TheInfoList

In
thermodynamics Thermodynamics is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in ot ...

, the chemical potential of a
species In biology, a species is the basic unit of biological classification, classification and a taxonomic rank of an organism, as well as a unit of biodiversity. A species is often defined as the largest group of organisms in which any two individu ...
is the
energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

that can be absorbed or released due to a change of the
particle number The particle number (or number of particles) of a thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All eve ...
of the given species, e.g. in a chemical reaction or
phase transition In chemistry Chemistry is the scientific Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is an occurrence in ...
. The chemical potential of a species in a mixture is defined as the rate of change of free energy of a
thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, whic ...
with respect to the change in the number of atoms or molecules of the species that are added to the system. Thus, it is the
partial derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of the free energy with respect to the amount of the species, all other species' concentrations in the mixture remaining constant. The molar chemical potential is also known as partial molar free energy. When both temperature and pressure are held constant, chemical potential is the partial molar
Gibbs free energy In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these qua ...
. At
chemical equilibrium In a chemical reaction A chemical reaction is a process that leads to the chemical transformation of one set of chemical substance A chemical substance is a form of matter In classical physics and general chemistry, matter is any sub ...
or in phase equilibrium the total sum of the product of chemical potentials and
stoichiometric coefficient Stoichiometry is the calculation of reactants and products in chemical reaction A chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classically, chemical A chemical s ...
s is zero, as the free energy is at a minimum. In
semiconductor A semiconductor material has an electrical conductivity Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric curre ...
physics, the chemical potential of a system of electrons at zero absolute temperature is known as the
Fermi energy The Fermi energy is a concept in quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contempl ...
.

# Overview

Particles tend to move from higher chemical potential to lower chemical potential. In this way, chemical potential is a generalization of such as
gravitational potential In classical mechanics, the gravitational potential at a location is equal to the work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical w ...

. When a ball rolls down a hill, it is moving from a higher gravitational potential (higher internal energy thus higher potential for work) to a lower gravitational potential (lower internal energy). In the same way, as molecules move, react, dissolve, melt, etc., they will always tend naturally to go from a higher chemical potential to a lower one, changing the
particle number The particle number (or number of particles) of a thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All eve ...
, which is
conjugate variable Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform dual (mathematics), duals, or more generally are related through Pontryagin duality. The duality relations lead naturally to an unc ...
to chemical potential. A simple example is a system of dilute molecules diffusing in a homogeneous environment. In this system, the molecules tend to move from areas with high
concentration In chemistry Chemistry is the scientific Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is an occurrence in t ...

to low concentration, until eventually, the concentration is the same everywhere. The microscopic explanation for this is based in
kinetic theory of the ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation o ...

and the random motion of molecules. However, it is simpler to describe the process in terms of chemical potentials: For a given temperature, a molecule has a higher chemical potential in a higher-concentration area and a lower chemical potential in a low concentration area. Movement of molecules from higher chemical potential to lower chemical potential is accompanied by a release of free energy. Therefore, it is a
spontaneous process In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quanti ...

. Another example, not based on concentration but on phase, is a glass of liquid water with ice cubes in it. Above 0 °C, an H2O molecule that is in the liquid phase (liquid water) has a lower chemical potential than a water molecule that is in the solid phase (ice). When some of the ice melts, H2O molecules convert from solid to liquid where their chemical potential is lower, so the ice cubes shrink. Below 0 °C, the molecules in the ice phase have the lower chemical potential, so the ice cubes grow. At the temperature of the
melting point The melting point (or, rarely, liquefaction point) of a substance is the temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can b ...

, 0 °C, the chemical potentials in water and ice are the same; the ice cubes neither grow nor shrink, and the system is in
equilibrium List of types of equilibrium, the condition of a system in which all competing influences are balanced, in a wide variety of contexts. Equilibrium may also refer to: Film and television * Equilibrium (film), ''Equilibrium'' (film), a 2002 scien ...
. A third example is illustrated by the chemical reaction of dissociation of a
weak acid Acid strength is the tendency of an acid An acid is a or capable of donating a (hydrogen ion H+) (a ), or, alternatively, capable of forming a with an (a ). The first category of acids are the proton donors, or s. In the special ca ...

