( «a ) 



6. The straight diameter. In order to luive at our disposal 

 some material to n^e for a (■onipiwlson in the derivation of the 

 coexisting liquid and vapour phases, and also with a view to a 

 )tiethoclicaI treatment of the problem which is very ditïicnlt in the 

 case of non-constant b, I will begin with the dei-ivation of the 

 equations referring to the problem for the case that b is constant. 

 Then va?4 uer Waals's equation of state holds in its original form, viz. 



RT a _ RT a 



V — 6 V V — V 



in which /■ refers to the liquid phase, v' to the vapour phase. 

 Then we ha\e further for the coexistence : 



1 /• RT v'—b a 



P — ', \ipdv— , log , . 



V — vj V — V v — b vv 



After substitution of /> = ?/>/, , T=inTf,, r^nuk, in which: 



I a S a 



^^ 27 /r '27 h 



we get the well-ivnown reduced relatioJis 



8m 3 Sm 3 8//t 3// — 1 3 



3^ — 1 ,/- 3// — 1 71- 3 {n' — n) ' 3n — 1 )in' 



or also, after introduction of the densities d ixnd d d^ and (/' = , : 



V // » J 



d d' dd' fd 3-d'\ 



8 = 8m - - - 3 ./•-• = 8m - 3d'' = 8m --^,^ log , ^- - - 3dd . (a) 



3-(/ 3-(7 3{d-d) \d S-dJ 



Equalisation of the tirst two equations (a) for 8 now yields: 



d d' _ 3 dr--d" 



3^d ~ 3^d' ~ IT m 



For this e(piation we can also write : 



3 {d-d') _ 3 d'—d" 

 {3-d){3-¥)~^"'7n ' 



or 



( 3-(i)(3-c/')(^/^c/') — 8m (1") 



If we equalize the tirst and the third, and also the second and 

 the third of the oriuiiial equations {a) for 8, we get : 



/ dd' d \ 



8m log = 3dd - 3(/- j 



\3{d—d') ^ 3—dJ I 



dd' ^' \ < /" 



lo(i = 3dd — 3(/ " 



3{d-d;) ' 3-d'^ 

 that is, after division of the tirst equation by J, and the second by r/': 



