2 8 2 D O C . 2 8 3 Q U A N T U M T H E O R Y O F I D E A L G A S
where the summation is taken over τ for all σ from 1 to
∞.
We can perform the summation over s, in that we substitute it by an integration
from 0 to
∞.
This is allowed owing to the slow variability of the exponential func-
tion with σ. We thus obtain:
(18b)
. (19b)
(18b) determines the degeneracy parameter λ as a function of V, T, and n; from this,
(19b) determines the energy and therefore also the pressure of the gas.
The general discussion of these equations can proceed such that one seeks the
function[13]
that expresses the sum in (19b) by the sum in (18b). Generally, one ob-
tains by division
. (22)
The mean energy of the gas molecule at this temperature (as well as the pressure)
therefore is always less than the classical value, namely, the factor expressing the
reduction the smaller, the larger the degeneracy parameter λ.
This itself is, according to (18b) and (21), a definite function of .
If λ is so small that may be neglected against 1, then one obtains
. (22a)[14]
We now still consider in what way the Maxwell distribution of states is influ-
enced by the quanta. If one develops (11) according to powers of λ, taking (20) into
account, one obtains
(23)
The parentheses express the influence of quanta on Maxwell’s distribution law.
One sees that the slow molecules are more frequent compared to the rapid ones than
would be the case according to Maxwell’s law.
n
3 π§
4
----------©
κT·
c
------¹
3 /2
3 2λτ /–
τ
¦τ
=
E c----------©
9 π§
8
κT·
c
------¹
5 /2
5 2λτ /–
τ
¦τ
=
E
n
---
3
2
--κT------------------------ -
5 2λτ /–
τ
¦τ
3 2λτ /–
τ
¦τ
=
V
n¹
---·
©
§
2 /3
mT
λ2
E
n
---
3
2
--κT - 1 0.0318h3---(
n
V
2πmκT) 3 2 /– –=
ns konst e
Es
κT§
------ -–
1 λe
Es
κT
------ -–
…·
+ +
© ¹
=
[p. 267]