Vapour Pressure and Affinity oj Isotopes. 175 



Now, if / is the restoring force which keeps each atom 

 in its position of equilibrium, the energy is obviously •'^ , 

 where I is the amplitude. At the melting-point therefore 



/ hn JX 1 m 



T" N ' 



and, since it was proved experimentally that T m is identical 

 and shown above that l m must be identical, one may conclude 

 that in isotopes the restoring force /is identical. 



Since the frequency v— o~~\/ ? where m is the mass 



of the atom, it follows from this that the frequency is 

 inversely proportional to the square root of the atomic 

 weight. Further, it is obvious that the compressibilities 

 must be identical if the restoring force is identical, and it 

 may be shown that it is unlikely that the coefficients of 

 thermal expansion vary appreciably. As was shown in the 

 paper referred to above *, it is impossible for the latent heats 

 at the absolute zero X to differ if the restoring force / is 

 mathematically identical over a finite range. In the instance 

 considered, the values of /are only equal to the second order, 

 and this fact has not been tested over a wide range, so that 

 the conclusion that X is identical is not quite binding. It 

 may, however, be regarded as extremely probable. 

 Now the vapour pressure p is given by 



l ogi>= _^+ j _g_J (C p - Cp )dT + i, 



where C p is the atomic heat of the gas and c p that of the 

 solid, whilst i is the chemical constant. Therefore unless 

 the atomic heats of the gaseous isotopes differ, which is 

 certainly not the case within a very wide range of tempera- 

 ture, the difference of vapour pressure is given by 



p i T dT r T 



F% Jo J^- 1 - ,'o 

 The atomic heat at constant pressure c„ — c v + — — , where 



V K 



c v is the atomic heat at constant volume, v the atomic volume, 

 a the expansion coefficient, and k the compressibility, v and 

 k. have been shown to be identical and u can only differ very 



* F. A. Lindeinann & F. W. Aston, Phil. Mag. xxxvii. p. 523 (1919). 



