174 Dr. F. A. Lindemann on the 



emit whole quanta of energy. At temperature T its average 

 energy is therefore 



hv _ R &v _ RT fi 



if Q h j3v pv 



P = r ^ = Y' K= %> 



in the ordinary equation of state of a gas and N is the 

 number of atoms in a .gramme-atom. This is the view first 

 put forward by Planck* and Einstein f. 



Planck's modified hypothesis % allowed an oscillator to 

 absorb energy continuously, but restricted its emission 

 to whole quanta. This theory, which is capable of much 

 more consistent treatment, entails that each oscillator pos- 

 sesses, on the average, half a quantum ^- over and above 

 the energy calculated above, i. e. 





hv hv _U / &v_ Bv\ _RT/_/*_ 



AT _ ]_ 



In calculating the relative vapour pressures or affinities of 

 isotopes, these two points of view lead to somewhat different 

 results. It will be convenient to consider the second hypo- 

 thesis before the first, since the consequences are simpler. 



In lead, as presumably in all other elements, the melting- 

 point, (T m = 600°), is large compared with (/3v = 95°), the 

 characteristic temperature §. One may therefore expand 



— ~~ in terms of ~ , and one finds for the energy of the 



e »» — 1 



atomic oscillation at the melting-point 



^"NJ 1 "' T + i "'i 1 ( 1} (2«)!\Tj + 2J- N ' 



if Tj,\7jr) and the higher powers are neglected in com- 

 12 \i m / 



parison with 1. 



* M. Planck, Vorlesungen ilber die Theorie der Warmestrahlunq (1906). 



+ A. Einstein, Ann d. Phtjsik, (4) ccxx. pp. 184 & 800 (1907). 



% M. Planck, Ann. d. Physik, (4) xxxi. p. 758 (1910), (4) p. 672 

 (1912); Sitzungsber. d. Akad. d. W. p. 723 (1911); and Verh. d. d. 

 Phys. Ges. xiii. p. 138 (1911). 



§ F. A. Lindemann, Diss. Berlin (1911). 



