232 



considered isolated. For vibrations with very small wave-lengths, 

 which are here the most important, it may, however, be assumed 

 to hold, in connection with their relation to heat motion, at least 

 as an approximate hypothesis for states which differ only little from 

 the state of equilibrium. It then follows^) that 



1 ( hv 1 . 



— — + -hv\ (2) 



kT , 

 e — 1 



As the different modes of vibration, which are possible in a gas 

 enclosed in a given vessel, must be supposed to have the same T 

 in the state of equilibrium, equation (2) at the same time represents 

 the part which each mode of \ibration in the gas in the state of 

 equilibrium at the temperature T contributes to the whole energy. 



We now assume that we obtain an approximately correct value 

 for the whole energy if, in a way corresponding to that which 

 Debi.te follows for a solid, we suppose the number of different 

 principal modes of vibration which are contained in the region 

 determined by the frequencies v and v -\- dv to be equal to ') 



c 



and cut off the so determined "spectrum" at v,,,, given by putting 

 the total number of modes of vibration equal to the number of 

 degrees óf freedom 'SN. V represents the volume of the gramme- 

 molecule of the gas, iV is the number of Avogaduo. We then obtain: 



9A^ i 



"J 



1';/) 



U-=^. I 1—4-7 + lhv\v^dv, .... (4) 







where Vm is determined bj* 



kT 



9 N 



--C' (5) 



4jr V 

 For the total entropy an expression can now also be easily given. 



1) As SoMMERFELD loc. cU. observes, the hypothesis mentioned above causes 

 as it were automatically tliat at high temperatures tlie mean energ-y per degree 

 of freedom becomes — '/o kT, as it must be for the molecular translatory motion. 

 For the difference of equation (2) from the corresponding one of Lenz cf. 

 p. 228 note 5. 



3) in accordance with a remark by Tetrode, I.e. this expression can be easily 

 deduced for a cubical vessel from the formula for the wave-lengtlis occurring in 

 it: Rayleigh, Theory of Sound II, 2nd ed. London 1896 p. 71. 



