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 cannot be used for small thermodynamic systems due to the influence of surface effects and other microscopic phenomena.

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