In thermodynamics, the Gibbs–Duhem equation describes the relationship between changes in chemical potential for components in a thermodynamical system:
Contents
where
The Gibbs−Duhem equation cannot be used for small thermodynamic systems due to the influence of surface effects and other microscopic phenomena.
Derivation
Deriving the Gibbs–Duhem equation from the fundamental thermodynamic equation is straightforward. The total differential of the Gibbs free energy
Since the Gibbs free energy is the Legendre transformation of the internal energy, the derivatives can be replaced by its definitions transforming the above equation into:
As shown in the Gibbs free energy article, the chemical potential is simply another name for the partial molar (or just partial, depending on the units of N) Gibbs free energy, thus the gibbs free energy of a system can be calculated by collecting moles together carefully at a specified T, P and at a constant molar ratio composition (so that the chemical potential doesn't change as the moles are added together), i.e.
The total differential of this expression is
By subtracting the two expressions for the total differential of the Gibbs free energy gives the Gibbs–Duhem relation:
Applications
By normalizing the above equation by the extent of a system, such as the total number of moles, the Gibbs–Duhem equation provides a relationship between the intensive variables of the system. For a simple system with
If multiple phases of matter are present, the chemical potentials across a phase boundary are equal. Combining expressions for the Gibbs–Duhem equation in each phase and assuming systematic equilibrium (i.e. that the temperature and pressure is constant throughout the system), we recover the Gibbs' phase rule.
One particularly useful expression arises when considering binary solutions. At constant P (isobaric) and T (isothermal) it becomes:
or, normalizing by total number of moles in the system
This equation is instrumental in the calculation of thermodynamically consistent and thus more accurate expressions for the vapor pressure of a fluid mixture from limited experimental data.