In the canonical ensemble, the Helmholtz free energy A(N,V,T) is a natural function of N, V and T. As usual, we perform a Legendre transformation to eliminate N in favor of :
It turns out that the free energy is the quantity -PV. We shall derive this result below in the context of the partition function. Thus,
To motivate the fact that PV is the proper free energy of the grand canonical ensemble from thermodynamic considerations, we need to introduce a mathematical theorem, known as Euler's theorem:
Euler's Theorem: Let be a function such that
Then f is said to be a homogeneous function of degree n. For example, the function is a homogeneous function of degree 2, is a homogeneous function of degree 3, however, is not a homogeneous function. Euler's Theorem states that, for a homogeneous function f,
Proof: To prove Euler's theorem, simply differentiate the the homogeneity condition with respect to lambda:
Then, setting , we have
which is exactly Euler's theorem.
Now, in thermodynamics, extensive thermodynamic functions are homogeneous functions of degree 1. Thus, to see how Euler's theorem applies in thermodynamics, consider the familiar example of the Gibbs free energy:
The extensive dependence of G is on N, so, being a homogeneous function of degree 1, it should satisfy
Applying Euler's theorem, we thus have
or, for a multicomponent system,
it can be seen that is consistent with the first law of thermodynamics.
Now, for the Legendre transformed free energy in the grand canonical ensemble, the thermodynamics are
the thermodynamics will be given by
Since, is a homogeneous function of degree 1, and its extensive argument is V, it should satisfy
Thus, applying Euler's theorem,
the assignment is consistent with the first law of thermodynamics. It is customary to work with PV, rather than -PV, so PV is the natural free energy in the grand canonical ensemble, and, unlike the other ensembles, it is not given a special name or symbol!