126 



SCIENCE 



[N. S. Vol. XXXVII. No. 943 



be so the total energy content of an atom 

 of a solid, in view of its three potential 

 and three kinetic degrees of freedom, must 

 be twice that of a molecule of a monatomie 

 gas. In other words, the atomic heats of 

 solids should be twice the molecular heats 

 of monatomie gases, i. e., they should be 6 

 calories, as in fact they are in most cases 

 found to be. But brilliant and successful as 

 was this stroke, it only made the abnormally 

 small values of the atomic heats of the ele- 

 ments of low atomic weight (C, Bo, Si) the 

 more inexplicable, especially after it was 

 found that these substances all behave nor- 

 mally at high enough temperatures. Now 

 the recent work of a number of experi- 

 menters, notably of Nernst^^ and his pupils, 

 shows that at sufficiently low temperatures 

 all substances show abnormally low atomic 

 heats, and that, in general, the lower the 

 atomic weight, the higher the temperature 

 at which the abnormality begins to appear. 

 This means that if a degree of rise in tem- 

 perature means a given increase in the 

 energy of vibration of the atoms of any 

 substance, then at low enough temperatures 

 only a fraction of the atoms take on their 

 normal energy load. But this is precisely 

 what the quantum theory demands. No 

 atom can take on any energy at all until 

 the impacts of the molecules of the sur- 

 rounding gas possess an energy as high as 

 hv, and hence the higher the v the higher the 

 temperature at which energy can begin to 

 be absorbed. Further, ceteris paribus, the 

 smaller the atomic weight, the higher the v 

 and hence the sooner, with decreasing tem- 

 perature, should atomic heats lower than 6 

 calories begin to appear. 



One can not withhold his admiration 

 from the beauty of the qualitative agree- 

 ment between this theory and experiment. 

 But can the theory stand a quantitative 



'* Nernst and Lindemann, /Site. Ber. d. Freuss. 

 Alcad., XIII., p. 306, 1910. 



test? Such a test has been made as fol- 

 lows. Lindemann, ^° by assuming simply 

 that a fixed relation holds at the melting 

 point, Ts, of any substance between the 

 amplitude of its atomic vibrations and the 

 distance between its atoms, that is, its atomic 

 volume, V, obtained without the aid of a 

 quantum theory, a formula of the form 





by which the frequency v in the solid state 

 of an atomic vibrator of atomic weight m 

 can be computed. This formula yields 

 results which agree fairly well with direct 

 measurements of v by means of "rest- 

 strahlen" wherever the latter have thus far 

 been made. With the aid of this formula, 

 then, we may first check our rough guess 

 that the order of diminishing frequencies 

 is the exact order in which atomic heats 

 begin with decreasing temperature to fall 

 below 6 calories, and, second, we may com- 

 pare the frequencies computed by Linde- 

 mann 's formula with those given by 

 Planck's equation and the observed de- 

 parture from Dulong and Petit 's law at 

 low temperatures. The method of doing 

 this was pointed out by Einstein.^" The 

 agreement is sufficiently good to warrant 

 the conclusion that the departures from 

 Dulong and Petit 's law are in fact funda- 

 mentally conditioned by atomic frequency. 

 But it can not be said that Planck's equa- 

 tion, as applied by Einstein to the com- 

 putation of the relation between atomic 

 heats and temperatures, succeeds in pre- 

 dicting very accurately the observed 

 curves.-" Furthermore, the departures are 

 in the same direction with all substances. 

 They may be explained by introducing 

 additional hypotheses into the quantiun 

 theory, as Nernst and Lindemann have 



" F. A. Lindemann, Phys. Zeit., 11, p. 509, 1910. 

 =» Einstein, Ann. der Phys., 22, p. 180, 1007. 



