4.1 Canonical Ensemble: Closed Systems
179
its two phases. 1 As we are working in terms of the number N of atoms or molecules
and the chemical potential μ associated with an individual atom or molecule, we
shall add μdN to the above expression for dU , to obtain
dU = T dS − P dV + μdN .
(4.1.23)
From the thermodynamic definition A ≡ U − T S of the Helmholtz energy A, we
obtain the total differential dA of A as
dA = dU − T dS − SdT ,
which, upon utilizing the form (4.1.23) for dU , gives
dA = −SdT − P dV + μdN .
(4.1.24)
This expression for the total differential of the characteristic thermodynamic
function for the canonical ensemble identifies the independent variables for the
thermodynamic system as T , V , and N and leads directly to the connecting relations
between S, P , μ, and the canonical partition function Z(N, T , V ), namely,
S(N, T , V ) = −
∂A
∂T
N,V
= k B ln Z(N, T , V ) + k B T
∂ ln Z
∂T
N,V
,
(4.1.25a)
P = −
∂A
∂V
N,T
= k B T
∂ ln Z
∂V
N,T
,
(4.1.25b)
and
μ(N, T , V ) =
∂A
∂N
V ,T
= −k B T
∂ ln Z
∂N
T ,V
.
(4.1.25c)
Now if we also recall that the internal energy U is given by
U = k B T
2
∂ ln Z
∂T
N,V
(4.1.26)
and that the enthalpy H and Gibbs energy G are related to U , P V , and T S by
H = U + P V ,
G = H − T S ,
(4.1.27)
we see how to proceed in principle to obtain all of thermodynamics from a
knowledge of the canonical partition function Z.
1 For a more extensive discussion of the role of the chemical potential, see Chap. 9 of Molecular
Thermodynamics, D. A. McQuarrie and J. D. Simon [University Science Books, Sausalito, CA,
1999].