Question

Show that PV is the characteristic thermodynamic function of Ξ (the grand canonical partition
function) then derive expressions for S, N, and P in terms of Ξ.

Answer 1

Grand canonical partition function for a grand canonical ensemble, a system that can exchange both heat and particles with the enviroment at constant temperature and chemical potential. In grand canonical ensemble the particles can flactuate at constant temperature T, chemical potential with respect to N number of particles.

In grand canonical ensemble, the Helmontz free energy A (N,V,T) in favour of = A/ N

A (, V, T) = A (N() , V , T) - N (A/ N)_V,T

A (, V, T) = A (N() , V , T) - N

It turns out that free energy A (, V, T) is a function of PV. And free energy is a function of grand canonical ensemble and it is related to grand canonical partition function.

So in total we can say that PV is themodynamic function of grand canonical partition function.

PV = A (N() , V , T) - N

We define grand canonical partition function

( , V , T) = 1/ N h^3N edxe^{-H (x, N)}

The normalization condition is
( , V , T) = e ^PV

ln ( , V , T) = PV / KT

Since PV is free energy of grand canonical ensemble and entropy S ( , V , T) is given by

S ( , V , T) =( (PV) / T) _{, N}

S ( , V , T) = K ln ( , V , T) - K ( / ln ( , V , T)) _{, N}