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 thermodynamical state function (depending on the experimental conditions, the characteristic thermodynamic state function is either: internal energy, enthalpy, Gibbs free energy, or Helmholtz free energy) per change in the number of molecules. 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.

Thermodynamic Chemical Potential

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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.

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.

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.

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,..., N_n:

$U\; \backslash equiv\; U(S,V,N\_1,..N\_n)$

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, μ_i is defined as the partial derivative

$\backslash mu\_i\; =\; \backslash left(\; \backslash frac\{\backslash partial\; U\}\{\backslash partial\; N\_i\}\; \backslash right)\_\{S,V,\; N\_\{j\; \backslash ne\; i\}\}$

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 use the Helmholtz free energy A, which is a function of the temperature T, volume, and particle numbers:

$A\; \backslash equiv\; A(T,V,N\_1,..N\_n)$

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

$\backslash mu\_i\; =\; \backslash left(\; \backslash frac\{\backslash partial\; A\}\{\backslash partial\; N\_i\}\; \backslash right)\_\{T,V,\; N\_\{j\; \backslash ne\; i\}\}$

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

$\backslash mu\_i=\backslash left(\backslash frac\{\backslash partial\; G\}\{\backslash partial\; N\_i\}\backslash right)\_\{T,p,N\_\{j\backslash neq\; i\}\}$

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).

Relativistic Chemical Potential

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For relativistic systems (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 charge is associated with a chemical potential. Thus, in a gas of photons and phonons, there is 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.

Electronic Chemical Potential

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The electronic chemical potential is the functional derivative of the density functional with respect to the electron density.

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

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

where $\backslash nu(\backslash mathbf\{r\})$ 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

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.,

$\backslash delta\backslash left\backslash \{E*-\backslash mu\backslash left(\backslash int\backslash rho(\backslash mathbf\{r\})d^3r-N\backslash right)\backslash right\backslash \}=0$

where $N$ is the number of electrons in the system and $\backslash mu$ 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.

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.,

where IP and EA are the ionization potential and electron affinity of the atom, respectively.

See also

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• Electrochemical potential

Thermodynamics

Chemisches Potential | Potentiel chimique | Chemische potentiaal | 化学ポテンシャル | Potencjał chemiczny | Химический потенциал | Kemijski potencial | Kemisk potential | Хімічний потенціал