where is the number of moles of component , the infinitesimal increase in chemical potential for this component, the entropy, the absolute temperature, volume and the pressure. It shows that in thermodynamics intensive properties are not independent but related, making it a mathematical statement of the state postulate. When pressure and temperature are variable, only of components have independent values for chemical potential and Gibbs' phase rule follows. The law is named after Josiah Willard Gibbs and Pierre Duhem.
As shown in the Gibbs free energy article, the chemical potential is just another name for the partial molar (or just partial, depending on the units of N) Gibbs free energy, thus
The total differential of this expression is
If one takes into account that for a system in thermodynamical equilibrium (and, by definition, in reversible condition) the infinitesimal change in G must be zero (dG = 0) and by subtracting the two expressions for the total differential of the Gibbs free energy gives the Gibbs–Duhem relation:
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 different components, there will be independent parameters or "degrees of freedom". For example, if we know a gas cylinder filled with pure nitrogen is at room temperature (298 K) and 25 MPa, we can determine the fluid density (258 kg/m3), enthalpy (272 kJ/kg), entropy (5.07 kJ/kg-K) or any other intensive thermodynamic variable. If instead the cylinder contains a nitrogen/oxygen mixture, we require an additional piece of information, usually the ratio of oxygen-to-nitrogen.
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.
or, normalizing by total number of moles in the system , substituting in the definition of activity coefficient and using the identity :
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.
See also 
- A to Z of Thermodynamics Pierre Perrot ISBN 0-19-856556-9
- Fundamentals of Engineering Thermodynamics, 3rd Edition Michael J. Moran and Howard N. Shapiro, p. 538 ISBN 0-471-07681-3
- Salzman, William R. (2001-08-21). "Open Systems". Chemical Thermodynamics. University of Arizona. Archived from the original on 2007-07-07. Retrieved 2007-10-11.
- Calculated using REFPROP: NIST Standard Reference Database 23, Version 8.0
- Fundamentals of Engineering Thermodynamics, 3rd Edition Michael J. Moran and Howard N. Shapiro, p. 710 ISBN 0-471-07681-3
- The Properties of Gases and Liquids, 5th Edition Poling, Prausnitz and O'Connell, p. 8.13, ISBN 0-07-011682-2
- Engineering and Chemical Thermodynamics, 1st Edition Milo D. Koretsky, p. 335, ISBN 978-0-471-38586-8