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4 Mean Values and Thermodynamics
which, upon substitution of ln p r i (N, V i ) from expression (4.3.13), gives S as
S =
1
T
i,r i
p r i (N, V i )E r i (N, V i )+
P
T
i,r i
p r i (N, V i )V i +k B ln
i,r i
p r i (N, V i ) .
The first sum/integral in this expression represents the internal energy, the second
sum/integral the ensemble average volume V [note: the overline does not represent
molar volume in this case], and the final sums simply give 1. Thus, we obtain the
basic relation
T S = U + P V + k B T ln
(4.3.14)
for the isothermal–isobaric ensemble. Equivalently, we may write
− k B T ln = U − T S + P V ≡ G ,
(4.3.15)
from which we see that the Gibbs energy, G, is related to the isothermal–isobaric
partition function T , P ) by
G(N, T , P ) = −k B T ln (N, T , P ) .
(4.3.16)
This relation thus identifies the Gibbs energy as the characteristic thermodynamic
function for the isothermal–isobaric ensemble.
The important thermodynamic relations
S = −
∂G
∂T
N,P
,
V =
∂G
∂P
N,T
(4.3.17)
arise from combining the basic thermodynamic expression
dG = −S dT + V dP + μ dN
(4.3.18)
for the total differential dG with the mathematical defining expression for the total
differential for a function G(N, T , P ). Employment of Eq. (4.3.16) in the relations
(4.3.17) provides us with expressions giving the entropy S(N, T , P ) and ensemble
averaged volume V (N, T , P ) in terms of the isothermal–isobaric partition function,
namely, as
S = k B ln + k B T
∂ ln
∂T
N,P
,
(4.3.19a)
and