Gibbs-Duhem and Maxwell relations

Gibbs-Duhem relation[edit | edit source]

We start from the differential of the Gibbs free energy:

(where we are considering a system composed of different types of particles and that can be characterized with different generalized displacements). However, differentiating we get:
and equating with the previous expression of we obtain:
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 -component system will have independent intensive thermodynamic variables; for a simple system the Gibbs-Duhem relation reduces to:

Maxwell relations[edit | edit source]

We have previously seen that for a system we have:

we also know (Schwartz's theorem) that (since is a sufficiently regular function) we must have:
which means:
or similarly, deriving with respect to and , or and :
These are called Maxwell relations. If we consider instead of , we get:
or, with :
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.