226 Isobaric ensembles
to eqn. (5.3.26). other thermodynamic quantities follow in a manner similar to the
canonical ensemble. The average volume is
〈V〉=−kT(
∂ln ∆(N,P,T)
∂P)
N,T, (5.3.27)
the chemical potential is given by
μ=−kT(
∂ln ∆(N,P,T)
∂N)
N,P, (5.3.28)
the heat capacity at constant pressureCPis
CP=(
∂H
∂T
)
N,P=kβ^2∂^2
∂β^2
ln ∆(N,P,T), (5.3.29)and the entropy is obtained from
S(N,P,T) =kln ∆(N,P,T) +H(N,P,T)
T
. (5.3.30)
The equivalence of the isobaric ensembles to the canonical and microcanonical ensem-
bles is explored in Problem 5.2 which examines the behavior of volume fluctuations as
a functions of system size.
5.4 Pressure and work virial theorems
In this chapter, we must distinguish between the external applied pressure, here de-
notedP, and the internal pressure of the system, denotedP(int), obtained by averaging
the estimatorP(r,p) in eqns. (4.6.57) and (4.6.58) over a canonical ensemble. In the
isobaric ensembles, the volume adjusts so that the volume-averaged internal pressure
〈P(int)〉is equal to the external applied pressureP. Recall that the internal pressure
P(int)at a particular volumeVis given in terms of the canonical partition function
by
P(int)=〈P(r,p)〉=kT∂lnQ
∂V=
kT
Q∂Q
∂V
. (5.4.1)
In order to determine the volume-averaged internal pressure, we need to average eqn.
(5.4.1) over an isothermal-isobaric distribution according to
〈P(int)〉=1
∆(N,P,T)
∫∞
0dVe−βPVQ(N,V,T)kT
Q(N,V,T)∂
∂V
Q(N,V,T)
=
1
∆(N,P,T)
∫∞
0dVe−βPVkT∂
∂V
Q(N,V,T). (5.4.2)
Integrating by parts in eqn. (5.4.2), we obtain
〈P(int)〉=1
∆
[
e−βPVkTQ(N,V,T)]
∣
∣
∣
∣
∞0−
1
∆
∫∞
0dV kT(
∂
∂V
e−βPV