HA (such as
acetic acid Acetic acid , systematically named ethanoic acid , is a colourless liquid organic compound with the chemical formula CH3COOH (also written as CH3CO2H, C2H4O2, or HC2H3O2). Vinegar is no less than 4% acetic acid by volume, making acetic acid ...

, A = CH3COO): :HA H+ + A
Vinegar Vinegar is an aqueous solution of acetic acid and trace compounds that may include flavorings. Vinegar typically contains 5–8% acetic acid by volume. Usually, the acetic acid is produced by a double fermentation; converting simple sugars to eth ...

contains acetic acid. When acid molecules dissociate, the concentration of the undissociated acid molecules (HA) decreases and the concentrations of the product ions (H+ and A) increase. Thus the chemical potential of HA decreases and the sum of the chemical potentials of H+ and A increases. When the sums of chemical potential of reactants and products are equal the system is at equilibrium and there is no tendency for the reaction to proceed in either the forward or backward direction. This explains why vinegar is acidic, because acetic acid dissociates to some extent, releasing
hydrogen ion A hydrogen ion is created when a hydrogen Hydrogen is the chemical element with the Symbol (chemistry), symbol H and atomic number 1. Hydrogen is the lightest element. At standard temperature and pressure, standard conditions hydrogen is ...
s into the solution. Chemical potentials are important in many aspects of
equilibrium chemistry Equilibrium chemistry is concerned with systems in chemical equilibrium In a chemical reaction A chemical reaction is a process that leads to the IUPAC nomenclature for organic transformations, chemical transformation of one set of chemical sub ...
, including
melting Melting, or fusion, is a physical process that results in the phase transition In chemistry Chemistry is the scientific Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiari ...

,
boiling Boiling is the rapid vaporization of a liquid, which occurs when a liquid is heated to its boiling point, the temperature at which the vapour pressure of the liquid is equal to the pressure exerted on the liquid by the surrounding atmosphere. Ther ...
,
evaporation Evaporation is a type of vaporization Vaporization (or vaporisation) of an element or compound is a phase transition from the liquid phase to vapor. There are two types of vaporization: evaporation and boiling. Evaporation is a surface phe ...

,
solubility In chemistry Chemistry is the scientific Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is an occurrence i ...

,
osmosis Osmosis (, ) is the spontaneous net movement or diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration In chemistry Chemistry is the ...

,
partition coefficient In the physical sciences Physical science is a branch of natural science that studies abiotic component, non-living systems, in contrast to life science. It in turn has many branches, each referred to as a "physical science", together called th ...
, liquid-liquid extraction and
chromatography In chemical analysis, chromatography is a laboratory technique for the Separation process, separation of a mixture into its components. The mixture is dissolved in a fluid solvent (gas or liquid) called the ''mobile phase'', which carries it ...

. In each case there is a characteristic constant which is a function of the chemical potentials of the species at equilibrium. In
electrochemistry Electrochemistry is the branch of physical chemistry Physical chemistry is the study of macroscopic The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying ...

,
ion An ion () is an atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ...
s do ''not'' always tend to go from higher to lower chemical potential, but they ''do'' always go from higher to lower ''
electrochemical potential In electrochemistry Electrochemistry is the branch of physical chemistry Physical chemistry is the study of macroscopic The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked e ...
''. The electrochemical potential completely characterizes all of the influences on an ion's motion, while the chemical potential includes everything ''except'' the
electric force Coulomb's law, or Coulomb's inverse-square law, is an experimental physical law, law of physics that quantifies the amount of force between two stationary, electric charge, electrically charged particles. The electric force between charged bodi ...
. (See
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
for more on this terminology.)

# Thermodynamic definition

The chemical potential ''μ''''i'' of species ''i'' (atomic, molecular or nuclear) is defined, as all
intensiveIn grammar, an intensive word form is one which denotes stronger, more forceful, or more concentrated action relative to the root on which the intensive is built. Intensives are usually lexical formations, but there may be a regular process for formi ...
quantities are, by the phenomenological fundamental equation of thermodynamics expressed in the form, which holds for both reversible and irreversible processes: : $\mathrmU = T\,\mathrmS - P\,\mathrmV\,$$+ \sum_^n \mu_i\,\mathrmN_i,$ where d''U'' is the infinitesimal change of
internal energy The internal energy of a thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that ca ...
''U'', d''S'' the infinitesimal change of
entropy Entropy is a scientific concept as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamic ...

