Chemical potential
Overview
 
Chemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system. In broadest terms, it is an analogue to electric potential
Electric potential
In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...

 or gravitational potential, utilizing the same idea of force fields as being the cause of things moving, be they charges, masses, or, in this case, chemicals.
Encyclopedia
Chemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system. In broadest terms, it is an analogue to electric potential
Electric potential
In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...

 or gravitational potential, utilizing the same idea of force fields as being the cause of things moving, be they charges, masses, or, in this case, chemicals. He defined it as follows:
Gibbs later noted also that for the purposes of this definition, any chemical element
Chemical element
A chemical element is a pure chemical substance consisting of one type of atom distinguished by its atomic number, which is the number of protons in its nucleus. Familiar examples of elements include carbon, oxygen, aluminum, iron, copper, gold, mercury, and lead.As of November 2011, 118 elements...

 or combination of elements in given proportions may be considered a substance, whether capable or not of existing by itself as a homogeneous body. Chemical potential is also referred to as partial molar Gibbs energy (see also partial molar property
Partial molar property
A partial molar property is a thermodynamic quantity which indicates how an extensive property of a solution or mixture varies with changes in the molar composition of the mixture at constant temperature and pressure, or for constant values of the natural variables of the extensive property...

). Chemical potential is measured in units of energy/particle or, equivalently, energy/mole
Mole (unit)
The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon-12 , the isotope of carbon with atomic weight 12. This corresponds to a value...

.

The term chemical potential can be used in thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

 and physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

 for any system undergoing change. Chemistry
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....

 usually restricts the term chemical potential to chemical change or to physical changes that might influence the course of a chemical reaction (phase, concentration, etc.) or to a property such as electron density. In modern statistical physics
Statistical physics
Statistical physics is the branch of physics that uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic...

 the chemical potential, divided by the temperature, is the Lagrange multiplier for the average particle constraint, when maximizing
Principle of maximum entropy
In Bayesian probability, the principle of maximum entropy is a postulate which states that, subject to known constraints , the probability distribution which best represents the current state of knowledge is the one with largest entropy.Let some testable information about a probability distribution...

 the entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

. This is the precise and abstract scientific definition, just like the temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

 is defined in terms of the Lagrange multiplier for the average energy constraint.

In some fields (particularly electrochemistry
Electrochemistry
Electrochemistry is a branch of chemistry that studies chemical reactions which take place in a solution at the interface of an electron conductor and an ionic conductor , and which involve electron transfer between the electrode and the electrolyte or species in solution.If a chemical reaction is...

), the term "chemical potential" is used to describe a fundamentally different (but related) concept, namely the "internal chemical potential"; see below for details.

The chemical potential of a system of electrons is also called the Fermi level
Fermi level
The Fermi level is a hypothetical level of potential energy for an electron inside a crystalline solid. Occupying such a level would give an electron a potential energy \epsilon equal to its chemical potential \mu as they both appear in the Fermi-Dirac distribution function,which...

.

Overview

Particles tend to move from higher chemical potential to lower chemical potential. In this way, chemical potential is a generalization of "potentials" in physics
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

 such as gravitational potential. When a ball rolls down a hill, it is moving from a higher gravitational potential (higher elevation) to a lower gravitational potential (lower elevation). 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.

A simple example is a system of dilute molecules diffusing in a homogeneous environment (animation at right). In this system, the molecules tend to move from areas with high concentration
Concentration
In chemistry, concentration is defined as the abundance of a constituent divided by the total volume of a mixture. Four types can be distinguished: mass concentration, molar concentration, number concentration, and volume concentration...

 to low concentration, until eventually the concentration is the same everywhere. The microscopic explanation for this is based in kinetic theory
Kinetic theory
The kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...

 and the random motion of molecules. However, it is simpler to describe the process in terms of chemical potentials: A molecule has a higher chemical potential in a higher-concentration area, and a lower chemical potential in a low concentration area. As always, the molecules move from higher chemical potential to lower chemical potential.

Another example is a glass of liquid water with ice cubes in it. Above 0°C, an H2O molecule has a lower chemical potential as part of the liquid than as part of an ice cube. The ice melts as the H2O molecules lower their chemical potential. Below 0°C, the ice has a lower chemical potential, so the ice cubes grow. At exactly 0°C, the chemical potential for water and ice are the same; the ice cubes will neither grow nor shrink, and the system is in equilibrium
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...

.

