Term information
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 species, z is the valence of ionic species, e is the elementary charge, kB 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. source: http://en.wikipedia.org/wiki/Nernst%E2%80%93Planck_equation
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mrow> <mfrac> <mrow> <mo>∂</mo> <mo>⁢</mo> <mi>c</mi> </mrow> <mrow> <mo>∂</mo> <mo>⁢</mo> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mrow> <mo>∇</mo> <mo>⁢</mo> <mo>⋅</mo> <mo>⁢</mo> <mfenced close="]" open="["> <mrow> <mrow> <mrow> <mi>D</mi> <mo>⁢</mo> <mo>∇</mo> <mo>⁢</mo> <mi>c</mi> </mrow> <mo>-</mo> <mrow> <mi>u</mi> <mo>⁢</mo> <mi>c</mi> </mrow> </mrow> <mo>+</mo> <mrow> <mfrac> <mrow> <mi>D</mi> <mo>⁢</mo> <mi>z</mi> <mo>⁢</mo> <mi>e</mi> </mrow> <mrow> <msub> <mi>k</mi> <mi>B</mi> </msub> <mo>⁢</mo> <mi>T</mi> </mrow> </mfrac> <mo>⁢</mo> <mi>c</mi> <mo>⁢</mo> <mo>∇</mo> <mo>⁢</mo> <mi>ϕ</mi> </mrow> </mrow> </mfenced> </mrow> </mrow> </math>
\frac{\partial c}{\partial t} = \nabla \cdot \left[ D \nabla c - u c + \frac{Dze}{k_B T}c\nabla \phi \right]
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <semantics> <mrow> <mfrac> <mrow> <mo>∂</mo> <mi>c</mi> </mrow> <mrow> <mo>∂</mo> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mo>∇</mo> <mo>⋅</mo> <mfenced close="]" open="["> <mrow> <mi>D</mi> <mo>∇</mo> <mi>c</mi> <mo>-</mo> <mi>u</mi> <mi>c</mi> <mo>+</mo> <mfrac> <mrow> <mi>D</mi> <mi>z</mi> <mi>e</mi> </mrow> <mrow> <msub> <mi>k</mi> <mi>B</mi> </msub> <mi>T</mi> </mrow> </mfrac> <mi>c</mi> <mo>∇</mo> <mi>ϕ</mi> </mrow> </mfenced> </mrow> <annotation encoding="SnuggleTeX">\[ \frac{\partial c}{\partial t} = \nabla \cdot \left[ D \nabla c - u c + \frac{Dze}{k_B T}c\nabla \phi \right] \]</annotation> </semantics> </math>
Term relations
- diffusion equation
- has_variable some (
variable and
is_about some time) - has_variable some (
variable and
is_about some velocity) - has_variable some (
variable and
is_about some concentration of) - has_constant value Boltzmann constant
- has_variable some (
variable and
is_about some electric charge) - has_variable some (
variable and
is_about some (
quality and
quality_of some diffusion)) - has_variable some (
variable and
is_about some temperature)