( 438 ) 



We shall namely see in what follows that theüretlaillii, both 

 according to the ideal equation of state of van der Waals and 

 according to the equation niodilied by association, the first term in 



the expansion of d — 1 or 1^ — d' is always of the order V 1 — m 

 and not (see among others van dkii Waals, (These Proc. Vol. XIII 

 p. 1259 and Vol. XIII p. 116 and 117) of the order 1' T— m. Bnt in 

 order to raise the coefücient of V^l — m from 2 (according to the 

 ideal equation of state) to about 3,6, it will be necessary to assume 

 association at the critical point to an amount of on an average 

 1' = 3 to 4 molecules (./;=: fro ui 2 to 3). For substances where the 

 coefficient is found greater than 3,(), a smaller mean complexity will 

 suffice; it may be observed that these complex molecules are always 

 decomposed at the critical poiul to an amount of about 0,95, so that 

 oidy Yso P^^i'* ^^^ '^" '''G molecules is slill complex. But these com- 

 [)lexes are then on an average (at 7/J ti'ij)le or quadriq)le ones. 

 Before, however, })roceeding lo the determination of the coefficients 

 a and h in the above expansions into series for iral substances, we 

 shall first carry out the operation — in order to serve for a later 

 comparison — for ideal substances, which would follow the simple 

 uumodilied equation of van dkr Waals. 



7. We can follow three courses for the determiuatiou of the 

 required expansion into series in the neighbourhood of the critical 

 point. In the first place the so-called asymmetrical method by making- 

 use of the equations (1) and (2), of which only the second contains 

 the logarithmic function. If we put : 



cZ = 1 + -lax + -Ibt^ + 'let'' + -Idr' -\- 2,'x''' + . . 



c/' = 1 — -lax 4- -Ihx"- — -lex' + 2dx' — 2ex' + . . . I' ' ' ^^^ 



in which t=V^1 — in, we have only to substitute these expressions 

 ill the mentioned equations to determine the coefficients a, b, etc. 

 immediately. That here d — 1 and 1 — d' are of the order J 1 — in 

 is immediately clear from (J"). And that the coefficients of the 

 expansions of d and (/' will be quite the same, with the exception 

 of the sign of the coefficients of the odd powers of t, is also clear 

 from the perfectly contmuous course of the saturation curve (/=ƒ(?/?) 

 through the critical point. For values of r beginning at the critical 

 point are measured both at d and at (/' iu the same direction, viz. 

 in both cases from the critical point, towards smaller values oi' ni ; 

 while the values of d and d/ are measured in opposite direction : 

 those of (/ from lower towards higher values, those of d^ just the 