In electrochemistry
Electrochemistry
Electrochemistry is a branch of chemistry that studies chemical reactions which take place in a solution at the interface of an electron conductor and an ionic conductor , and which involve electron transfer between the electrode and the electrolyte or species in solution.If a chemical reaction is...

, ion
Ion
An ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge. The name was given by physicist Michael Faraday for the substances that allow a current to pass between electrodes in a...

s do not always tend to go from higher to lower chemical potential, but they do always go from higher to lower electrochemical potential. The electrochemical potential completely characterizes all of the influences on an ion's motion, while the chemical potential includes everything except the electric force. (See below for more on this terminology.)

History

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
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

internal energy
Internal energy
In thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal...

 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
Internal energy
In thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal...

 of the body, η refers to the entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

 of the body, and ν is the volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

 of the body.

Related terms

The precise meaning of the term chemical potential depends on the context in which it is used.
  • When speaking of thermodynamic systems, chemical potential refers to the thermodynamic chemical potential. In this context, the chemical potential is the change in a characteristic thermodynamic state function
    State function
    In thermodynamics, a state function, function of state, state quantity, or state variable is a property of a system that depends only on the current state of the system, not on the way in which the system acquired that state . A state function describes the equilibrium state of a system...

     per change in the number of molecules. Depending on the experimental conditions, the characteristic thermodynamic state function is either: internal energy
    Internal energy
    In thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal...

    , enthalpy
    Enthalpy
    Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...

    , Gibbs energy
    Gibbs free energy
    In thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or process-initiating work obtainable from a thermodynamic system at a constant temperature and pressure...

    , or Helmholtz energy
    Helmholtz free energy
    In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the “useful” work obtainable from a closed thermodynamic system at a constant temperature and volume...

    . This particular usage is most widely used by experimental chemists, physicists, and chemical engineers.

  • Theoretical chemists and physicists often use the term chemical potential in reference to the electronic chemical potential
    Fermi level
    The Fermi level is a hypothetical level of potential energy for an electron inside a crystalline solid. Occupying such a level would give an electron a potential energy \epsilon equal to its chemical potential \mu as they both appear in the Fermi-Dirac distribution function,which...

    , which is related to the functional derivative of the density functional, sometimes called the energy functional, found in Density Functional Theory
    Density functional theory
    Density functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...

    . This particular usage of the term is widely used in the field of electronic structure theory
    Electron configuration
    In atomic physics and quantum chemistry, electron configuration is the arrangement of electrons of an atom, a molecule, or other physical structure...

    .

  • Physicists sometimes use the term chemical potential in the description of relativistic systems of fundamental particles.

Thermodynamic chemical potential

The chemical potential of a thermodynamic
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

 system is the amount by which the energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

 of the system would change if an additional particle were introduced, with the entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

 and volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

 held fixed. If a system contains more than one species of particle, there is a separate chemical potential associated with each species, defined as the change in energy when the number of particles of that species is increased by one. The chemical potential is a fundamental parameter in thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

 and it is conjugate
Conjugate variables (thermodynamics)
In thermodynamics, the internal energy of a system is expressed in terms of pairs of conjugate variables such as temperature/entropy or pressure/volume. In fact all thermodynamic potentials are expressed in terms of conjugate pairs....

 to the particle number
Particle number
The particle number of a thermodynamic system, conventionally indicated with the letter N, is the number of constituent particles in that system. The particle number is a fundamental parameter in thermodynamics which is conjugate to the chemical potential. Unlike most physical quantities, particle...

.

The chemical potential is particularly important when studying systems of reacting particles. Consider the simplest case of two species, where a particle of species 1 can transform into a particle of species 2 and vice versa. An example of such a system is a saturated mixture of water liquid (species 1) and water vapor (species 2). If the system is at equilibrium, the chemical potentials of the two species must be equal. Otherwise, a net release of energy in the form of heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

 would occur (see second law of thermodynamics
Second law of thermodynamics
The second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...

) when the species of higher potential transforms into the other species, and a net gain of energy (again in the form of heat) would occur for the reverse transformation. In chemical reaction
Chemical reaction
A chemical reaction is a process that leads to the transformation of one set of chemical substances to another. Chemical reactions can be either spontaneous, requiring no input of energy, or non-spontaneous, typically following the input of some type of energy, such as heat, light or electricity...

s, the equilibrium conditions are generally more complicated because more than two species are involved. In this case, the relation between the chemical potentials at equilibrium is given by the law of mass action.

