4 2 2 D O C . 4 2 7 Q U A N T U M T H E O R Y O F I D E A L G A S
We now look at the entropy of the gas whose state distribution is given by (5).
In doing so we assume that this entropy is additively composed of parts corre-
sponding to the individual energy regions dL. In radiation theory this hypothesis is
analogous to the one that the entropy of radiation is composed additively from its
quasi-monochromatic
parts.[9]
It is equivalent to the assumption that one is permit-
ted to introduce semipermeable walls for molecules of different velocity
regions.2)
According to this hypothesis, we must ascribe to a gas whose molecules are distrib-
uted isotropically and belong to the momentum region dΦ, the entropy
(12)
where s signifies an initially unknown function of two variables.
Under the adiabatic compression considered before, this entropy must remain
unchanged; hence,
or because of (7) and (10)
From this follows, because of (11)
(13)
s is thus a function of ρ alone.
Now we set the condition that a gas be in thermodynamic equilibrium with re-
gard to the velocity distribution. For that, the entropy
must be a maximum with regard to all variations of ρ that satisfy both conditions
and
.
Performing the variation yields the condition
2)
One can imagine such semipermeable walls realized as conservative force fields.
dS
κ
----- -
V
h3
---- -s( ρ, L)dΦ =
[p. 22]
ΔdS 0 =
0 Δs
ρ∂
∂s
Δρ
∂L
∂s
ΔL. + = =
∂L
∂s
0, =
S
κ
-- -
V
h3
----
-³sdΦ
=
δ®
V
h3
----
-³ρ
dΦ
¯ ¿
¾
­ ½
0 =
δ®
V
h3
----
-³Lρ
dΦ
¯ ¿
¾
­ ½
0 =