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) + + =