4.2 Grand Ensemble: Open Systems
189
and, since dA = −S dT − P dV + μ dN , dG can be rewritten as
dG = −S dT + V dP + μ dN ,
from which we see that the chemical potential for a pure system is given by
∂G
∂N
T ,P
= μ .
(4.2.25)
Now, G is an extensive thermodynamic variable, as is N, while T and P are both
intensive thermodynamic variables, so that if we write G(N, P , T ) as
G(N, P , T ) = Ng(P , T ) ,
we obtain the result that
∂G
∂N
T ,P
= g(P , T ) = μ ,
and hence
G = Nμ .
(4.2.26)
We thus see that the chemical potential is simply the Gibbs energy per particle. This
result, together with Eq. (4.2.23), gives us the relation
k B T ln = Nμ − A = G − A = P V ,
(4.2.27)
from which we see that P V is the characteristic function for the grand ensemble.
From expression (4.2.27), we see that the differential of P V takes the form
d(P V ) = dG − dA = N dμ + S dT + P dV ,
(4.2.28)
from which we may conclude that
= (T , V , λ) = (T , V , μ) ,
(4.2.29)
and that N , S, P , and U are given by
N =
∂(P V )
∂μ
T ,V
= k B T
∂ ln
∂μ
T ,V
,
(4.2.30a)