1169 
substance in this respect differs more or less from a real substance, 
yet. this will not cause a very great error in the final result, seeing 
that the difference is comparatively small. It seems even probable 
to me that when we consider a system approaching reality still more 
closely whose vibrations are not purely sine shaped, an accurate 
knowledge of the thermodynamic behaviour of it at low temperatures 
would lead us to no other final result for the entropy constant of 
the gas, that therefore the expressions to be found possess universal 
validity. 
The thermal energy of a solid substance as assumed by us according 
to a formula which has been accepted and confirmed of late years, 
is given by: 
hy; 
-—— + const., 
Io, 
kT — J] 
in which A and / are the constants of PLAaNck, while the summation 
is to be extended over all the degrees of freedom of the system, 
each of a frequency »;. The entropy is then given by : 
1 dU hy; 1 ane 
sl en ( ed . a L > log 1 bm kT (2 
T AT aD; ; B 
which for the higher temperatures, with which we shall be exclu. 
sively occupied in what follows, passes into : 
hy; hv 
Sia ks Jom ob en Oe rn ee 
(trg) = & (1 = toy (3 
in which 2 is the number of degrees of freedom, and the line 
expresses the mean value. 
§ 3. Calculation of the vapour pressure of a monatomic solid 
substance and of the entropy constant of the gas *). 
Let us now consider a gramme molecule of a monatomic substance 
consisting of NV molecules, inclosed within the invariable volume V 
and in temperature equilibrium with its surroundings. This system 
may then be considered as part of a much larger one of the same 
temperature. If qi,....gsn are the coordinates of the molecules, 
1) A similar calculation with the correct final result has already been published 
by O. Srern (Phys. Zeitschr. 14, 629 (1913)), however with an imaginary solid 
substance, which perhaps departs somewhat too much from reality. Nor is in this 
way the occurrence of N! in the general formula (see below), caused by the 
exchangeability of the molecules, made clear. 
wry 
bj 
