Gibbs–Duhem equation

The Gibbs–Duhem equation relates changes in a system’s temperature, pressure, and component chemical potentials. At fixed temperature and pressure, it requires composition-weighted chemical-potential changes to sum to zero.

Chemical potential: The change in a system’s thermodynamic potential when the amount of a component increases, under specified conditions. The equation constrains how each component’s chemical potential can change.
Activity coefficient: A factor correcting a component’s concentration to represent its effective thermodynamic activity. The equation constrains composition-dependent activity coefficients in a mixture.
Thermodynamic system: A quantity of matter or region of space selected for thermodynamic analysis. The equation applies to a system described by a thermodynamic potential and component amounts.
Maxwell relations: Identities equating cross-derivatives of thermodynamic potentials with respect to their natural variables. Both follow from thermodynamic potentials, but Maxwell relations connect mixed derivatives rather than mixture chemical-potential changes.
Extensive and intensive properties: Extensive properties scale with system size; intensive properties do not. The equation arises because thermodynamic potentials are extensive while their conjugate variables are intensive.
Fugacity coefficient: The ratio of a real substance’s fugacity to its pressure, measuring deviation from ideal-gas behavior. For mixtures, its composition derivatives obey the equation’s chemical-potential constraint.
Chemical component: An independently variable chemical constituent used to describe a system’s composition. Each component contributes a chemical potential to the equation’s composition sum.
Phase rule: A rule giving the number of independent intensive variables in a system at phase equilibrium. It counts degrees of freedom; the Gibbs–Duhem equation instead constrains differential changes in thermodynamic variables.
Euler relation for homogeneous functions: A mathematical identity relating a differentiable homogeneous function to its partial derivatives. Applying this identity to an extensive thermodynamic potential yields the relation’s differential constraint.
Binary mixture: A mixture containing two chemical components. Its single independent composition variable makes the constraint especially simple to apply.

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