Euler relation for extensive thermodynamic potentials

For an extensive fundamental relation U(S,V,N), Euler’s theorem gives U = TS − pV + μN (and multicomponent variants).

Euler relation for extensive thermodynamic potentials

Statement

Let a thermodynamic system be in thermodynamic equilibrium and admit a differentiable fundamental relation for the internal energy of the form $U=U(S,V,N)$ , where $S,V,N$ are extensive variables.
Assume $U$ is extensive, i.e. homogeneous of degree $1$ :

U(\lambda S,\lambda V,\lambda N)=\lambda\,U(S,V,N)\qquad(\lambda>0).

Define the intensive variables in the usual way:

T=\left(\frac{\partial U}{\partial S}\right)_{V,N},\qquad p=-\left(\frac{\partial U}{\partial V}\right)_{S,N},\qquad \mu=\left(\frac{\partial U}{\partial N}\right)_{S,V}.

Then the Euler relation holds:

U = TS - pV + \mu N.

More generally, for a multicomponent system with $N=(N_1,\dots,N_r)$ and chemical potentials $\mu_i$ , one has

U = TS - pV + \sum_{i=1}^r \mu_i N_i.

Key hypotheses

A well-defined thermodynamic state in equilibrium with differentiable $U(S,V,N)$ .
Extensivity (first-order homogeneity) of $U$ in the extensive variables.
Standard identifications of $T,p,\mu$ as partial derivatives (see temperature , pressure , chemical potential ).

Conclusions

The internal energy decomposes into intensive–extensive pairings: $U=TS-pV+\mu N$ (or the multicomponent sum).
Differentiating the Euler relation and comparing with the first law yields the Gibbs–Duhem theorem .

Proof idea / significance

This is an application of Euler’s theorem for homogeneous functions: if $U$ is differentiable and homogeneous of degree $1$ , then

U = S\frac{\partial U}{\partial S}+V\frac{\partial U}{\partial V}+N\frac{\partial U}{\partial N}.

Substituting the thermodynamic identifications of the derivatives gives $U=TS-pV+\mu N$ . Conceptually, the theorem encodes extensivity (scaling with system size) and is the starting point for intensive-variable constraints (Gibbs–Duhem) and for Legendre duality between thermodynamic potentials.