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 the mass of the system must remain constant over time. The law implies that mass can neith ...

equation used to describe the motion of a charged

chemical species Chemical species are a specific form of chemical substance or chemically identical molecular entities that have the same molecular energy level at a specified timescale. These entities are classified through bonding types and relative abundance of ...

in a fluid medium. It extends

Fick's law of diffusion Fick's laws of diffusion describe diffusion and were first posited by Adolf Fick in 1855 on the basis of largely experimental results. They can be used to solve for the Mass diffusivity, diffusion coefficient, . Fick's first law can be used to ...

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

Walther Nernst Walther Hermann Nernst (; 25 June 1864 – 18 November 1941) was a German physical chemist known for his work in thermodynamics, physical chemistry, electrochemistry, and solid-state physics. His formulation of the Nernst heat theorem helped ...

and

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

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 in physics. For transport phe ...

. 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 fields 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 (material), conductor due to the momentum transfer between conducting electrons and diffusing metal atoms. The effect is important in applicat ...

. This implies that the concentration is affected by an ionic

concentration gradient Fick's laws of diffusion describe diffusion and were first posited by Adolf Fick in 1855 on the basis of largely experimental results. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second ...

\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 f ...

, and an

electric field An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...

: :

= -\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 a fundamental physical constant, defined as the electric charge carried by a single proton (+1 ''e'') or, equivalently, the magnitude of the negative electric charge carried by a single electron, ...

k_\text

is the

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

, and

T

is the

absolute temperature Thermodynamic temperature, also known as absolute temperature, is a physical quantity which measures temperature starting from absolute zero, the point at which particles have minimal thermal motion. Thermodynamic temperature is typically expres ...

. The electric field may be further decomposed as: :

= -\nabla \phi -

where

\phi

is the

electric potential Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work (physic ...

and

is the

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

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

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 slow-moving or stationary electric charges. Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word (), mean ...

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 (symbol , sometimes stylized as ℱ) is a physical constant defined as the quotient of the total electric charge () by the amount () of elementary charge carriers in any given sample of matter: it ...

equal to

N_\texte

; the product of

Avogadro constant The Avogadro constant, commonly denoted or , is an SI defining constant with an exact value of when expressed in reciprocal moles. It defines the ratio of the number of constituent particles to the amount of substance in a sample, where th ...

and the elementary charge.

Applications

The Nernst–Planck equation is applied in describing the ion-exchange

kinetics Kinetics (, ''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 kinetics, the study of chemical ...

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. Bi ...

electrochemistry Electrochemistry is the branch of physical chemistry concerned with the relationship between Electric potential, electrical potential difference and identifiable chemical change. These reactions involve Electron, electrons moving via an electronic ...

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