Transport. http://www.kirbyresearch.com/index.cfm/wrap/textbook/microfluidicsnanofluidicsch11.html. 
2. ^ Probstein R (1994). Physicochemical Hydrodynamics. 
3. ^ Sparks, D.L. (1988). Kinetics of Soil Chemical Processes.. Academic Press, New York. pp. 101ff