Gibbs–Duhem equation

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

where ${\displaystyle N_{i}\,}$ is the number of moles of component ${\displaystyle i\,}$, ${\displaystyle \mathrm {d} \mu _{i}\,}$ 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 law is named after Josiah Willard Gibbs and Pierre Duhem.

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