371 



From the equations (7^ it also follows that 



1 R{Tk)oL . „ 

 == a constant ^) (7 ) 



6. We shall now consider the equation (4) as representing the 



^) This constant, the crilicai viiial coefficient K^, (vid. Kamerlingh Onnes and 

 Keesom, loc. cit., p. 752 (138)), again has a value which changes with k. For: 



^ = 1 V. - '/, - I 



K^= I = 2,67 2,69 2,73 2,80 3 



The change of Ki with k is seen to be comparatively small, and the relation 

 which we have assumed between b and v does not raise the coeflicient ^^4 to the 

 experimental value of 3.6 about, holding for the so-called normal substances, or 

 the still higher value required by the associating substances; it is even doubtful, 

 whether this could be attained with any other relation b = ip(v), unless a very spe- 

 cial form were purposely chosen (comp. on this point van der Waals. These 

 Proceedings, Xill (2), p. 1211 ; see also Kamerlingh Onnes and W. H. Keesom, 

 loc. cit., p. 7o2) It is true that v. d. Waals (Boltzmann-Festschrift, 1904, p. 805) 



r b \ 



found the correct value of ^4. by putting 6 — 6^ 1 — k—^ and /c = f, but it 



is easily seen, that this expression for b cannot hold down to small volumes : in 

 fact for viim = biim it gives an imaginary value. 



In this connection it may be of interest to remark, that the author (vid. these 

 Proceedings, II, p. 558 and Arch. Néerl. d. sc. ex. et iiat., (2), 650 1901) esta- 

 blished a purely empirical form of the equation of state 



''=P' + '^(S ^0 ^ "C^'~ ' 



according to which 



RT { b 

 K, = = ra- 1 



Pk vk V ^fc. 



where it appeared that approximately «jt = + and b = 0,1 v k, which gives a value 

 for K^ agreeing well with the experimental value. 



Batschinski (ZS. physik. Chem., 40 (1902) p. 629; Bull. soc. imp. nat. 

 Moscou, 1903, p. 188) has made the relation il") with ^4 = |- the basis of a 

 determination of the degree of association for associating substances, in particular 

 for acetic acid at different temperatures, assuming that the degree of association 

 varies with the temperature, but not with the volume (an assumption which is, 

 however, in contradiction with the well known laws of dissociation and also with 

 determinations of the vapour-density). By comparing the separate isolhermals of 

 the associating substance with the set of isothermals of a normal standard substance, 

 Batschinski determines the critical constants appertaining to each isothermal and 

 then calculates h by means of {!"). In this manner he finds, that the value of 

 {vk)y. is very much the same for all isothermals, whereas {Tkh diminishes a little 

 towards the higher temperatures, whereby a also gels smaller and {pk)y. increases 

 in the same direction ; this agrees in the main with what we have just derived 

 about (!;;fc)a, (pA)« and [Tkh from the equation of state. 



