Statistical Mechanics · Problem 3.5

The Grand Canonical Ensemble — R. K. Pathria

Physics — Statistical Mechanics · helmholtz-free-energy-extensive

Problem

Making use of the fact that the Helmholtz free energy

A(N, V, T)

of a thermodynamic system is an extensive property of the system, show that

N\left(\frac{\partial A}{\partial N}\right)_{V,T} + V\left(\frac{\partial A}{\partial V}\right)_{N,T} = A.

[Note that this result implies the well-known relationship:

N\mu = A + PV\ (\equiv G)