274 Grand canonical ensemble
6.5 Illustration of the grand canonical ensemble: The ideal gas
In Chapter 11, the grand canonical ensemble will be used to derive the properties of
the quantum ideal gases. It will be seen that the use of the grand canonical ensemble
greatly simplifies the treatment over the canonical ensemble. Thus, in order to prepare
for this analysis, it is instructive to illustrate the grand canonical procedure for deriving
the equation of state with a simple example, namely, the classical ideal gas. Since the
partition function of the grand canonical ensemble is given by eqn. (6.4.25), we can
start by recalling the expression of the canonical partition function of the classical
ideal gas
Q(N,V,T) =1
N!
[
V
(
2 πm
βh^2) 3 / 2 ]N
=
1
N!
(
V
λ^3)N
. (6.5.1)
Substituting this expression into eqn. (6.4.25) gives
Z(ζ,V,T) =∑∞
N=0ζN1
N!
(
V
λ^3)N
=
∑∞
N=01
N!
(
V ζ
λ^3)N
. (6.5.2)
Eqn. (6.5.2) is in the form of a Taylor series expansion for the exponential:
ex=∑∞
k=0xk
k!. (6.5.3)
Eqn. (6.5.2) can, therefore, be summed overNto yield
Z(ζ,V,T) = eV ζ/λ3. (6.5.4)
The procedure embodied in eqns. (6.4.28) and (6.4.26) requires firstthe calculation of
ζas a function of〈N〉. From eqn. (6.4.28),
〈N〉=ζ∂
∂ζlnZ(ζ,V,T) =V ζ
λ^3. (6.5.5)
Thus,
ζ(〈N〉) =〈N〉λ^3
V. (6.5.6)
From eqn. (6.5.4), we have
PV
kT= lnZ(ζ,V,T) =
V ζ
λ^3. (6.5.7)
By substitutingζ(〈N〉) into eqn. (6.5.7), the expected equation of state results:
PV
kT