3 8 2 D O C . 3 8 5 Q U A N T U M T H E O R Y O F I D E A L G A S I I
must assume very large free lengths of path (in order of magnitude cm) be-
cause of the very low volume density of the electrons that, according to our results,
are participating in the thermal agitation. Nor does it seem possible on the basis of
this theory to comprehend the behavior of metals toward infrared radiation (reflec-
tion, emission).
§11. Equation of State for Unsaturated Gas
We now want to take a closer look at the deviation of the equation of state for an
ideal gas from the classical equation of state in the unsaturated region. For this we
return to equations (15), (18b), and (19b).
To abbreviate, we set
and set ourselves the task of expressing z as a function of y The solu-
tion to this problem, which I owe to Mr. J. Grommer, is based on the following gen-
eral theorem
(Lagrange):[23]
Under the condition, satisfied in our case, that y and z vanish for λ = 0, and that
y and z in a certain region around absolute zero are regular functions of λ, for suf-
ficiently small y the Taylor expansion
(43)
applies, where the coefficients can, by means of the recursion formula, be denoted
from the functions y(λ) and z(λ) as
(44)
One thus obtains in our case the expansion,
[24]
10–3
τ 3 2λτ /–
τ 1 =
τ = ∞
¦
y(λ) =
τ 5 2λτ /–
τ 1 =
τ = ∞
¦
z(λ) =
z Φ(y)). = (
z
ν
dyν¹
d z
©
¨ ¸
§ ·
λ 0 =
yν
ν!
----, -
ν 1 =
ν = ∞
¦
=
ν
dyν
d
z) (
dλ©
d
1 –
ν 1) – (
dyν
d z
¹
¨ ¸
§ ·
dλ
dy
----------------------------------. =
z y 0.1768y2 – 9.9934y3 – 0.0005y4 – =