Since the chemical potential is a thermodynamic quantity, it is defined independently of the microscopic behavior of the system, i.e. the properties of the constituent particles. However, some systems contain important variables that are equivalent to the chemical potential. In Fermi gas
Fermi gas
A Fermi gas is an ensemble of a large number of fermions. Fermions, named after Enrico Fermi, are particles that obey Fermi–Dirac statistics. These statistics determine the energy distribution of fermions in a Fermi gas in thermal equilibrium, and is characterized by their number density,...

es and Fermi liquid
Fermi liquid
Fermi liquid theory is a theoretical model of interacting fermions that describes the normal state of most metals at sufficiently low temperatures. The interaction between the particles of the many-body system does not need to be small...

s, the chemical potential at zero temperature
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

 is equivalent to the Fermi energy
Fermi energy
The Fermi energy is a concept in quantum mechanics usually referring to the energy of the highest occupied quantum state in a system of fermions at absolute zero temperature....

. In electronic
Electronics
Electronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...

 systems, the chemical potential is related to an effective electrical potential.

A way to understand the chemical potential is to consider one mole of methane and 2 moles of oxygen. If a flame is brought near this mixture, the following reaction will occur:
and energy (heat) will be released. This energy comes from the difference in chemical potential between CH4 and O2 on one hand (higher potential) and CO2 and H2O on the other hand (lower). The whole energy that will be released will be given by

Similar examples can be found within batteries where chemical energy is converted into electrical energy.

Precise definition

Consider a thermodynamic system containing n constituent species. Its total internal energy U is postulated to be a function of the entropy S, the volume V, and the number of particles of each species N1, ..., Nn


By referring to U as the internal energy, it is emphasized that the energy contributions resulting from the interactions between the system and external objects are excluded. For example, the gravitational potential energy of the system with the Earth are not included in U.

The chemical potential of the i-th species, μi is defined as the partial derivative
Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...




where the subscripts simply emphasize that the entropy, volume, and the other particle numbers are to be kept constant.

A similar expression for the chemical potential can be written in terms of partial derivative of the enthalpy
Enthalpy
Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...

 (under conditions of constant entropy and pressure).


In real systems, it is usually difficult to hold the entropy fixed, since this involves good thermal insulation
Thermal insulation
Thermal insulation is the reduction of the effects of the various processes of heat transfer between objects in thermal contact or in range of radiative influence. Heat transfer is the transfer of thermal energy between objects of differing temperature...

. It is therefore more convenient to define the Helmholtz energy
Helmholtz free energy
In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the “useful” work obtainable from a closed thermodynamic system at a constant temperature and volume...

 A, which is a function of the temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

 T, volume, and particle numbers:


In terms of the Helmholtz energy
Helmholtz free energy
In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the “useful” work obtainable from a closed thermodynamic system at a constant temperature and volume...

, the chemical potential is


Laboratory experiments are often performed under conditions of constant temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

 and pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

. Under these conditions, the chemical potential is the partial derivative of the Gibbs energy
Gibbs free energy
In thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or process-initiating work obtainable from a thermodynamic system at a constant temperature and pressure...

 with respect to number of particles


Here, the chemical potential has been defined as the energy per molecule. A variant of this definition is to define the chemical potential as the energy per mole.

Electronic chemical potential

The electronic chemical potential is the functional derivative
Functional derivative
In mathematics and theoretical physics, the functional derivative is a generalization of the gradient. While the latter differentiates with respect to a vector with discrete components, the former differentiates with respect to a continuous function. Both of these can be viewed as extensions of...

 of the density functional with respect to the electron density
Electron density
Electron density is the measure of the probability of an electron being present at a specific location.In molecules, regions of electron density are usually found around the atom, and its bonds...

.
Formally, a functional derivative
Functional derivative
In mathematics and theoretical physics, the functional derivative is a generalization of the gradient. While the latter differentiates with respect to a vector with discrete components, the former differentiates with respect to a continuous function. Both of these can be viewed as extensions of...

 yields many functions, but is a particular function when evaluated about a reference electron density—just as a derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 yields a function, but is a particular number when evaluated about a reference point.
The density functional is written as
where ν(r) is the external potential, e.g., the electrostatic potential
Electric potential
In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...

 of the nuclei and applied fields, and F is the universal functional, which describes the kinetic energy of electrons and electron–electron interactions, e.g. kinetic energy, electron Coulomb repulsion, and the non-classical effects of exchange
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...

 and correlation
Electronic correlation
Electronic correlation is the interaction between electrons in the electronic structure of a quantum system.- Atomic and molecular systems :...

