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Chemistry. — ''The shape of the spinodal and plaitpoint curves 

 for binary mixtures of normal substances." (Fourth communi- 

 cation : The longitudinal jjlait.) By J. J. van Laar. (Com- 

 municated by Prof. H. A. Lorentz.) 



1. In order to facilitate the survey of what has been discussed 

 by me up to now, I shall shortly resume what has been communi- 

 cated on this subject in four papers in These Proceedings and in 

 two papers in the Arch. Teyler. 



a. In the first paper in These Proceedings (22 April 1905) the 

 equation : 



2 

 RT = - [w (1— .v) {av-^l/aY + a{v—by] . . . (1 ) 



was derived for the spinodal lines for mixtures of ?2örma/ substances, 



on the supposition that a and b are independent of v and T, and 



that 01,=! 1/^1^2, while 



{€w-^]/ay [(1— 2.tO v—^x (1— A')/?] 4- 



a{v—h){v — 36) 

 ^av—^\/a){av—2^\/a)^ 



-^Vaiv-by 



= (2) 



.v{l — ,v) 



was found for the ?;,A'-projection of the p)^<^'dpoint line, when 

 a—{/a^ — \/a^ and ^ = b^—bi. 



b. In the second paper in These Proceedings (27 May 1905) the 

 shape of these lines for different cases was subjected to a closer examina- 

 tion. For the simplification of the calculations ^^0,i.e. b^:=b^, was 

 assumed, so that then the proportion ^ of the critical temperatures of 

 the two components is equal to the proportion Jt of the two critical 



[/a, b T 



pressures. If we then put =(p, -=:to, — = t (where 7 „ is the 



"third" critical temperature, i. e. the plaitpoint temperature for 

 V = b), the two preceding equations become : 



T = 4a> [41-;iO + {(f + .xy (l-^ri . . . . (la) 



(rp +.7-)»(l-to)='(l-3a>) ^ ^ ^ 

 (1-2.-) + 3{cp 4- ..) {l-o>y + ^'^ ^ ; -^ -^ = 0. (2«) 



It now appeared that the plaitpoint curve has a double point, 

 when cf = 1,43, i.e. = ji = 2,89. If <9 > 2,89, the (abnormal) case 

 of fig. 1 (loc. cit.) presents itself (construed for (p = l, ^ = {l-{- 7^)^ = 4); 

 if on the other hand 6 < 2,89, we find the (normal) case of fig. 2 

 (loc. cit.) (construed for (f := 2, <9 = 2^/^). 



At the same time the possibility was pointed out of the appearance 

 of a third case (fig. 3, loc. cit.), in which the branch of the plaitpoint 



