418 Quantum ideal gases
PV
kT=g∫
dnln(
1 +ζe−βεn)
=g∫
dnln(
1 +ζe−^2 π(^2) β ̄h (^2) |n| (^2) /mL 2 )
= 4πg
∫∞
0dn n^2 ln(
1 +ζe−^2 π(^2) β ̄h (^2) |n| (^2) /mL 2 )
, (11.5.1)
where, in the last line, we have transformed to spherical polar coordinates. Next, we
introduce a change of variables
x=
√
2 π^2 β ̄h^2
mL^2n, (11.5.2)which gives
PV
kT= 4πgV(
m
2 π^2 β ̄h^2) 3 / 2 ∫∞
0dx x^2 ln(
1 +ζe−x2 )
=
4 V g
√
πλ^3∫∞
0dx x^2 ln(
1 +ζe−x2 )
. (11.5.3)
The remaining integral can be evaluated by expanding the log in a power series and
integrating the series term by term. Using the fact that
ln(1 +y) =∑∞
l=1(−1)l+1
yl
l, (11.5.4)
we obtain
ln(
1 +ζe−x2 )
=
∑∞
l=1(−1)l+1ζl
l
e−lx2PV
kT=
4 V g
√
πλ^3∑∞
l=1(−1)l+1ζl
l∫∞
0dx x^2 e−lx2=
V g
λ^3∑∞
l=1(−1)l+1ζl
l^5 /^2. (11.5.5)
In the same way, it can be shown that the average particle number〈N〉is given by
the expression
〈N〉=V g
λ^3∑∞
l=1(−1)l+1ζl
l^3 /^2. (11.5.6)
Multiplying eqns. (11.5.5) and (11.5.6) by 1/V, we obtain