GibbsDuhem

Gibbs-Duhem is a fundamental thermodynamic relation that constrains the chemical potentials of components in a single-phase system. It is named after Josiah Willard Gibbs and Pierre Duhem, who contributed to the development of chemical thermodynamics in the late 19th century. The relation expresses that, for a system with N chemical species at a fixed temperature and pressure, the differential form sum_i N_i dμ_i = 0 holds, where N_i is the number of moles of species i and μ_i is its chemical potential.

The Gibbs-Duhem relation follows from expressing the Gibbs energy G as G = sum_i N_i μ_i and using the fact that G is a first-order homogeneous function of the extensive variables N_i. By Euler’s theorem, dG = sum_i μ_i dN_i. On the other hand, the general differential form of G gives dG = sum_i μ_i dN_i + sum_i N_i dμ_i. Equating these expressions yields sum_i N_i dμ_i = 0. This demonstrates that the chemical potentials are not independently variable; a change in one μ_i must be offset by changes in the others to satisfy the constraint.

In practical terms, the Gibbs-Duhem equation reduces the degrees of freedom in phase-equilibrium and solution theory. For a binary mixture, it implies a relation such as x1 dμ1 + x2 dμ2 = 0, linking composition changes to chemical-potential changes and constraining activity coefficients. The relation remains a central tool in physical chemistry, materials science, and related fields for analyzing mixtures and phase behavior.