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 5
The first term of this expression does not depend on the choice of macroscopic dis-
tribution, but only on the total number of molecules. When comparing the entropies
of different macroscopic states of the same gas, this term plays the role of an incon-
sequential constant that we can leave out. We must leave it out if—as is customary
in thermodynamics—we want to achieve that the entropy be proportional to the
number of molecules at a given inner state of the gas. We thus have to set
(29c)
One usually tends to justify this omission of the factor n! in W for gases by regard-
ing complexions arising from the mere exchange of molecules of the same kind as
not different and, therefore, as being counted only once.
Now we have to find for both cases the maximum of S under the secondary con-
ditions
In cas (a), we obtain
(30a)
which agrees with
(13),[7]
disregarding the manner of notation. In case (b), we
obtain
(30b)
In both cases here.
One sees furthermore that in case (b), Maxwell’s distribution law results. The
quantum structure does not make itself noticeable here (at least not for an infinitely
large total volume of the gas).[8] It is now easy to see that case (b) is irreconcilable
with Nernst’s theorem. For, in order to calculate the entropy value at absolute zero
temperature for this case, one must calculate (29c) for absolute zero. There all the
molecules will be in the first quantum state. We therefore have to set
(29c) hence yields for T = 0
(31)
S
κ¦nν(
lg zν lg nν) –
ν
=
[p. 7]
E
¦Eνnν
const. = =
n
¦nν
const. = =
nν
zν
eα βE + 1 –
------------------------, =
nν zνe– α βE – =
βκT 1 =
nν 0 for ν 1 ≠ =
n1 n =
z1 1. =
S n lg n. –=