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Encyclopedia > Chemical potential

In thermodynamics and chemistry, chemical potential, symbolized by μ, is a term introduced in 1876 by the American mathematical physicist Willard Gibbs, which he defined as follows: Thermodynamics (from the Greek Î¸ÎµÏÎ¼Î·, therme, meaning heat and Î´Ï…Î½Î±Î¼Î¹Ï‚, dunamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... This article or section includes a list of works cited or a list of external links, but its sources remain unclear because it lacks in-text citations. ... Josiah Willard Gibbs (February 11, 1839 â€“ April 28, 1903) was an American mathematical physicist who contributed much of the theoretical foundation that led to the development of chemical thermodynamics and was one of the founders of vector analysis. ...

 “ If to any homogeneous mass in a state of hydrostatic stress we suppose an infinitesimal quantity of any substance to be added, the mass remaining homogeneous and its entropy and volume remaining unchanged, the increase of the energy of the mass divided by the quantity of the substance added is the potential for that substance in the mass considered. ”

Gibbs noted also that for the purposes of this definition, any chemical element 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 article needs to be cleaned up to conform to a higher standard of quality. ... Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ... The periodic table of the chemical elements A chemical element, or element for short, is a type of atom that is defined by its atomic number; that is, by the number of protons in its nucleus. ...

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-entropy-internal energy 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 which are chemically different from each other. In his own words, to summarize his results in 1873, Gibbs states: The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ... Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of...

 If we wish to express in a single equation the necessary and sufficient condition of thermodynamic equilibrium for a substance when surrounded by a medium of constant pressure P and temperature T, this equation may be written: δ(ε − Tη + Pν) = 0 when δ refers to the variation produced by any variations in the state of the parts of the body, and (when different parts of the body are in different states) in the proportion in which the body is divided between the different states. The condition of stable equilibrium is that the value of the expression in the parenthesis shall be a minimum.

In this description, as used by Gibbs, ε refers to the internal energy of the body, η refers to the entropy of the body, and υ is the volume of the body. In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ...

## 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 per change in the number of molecules. Depending on the experimental conditions, the characteristic thermodynamic state function is either: internal energy, enthalpy, Gibbs free energy, or Helmholtz free energy. 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, which is related to the functional derivative of the density functional, sometimes called the energy functional, found in Density Functional Theory. This particular usage of the term is widely used in the field of electronic structure theory.
• Physicists sometimes use the term chemical potential in the description of relativistic systems of fundamental particles.

In thermodynamics, a state function, or state quantity, is a property of a system that depends only on the current state of the system, not on the way in which the system got to that state. ... In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... In thermodynamics and molecular chemistry, the enthalpy or heat content (denoted as H or Î”H, or rarely as Ï‡) is a quotient or description of thermodynamic potential of a system, which can be used to calculate the useful work obtainable from a closed thermodynamic system under constant pressure. ... In thermodynamics, the Gibbs free energy is a thermodynamic potential which measures the useful work obtainable from a closed thermodynamic system at a constant temperature and pressure. ... In thermodynamics, the Helmholtz free energy is a thermodynamic potential which measures the â€œusefulâ€ work obtainable from a closed thermodynamic system at a constant temperature. ... Density functional theory (DFT) is a quantum mechanical method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular molecules and the condensed phases. ... ...

## Thermodynamic Chemical Potential

Conjugate variables
of thermodynamics
Pressure Volume
(Stress) (Strain)
Temperature Entropy
Chem. potential Particle no.

