We start from the differential of the Gibbs free energy:

${\begin{aligned}dG=\sum _{i}{\frac {\partial G}{\partial x_{i}}}_{|T,N}dx_{i}+{\frac {\partial G}{\partial T}}_{|P,N}dT+\sum _{i}{\frac {\partial G}{\partial N_{i}}}_{|P,T,N_{j\neq i}}dN_{i}=\\=\sum _{i}X_{i}dx_{i}-SdT+\sum _{j}\mu _{j}dN_{j}\end{aligned}}$

(where we are considering a system composed of different types of particles and that can be characterized with different generalized displacements). However, differentiating

${\textstyle G(T,\lbrace X_{i}\rbrace ,\lbrace N_{j}\rbrace )=\sum _{j}\mu _{j}N_{j}}$ we get:

$dG=\sum _{j}N_{j}d\mu _{j}+\sum _{j}\mu _{j}dN_{j}$

and equating with the previous expression of

$dG$ we obtain:

$SdT-\sum _{i}X_{i}dx_{i}+\sum _{j}N_{j}d\mu _{j}=0$

which is called

*Gibbs-Duhem relation*. We can see that as a consequence of this relation the intensive variables of the system are not all independent from each other: in particular, an

$r$-component

$PVT$ system will have

$r+1$ independent intensive thermodynamic variables; for a simple

$PVT$ system the Gibbs-Duhem relation reduces to:

$SdT-VdP+Nd\mu =0$

We have previously seen that for a $PVT$ system we have:

$T={\frac {\partial U}{\partial S}}_{|V,N}\quad \qquad -P={\frac {\partial U}{\partial V}}_{|S,N}$

we also know (Schwartz's theorem) that (since

$U$ is a sufficiently regular function) we must have:

${\frac {\partial ^{2}U}{\partial V\partial S}}={\frac {\partial ^{2}U}{\partial S\partial V}}$

which means:

${\frac {\partial T}{\partial V}}_{|S,N}=-{\frac {\partial P}{\partial S}}_{|V,N}$

or similarly, deriving

$U$ with respect to

$S$ and

$N$, or

$V$ and

$N$:

${\frac {\partial T}{\partial N}}_{|S,V}={\frac {\partial \mu }{\partial S}}_{|N,V}\qquad -{\frac {\partial P}{\partial N}}_{|S,V}={\frac {\partial \mu }{\partial V}}_{|S,N}$

These are called

*Maxwell relations*.
If we consider

$F$ instead of

$U$, we get:

${\frac {\partial S}{\partial V}}_{|T,N}={\frac {\partial P}{\partial T}}_{|V,N}\qquad {\frac {\partial S}{\partial N}}_{|V,T}={\frac {\partial \mu }{\partial P}}_{|T,N}\qquad {\frac {\partial S}{\partial N}}_{|P,T}=-{\frac {\partial \mu }{\partial T}}_{|P,N}$

or, with

$G$:

${\frac {\partial S}{\partial P}}_{|T,N}=-{\frac {\partial V}{\partial T}}_{|P,N}\qquad {\frac {\partial S}{\partial N}}_{|P,T}=-{\frac {\partial \mu }{\partial T}}_{|P,N}\qquad {\frac {\partial V}{\partial N}}_{|P,T}={\frac {\partial \mu }{\partial P}}_{|T,N}$

other similar relations can be found using other thermodynamic potentials or also for magnetic systems.
The usefulness of these relations will become clear in the next section.