2 7 6 D O C . 2 8 3 Q U A N T U M T H E O R Y O F I D E A L G A S

283. “Quantum Theory of the Monatomic Ideal Gas”

[Einstein 1924o]

Presented 10 July 1924

Published 20 September 1924

In: Preußische Akademie der Wissenschaften (Berlin). Physikalisch-mathematische Klasse.

Sitzungsberichte (1924): 261–267.

A quantum theory for the monatomic ideal gas free from arbitrary assumptions

still does not exist today. This lacuna ought to be filled in the following on the

grounds of a new approach derived by Mr. D.

Bose,[1]

upon which this author has

based a highly noteworthy derivation of Planck’s radiation

formula.1)

The path to be taken below, following Bose, is to be described thus: The phase

space of an elementary structure (here of a monatomic molecule) is divided, with

reference to a given (three-dimensional) volume, into “cells” of extension . If

many elementary structures are present, then their (microscopic) distribution as re-

gards thermodynamics is characterized by the ways and means by which the ele-

mentary entities are distributed across these cells. The “probability” of a macro-

scopically defined state (in Planck’s

sense)[2]

is equal to the number of different

microscopic states by which the macroscopic state can be thought to be realized.

The entropy of the macroscopic state, and therefore the statistical and thermody-

namical behavior of the system, is then determined by Boltzmann’s principle.

§1. The Cells

The phase volume, which belongs to a certain region of the coordinates x, y, z

and the associated moments of a monatomic molecule, is expressed by

the integral

. (1)

If V is the volume available to the molecule, then the phase volume of all states

whose energy is smaller than a certain value E, is given by

1)

Appearing imminently in the Zeitschr. für Physik.

[p. 261]

h3

px, py, pz

Φ x d yd zd d pxd pydpz

³

=

E

1

2m

-------( px 2 py 2 pz 2) + + =