The chemical potential of a thermodynamic system is the amount by which the energy of the system would change if an additional particle were introduced, with the entropy and volume 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 and it is conjugate to the particle number. This article needs to be cleaned up to conform to a higher standard of quality. ... Thermodynamic potentials Maxwell relations Bridgmans equations Exact differential (edit) In thermodynamics, the internal energy of a system is expressed in terms of pairs of conjugate variables such as pressure/volume or temperature/entropy. ... The use of water pressure - the Captain Cook Memorial Jet in Lake Burley Griffin, Canberra. ... The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ... Stress is the internal distribution of force per unit area that balances and reacts to external loads applied to a body. ... This article is about the deformation of materials. ... Fig. ... Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... The particle number, N, is the number of so called elementary particles (or elementary constituents) of a thermodynamical system. ... Thermodynamics (from the Greek Î¸ÎµÏÎ¼Î·, therme, meaning heat and Î´Ï…Î½Î±Î¼Î¹Ï‚, dunamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ... Thermodynamics (from the Greek Î¸ÎµÏÎ¼Î·, therme, meaning heat and Î´Ï…Î½Î±Î¼Î¹Ï‚, dunamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... Thermodynamic potentials Maxwell relations Bridgmans equations Exact differential (edit) In thermodynamics, the internal energy of a system is expressed in terms of pairs of conjugate variables such as pressure/volume or temperature/entropy. ... The particle number, N, is the number of so called elementary particles (or elementary constituents) of a thermodynamical system. ...

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 supersaturated 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, any increase in one chemical potential would result in an irreversible net release of energy of the system in the form of heat (see second law of thermodynamics) when that species of increased potential transformed into the other species, or a net gain of energy (again in the form of heat) if the reverse transformation took place. In chemical reactions, 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. In physics, heat, symbolized by Q, is defined as transfer of thermal energy [1] Generally, heat is a form of energy transfer associated with the different motions of atoms, molecules and other particles that comprise matter when it is hot and when it is cold. ... The second law of thermodynamics is an expression of the universal law of increasing entropy. ... Vapours of hydrogen chloride in a beaker and ammonia in a test tube meet to form a cloud of a new substance, ammonium chloride A chemical reaction is a process that results in the interconversion of chemical substances. ... Mass action in science is the idea that a large number of small units (especially atoms or molecules) acting randomly by themselves can in fact have a larger pattern. ...

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 gases and Fermi liquids, the chemical potential at zero temperature is equivalent to the Fermi energy. In electronic systems, the chemical potential is related to an effective electrical potential. A Fermi gas is a collection of non-interacting fermions. ... A Fermi liquid is a generic term for a quantum mechanical liquid of fermions that arises under certain physical conditionsâ€”when the temperature is sufficiently low, and when the system is translationally invariant. ... Absolute zero is the lowest possible temperature where nothing could be colder, and no heat energy remains in a substance. ... The Fermi energy is a concept in quantum mechanics referring to the energy of the highest occupied quantum state in a system of fermions at zero temperature. ... // Electronics is the study of electron mechanics. ... Electrical potential is the potential energy per unit charge associated with a static (time-invariant) electric field, also called the electrostatic potential or the electric potential, typically measured in volts. ...

### 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: This article or section does not adequately cite its references or sources. ...

U = U(S,V,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 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 (as opposed to the total derivative, in which all variables are allowed to vary). ...

$mu_i = left( frac{partial U}{partial N_i} right)_{S,V, N_{j ne i}}$

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

In real systems, it is usually difficult to hold the entropy fixed, since this involves good thermal insulation. It is therefore more convenient to define the Helmholtz free energy A, which is a function of the temperature T, volume, and particle numbers: Thermal insulation on the Huygens probe The term thermal insulation can refer to materials used to reduce the rate of heat transfer, or the methods and processes used to reduce heat transfer. ... In thermodynamics, the Helmholtz free energy is a thermodynamic potential which measures the â€œusefulâ€ work obtainable from a closed thermodynamic system at a constant temperature. ... Fig. ...

