https://doi.org/10.1351/goldbook.15329
Equation relating the intensive variables \(T\), \(p\) and the \(C\) chemical potentials \(\mu_{i}\) in a phase \[S{\rm{d}}T - V{\rm{d}}p + \sum \limits_{i=1}^{C} n_{i} {\rm{d}}\mu_{i} = 0\] where \(C\) is the total number of components, \(i\) in a phase.
Notes:
- Note that the variables in this equation are the intensive quantities \(T\), \(p\) and \(\mu_{i}\).
- The Gibbs–Duhem equation may be written in terms of intensive quantities, \[\sum \limits_{i=1}^{C} n_{i}(S_{i}{\rm{d}}T - V_{i}{\rm{d}}p + {\rm{d}}\mu_{i}) = 0\] where \(S_{i}\), \(V_{i}\), \(x_{i}\) are the respective partial molar entropy, partial molar volume, and mole fraction of component \(i\).
- There is a Gibbs–Duhem equation for each phase in a system exhibiting multiphase equilibria. Application of the conditions for an equilibrium state leads to the phase rule as one example of the application of this equation. When equilibrium conditions are applied, \(T\), \(p\) and \(\mu\) are equal in all phases of an equilibrated system, while \(S_{i}\), \(V_{i}\), and \(x_{i}\) are not.