The Nernst–Planck equation is a

conservation of mass In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's mass ca ...

equation used to describe the motion of a charged

chemical species A chemical species is a chemical substance or ensemble composed of chemically identical molecular entities that can explore the same set of molecular energy levels on a characteristic or delineated time scale. These energy levels determine the wa ...

in a fluid medium. It extends

Fick's law of diffusion Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion eq ...

for the case where the diffusing particles are also moved with respect to the fluid by electrostatic forces. It is named after Walther Nernst and

Max Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical ...

Equation

The Nernst–Planck equation is a

continuity equation A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. ...

for the time-dependent concentration

c(t,)

of a chemical species: :

+ \nabla \cdot  = 0

where

is the

flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ...

. It is assumed that the total flux is composed of three elements:

diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...

advection In the field of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is a ...

, and

electromigration Electromigration is the transport of material caused by the gradual movement of the ions in a conductor due to the momentum transfer between conducting electrons and diffusing metal atoms. The effect is important in applications where high direc ...

. This implies that the concentration is affected by an ionic

concentration gradient Molecular diffusion, often simply called diffusion, is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size (mass) o ...

\nabla c

flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the ...

, and an electric field

: :

= -\underbrace_ + \underbrace_ +\underbrace_

where

D

is the

diffusivity Diffusivity is a rate of diffusion, a measure of the rate at which particles or heat or fluids can spread. It is measured differently for different mediums. Diffusivity may refer to: *Thermal diffusivity, diffusivity of heat *Diffusivity of mass: ...

of the chemical species,

z

is the valence of ionic species,

e

is the

elementary charge The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a funda ...

k_\text

is the

Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas consta ...

, and

T

is the

absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic ...

. The electric field may be further decomposed as: :

= -\nabla \phi -

where

\phi

is the

electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...

and

is the

magnetic vector potential In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the magnetic ...

. Therefore, the Nernst–Planck equation is given by: :

= \nabla \cdot \left D\nabla c - c + c \left( \nabla \phi +  \right) \right /math>

Simplifications

Assuming that the concentration is at equilibrium

(\partial c/\partial t = 0)

and the flow velocity is zero, meaning that only the ion species moves, the Nernst–Planck equation takes the form: :

\nabla \cdot \left\ = 0

Rather than a general electric field, if we assume that only the

electrostatic Electrostatics is a branch of physics that studies electric charges at rest (static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for am ...

component is significant, the equation is further simplified by removing the time derivative of the magnetic vector potential: :

\nabla \cdot \left D\left(\nabla c + c \nabla \phi \right) \right = 0

Finally, in units of mol/(m²·s) and the

gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment p ...

R

, one obtains the more familiar form: :

\nabla \cdot \left D\left(\nabla c + c \nabla \phi \right) \right = 0

where

F

is the

Faraday constant In physical chemistry, the Faraday constant, denoted by the symbol and sometimes stylized as ℱ, is the electric charge per mole of elementary charges. It is named after the English scientist Michael Faraday. Since the 2019 redefinition of S ...

equal to

N_\texte

; the product of

Avogadro constant The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining con ...

and the elementary charge.

Applications

The Nernst–Planck equation is applied in describing the

ion-exchange Ion exchange is a reversible interchange of one kind of ion present in an insoluble solid with another of like charge present in a solution surrounding the solid with the reaction being used especially for softening or making water demineralised, ...

kinetics Kinetics ( grc, κίνησις, , kinesis, ''movement'' or ''to move'') may refer to: Science and medicine * Kinetics (physics), the study of motion and its causes ** Rigid body kinetics, the study of the motion of rigid bodies * Chemical k ...

in soils. It has also been applied to

membrane A membrane is a selective barrier; it allows some things to pass through but stops others. Such things may be molecules, ions, or other small particles. Membranes can be generally classified into synthetic membranes and biological membranes. ...

electrochemistry Electrochemistry is the branch of physical chemistry concerned with the relationship between electrical potential difference, as a measurable and quantitative phenomenon, and identifiable chemical change, with the potential difference as an outc ...

References

{{DEFAULTSORT:Nernst-Planck equation Walther Nernst Diffusion Physical chemistry Electrochemical equations Statistical mechanics Max Planck Transport phenomena Electrochemistry

Equation

Simplifications

Applications

See also

References