.
With this general definition of the density functional, the chemical potential is written as
Thus, the electronic chemical potential is the effective electrostatic potential experienced by the electron density.

The ground state electron density is determined by a constrained variational optimization of the electronic energy
Variational principle
A variational principle is a scientific principle used within the calculus of variations, which develops general methods for finding functions which minimize or maximize the value of quantities that depend upon those functions...

. The Lagrange multiplier enforcing the density normalization constraint is also called the chemical potential, i.e.,
where is the number of electrons in the system and is the Lagrange multiplier enforcing the constraint. When this variational statement is satisfied, the terms within the curly brackets obey the property
where the reference density is the density that minimizes the energy. This expression simplifies to
The Lagrange multiplier enforcing the constraint is, by construction, a constant; however, the functional derivative is, in the formal sense, a function. Therefore, when the density minimizes the electronic energy, the chemical potential has the same value at every point in space. The gradient of the chemical potential is an effective electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

. An electric field describes the force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

 per unit charge as a function of space. Therefore, when the density is the ground state density, the electron density is stationary, because the gradient of the chemical potential (which is invariant with respect to position) is zero everywhere; i.e., all forces are balanced. As the density undergoes a change from a non-ground state density to the ground state density, it is said to undergo a process of chemical potential equalization.

The chemical potential of an atom is sometimes said to be the negative of the atom's electronegativity
Electronegativity
Electronegativity, symbol χ , is a chemical property that describes the tendency of an atom or a functional group to attract electrons towards itself. An atom's electronegativity is affected by both its atomic number and the distance that its valence electrons reside from the charged nucleus...

. Likewise, the process of chemical potential equalization is sometimes referred to as the process of electronegativity equalization. This connection comes from the Mulliken definition of electronegativity. By inserting the energetic definitions of the ionization potential
Ionization potential
The ionization energy of a chemical species, i.e. an atom or molecule, is the energy required to remove an electron from the species to a practically infinite distance. Large atoms or molecules have a low ionization energy, while small molecules tend to have higher ionization energies.The property...

 and electron affinity
Electron affinity
The Electron affinity of an atom or molecule is defined as the amount of energy released when an electron is added to a neutral atom or molecule to form a negative ion....

 into the Mulliken electronegativity, it is possible to show that the Mulliken chemical potential is a finite difference approximation of the electronic energy with respect to the number of electrons., i.e.,
where IP and EA are the ionization potential and electron affinity of the atom, respectively.

The values of the chemical potential

For standard conditions (T = 298.15 K
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

; p = 1 atm
Atmosphere (unit)
The standard atmosphere is an international reference pressure defined as 101325 Pa and formerly used as unit of pressure. For practical purposes it has been replaced by the bar which is 105 Pa...

) the values of the chemical potential are tabulated, see under "External links". If the chemical potential is known in a certain state (e.g., for standard conditions), then it can be calculated in linear approximation for pressures and temperatures in the vicinity of this state:
and
Here

is the temperature coefficient and


is the pressure coefficient.

With the Maxwell relations
Maxwell relations
Maxwell's relations are a set of equations in thermodynamics which are derivable from the definitions of the thermodynamic potentials. The Maxwell relations are statements of equality among the second derivatives of the thermodynamic potentials. They follow directly from the fact that the order of...




and


it follows that the temperature coefficient is equal to the negative molar entropy and the pressure coefficient is equal to the molar volume.

Fundamental particle chemical potential

In recent years, thermal physics
Thermal physics
Thermal physics is the combined study of thermodynamics, statistical mechanics, and kinetic theory. This umbrella-subject is typically designed for physics students and functions to provide a general introduction to each of three core heat-related subjects...

 has applied the definition of chemical potential to systems in particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

 and its associated processes. In general, chemical potential measures the tendency of particles to diffuse. This characterization focuses on the chemical potential as a function of spatial location. Particles tend to diffuse from regions of high chemical potential to those of low chemical potential. Being a function of internal energy
Internal energy
In thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal...

, chemical potential applies equally to both fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

 and boson
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

 particles. That is, in theory, any fundamental particle can be assigned a value of chemical potential, depending upon how it changes the internal energy of the system into which it is introduced. The application of chemical potential concepts for systems at absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

 has significant appeal.