''S'', and d''V'' is the infinitesimal change of
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...
''V'' for a
thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, whic ...
in thermal equilibrium, and d''N''''i'' is the infinitesimal change of particle number ''N''''i'' of species ''i'' as particles are added or subtracted. ''T'' is
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of ...
, ''S'' is
entropy Entropy is a scientific concept as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamic ...

, ''P'' is pressure, and ''V'' is volume. Other work terms, such as those involving electric, magnetic or gravitational fields may be added. From the above equation, the chemical potential is given by : $\mu_i = \left\left(\frac \right\right)_.$ This is an inconvenient expression for condensed-matter systems, such as chemical solutions, as it is hard to control the volume and entropy to be constant while particles are added. A more convenient expression may be obtained by making a
Legendre transformation In mathematics and physics, the Legendre transformation, named after Adrien-Marie Legendre, is an involution (mathematics), involutive List of transforms, transformation on the real number, real-valued convex functions of one real variable. In phys ...
to another
thermodynamic potential A thermodynamic potential (or more accurately, a thermodynamic potential energy)ISO/IEC 80000-5, Quantities an units, Part 5 - Thermodynamics, item 5-20.4 Helmholtz energy, Helmholtz functionISO/IEC 80000-5, Quantities an units, Part 5 - Thermodyn ...
: the
Gibbs free energy In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these qua ...
$G = U + PV - TS$. From the differential $\mathrmG = \mathrmU + P\,\mathrmV + V\,\mathrmP - T\,\mathrmS - S\,\mathrmT$ and using the above expression for $\mathrmU$, a differential relation for $\mathrmG$ is obtained: : $\mathrmG = -S\,\mathrmT + V\,\mathrmP + \sum_^n \mu_i\,\mathrmN_i.$ As a consequence, another expression for $\mu_i$ results: : $\mu_i = \left\left(\frac \right\right)_,$ and the change in Gibbs free energy of a system that is held at constant temperature and pressure is simply : $\mathrmG = \sum_^n \mu_i\,\mathrmN_i.$ In thermodynamic equilibrium, when the system concerned is at constant temperature and pressure but can exchange particles with its external environment, the Gibbs free energy is at its minimum for the system, that is $\mathrmG = 0$. It follows that : $\mu_1\,\mathrmN_1 + \mu_2\,\mathrmN_2 + \dots = 0.$ Use of this equality provides the means to establish the
equilibrium constant The equilibrium constant of a chemical reaction is the value of its reaction quotientIn chemical thermodynamics Chemical thermodynamics is the study of the interrelation of heat In thermodynamics, heat is energy in transfer to or from a ...
for a chemical reaction. By making further Legendre transformations from ''U'' to other thermodynamic potentials like the
enthalpy Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant p ...

$H = U + PV$ and
Helmholtz free energy In thermodynamics Thermodynamics is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order ...
$F = U - TS$, expressions for the chemical potential may be obtained in terms of these: : $\mu_i = \left\left(\frac\right\right)_,$ : $\mu_i = \left\left(\frac\right\right)_.$ These different forms for the chemical potential are all equivalent, meaning that they have the same physical content and may be useful in different physical situations.

# Applications

The
Gibbs–Duhem equationIn thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, radiation, and physical properties of matter. The behavior of these quantities is governed b ...
is useful because it relates individual chemical potentials. For example, in a binary mixture, at constant temperature and pressure, the chemical potentials of the two participants are related by : $d\mu_\text = -\frac\,d\mu_\text.$ Every instance of phase or chemical equilibrium is characterized by a constant. For instance, the melting of ice is characterized by a temperature, known as the
melting point The melting point (or, rarely, liquefaction point) of a substance is the temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can b ...

at which solid and liquid phases are in equilibrium with each other. Chemical potentials can be used to explain the slopes of lines on a
phase diagram A phase diagram in physical chemistry Physical chemistry is the study of macroscopic The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical ins ...

by using the Clapeyron equation, which in turn can be derived from the Gibbs–Duhem equation. They are used to explain
colligative properties In chemistry Chemistry is the scientific Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is an occurrence in th ...
such as melting-point depression by the application of pressure. Both
Raoult's law Raoult's law ( law) is a relation of physical chemistry Physical chemistry is the study of macroscopic The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifyi ...
and
Henry's law In physical chemistry, Henry's law is a Gas laws, gas law that states that the amount of dissolved gas in a liquid is proportional to its partial pressure above the liquid. The proportionality factor is called Henry's law constant. It was formulate ...
can be derived in a simple manner using chemical potentials.