A = A(T,V,N1,..Nn)

In terms of the Helmholtz free energy, the chemical potential is

$mu_i = left( frac{partial A}{partial N_i} right)_{T,V, N_{j ne i}}$

Laboratory experiments are often performed under conditions of constant temperature and pressure. Under these conditions, the chemical potential is the partial derivative of the Gibbs free energy with respect to number of particles Fig. ... The use of water pressure - the Captain Cook Memorial Jet in Lake Burley Griffin, Canberra. ... In thermodynamics, the Gibbs free energy is a thermodynamic potential which measures the useful work obtainable from a closed thermodynamic system at a constant temperature and pressure. ...

$mu_i=left(frac{partial G}{partial N_i}right)_{T,p,N_{jneq i}}$

A similar expression for the chemical potential can be written in terms of partial derivative of the enthalpy (under conditions of constant entropy and pressure). In thermodynamics and molecular chemistry, the enthalpy or heat content (denoted as H or Î”H, or rarely as Ï‡) is a quotient or description of thermodynamic potential of a system, which can be used to calculate the useful work obtainable from a closed thermodynamic system under constant pressure. ...

## Electronic Chemical Potential

The electronic chemical potential is the functional derivative of the density functional with respect to the electron density. In mathematics and theoretical physics, the functional derivative is a generalization of the directional derivative. ... Generally, functional refers to something with and able to fulfill its purpose or function. ... Electron density is the measure of the probability of an electron being present at a specific location. ...

Formally, a functional derivative yields many functions, but is a particular function when evaluated about a reference electron density - just as a derivate yields a function, but is a particular number when evaluated about a reference point. The density functional is written as ...

$E[rho] = int rho(mathbf{r})nu(mathbf{r})d^3r + F[rho]$

where $nu(mathbf{r})$ is the external potential, e.g., the electrostatic potential of the nuclei and applied fields, and F is the Universal functional, which describes the electron-electron interactions, e.g., electron Coulomb repulsion, kinetic energy, and the non-classical effects of exchange and correlation. With this general definition of the density functional, the chemical potential is written as Electric potential is the potential energy per unit of charge associated with a static (time-invariant) electric field, also called the electrostatic potential, typically measured in volts. ... The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. ... Electronic correlation refers to the interaction between electrons in a quantum system whose electronic structure is being considered. ...

$mu(mathbf{r}) = nu(mathbf{r})+left[frac{delta F[rho]}{deltarho(mathbf{r})}right]_{rho=rho_{ref}}$

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. The Lagrange multiplier enforcing the density normalization constraint is also called the chemical potential, i.e., A variational principle is a principle in physics which is expressed in terms of the calculus of variations. ... In mathematical optimization problems, Lagrange multipliers are a method for dealing with constraints. ...

$deltaleft{E[rho]-muleft(intrho(mathbf{r})d^3r-Nright)right}=0$

where N is the number of electrons in the system and μ 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, formally, 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. An electric field describes the force 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. It has been suggested that optical field be merged into this article or section. ... In physics, the force experienced by a body is defined as the rate of change of momentum with time. ...

The chemical potential of an atom is sometimes said to be the negative of the atom's electronegativity. Similarly 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 and electron affinity 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., Electronegativity is a measure of the ability of an atom or molecule to attract electrons in the context of a chemical bond. ... The Mulliken scale (also called Mulliken-Jaffe scale) is a scale for the electronegativity of chemical elements. ... The ionization potential, or ionization energy, of an atom or molecule is the energy required to strip it of an electron. ... The electron affinity, Eea, of an atom or molecule is the energy required to detach an electron from a singly charged negative ion, that is the energy change for the process X- â†’ X + e- An equivalent definition is the energy released (Einitial âˆ’ Efinal) when an electron is attached to a...

$mu_{Mulliken}=-chi_{Mulliken}=-frac{IP+EA}{2}=left[frac{delta E[N]}{delta N}right]_{N=N_0}$

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; p = 101,325 kPa) the values of the chemical potential are tabulated, see under "Weblinks". 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:
μ(T) = μ(T0) + α(TT0)
and
μ(p) = μ(p0) + β(pp0)
Here Temperature and air pressure can vary from one place to another on the Earth, and can also vary in the same place with time. ...

