Gibbs–Duhem equation

In thermodynamics, the Gibbs–Duhem equation describes the relationship between changes in chemical potential for components in a thermodynamic system:

$${\displaystyle \sum _{i=1}^{I}N_{i}\mathrm {d} \mu _{i}=-S\mathrm {d} T+V\mathrm {d} p}$$

where ${\displaystyle N_{i}}$ is the number of moles of component ${\displaystyle i,\mathrm {d} \mu _{i}}$ the infinitesimal increase in chemical potential for this component, ${\displaystyle S}$ the entropy, ${\displaystyle T}$ the absolute temperature, ${\displaystyle V}$ volume and ${\displaystyle p}$ the pressure. ${\displaystyle I}$ is the number of different components in the system. This equation 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 ${\displaystyle I-1}$ of ${\displaystyle I}$ components have independent values for chemical potential and Gibbs' phase rule follows.

The Gibbs−Duhem equation applies to homogeneous thermodynamic systems. It does not apply to inhomogeneous systems such as small thermodynamic systems, systems subject to long-range forces like electricity and gravity, or to fluids in porous media.

The equation is named after Josiah Willard Gibbs and Pierre Duhem.