For relativistic
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....

 systems (i.e., systems in which the rest mass is much smaller than the equivalent
Mass-energy equivalence
In physics, mass–energy equivalence is the concept that the mass of a body is a measure of its energy content. In this concept, mass is a property of all energy, and energy is a property of all mass, and the two properties are connected by a constant...

 thermal energy
Thermal energy
Thermal energy is the part of the total internal energy of a thermodynamic system or sample of matter that results in the system's temperature....

) the chemical potential is related to symmetries and charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

s. Each conserved quantity is associated with a chemical potential.

In a gas of photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

s in equilibrium with massive particles, the number of photons is not conserved, and, so, in this case, the chemical potential is zero. Likewise, for a gas of phonon
Phonon
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...

s, there is also no chemical potential. However, if the temperature of such a system were to rise above the threshold for pair production
Pair production
Pair production refers to the creation of an elementary particle and its antiparticle, usually from a photon . For example an electron and its antiparticle, the positron, may be created...

 of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s, then it might be sensible to add a chemical potential for the electrical charge. This would control the electric charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

 density of the system, and hence the excess of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s over positron
Positron
The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1e, a spin of ½, and has the same mass as an electron...

s, but not the number of photons. In the context in which one meets a phonon
Phonon
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...

 gas, temperatures high enough to pair produce other particles are seldom relevant. QCD matter
QCD matter
Quark matter or QCD matter refers to any of a number of theorized phases of matter whose degrees of freedom include quarks and gluons. These theoretical phases would occur at extremely high temperatures and densities, billions of times higher than can be produced in equilibrium in laboratories...

 is the prime example of a system in which many such chemical potentials appear.

Internal, external, and total chemical potential

When there is a chemical potential difference between two locations, some of it may be due to potentials associated with "external" force fields (Electric potential energy
Electric potential energy
Electric potential energy, or electrostatic potential energy, is a potential energy that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system...

 differences, gravitational potential energy differences, etc.), while the rest would be due to "internal" factors (density, temperature, etc.) The true chemical potential, also called total chemical potential, can be split into internal chemical potential and external chemical potential
where
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, respectively, V and h are the voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

 and height of the container, respectively, and g is the acceleration due to gravity
Standard gravity
Standard gravity, or standard acceleration due to free fall, usually denoted by g0 or gn, is the nominal acceleration of an object in a vacuum near the surface of the Earth. It is defined as precisely , or about...

.)

Although the phrase "chemical potential" usually refers to "total chemical potential", this is not universally observed. In some fields, in particular electrochemistry
Electrochemistry
Electrochemistry is a branch of chemistry that studies chemical reactions which take place in a solution at the interface of an electron conductor and an ionic conductor , and which involve electron transfer between the electrode and the electrolyte or species in solution.If a chemical reaction is...

, the term "chemical potential" is instead used to mean internal chemical potential, whereas the term electrochemical potential is used to mean total chemical potential. However, in solid-state physics, precisely the opposite convention is often used in the context of electrons, with "chemical potential" meaning total chemical potential, and "electrochemical potential" meaning internal chemical potential.

See also

  • Activity (chemistry)
    Activity (chemistry)
    In chemical thermodynamics, activity is a measure of the “effective concentration” of a species in a mixture, meaning that the species' chemical potential depends on the activity of a real solution in the same way that it would depend on concentration for an ideal solution.By convention, activity...

  • Chemical equilibrium
    Chemical equilibrium
    In a chemical reaction, chemical equilibrium is the state in which the concentrations of the reactants and products have not yet changed with time. It occurs only in reversible reactions, and not in irreversible reactions. Usually, this state results when the forward reaction proceeds at the same...

  • Electrochemical potential
  • Excess chemical potential
    Excess chemical potential
    The excess chemical potential is defined as the difference between the chemical potential of a given species and that of an ideal gas under the same conditions ....

  • Fugacity
    Fugacity
    In chemical thermodynamics, the fugacity of a real gas is an effective pressure which replaces the true mechanical pressure in accurate chemical equilibrium calculations. It is equal to the pressure of an ideal gas which has the same chemical potential as the real gas. For example, nitrogen gas ...

  • Partial molar property
    Partial molar property
    A partial molar property is a thermodynamic quantity which indicates how an extensive property of a solution or mixture varies with changes in the molar composition of the mixture at constant temperature and pressure, or for constant values of the natural variables of the extensive property...

  • Thermodynamic equilibrium
    Thermodynamic equilibrium
    In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...


External links

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