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 3 7 3
or—as is required for the coexistence of the saturated ideal gas with the condensed
substance—
(27)
Thus, we obtain the following theorem:
According to the developed equation of the state of ideal gas, a maximum den-
sity for molecules is in agitation at any temperature. Upon surpassing this density,
the excess molecules drop out as immobile (they “condense” without attractive
forces). The remarkable thing is that the “saturated ideal gas” not only represents
the state of maximal density possible for moving gas molecules, but also represents
that density at which the gas is at thermodynamical equilibrium with the “conden-
sate.” So, nothing analogous to “supersaturated vapor” exists for the ideal gas.
§7. Comparison of the Developed Theory of Gases with That Which Follows
from the Mutual Statistical Independence of Gas Molecules
Mr. Ehrenfest and other colleagues have faulted Bose’s theory of radiation and
my analogous one for ideal gases for not treating quanta, or molecules, as statisti-
cally mutually independent structures, without specifically pointing out this cir-
cumstance in our
papers.[6]
This is entirely correct. If one treats the quanta as sta-
tistically independent of one another in their localizations, one arrives at Wien’s
radiation law; if one treats the gas molecules analogously, one arrives at the classi-
cal equation of state for ideal gases, even if one otherwise proceeds exactly as Bose
and I have done. Here I will compare both considerations for gases with each other
in order to clearly bring out the difference, and to be able to compare our results
easily with those of the theory for independent molecules.
According to both theories, the number z of “cells” that belong in the infinitesi-
mal region ΔE of the molecular energy (in the following called “elementary re-
gion”) is given by
(2a)
Let the state of the gas be (macroscopically) defined by the indication of how many
molecules n lie in each of such infinitesimal regions. The number W of realizable
possibilities (Planck’s probability) of the thus defined state is to be calculated,
a) according to Bose:
A state is defined microscopically by indicating how many molecules are sitting
inside each cell (complexion). The number of complexions for the ν th infinitesi-
mal region is then
S
E pV +
T
---------------- -=
Φ 0. =
[p. 5]
zν 2π----
V
h3
-(2m)3 /2E1 /2ΔE. =