Nernst–Planck equation - Alchetron, The Free Social Encyclopedia

The time dependent form of the Nernst–Planck equation is a conservation of mass equation used to describe the motion of a charged chemical species in a fluid medium. It describes the flux of ions under the influence of both an ionic concentration gradient ∇c and an electric field E = −∇φ −∂A/∂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:

∂ c ∂ t = − ∇ ⋅ J J = − [ D ∇ c − u c + D z e k B T c ( ∇ ϕ + ∂ A ∂ t ) ] ∂ c ∂ t = ∇ ⋅ [ D ∇ c − u c + D z e k B T c ( ∇ ϕ + ∂ A ∂ t ) ]

Where

t is time,

D is the diffusivity of the chemical species,

c is the concentration of the species

z is the valence of ionic species,

e is the elementary charge,

k_B is the Boltzmann constant,

T is the temperature,

u is velocity of fluid.

If the diffusing particles are themselves charged they are influenced by the electric field. Hence the Nernst–Planck equation is applied in describing the ion-exchange kinetics in soils.

Setting time derivatives to zero, and the fluid velocity to zero (only the ion species moves),

J = − [ D ∇ c + D z e k B T c ( ∇ ϕ + ∂ A ∂ t ) ]

In the static electromagnetic conditions, one obtains the steady state Nernst–Planck equation

J = − [ D ∇ c + D z e k B T c ( ∇ ϕ ) ]

Finally, in units of mol/(m²·s) and the gas constant R, one obtains the more familiar form:

J = − D [ ∇ c + F z R T c ( ∇ ϕ ) ]

where F is the Faraday constant equal to N_Ae.

References

Nernst–Planck equation Wikipedia

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