# History

Chemical potential was first described by the American engineer, chemist and mathematical physicist
Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics Thermodynamics is ...

. He defined it as follows: Gibbs later noted also that for the purposes of this definition, any
chemical element In chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo du ...
or combination of elements in given proportions may be considered a substance, whether capable or not of existing by itself as a homogeneous body. This freedom to choose the boundary of the system allows the chemical potential to be applied to a huge range of systems. The term can be used in
thermodynamics Thermodynamics is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in ot ...

and
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

for any system undergoing change. Chemical potential is also referred to as partial molar Gibbs energy (see also partial molar property). Chemical potential is measured in units of energy/particle or, equivalently, energy/
mole Mole (or Molé) may refer to: Animals * Mole (animal) or "true mole", mammals in the family Talpidae, found in Eurasia and North America * Golden moles, southern African mammals in the family Chrysochloridae, similar to but unrelated to Talpidae ...
. In his 1873 paper ''A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces'', Gibbs introduced the preliminary outline of the principles of his new equation able to predict or estimate the tendencies of various natural processes to ensue when bodies or systems are brought into contact. By studying the interactions of homogeneous substances in contact, i.e. bodies, being in composition part solid, part liquid, and part vapor, and by using a three-dimensional
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

entropy Entropy is a scientific concept as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamic ...

internal energy The internal energy of a thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that ca ...
graph, Gibbs was able to determine three states of equilibrium, i.e. "necessarily stable", "neutral", and "unstable", and whether or not changes will ensue. In 1876, Gibbs built on this framework by introducing the concept of chemical potential so to take into account chemical reactions and states of bodies that are chemically different from each other. In his own words, to summarize his results in 1873, Gibbs states: In this description, as used by Gibbs, ''ε'' refers to the
internal energy The internal energy of a thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that ca ...
of the body, ''η'' refers to the
entropy Entropy is a scientific concept as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamic ...

of the body, and ''ν'' is the
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

of the body.

# Electrochemical, internal, external, and total chemical potential

The abstract definition of chemical potential given above—total change in free energy per extra mole of substance—is more specifically called total chemical potential.
Thermal Physics
' by Kittel and Kroemer, second edition, page 124.
If two locations have different total chemical potentials for a species, some of it may be due to potentials associated with "external" force fields (
Electric potential energy Electric potential energy, is a potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of ...

differences,
gravitational potential energy Gravitational energy or gravitational potential energy is the potential energy In physics, potential energy is the energy In , energy is the that must be to a or to perform on the body, or to it. Energy is a ; the law of stat ...
differences, etc.), while the rest would be due to "internal" factors (density, temperature, etc.) Therefore, the total chemical potential can be split into internal chemical potential and external chemical potential: : $\mu_\text = \mu_\text + \mu_\text,$ where : $\mu_\text = qV + mgh + \cdots,$ i.e., the external potential is the sum of electric potential, gravitational potential, etc. (''q'' and ''m'' are the charge and mass of the species, ''V'' and ''h'' are the
voltage Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is the ...

and height of the container, and ''g'' is the acceleration due to gravity). The internal chemical potential includes everything else besides the external potentials, such as density, temperature, and enthalpy. This formalism can be understood by assuming that the total energy of a system, $U$, is the sum of two parts: an internal energy, $U_\text$, and an external energy due to the interaction of each particle with an external field, $U_\text = N \left(qV + mgh + \cdots\right)$. The definition of chemical potential applied to $U_\text + U_\text$ yields the above expression for $\mu_\text$. The phrase "chemical potential" sometimes means "total chemical potential", but that is not universal. In some fields, in particular
electrochemistry Electrochemistry is the branch of physical chemistry Physical chemistry is the study of macroscopic The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying ...

