Chemistry:Nernst–Planck equation
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 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] It is named after Walther Nernst and Max Planck.
Equation
The Nernst–Planck equation is a continuity equation for the time-dependent concentration [math]\displaystyle{ c(t,{\bf x}) }[/math] of a chemical species:
- [math]\displaystyle{ {\partial c\over{\partial t}} + \nabla \cdot {\bf J} = 0 }[/math]
where [math]\displaystyle{ {\bf J} }[/math] is the flux. It is assumed that the total flux is composed of three elements: diffusion, advection, and electromigration. This implies that the concentration is affected by an ionic concentration gradient [math]\displaystyle{ \nabla c }[/math], flow velocity [math]\displaystyle{ {\bf v} }[/math], and an electric field [math]\displaystyle{ {\bf E} }[/math]:
- [math]\displaystyle{ {\bf J} = -\underbrace{D\nabla c}_{\text{Diffusion}} + \underbrace{c{\bf v}}_{\text{Advection}} +\underbrace{{Dze\over{k_\text{B}T}}c{\bf E}}_{\text{Electromigration}} }[/math]
where [math]\displaystyle{ D }[/math] is the diffusivity of the chemical species, [math]\displaystyle{ z }[/math] is the valence of ionic species, [math]\displaystyle{ e }[/math] is the elementary charge, [math]\displaystyle{ k_\text{B} }[/math] is the Boltzmann constant, and [math]\displaystyle{ T }[/math] is the absolute temperature. The electric field may be further decomposed as:
- [math]\displaystyle{ {\bf E} = -\nabla \phi - {\partial {\bf A}\over{\partial t}} }[/math]
where [math]\displaystyle{ \phi }[/math] is the electric potential and [math]\displaystyle{ {\bf A} }[/math] is the magnetic vector potential. Therefore, the Nernst–Planck equation is given by:
[math]\displaystyle{ \frac{\partial c}{\partial t} = \nabla \cdot \left[ D\nabla c - c\mathbf{v} + \frac{Dze}{k_\text{B}T} c \left( \nabla \phi + {\partial {\bf A}\over{\partial t}} \right) \right] }[/math]
Simplifications
Assuming that the concentration is at equilibrium [math]\displaystyle{ (\partial c/\partial t = 0) }[/math] and the flow velocity is zero, meaning that only the ion species moves, the Nernst–Planck equation takes the form:
- [math]\displaystyle{ \nabla \cdot \left\{ D\left[\nabla c + {ze\over{k_\text{B}T}}c \left( \nabla \phi + {\partial {\bf A}\over{\partial t}} \right) \right] \right\} = 0 }[/math]
Rather than a general electric field, if we assume that only the electrostatic component is significant, the equation is further simplified by removing the time derivative of the magnetic vector potential:
- [math]\displaystyle{ \nabla \cdot \left[ D\left(\nabla c + {ze\over{k_\text{B}T}}c \nabla \phi \right) \right] = 0 }[/math]
Finally, in units of mol/(m2·s) and the gas constant [math]\displaystyle{ R }[/math], one obtains the more familiar form:[3][4]
- [math]\displaystyle{ \nabla \cdot \left[ D\left(\nabla c + {zF\over{RT}}c \nabla \phi \right) \right] = 0 }[/math]
where [math]\displaystyle{ F }[/math] is the Faraday constant equal to [math]\displaystyle{ N_\text{A}e }[/math]; the product of Avogadro constant and the elementary charge.
Applications
The Nernst–Planck equation is applied in describing the ion-exchange kinetics in soils.[5] It has also been applied to membrane electrochemistry.[6]
See also
References
- ↑ Kirby, B. J. (2010). Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices: Chapter 11: Species and Charge Transport. http://www.kirbyresearch.com/index.cfm/wrap/textbook/microfluidicsnanofluidicsch11.html.
- ↑ Probstein, R. (1994). Physicochemical Hydrodynamics.
- ↑ Hille, B. (1992). Ionic Channels of Excitable Membranes (2nd ed.). Sunderland, MA: Sinauer. p. 267. ISBN 9780878933235. https://archive.org/details/ionicchannelsofe00hill.
- ↑ Hille, B. (1992). Ionic Channels of Excitable Membranes (3rd ed.). Sunderland, MA: Sinauer. p. 318. ISBN 9780878933235. https://archive.org/details/ionicchannelsofe00hill.
- ↑ Sparks, D. L. (1988). Kinetics of Soil Chemical Processes. Academic Press, New York. pp. 101ff.
- ↑ Brumleve, Timothy R.; Buck, Richard P. (1978-06-01). "Numerical solution of the Nernst-Planck and poisson equation system with applications to membrane electrochemistry and solid state physics" (in en). Journal of Electroanalytical Chemistry and Interfacial Electrochemistry 90 (1): 1–31. doi:10.1016/S0022-0728(78)80137-5. ISSN 0022-0728. https://www.sciencedirect.com/science/article/pii/S0022072878801375.
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