Nernst–Planck equation

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The NernstPlanck 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 - \frac{\partial \mathbf A}{\partial 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]

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


  • 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
  • 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]

See also[edit]


  1. ^ Kirby BJ. (2010). Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices: Chapter 11: Species and Charge Transport. 
  2. ^ Probstein R (1994). Physicochemical Hydrodynamics. 
  3. ^ Sparks, D.L. (1988). "Kinetics of Soil Chemical Processes.". Academic Press, New York. pp. 101ff.