377 



10. We now return to the more general equation of slate (6"j, 

 where « has the vahie given by (12). 



As at very low temperatures e^ becomes very large (infinite of 

 infinitely high degree), it is easily seen that, provided v itself is 

 not infinite, 



^— =0 and even -- h^O.') 



¥ov the free energy by introducing (11) into (8) we find the 

 expression : 



F= — RTlog{\-^e-^) 



RT 



{\-8}" 



+ E-E~T{H-H), (17) 



and therefore, (foi- the sake of simplicity assuming a and h inde- 

 pendent of T) '), 



5r= 





M--(i+^, 



%(l-f.--^)-fi?( l + A )X,-^Y_ 



2a 



V 



E 

 RT~' 





+ ^. 



H 



(18) 



1 " 



At very low temperatures and moderately large volumes we 

 thus have : 



S^H, — H and U = F + TSz=: E,- E ~ ^{l -ef 



(19) 



It will, therefore, be seen that at low temperature the entropy 

 on the side of the condensed condition of matter is no longer a 

 function of the volume for of the pressure)'), so that there can only 

 he question of one specific heat 



dT dT dT 



(20) 



^) This agrees with what may be derived from the theory of quanta at low 

 temperatures (vid. e.g. P. Langevin et M. de Brogue : La theorie du rayonnement 

 et les quanta, Paris, 1912, p 284). 



") Corap. VAN DER Waals, these Proceedings, XIII (2), p. 1213. 



3) This follows also from the relation 



bpjT 



dT 



, and therefore = 0; 



similarly ^ — 



— I =0. This is again in accordance with modern views. 

 dvjT 



*) In our equations Ei and E are indeterminate functions of the temperature 

 and we cannot, therefore, conclude from them that C approaches zero with T. 

 It is clear why this is so. The dependence of C on the temperature is not solely 

 determined by the equation of state of the system of separate molecules, but 

 also by the internal mechanism of the molecule itself, that is at low temperature 

 and small volume the internal mechanism of the amorphous-solid body consisting 



25 



Proceedings Royal Acad. Amsterdam, Vol. XX. 



