D O C . 4 2 7 Q U A N T U M T H E O R Y O F I D E A L G A S 4 1 9

is sought. There dn means the number of molecules whose orthogonal components

of momentum p1, p2, p3 lie within the limits indicated by dp1, dp2, dp3. L means

the kinetic energy of the molecule ; for, owing to the self-explan-

atory isotropy condition, p1, p2, p3 can only occur in ρ in connection with L. ρ is

an erstwhile unknown function of the indicated four variables. If the density func-

tion ρ is known, then the equation of state is also known, of course, because there

can be no doubt that the mechanical calculation out of the molecules’ collisions

with the wall is defining for the pressure. However, we may not presume that the

molecules’ collisions among themselves follow the rules of mechanics; otherwise

we would, of course, arrive at Maxwell’s distribution law and the classical gas

equation.[5]

§2. Why Does the Classical Equation of State Not Fit into Quantum Theory?

Since Planck’s first papers on quantum theory, one interprets the magnitude W

in Boltzmann’s principle

as a whole number.[6] It indicates in how many discrete ways (in the sense of quan-

tum theory) the condition of entropy S that interests us can be realized. Although

in most cases it is not possible to calculate W theoretically without arbitrariness,

this manner of conceiving it leads to the conviction that S does not contain an arbi-

trary additive constant, but rather is completely determined in the sense of quantum

theory and is always positive. With Nernst’s theorem, this interpretation according

to Planck becomes almost a necessity. At absolute zero, every disorder generated

by thermal agitation stops, and the state being considered can be realized in just one

way (W = 1), which only means that Nernst’s theorem (S = 0 for T = 0) is

valid.[7]

This simple explanation of Nernst’s theorem by Planck’s conception of Boltz-

mann’s principle argues persuasively for the general correctness of this interpreta-

tion. It specifically leads us to the conviction that entropy cannot become negative.

According to the classical equation of state of ideal gases, the entropy of a mole

contains the additive term R lg V, which expresses its dependence on the volume at

constant temperature. This term can be made arbitrarily strongly negative by reduc-

ing V such that the entropy itself becomes negative. In real gases, these values for

V lie far below the critical volume for those gases, so one does not need to conclude

that negative entropy values are attained in real gases, for the reason indicated.

Nevertheless, we are certainly allowed to believe that the fabrication of gases that

approach ideal gas more closely than gases really present in nature must not lead

to a violation of the general thermal laws. However, according to the classical

1

2m

-------( p1 2 p2 2 p3 2) + +

[p. 19]

S κlgW =