B()3 



Instead of a regular linear decrease with 2 : ('lj-|-/?), i.e. with 

 1 -{- .V, vahies are even seen to appear <:^ J in the neighbonrhood 

 of [? = 1 (all the molecules single), with a min iviiim at ahoui [j^^: 0,8 

 (accurately at i3=r 0,8015), and a horizontal tinal direction, i.e. 

 d /s 



, . 0. 



diB V^o 



On increasing association ((O' from 1 to 0), s will therefore tirst 

 become somewhat smaller than s^ (= 3,77 for "ordinary" substances), 

 and then (from /? = 0,7) s : .^u will become greater than 1, and increase 

 to 2 for /5 = 0, when the association to double molecules is perfect. 



A straight line for .v : .y„ (as van der Waat.s thinks) therefore 

 replaced by a line that is pretty considerably curved downward 

 between the values 2 and 1 with a minimum close to 1, so that 

 s : s„ at first decreases there instead of increasing. 



What consequences this behaviour will have with respect to the 

 degree of association ^, calculated from the value found for s for 

 metkylalcohol, viz. 4,52, may appear from what follows. 



As ó':6'o — 4,52 : 3,77 = 1,2, we shonld find about /5=r 0,67 or 

 ,t' = 0,2 for ^, according to the second column of the above tal)le, 

 when we were led by a supposed linear dependence. But when we 

 also take account of the "factor" by the side of 2 : (l-f-/^), we find 

 about /3=rO,35 or ,r = 0,5 from the last column for the value for 

 /i answ^ering to the ratio s -. s,^ = 1,2. 



A difference, in fact, too large to be neglected. Instead of 0,8 

 single molecules to 0,2 double molecules, as van der Waals would 

 find with his linear dependence, we find more accurately 0,5 single 

 molecules to 0,5 double ones. The relation .c : (1 — .-i') has become 

 1 instead of 4. 



4. The second quantity which plays a ]»art in the cited [)aper by 

 van der Waals, is the quantity 7'k : pL- , which may be put |)roj)Or- 

 tional to the molecule size for non-associating substances. We now 

 find for it : 



^ ^ 1 ^ J__ (i-f/?)(i-V.(H7.^^T 



PL li ^'l-f/i^(l-V,p)(l-+l^— 3/i-^ + V./^^) ' • • ^ ''^ 

 which with lib = jiasses into 

 Ti, 8 , 2 



z=: — bk. X (8) 



pf. E l-\-ir{3m' — 2n){4:ti — dm) ^' 



We shall not discuss the course of this again, but solve from this 



