90 Microcanonical ensemble
Ω(N,V,E) =
E 0 (2m)^3 N/^2 VN
N!h^3 N2 π^3 N/^2
Γ(3N/2)1
2
√
E
E(3N−1)/^2
=
E 0
E
1
N!
1
Γ(3N/2)
[
V
(
2 πmE
h^2) 3 / 2 ]N
. (3.5.18)
The prefactor of 1/Ecauses the actual dependence of Ω(N,V,E) onEto beE^3 N/^2 −^1.
In the thermodynamic limit, we may approximate 3N/ 2 − 1 ≈ 3 N/2, in which case,
we may simply neglect theE 0 /Eprefactor altogether. We may further simplify this
expression by introducingStirling’s approximationfor the factorial of a large number:
N!≈e−NNN, (3.5.19)so that the Gamma function can be written as
Γ
(
3 N
2
)
=
(
3 N
2
− 1
)
!≈
(
3 N
2
)
!≈e−^3 N/^2(
3 N
2
) 3 N/ 2
. (3.5.20)
Substituting eqn. (3.5.20) into eqn. (3.5.18) gives the partition function expression
Ω(N,V,E) =
1
N!
[
V
h^3(
4 πmE
3 N) 3 / 2 ]N
e^3 N/^2. (3.5.21)We have intentionally not applied Stirling’s approximation to the prefactor 1/N!
because, as was discussed earlier, this factor is appendeda posteriori in order to
account for the indistinguishability of the particles not treated in a classical description.
Leaving this factor as is will allow us to assess its effects on the thermodynamics of
the ideal gas, which we will do shortly.
Let us now use the machinery of statistical mechanics to compute the temperature
of the ideal gas. From eqn. (3.2.21), we obtain
1
kT=
(
∂ln Ω
∂E)
N,V. (3.5.22)
Since Ω∼E^3 N/^2 , ln Ω∼(3N/2) lnEso that the derivative yields
1
kT=
3 N
2 E
, (3.5.23)
orkT=2 E
3 N
E=
3
2
NkT, (3.5.24)which expresses the familiar relationship between temperature andinternal energy
from kinetic theory. Similar, the pressure of the ideal gas is given by