12-4 On the Statistical Theory of Heat Radiation. 



E 

 we have p = jr-, where E denotes the total energy of the gas 



per c.c. Also for an adiabatic expansion of such a gas 

 pv i/3 =: const. These two equations also hold for full radiation, 

 which suggests that the elements of disturbance ought to 

 have energy corresponding to six degrees of freedom instead 

 of only 5*4, but the energy is not distributed among" the 

 elements in the same way as among the gas molecules. 



Consider the free expansion of full radiation from a volume 

 i\ to a volume v 2 . The chance that an element is in v x \\ hen 

 the volume is v 2 is v,/r 2 . Thus the chance that all the S¥vi 



- l ) ; hence the increase of entropy 



S 2 — S x due to the free expansion is k^i^log-. If r 2 - r 2 is 



v i 



very small, say dv, this becomes k&dv = d$. Now 



,^_ dXJ + -pdv . 

 t ' 



so that for an infinitesimal free expansion, if for the moment 

 we regard t as unaffected, we have 



,n_ pdv __ ffl^edv 



~~r Wt 



Hence , GUl $V~e , 



ot 



or € = Mt instead of e =. Ut "• 



a. 



This makes e equal to the energy of six degrees of freedom, 

 but the supposed infinitesimal free expansion alters the tem- 

 perature of the radiation by different infinitely small amounts 

 for the energy of different wave-lengths. Consequently it is 

 not clear that even after only an infinitesimal free expansion 

 the radiation can be regarded as having a definite temperature 

 differing infinitely little from t. 



In the case of the gas we have in the same way for a free 

 expansion 



t 3 t 



where m is the mass of a molecule and u 2 the average square 

 of the velocity of the molecules. Hence |^ = J??ur, which 

 gives the value of h due to Planck, The known equation 