,
semiconductor physics A semiconductor material has an electrical conductivity Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric curre ...
, and
solid-state physics Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the l ...
, the term "chemical potential" means ''internal'' chemical potential, while the term
electrochemical potential In electrochemistry Electrochemistry is the branch of physical chemistry Physical chemistry is the study of macroscopic The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked e ...
is used to mean ''total'' chemical potential.

# Systems of particles

## Electrons in solids

Electrons in solids have a chemical potential, defined the same way as the chemical potential of a chemical species: The change in free energy when electrons are added or removed from the system. In the case of electrons, the chemical potential is usually expressed in energy per particle rather than energy per mole, and the energy per particle is conventionally given in units of
electronvolt In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular suc ...
(eV). Chemical potential plays an especially important role in
solid-state physics Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the l ...
and is closely related to the concepts of
work function In solid-state physics Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state phy ...
,
Fermi energy The Fermi energy is a concept in quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contempl ...
, and
Fermi level The Fermi level of a solid-state body is the thermodynamic work In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, radiation, and physi ...
. For example, n-type silicon has a higher internal chemical potential of electrons than p-type silicon. In a
p–n junction A p–n junction is a boundary or interface between two types of semiconductor material A semiconductor material has an electrical conductivity Electrical resistivity (also called specific electrical resistance or volume resistivity) ...
diode at equilibrium the chemical potential (''internal'' chemical potential) varies from the p-type to the n-type side, while the total chemical potential (electrochemical potential, or,
Fermi level The Fermi level of a solid-state body is the thermodynamic work In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, radiation, and physi ...
) is constant throughout the diode. As described above, when describing chemical potential, one has to say "relative to what". In the case of electrons in semiconductors, internal chemical potential is often specified relative to some convenient point in the band structure, e.g., to the bottom of the conduction band. It may also be specified "relative to vacuum", to yield a quantity known as
work function In solid-state physics Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state phy ...
, however, work function varies from surface to surface even on a completely homogeneous material. Total chemical potential, on the other hand, is usually specified relative to
electrical ground In electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics The field of electronics is a branch of ...
. In atomic physics, the chemical potential of the electrons in an atom is sometimes said to be the negative of the atom's
electronegativity Electronegativity, symbolized as '' χ'', is the tendency for an atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume ...

. Likewise, the process of chemical potential equalization is sometimes referred to as the process of electronegativity equalization. This connection comes from the Mulliken electronegativity scale. By inserting the energetic definitions of the
ionization potential In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and forc ...
and
electron affinity The electron affinity (''E''ea) of an atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that c ...
into the Mulliken electronegativity, it is seen that the Mulliken chemical potential is a finite difference approximation of the electronic energy with respect to the number of electrons., i.e., :

## Sub-nuclear particles

In recent years,
thermal physics Example of a thermal column between the ground and a cumulus A thermal column (or thermal) is a column of rising air in the lower altitudes of Earth's atmosphere File:Atmosphere gas proportions.svg, Composition of Earth's atmosphere by vo ...
has applied the definition of chemical potential to systems in
particle physics Particle physics (also known as high energy physics) is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which rel ...
and its associated processes. For example, in a
quark–gluon plasma Quark–gluon plasma or QGP is an interacting localized assembly of quark A quark () is a type of elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of othe ...
or other
QCD matter Quark matter or QCD matter ( quantum chromodynamic) refers to any of a number of hypothetical phases of matter whose degrees of freedom Degrees of Freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters ...
, at every point in space there is a chemical potential for
photon The photon ( el, φῶς, phōs, light) is a type of elementary particle In , an elementary particle or fundamental particle is a that is not composed of other particles. Particles currently thought to be elementary include the fundamental s ...

s, a chemical potential for electrons, a chemical potential for baryon number,
electric charge Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectively). Like c ...
, and so forth. In the case of photons, photons are
boson In quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinkin ...

