Statistical Mechanics - Physics at Oregon State University

3.7. IDEAL GAS IN FIRST APPROXIMATION. 61
the more rigorous treatment which will follow will indeed justify it. Before this
more rigorous theory had been developed, however, experiments had already
shown that one needed this factor! Using this formula we obtain the results for
an ideal gas:
Z(T, V, N) = 1
N! (V nQ(T )) N
U = kBT 2
∂ log(Z)
∂T
V,N
(3.95)
= 3
2 NkBT (3.96)
F = −kBT log(Z) = kBT log(N!) − NkBT log(nQV ) (3.97)
Using Stirling’s formula and only keeping terms in F which are at least proportional
to N, we obtain
n
F = NkBT log( ) − 1
(3.98)
nQ(T )
∂F
p = −
=
∂V T,N
NkBT
V
∂F
S = −
= NkB
∂T V,N
log( nQ(T
) 5
+
n 2
(3.99)
(3.100)
This last formula is called the Sackur-Tetrode relation.
If one ignores the N! in the definition of the partition function for N particles,
several problems appear. First of all, the argument of the logarithm in all
formulas would be nQ(T )V , and quantities like F and S would not be extensive,
i.e. they would not be proportional to N (but to N log N)! Also, the second
term in S would be 3
2NkB. S can be measured precise enough to show that we
really need 5
2
Hence these experiments showed the need of a factor N! (or at
least for an extra N log N − N).
The formula for the entropy is valid for an ideal gas, or for a gas in the
classical limit. Nevertheless, it contains factors (through nQ(T )). That is
really surprising. In many cases the classical limit can be obtained by assuming
that is small. Here that does not work. So why does show up in a classical
formula? Interesting indeed.
The chemical potential follows from the Helmholtz free energy:
n
µ = kBT log( ) (3.101)
nQ(T )
A calculation of the grand partition function is also straightforward. We
find
Z(T, µ, V ) =
ˆN
z ˆ N 1
ˆN! (Z1) ˆ N = e zZ1 (3.102)