# Chemical potential

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 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). In equilibrium, the chemical potentials of the two species must be equal, because any increase in one chemical potential would allow particles of that species to transform into the other species with the net emission of heat (see second law of thermodynamics.) 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 equivalent to the negative of the electrical potential.

For relativistic systems (systems in which the rest mass is much smaller than the 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.

## Precise definition

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

[itex]E \equiv E(S,V,N_1,..N_n)[itex]

The chemical potential of the j-th species, μj is defined as the partial derivative

[itex]\mu_j = \left( \frac{\partial E}{\partial N_j} \right)_{S,V, N_{i \ne j}}[itex]

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 F, which is a function of the temperature T, volume, and particle numbers:

[itex]F \equiv F(T,V,N_1,..N_n)[itex]

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

[itex]\mu_j = \left( \frac{\partial F}{\partial N_j} \right)_{T,V, N_{i \ne j}}[itex]

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