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) = (