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
of a chemical species:
:
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 ...
,
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 ...
:
:
where
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,
is the valence of ionic species,
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, ...
,
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
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:
:
where
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
and the flow velocity is zero, meaning that only the ion species moves, the Nernst–Planck equation takes the form:
:
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:
:
Finally, in units of mol/(m
2·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 ...
, one obtains the more familiar form:
:
where
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
; 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 ...
.
See also
*
Goldman–Hodgkin–Katz equation
*
Bioelectrochemistry
Bioelectrochemistry is a branch of electrochemistry and biophysical chemistry concerned with electrophysiological topics like cell electron-proton transport, cell membrane potentials and electrode reactions of redox enzymes.
History
The beginnin ...
References
{{DEFAULTSORT:Nernst-Planck equation
Walther Nernst
Diffusion
Physical chemistry
Electrochemical equations
Statistical mechanics
Max Planck
Transport phenomena
Electrochemistry