369 



If all k wei^ equal to 1, a, would be equal to a and the equation 



would be 



1 7^7' a 



a v — h V* 



If L were equal to - '), we should have 

 n 



(4') 



P= 1. „»..» 



(4") 



" "^ c /A ^ c ^ n . f;n that S k„.v„ = 1 « ; - (5) 



1 RT a 



n V — b 

 A more general assumption would be that 



/.^^ — 1 _ " ~ f (0 < f < 1) . so that S k„.c„ — 1 



in that case *) : 



2 i^ _ filf) , where , («) = (l - '^ ^ ■ '5') 



4 The manner in which h changes with y, independently of all 

 association, has also been fully discussed by van der Waals. The 

 law of dependence sketched ont by liim ') may be very well 

 represented by the following expression: 



{b^-b){v-kb^) = i{l-fcrbl or 6 = 6«-i-— ^^ , (Ö) 



where b is the value of b for an infinite volume, whereas at the 

 smallest possible volume: 



t,,,„ = b,„, = i (1 +k)b^, where < 6,,„ < 6^ or - 1 < /: < + 1 ^). (6') 

 On this assumption as regards the relation between b and v the 

 equation of state (5') becomes: 



= a [ I — "K ) is arrived 



') The special form proposed by van deb Waals as -a M — 2 



at by putting kn (n = = 1) constant and equal to V,; even at the highest degree 

 of association (x=l) «x would still be i a. It seems more natural, however, to assume 

 that as n increases, fc, becomes smaller; if the diminution of A:„ went so far, that 

 it approached zero, ax would ultimately become zero. 



2) Although a given mean degree of association may be obtamed m an infinite 

 number of ways, still according to the assumption (5) the cohesion (and thereby 

 the equation of state for a given a) is no longer dependent on the special way in 

 which the molecules are grouped. 



If e-- i, so that <f:(a) = i(\ + j]\ the limiting value of a^- will be equal 



to { a, as according to van dkr Waals's assumption. 



8) These Proceedings, XV (2), p. 1131. 



4, Uk = l, 0= constant =/^,, as in the original equation of van der Waals. 

 With A; = 1 we should have viim = blim = 0. 



