Gibbs–Duhem Relation

Definition and physical interpretation

For a single-phase extensive thermodynamic system , the intensive variables are constrained by the Gibbs–Duhem relation. For a simple, single-component system with variables $(S,V,N)$, it reads

where $T$ is the temperature , $P$ the pressure , and $\mu$ the chemical potential ; $S$, $V$, and $N$ are the corresponding extensive variables.

More generally, for multiple species with particle numbers $N_i$ and chemical potentials $\mu_i$,

Physically, this expresses that intensities cannot vary independently once the system is extensive: the freedom to scale the system size (doubling $(S,V,N_i)$) does not introduce an independent way to scale the conjugate intensities. For a one-component system, it implies that specifying $(T,P)$ fixes $\mu$ (up to phase coexistence subtleties).

How it arises

Starting from the Euler relation for the internal energy $U(S,V,N)$,

differentiate and compare with the exact differential from the fundamental thermodynamic differential

The difference between these two expressions eliminates $dU$, $dS$, $dV$, and $dN$ and leaves the Gibbs–Duhem constraint.

Useful rearrangements

• Dividing by $N$ introduces specific (per-particle) quantities $s=S/N$ and $v=V/N$:

  $$d\mu = v\,dP - s\,dT.$$

  This makes explicit that for a one-component phase, $\mu$ is a function of $(T,P)$.

• Using Gibbs free energy with $G=\mu N$ (from Euler’s relation for a simple system) gives the same constraint by comparing

  $$dG = -S\,dT + V\,dP + \mu\,dN$$

  with $d(\mu N)=\mu\,dN+N\,d\mu$.