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 1
385. “Quantum Theory of the Monatomic Ideal Gas.
Second Paper”
[Einstein 1925f]
Dated December 1924
Presented 8 January 1925
Published 9 February 1925
In: Preußische Akademie der Wissenschaften (Berlin).Physikalisch-mathematische Klasse.
Sitzungsberichte (1925): 3–14.
In an article recently published in these Proceedings (XXII 1924, p.
261),[1]
a
theory of the “degeneration” of ideal gases was presented when applying a method
conceived by Mr. D. Bose to derive Planck’s radiation
formula.[2]
The interesting
thing about this theory is that it is based on the hypothesis of a far-reaching formal
relationship between radiation and gas. According to this theory, degenerate gas
deviates from gas of mechanical statistics in an analogous manner to how radiation
according to Planck’s law deviates from radiation according to Wien’s law. If
Bose’s derivation of Planck’s radiation formula is taken seriously, the one cannot
disregard this theory of ideal gas, either; because, if one can justifiably conceive of
radiation as a quantum gas, then the analogy between a quantum gas and a molec-
ular gas is complete. In what follows, the earlier considerations will be supplement-
ed by a few new ones that seem to me to increase the interest in this subject. For
convenience, I shall write the following formally as a continuation of the cited
paper.[3]
§6. Saturated Ideal Gas
In the theory of ideal gas, it seems a self-evident requisite that the volume and
temperature of an amount of gas can be chosen at will. The theory then determines
the energy, i.e., the pressure of the gas. A study of the equation of state contained
in equations (18), (19), (20), and (21) shows, however, that for a given number n of
molecules and at a given temperature T, the volume cannot be made arbitrarily
small. Since equation (18) demands that, for all s, , which, according to (20),
must mean that . This means that in equation (18b), valid in this case, λ
must lie between 0 and 1. From (18b) it accordingly follows that the num-
ber of molecules in such a gas at the given volume V cannot be greater than
[p. 3]
αs 0 ≥
A 0 ≥
e–A) = (