228 Isobaric ensembles
The quantity〈P(int)V〉can be defined for any ensemble. However, because the
volume can fluctuate in an isobaric ensemble, we can think of the volume as an ad-
ditional degree of freedom that is not present in the microcanonical and canonical
ensembles. If energy is equipartitioned, there should be an additionalkTof energy in
the volume motion, giving rise to a difference ofkTbetween〈P(int)V〉andP〈V〉. Since
the motion of the volume is driven by an imaginary “piston” that acts to adjust the
internal pressure to the external pressure, this piston also adds an amount of energy
kTto the system so that eqn. (5.4.8) is satisfied.
5.5 An ideal gas in the isothermal-isobaric ensemble
As an example application of the isothermal-isobaric ensemble, we compute the par-
tition function and thermodynamic properties of an ideal gas. Recall from Section 4.5
that canonical partition function for the ideal gas is
Q(N,V,T) =
VN
N!λ^3 N, (5.5.1)
whereλ=
√
βh^2 / 2 πm. Substituting eqn. (5.5.1) into eqn. (5.3.22) gives the isothermal-
isobaric partition function
∆(N,P,T) =
1
V 0
∫∞
0dVe−βPVVN
N!λ^3 N=
1
V 0 N!λ^3 N∫∞
0dVe−βPVVN. (5.5.2)The volume integral can be rendered dimensionless by lettingx=βPV, leading to
∆(N,P,T) =
1
V 0 N!λ^3 N1
(βP)N+1∫∞
0dx xNe−x. (5.5.3)The value of the integral is justN!. Hence, the isothermal-isobaric partition function
for an ideal gas is
∆(N,P,T) =1
V 0 λ^3 N(βP)N+1. (5.5.4)
The thermodynamics of the ideal gas follow from the relations derived in Sec-
tion 5.3. For the equation of state, we obtain the average volume from
〈V〉=−kT(
∂ln ∆
∂P)
=
(N+ 1)kT
P, (5.5.5)
or
P〈V〉= (N+ 1)kT≈NkT, (5.5.6)
where the last expression follows from the thermodynamic limit. Usingeqn. (5.4.8),
we can express eqn. (5.5.6) in terms of the averageP(int)V product:
〈P(int)V〉=NkT. (5.5.7)Eqn. (5.5.7) is generally true even away from the thermodynamic limit.The average
enthalpy of the ideal gas is given by