$alpha = left( frac{partial mu}{partial T} right)_{p,n}$

is the temperature coefficient and

$beta =left( frac{partial mu}{partial p} right)_{T,n}$

is the pressure coefficient.
With the Maxwell relations Maxwells relations are a set of equations in thermodynamics which are derivable from the definitions of the thermodynamic potentials. ...

$left( frac{partial mu}{partial T} right)_{p,n} =- left( frac{partial S}{partial n} right)_{T,p}$

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 has applied the definition of chemical potential to systems in particle physics 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. [1] Being a function of internal energy, chemical potential applies equally to both fermion and boson 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 has significant appeal. Thermal physics is the combined study of thermodynamics, statistical mechanics, and kinetic theory. ... Thousands of particles explode from the collision point of two relativistic (100 GeV per ion) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... In particle physics, fermions are particles with half-integer spin. ... In particle physics, bosons, named after Satyendra Nath Bose, are particles having integer spin. ... In particle physics, an elementary particle is a particle of which other, larger particles are composed. ... Absolute zero is the lowest possible temperature where nothing could be colder, and no heat energy remains in a substance. ...

For relativistic systems, i.e., systems in which the rest mass is much smaller than the equivalent thermal energy, the chemical potential is related to symmetries and charges. Each conserved quantity is associated with a chemical potential. Albert Einsteins theory of relativity is a set of two theories in physics: special relativity and general relativity. ... The term mass in special relativity is used in a couple of different ways, occasionally leading to a great deal of confusion. ... Mass-energy equivalence is where mass has an energy equivalence, and energy has a mass equivalence. ... This article is about the atmospheric phenomenon. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ...

In a gas of photons in equilibrium with massive particles, the number of photons is not conserved, and so in this case, the chemical potential is zero. Similarly, for a gas of phonons, there is also no chemical potential. However, if the temperature of such a system were to rise above the threshold for pair production of electrons, then it might be sensible to add a chemical potential for the electrical charge. This would control the electric charge density of the system, and hence the excess of electrons over positrons, but not the number of photons. In the context in which one meets a phonon gas, temperatures high enough to pair produce other particles are seldom relevant. QCD matter is the prime example of a system in which many such chemical potentials appear. The word light is defined here as electromagnetic radiation of any wavelength; thus, X-rays, gamma rays, ultraviolet light, microwaves, radio waves, and visible light are all forms of light. ... Normals modes of vibration progression through a crystal. ... Pair production refers to the creation of an elementary particle and its antiparticle, usually from a photon (or another neutral boson). ... e- redirects here. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... e- redirects here. ... The first detection of the positron in 1932 by Carl D. Anderson The positron is the antiparticle or the antimatter counterpart of the electron. ... The word light is defined here as electromagnetic radiation of any wavelength; thus, X-rays, gamma rays, ultraviolet light, microwaves, radio waves, and visible light are all forms of light. ... Normals modes of vibration progression through a crystal. ... Quark matter or QCD matter refers to any of a number of phases of matter whose degrees of freedom include quarks and gluons. ...

Electrochemical potential is a thermodynamic measure that reflects energy from entropy and electrostatics and is typically invoked in molecular processes that involve diffusion. ... In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, and chemical equilibrium. ... Chemical equilibrium is the state in which the concentrations of the reactants and products have no net change over time. ...

## References

1. ^ Baierlein, Ralph (2003). Thermal Physics. Cambridge University Press. ISBN 0-521-65838-1.
• Kaplan, T. A. (March 2006). "The Chemical Potential". Journal of Statistical Physics 122 (6): 1237-1260. DOI:10.1007/s10955-005-8067-x. Retrieved on 2006-12-20.
• Job, G.; Herrmann, F. (February 2006). "Chemical potential–a quantity in search of recognition" (PDF). European Journal of Physics 27: 353-371. DOI:10.1088/0143-0807/27/2/018.

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