Entropy (statistical thermodynamics)

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 $\mathrm{\nabla }c$ and an electric field $E=-\mathrm{\nabla }\varphi -\frac{\partial \mathbf{A}}{\partial t}$. 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:

$\frac{\partial c}{\partial t}=\mathrm{\nabla }\cdot [D\mathrm{\nabla }c-uc+\frac{Dze}{k_{B}T}c(\mathrm{\nabla }\varphi +\frac{\partial \mathbf{A}}{\partial t})]$

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
• 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.

See also

• Goldman-Hodgkin-Katz equation
• Bioelectrochemistry

Notes

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