s and can very easily and rapidly appear or disappear. Therefore, at thermodynamic equilibrium, the chemical potential of photons is always and everywhere zero. The reason is, if the chemical potential somewhere was higher than zero, photons would spontaneously disappear from that area until the chemical potential went back to zero; likewise, if the chemical potential somewhere was less than zero, photons would spontaneously appear until the chemical potential went back to zero. Since this process occurs extremely rapidly (at least, it occurs rapidly in the presence of dense charged matter), it is safe to assume that the photon chemical potential is never different from zero. Electric charge is different because it is conserved, i.e. it can be neither created nor destroyed. It can, however, diffuse. The "chemical potential of electric charge" controls this diffusion: Electric charge, like anything else, will tend to diffuse from areas of higher chemical potential to areas of lower chemical potential. Other conserved quantities like baryon number are the same. In fact, each conserved quantity is associated with a chemical potential and a corresponding tendency to diffuse to equalize it out.Hadrons and Quark-Gluon Plasma
by Jean Letessier, Johann Rafelski, p. 91.
In the case of electrons, the behaviour depends on temperature and context. At low temperatures, with no
positron The positron or antielectron is the antiparticle s (left) and antiparticles (right). From top to bottom; electron The electron is a subatomic particle In physical sciences, subatomic particles are smaller than atom An atom is ...

s present, electrons cannot be created or destroyed. Therefore, there is an electron chemical potential that might vary in space, causing diffusion. At very high temperatures, however, electrons and positrons can spontaneously appear out of the vacuum (
pair production Pair production is the creation of a subatomic particle and its antiparticle from a electric charge, neutral boson. Examples include creating an electron and a positron, a muon and an antimuon, or a proton and an antiproton. Pair production often ...

), so the chemical potential of electrons by themselves becomes a less useful quantity than the chemical potential of the conserved quantities like (electrons minus positrons). The chemical potentials of
boson In quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinkin ...

s and
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics and generally has half odd integer spin: spin 1/2, Spin (physics)#Higher spins, spin 3/2, etc. These particles obey the Pauli exclusion principle. Fermions include ...
s is related to the number of particles and the temperature by
Bose–Einstein statistics In quantum statistics, Bose–Einstein (B–E) statistics describe one of two possible ways in which a collection of non-interacting, indistinguishable particles may occupy a set of available discrete Energy level, energy states at thermodynamic e ...
and
Fermi–Dirac statistics Fermi-Dirac statistics is a type of quantum statistics Particle statistics is a particular description of multiple particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:loca ...
respectively.

## Ideal vs. non-ideal solutions

Generally the chemical potential is given as a sum of an ideal contribution and an excess contribution: : $\mu_i = \mu_i^\text + \mu_i^\text,$ In an ideal solution, the chemical potential of species ''i'' (μ''i'') is dependent on temperature and pressure. μ''i''0(''T'', ''P'') is defined as the chemical potential of pure species ''i''. Given this definition, the chemical potential of species ''i'' in an ideal solution is : $\mu_i^\text \approx \mu_ + RT \ln\left(x_i\right),$ where ''R'' is the gas constant, and $x_i$ is the mole fraction of species ''i'' contained in the solution. Note that the approximation is only valid for $x_i$ not approaching zero. This equation assumes that $\mu_i$ only depends on the mole fraction ($x_i$) contained in the solution. This neglects intermolecular interaction between species ''i'' with itself and other species 'i''–(''j''≠''i'') This can be corrected for by factoring in the coefficient of activity of species ''i'', defined as γ''i''. This correction yields : $\mu_i = \mu_\left(T, P\right) + RT \ln\left(x_i\right) + RT \ln\left(\gamma_i\right) = \mu_\left(T, P\right) + RT \ln\left(x_i \gamma_i\right).$ The plots above give a rough picture of the ideal and non-ideal situation.

# See also

* Chemical equilibrium * Electrochemical potential * Equilibrium chemistry * Excess chemical potential * Fugacity * Partial molar property * Thermodynamic activity * Thermodynamic equilibrium

# External links

Chemical Potential

Values of the chemical potential of 1300 substances

G. Cook and R. H. Dickerson "Understanding the chemical potential", American Journal of Physics 63 pp. 737-742 (1995)

T. A. Kaplan "The Chemical Potential", Journal of Statistical Physics 122 pp. 1237-1260 (2006)
{{Authority control Physical chemistry Potentials Chemical thermodynamics Thermodynamic properties Chemical engineering thermodynamics