2 8 0 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
. (14)
From this results for the pressure p of the gas:
.. (15)
[8]
Thus one obtains the remarkable result that the relation between the kinetic energy
and the pressure is exactly the same as in the classical theory, where it is derived
from the virial theorem.
§4. The Classical Theory as a Limiting Case
If one neglects 1 as compared to one obtains the results of the classical the-
ory; from the following it will soon emerge under which conditions this neglect is
legitimate. According to (11), (9), and (13), the mean number of molecules per cell
is then given by
. (11a)
The number of molecules whose energy lies in the elementary range of is, ac-
cording to (8), hence given by
, (11b)
in conformity with the classical theory. Equation (6) consequently yields, upon ap-
plication of the same negligence,
. (16)
For hydrogen gas at atmospheric pressure this quantity is approximately equal to
[9] thus very large compared to 1. Classical theory thus still delivers a quite
good approximation here. The error increases substantially, however, with rising
density and dropping temperature and is quite considerable for helium in the vicin-
ity of the critical state; albeit then there certainly cannot be any question of an ideal
gas anymore.
We now calculate from (12) the entropy for our limiting case. By substituting
by in (12), and this by , one obtains, taking (6a) into
account,
F E TS – κT® lg 1
e–αs)
– An –
s
¦(
¯ ¿
¾
­ ½
= =
p
∂V
∂F
–
E∂B
c ∂V
–κT---
∂V
∂
lg c –E
2E
3V
----- - = = = =
eαs
ns
ns
e–αs
e–A e
Es
κT
------ -–
⋅ = =
dEw
3
2
--e - 3 2e–Ae /–
E
κTE1
------ -–
/2dE
[p. 265]
eA π3 /2h–3---(
V
n
2mκT)1 /2 =
6 104 ⋅
lg 1 e–αs) – ( –e–αs
1
eαs
1 –
–----------------