224 Isobaric ensembles
Q 2 (N 2 ,V−V 1 ,T) = e−βA(N^2 ,V−V^1 ,T)Q(N,V,T) = e−βA(N,V,T)Q 2 (N 2 ,V−V 1 ,T)
Q(N,V,T)
= e−β[A(N−N^1 ,V−V^1 ,T)−A(N,V,T)]. (5.3.14)Recalling thatN≫N 1 andV ≫V 1 , the free energyA(N−N 1 ,V−V 1 ,T) can be
expanded to first order aboutN 1 = 0 andV 1 = 0, which yields
A(N−N 1 ,V−V 1 ,T)≈A(N,V,T)−N 1
(
∂A
∂N
)∣
∣
∣
∣
N 1 =0,V 1 =0−V 1
(
∂A
∂V
)∣
∣
∣
∣
N 1 =0,V 1 =0.(5.3.15)
The minus signs arise from differentiating with respect toNandVinstead ofN 1 and
V 1. Using the relationsμ=∂A/∂NandP=−∂A/∂V, eqn. (5.3.15) becomes
A(N−N 1 ,V−V 1 ,T)≈A(N,V,T)−μN 1 +PV 1. (5.3.16)Substituting eqn. (5.3.16) into eqn. (5.3.12) yields the distribution
f 1 (x 1 ,V 1 ) =g(N,N 1 ,N 2 )eβμN^1 e−βPV^1 e−βH^1 (x^1 ). (5.3.17)System 2 has now been eliminated, and we can drop the extraneous “1” subscript.
Rearranging eqn. (5.3.17), integrating both sides, and taking the thermodynamic limit,
we obtain
e−βμN∫∞
0dV∫
dxf(x,V) =IN∫∞
0dV∫
dx e−β(H(x)+PV). (5.3.18)Eqn. (5.3.18) defines the partition function of the isothermal-isobaric ensemble as
∆(N,P,T) =IN∫∞
0dV∫
dx e−β(H(x)+PV), (5.3.19)where the definition of the prefactorINis analogous to the microcanonical and canon-
ical ensembles but with an additional reference volume to make the overall expression
dimensionless:
IN=1
V 0 N!h^3 N. (5.3.20)
As noted in Sections 3.2 and 4.3, the factorsINandMN(see eqn. (5.3.4)) should be
generalized toI{N}andM{N}for multicomponent systems.
Eqn. (5.3.18) illustrates an important point. Since eqn. (5.3.13) is true in the limit
V→ ∞, and ∆(N,P,T) = exp(−βG(N,P,T)) (we will prove this shortly), it follows
that
e−βμN= e−βG(N,P,T), (5.3.21)
orG(N,P,T) =μN. This relation is a special case of a more general result known
asEuler’s theorem(see Section 6.2). Euler’s theorem implies that if a thermodynamic