524
10 Quantum Statistics
we obtain
N(T , V , α) = V T
3
2
1
k B
ϕ
(α) .
(10.2.29)
We are now set to consider an adiabatic process, for which both the number of
particles N and the entropy S are held constant. In particular, this means that the
ratio
S
Nk B
=
5
2
ϕ(α)
ϕ (α)
− α
must be constant which, in turn, requires that α, and hence ϕ(α) and ϕ (α), be
constant. The constancy of ϕ(μ/T ), taken together with Eq. (10.2.27) for the
pressure, then requires that P T
−
5
2 be constant. Similarly, the constancy of N,
together with the constancy of ϕ (μ/T ) and Eq. (10.2.29) for N, requires that V T
3
2
be constant. Hence, if we now raise V T
3
2 to the power 5/3, and multiply it by P T
−
5
2
(in order to eliminate temperature) that product must also be constant: we thus have
the requirement
P V
5
3 = constant
(10.2.30)
for an adiabatic process in a quantum mechanical gas. This requirement is the same
as that for an adiabatic process in a classical gas. Note, however, that in this case,
it does not represent P V γ , with γ = C P /C V , as C P /C V = 5/3 even for ideal BE
and FD quantum gases.
10.3 Connection to Maxwell–Boltzmann Statistics
We have seen that when NN 3 /V is small we may apply the classical Maxwell–
Boltzmann statistics and that when NN 3 /V is of order 1 or larger we must apply
one of the quantum statistics. In particular, we note that NN 3 (T )/V will be small
for
• fixed temperature when N/V is small: if the container is also of fixed volume,
then N/V small implies that N is small, which in turn implies via Eq. (10.2.20)
that the individual n k values must be very small, i.e.,
n k =
1
e β(( k −μ) ± 1
1 .
This condition holds when e β(( k −μ) 1 or, equivalently, when k μ, in which
case n k can be approximated as
n k e
−β(( k −μ)
= λe
−ββ k .