930 



relations (24) by division, llial also oaU'ide the critical point 



b" _ h' 

 " \~l>' ~ h-h, ' 



one would easily tind back (25) after integi-ation of this differential 

 equation. But we know that this equation does not satisfy. Also 

 other obvious suppositions about h" and //, which satisfy (24) at the 

 critical point, lead to such impossible tinal results. 



10. We have now come to the forms, which lead to possible, 

 and at the same time not too intricate results, also as far as the 

 convergency point I'o. ^o '^ concerned. In all these forms the relation 

 (6 — ^o) • (^ — ''o) or also (/> — h^ -. {v — 0) occurs by the side of 

 {.b — bg) : {b,f — bj. In this respect the general form of van der 

 Waals's relation (27) is the best that can be assumed. Here every- 

 thing is reached that can be desired. The rehition (6 — b^) : iv — b) 

 approaches to a finite limiting value ƒ at u = b =^ i\, when in the 

 second member b =^ b^, so that the convergency at i\, b^ has been 

 properly warranted beforehand. Further b becomes b = bg for 

 v=:c. But as has been said before — in order properly to obtain 

 the values given by (24) at vk, exceediiighj intricate expressions must 

 be assi'gned to ƒ and n, in which for the case b =: constant, i.e. 

 bk — b, or bj — b^ = (7 = \ ,; ƒ approaches to and n to ac. 



This is of course in itself nothing particular, as it is e.g. by no 

 means necessary that, as van der Waals desires, Lim {b — /;„) : 

 (y — yj^ is 3r= J or assumes another finite value ; it can very well 

 become = 0, as for values of r close to v^ (or b) b can long have 

 assumed a value close to the limiting value b^. This is the more 

 apparent when we consider the case that A no longer changes at all. 

 or hardly changes (at 7 = \/J. The xalue of 6 can then be put 

 about = 60 throughout its course, from ?; = go to r z= v^, so that 

 the numerator of {b — b„) : {c — v\) approaches much more rapidly 

 (or infinitely more rapidly! to than the denominator. Xor is a 

 very large value for n with small values of bk — b^ impossible in 

 itself. 



But it is the exceeding intricacy of the expressions for ƒ and n 

 that make us reject equation (27) in that form. And these intricate 

 results remain, so long as the exponent of {b — /;J : {v — b), which 

 is always =1 in (27), differs from that of ^6 — ^J : (^^ — 60), viz. ?2. 



Let us take generally : 



h—h. 



h 



1 — 





(27«) 



