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| The '''[[Nernst]]–[[Max Planck|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 [[ion]]s under the influence of both an ionic [[concentration gradient]] <math>\nabla c</math> and an [[electric field]] <math>E=-\nabla \phi - \frac{\partial \mathbf A}{\partial t}</math>. 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:<ref name=Kirby>{{cite book | author=Kirby BJ. | title=Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices: Chapter 11: Species and Charge Transport |
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| url=http://www.kirbyresearch.com/index.cfm/wrap/textbook/microfluidicsnanofluidicsch11.html | year=2010}}</ref><ref name=Probstein>{{cite book | author=Probstein R | title=Physicochemical Hydrodynamics |
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| year=1994}}</ref>
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| <math>\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]</math>
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| Where
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| * t is time,
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| * D is the [[Mass diffusivity|diffusivity]] of the chemical species,
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| * c is the concentration of the species, and u is the velocity of the fluid,
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| * ''z'' is the valence of ionic species,
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| * ''e'' is the [[elementary charge]],
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| * <math>k_B</math> is the [[Boltzmann constant]]
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| * ''T'' is the temperature.
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| 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]] [[chemical kinetics|kinetics]] in soils.<ref name=Sparks1988>{{Cite document | last = Sparks | first = D.L. | year = 1988 | title = Kinetics of Soil Chemical Processes. | publisher=Academic Press, New York | pages = 101ff | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->
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| }}</ref>
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| ==See also==
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| *[[Goldman-Hodgkin-Katz equation]]
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| *[[Bioelectrochemistry]]
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| ==Notes==
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| {{reflist}}
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| {{DEFAULTSORT:Nernst-Planck equation}}
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| [[Category:Articles with inconsistent citation formats]]
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| [[Category:Diffusion]]
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| [[Category:Physical chemistry]]
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| [[Category:Electrochemical equations]]
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| [[Category:Statistical mechanics]]
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