376 



and to a volume 



vk=b{2—u -[- [/l -u-^-u'} , ... (15) 



whereas equations (12') and (J 3') determine the critical degree of 

 association au and the critical pressure pjc. Moreover 

 pkvic 2 — ti -\- V\ — u -f u'' 



log 



V \—u^V\-uA-u'J 



2— w-H^T— mT^ 

 ■ ==r . (16) 



2(1— M+ |/l— M + M^)(l + V\—U t m') 



As u increases from to ao (that is a^ from 1 to oo ), the 

 expression (16) diminishes slowly but steadily from | to 0. Now 

 it is well-known, that for most (so-called normal) substances the 



value of — — is about .28, whereas for the so-called abnormal 



RTk 

 substances it is even smaller. According to (16) this might be 



explained bj assuming, that even for the so-called normal substances 



association for quasi-association) exists at the critical point, to such 



a degree that n = 7 about, giving Vk = i.5b and «^ = 5. ^) 



1) If a and b are functions of the volume this value will probably be considerably 

 lowered (van der Waals, these Proceedings, Xlll (2), 1257 etc.). 



Similarly Schames (Ann. d. Phys. (4), 39, (1912) p. 887; Verb. d. D. Physik. 



Ges. 15, (1913) p. 1017.) tries to explain the deviations of experiment from 



VAN DER Waals's Original equation by association of simple to double molecules. 



He starts from equation (4'), in which, however he considers 6 and a as functions 



ajc Tjc a — 1 



of a, in such a manner that b = bk — a.ud a = ak-pf; ; he thus establishes 



a 1 ajc—i 



the equation 



«A/ , «-1 aj, Tk\ RT 



a J \ afc — } v' T J a 



in which « changes from 1 to 2 according to a law which is not specified, as v 



decreases from oo to its smallest value. He further assumes the relations (7) (with 



RT/c 8 

 a = ajc and 6oo = bk and (p (a) = 1), which involves the relation — — =7r«/,-. Owing 



pkvk 3 



RTk 



to these assumptions Schames finds for the experimental value 3.6 by taking 



PkVk 



«;^=1,4 and for the smallest volume the value ^V)i- instead oï ^vk- The fact, that 



Tkf^p\ 

 for the critical coefficient — I - — he also obtains the experimental value 7 is 



Pk\M^)k 

 not due to his association-hypothesis, but is the consequence of the further fact, 

 that he makes a change inversely as T (for this well-known rule vid. e.g. J. P. 

 KuENEN, Die Zustandsgleichung etc., 1907, p. 194). 



