435 



platinum, and the other substances which have been investigated in 

 liquid hydrogen show a siinihxr tendency. There is therefore every 

 reason for the assumption of tlie temperature-function (1) with the 

 inclusion of a zero-point energy. 



§ 5. It was further tried for a number of substances, whether the 

 dependence of the susceptibility on the temperature is in quantita- 

 tive agreement with the above hypothesis. In the calculations the 

 frequency v which occurs in the formula for U was not taken in- 

 dependent of the temperature (r independent of T gives the curve 2 

 but following Einstein and Stern it was assumed that 



U 



^■^ = 2^0^ • • (2) 



The change of U I or — J with T according to the relations (1) 



and (2) is represented by curve 4 ; i\ is the value which v assumes 

 at very low temperatures -. it is related to the moment of inertia I 

 of the molecule by the formula 



"" = 4^7 (»)'» 



In the first place it may now be observed that for temperatures, 

 which are not too low, the relation between ü and T expressed 

 by (1) and (2) leads to the empirical relation •/ (7"-]- A) =: const., 

 which was deduced from the observations (comp. ^ 1). This is seen 

 by developing (J) in a series and neglecting the terms beyond the 

 third which is certainlj^ allowed for high values of 7^; this gives 

 rr . / ^ 1 I hv\ 1 



1 h'v' 



U=kT -\ 



^ 12 kT 



U 

 When we substitute : v"^ = 2v\ — in this, we find 



h 



1 hv, 



U=kT-\ ° U 



6 kT 



^) The relations (2) and {3) are at once arrived at when it is remembered that 

 C7=— 7(2tv)2. For T=0 this gives: — h^o = ~ I {2,rv,)^ or v, ^' 



2 ^"^"- 2 " 2 ^'"'"" "^ '"~4^'r 



When this is introduced into U= -—1(2^.)- the result is (2). 



28* 



