4 2 4 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
From this follows, taking the molecule’s equations of motion into consideration,
in the familiar way
(16a)
ρ is therefore constant along the curve of the trajectory. Because of the isotropy of
the equilibrium distribution, ρ can contain only the pi’s in the combination L, and
then ρ must be representable in the form
(17)
As equilibrium distributions prevail in the different places of our gas, which corre-
spond to the different values for V at the same temperature, equation (17) also ex-
presses the form of dependence of the phase density ρ on V, in that Π is a function
of V.
§6. Conclusions about the Equation of State of the Ideal Gas
If we write down in detail the results of the analyses of the last two sections as
regards the problem of the equation of state, instead of (15a) and (17) we must write
(15b)
(17b)
A, B, and Π are here still unknown universal functions of h, m, κT, and V. In this
notation Ψ and Ψ* are dimensionless universal functions. Each of these results
shows now that equation (2) obtained from the dimensional consideration must be
specialized in the following way:
(18)
Here ψ and χ are two universal functions, each of one dimensionless variable. Both
functions ψ and χ are connected by (3), such that in reality the result just contains
the unknown function ψ. For, from (2), (3), and (4) one obtains the relation
xi
·
1
m
--- -pi =
pi
·
∂xi
∂Π
–=
∂xi
∂ρ
xi
·
∂pi
∂ρ
pi
·
+ 0. =
ρ Ψ*( L Π). + =
[p. 24]
ρ
Ψ§
h, m,
L
κT
------
B·
+
© ¹
, =
ρ Ψ*( h, m, κT, L Π). + =
ρ ψ¨
L
κT
------ χ¨
m§
V
N¹
---·
-
©
2 /3
κT··
h2
----------------------------¸¸
© ¹¹
¨ ¸¸
¨ ¸¸
§
+
©
¨
¨
§
. =