Chemical Potential

Chemical potential, usually denoted by $\mu$, is an intensive quantity in thermodynamics that describes how a thermodynamic potential changes when the amount of a chemical species changes under specified constraints. In chemical applications, it is the partial molar Gibbs free energy: the change in Gibbs energy per infinitesimal amount of substance added at fixed temperature, pressure, and amounts of other components. It provides a common framework for describing material transfer, phase coexistence, and reacting mixtures. (goldbook.iupac.org)

Thermodynamic definition

For a mixture containing amounts $n_1,n_2,\ldots$, the molar chemical potential of component $i$ is

where $G$ is Gibbs energy, $T$ is temperature, and $p$ is pressure. The partial derivative specifies that the amounts of all other components remain fixed. For a pure, homogeneous bulk substance, $\mu=G/n$, its molar Gibbs energy. In a mixture, however, $\mu_i$ is generally not the total Gibbs energy divided by the amount of component $i$. (goldbook.iupac.org)

For a simple multicomponent system with only pressure–volume work, the fundamental differential is

where $U$ is internal energy, $S$ is entropy, and $V$ is volume. Consequently, the same chemical potential can be obtained by differentiating $U$ at fixed entropy and volume, or Helmholtz free energy at fixed temperature and volume. The constraints are essential: changing them changes the meaning of an energy derivative. (live.ocw.mit.edu)

Molar chemical potentials have units of joules per mole. In physics, particle numbers $N_i$ often replace mole amounts, giving chemical potentials in energy units per particle. The molar value is the per-particle value multiplied by the Avogadro constant. Chemical potential is intensive: scaling a homogeneous system while preserving temperature, pressure, and composition does not change it. (damtp.cam.ac.uk)

Material transfer and phase equilibrium

Consider transferring a small amount $dn_i$ from region $\alpha$ to region $\beta$ at common temperature and pressure. In the absence of relevant external-field contributions,

Transfer from higher to lower chemical potential decreases the total Gibbs energy. At thermodynamic equilibrium, every species able to pass between the regions must therefore satisfy

Equal concentrations are not required, because the same substance can have different interactions and densities in different environments. (live.ocw.mit.edu)

This criterion explains phase coexistence and phase transitions. For example, ice and liquid water coexist when their water chemical potentials are equal. It also underlies diffusion and osmosis: solvent transfer through a selectively permeable membrane is governed by the solvent’s chemical-potential difference, rather than by solute concentration alone. A pressure difference can counterbalance the effect of composition on solvent chemical potential. (live.ocw.mit.edu)

Activity and composition

The composition dependence of molar chemical potential is expressed through thermodynamic activity:

Here $a_i$ is dimensionless, $R$ is the gas constant, and $\mu_i^\circ$ is the chemical potential in a specified standard state. Activity incorporates departures from ideal behavior; its numerical value depends on the chosen reference convention. A standard state is a reference condition, not necessarily the actual state of the sample. (publications.iupac.org)

For a mixture of ideal gases,

where $p_i$ is the component’s partial pressure and $p^\circ$ is standard pressure. For an ideal solution using the pure component as reference, the corresponding composition term is $RT\ln x_i$, where $x_i$ is mole fraction. The logarithmic dependence shows why dilution lowers a component’s chemical potential relative to its pure state under otherwise equivalent conditions. (live.ocw.mit.edu)

Chemical reactions

For a chemical reaction, assign positive stoichiometric coefficients $\nu_i$ to products and negative coefficients to reactants. The reaction Gibbs energy is

At fixed temperature and pressure, a negative value favors forward reaction, while a positive value favors reverse reaction. At chemical equilibrium, $\Delta_rG=0$; individual chemical potentials need not be zero or equal across different species. (live.ocw.mit.edu)

Substituting the activity expression gives

where $Q$ is the reaction quotient. At equilibrium, $Q=K$ and $\Delta_rG^\circ=-RT\ln K$. Chemical potential thus connects composition-dependent reaction direction with the equilibrium constant. These relations describe thermodynamic driving forces, not the reaction rates studied by chemical kinetics. (live.ocw.mit.edu)

Statistical and electrochemical descriptions

In statistical mechanics, the grand canonical ensemble describes a system exchanging energy and particles with a reservoir. A microstate $r$, with energy $E_r$ and particle number $N_r$, has probability

where $k_B$ is the Boltzmann constant and $\Xi$ is the grand partition function. Chemical potential controls the mean particle number, while temperature controls energy exchange. Here $\mu$ is expressed per particle. (damtp.cam.ac.uk)

For an ideal Fermi gas, the chemical potential approaches the Fermi energy at zero temperature; at finite temperature, the two generally differ. For equilibrium blackbody radiation, photon number is not constrained, and the occupation distribution corresponds to zero photon chemical potential. (damtp.cam.ac.uk)

For charged species, electrical contributions must also be included. The electrochemical potential combines chemical and electrical terms, commonly written in molar form as $\widetilde{\mu}_i=\mu_i+z_iF\phi$. Here $z_i$ is charge number, $F$ is the Faraday constant, and $\phi$ is electric potential. Equilibrium transfer of charged species requires equality of electrochemical potentials, rather than chemical terms alone. (media.iupac.org)