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Gibbs–Duhem theorem

Intensive variables are not independent: for a simple system, S dT − V dp + N dμ = 0 (and multicomponent generalizations).

Gibbs–Duhem theorem

Statement

For an equilibrium thermodynamic system with extensive fundamental relation $U=U(S,V,N)$ , assume the hypotheses of the Euler relation theorem so that

U = TS - pV + \mu N.

Then the Gibbs–Duhem relation holds:

S\,dT - V\,dp + N\,d\mu = 0.

For a multicomponent system with particle numbers $N_1,\dots,N_r$ and chemical potentials $\mu_1,\dots,\mu_r$ ,

S\,dT - V\,dp + \sum_{i=1}^r N_i\,d\mu_i = 0.

Key hypotheses

Thermodynamic equilibrium and differentiability of the fundamental relation.
Extensivity (first-order homogeneity), used via Euler’s relation .
Validity of the differential form of the first law : $dU = T\,dS - p\,dV + \mu\,dN$ (or the multicomponent variant).

Conclusions

The intensive variables $T,p,\mu$ (or $T,p,\mu_i$ ) cannot be chosen independently; they satisfy one linear differential constraint.
Along a one-phase equilibrium manifold, specifying two of $(T,p,\mu)$ locally determines the third (subject to regularity).

Proof idea / significance

Differentiate the Euler relation:

dU = T\,dS + S\,dT - p\,dV - V\,dp + \mu\,dN + N\,d\mu.

Subtract the first-law identity for $dU$ to eliminate $T\,dS - p\,dV + \mu\,dN$ , leaving $S\,dT - V\,dp + N\,d\mu=0$ .
This relation is central in thermodynamic manipulations (e.g., changing independent variables) and underlies many standard identities involving temperature , pressure , and chemical potential .