548 
An estimate of the uncertainty of this value is obtained by cal- 
culating the number of elementary cells which lie inside the hy per- 
cube inscribed in the energy-hypersphere. This number will be a 
lower limit for the quantity in question; it is equal to 
(12) 3N — about 10'9,9.10% — about Z0,77. 
The possible error in the thermodynamic probability, calculated 
from the ratio of the two volumes, is thus 7°?3 at the most, from 
which it follows that that part of the entropy which depends on / may . 
be incorrect by 23°/, at the most. It is clear, however, that this 
estimate of the error is very much exaggerated, and also that the 
error diminishes rapidly on increasing /. If / is taken say 100 
times larger, it would seem, that the deviations from the ideal 
gaseous condition may be neglected without scruple. 
The limiting temperature 7’, corresponding to £=—=100 X 1.055 >< 
<x 10-§ = 1.055 « 10-4 is equal to 0.85 Xx 10-12 or rounded off 
T, = 10-12 of a degree. ') 
§ 3. The cause of the gas-model remaining ideal at temperatures 
which are too low must be looked for in the small frequency of 
the vibrations. ScHERRER takes, as the periods in fhe quanta-formulae, 
times of the order of those required by a molecule to go backwards 
and forwards between the walls of the vessel. In order to obtain 
admissible values of the entropy-constant the following procedure 
suggested by Dr. Kurrsom might be followed: each molecule is allowed 
the NM part of the total volume as its “vibrational space’’’). If for 
the positional coordinates the original margin is retained, the elementary 
cell is then to be multiplied by NY’) and the values of /, and 7’, 
by N*s= 72 10", so that the contradiction with experiment 
ceases; S, becomes: 
S, = —13,3.R+«. 
It is quite possible, that the value of ¢ is such, that S, in this 
equation obtains a suitable value of about 3/ (see above $ 1). 
1) If the volume V is not 1 ce. the value of 7) depends on V as follows 
Lo V = ToD Fis eV 
The entropy at temperature 7 and volume V is then given by: 
ATV) =3/2. RUIT +h lg VR igi0—2 S =: 
The ordinary entropy-formula is.thus obtained, although still with too large an 
additive constant. 
2\ This means, that the time in which the mean distance between two molecules 
is described to and fro is taken as the time of vibration. 
3) If in ScHERRER’s formula for the absolute value of the entropy the factor NV! 
is omitted and replaced by NA, his entropy-constant becomes smaller by ZP, and 
thereby agrees even beiter with the experimental values (l.c. p. 6). 
Ss 
