Atomic Frequency and Atomic Numher. 479 



§ 2. Frequency Formula with Empirical Constants. 



Various formulae have been suggested for the purpose of 

 •expressing the atomic frequency in terms of other physical 

 properties. Several of these have been discussed and 

 compared in a paper by Blom *. They are based on grounds 

 partly theoretical, partly empirical, but are all variants of 

 the expression lor the frequency in simple harmonic motion 



1 /D 



2^V A> 



where A is the atomic mass, and D is the restoring force 

 for unit displacement from the equilibrium position. 



It has been overlooked by later writers that the first 

 formula of this kind was given by Sutherland f, who claimed 

 that he had found simple relations between the periods of 

 vibration of elements at the melting point for several 

 chemical families. His reasoning may be summarized as 

 follows : — When a molecule, mass M, and specific heat 

 (mean) c, is heated from rest at absolute zero to its melting- 

 point T 5 , it receives heat McT s , which is taken to be propor- 

 tional to its mean kinetic energy plw 2 . Thus v is propor- 

 tional to -y/fMcTyM). The length or amplitude of the 

 vibration is assumed to be equal to (or proportional to) 

 aT s (M//o) 3 , where a is the mean coefficient of linear expansion 

 and p is the density. Hence the periodic time is propor- 

 tional to 



aT s (M/ /3 )7 v /(McT,/M). 



Now it is a characteristic feature of Sutherland's theory 

 of the process of fusion, that melting occurs when the space 

 between the molecules attains a certain value relatively to 

 the size of the molecules. Thus aT s is constant, a relation 

 that has been verified by Griineisen J for monatomic elements. 

 Again, according to the law of Dulong and Petit, Mc is 

 constant. Hence the periodic time is proportional to 



(M/p)V(T,/M). 



Putting M//3 = V, the molecular (or atomic) volume, the 

 frequency is proportional to 



V (mv»)- 



* Blom, Ann. d. Fhysik, vol. xlii. p. 1402 (1913). 



t Sutherland, Phil. Mag. vol. xxx. p. 318 (1890) ; vol. xxxii. p. 524 

 <1891). 



| Griineisen, Ann. d. Physik, vol. xxxix. p. 257 (1912). 



2M2 



