46 WILSON ART. c 



the various forms depends on the particular inquiry to which 

 they are applied. 



Lecture XXVIII. Returning to van der Waals' law, 



( 



dp\ R 



dt/v V — 



This is not quite true, of course, but it is surprisingly correct 

 in many cases over a very wide range. For very great densities 

 it cannot be expected to hold, and we have to exclude dissocia- 

 tion at very high temperatures, and those states in which the 

 substance is congealed. Now in the -pt plane a line of constant 

 volume becomes straight. It is easy to determine correspond- 

 ing values of p and t under conditions of constant volume and 

 observe how straight the curves in p against t are. At the limit 

 of stability we had {dp/dv)t = 0, i.e., maxima or minima of the 

 isothermals in the yv plane. Keeping t constant in the p^-dia- 

 gram corresponds to a vertical displacem.ent. If {dip/dv) « > it 

 is seen that the lines of increasing volume on the p^-diagram lie 

 one above the other in the direction of increasing pressure; in 

 the limit when {dp/dv)t = the successive lines of constant 

 volume intersect. These lines will therefore envelop a locus 

 which consists of points pv for which (dp/dv)t = 0, i.e., for states 

 at the limit of stability. This locus has a cusp which is the crit- 

 ical point. In the region within the cusp and near to it there 

 are three tangent lines of the envelope through each point, i.e., 

 for a given pair of values p, t there are three lines of constant 

 volume along which one may proceed. Taking van der Waals' 

 equation in the form (19), the equations 



8 8^ 



- T -T 



V V - 1/3' UfA ~ Y^ {V - 1/3)2 



will give the cuspidal locus on elimination of V from 



9(V-l/3)'' ^ 3 7-1/3 3 2 



4 73 ' 72 "■ " 73 72 73 



The plot of P against T is more readily made from this para- 

 metric form than from the equation obtained by eliminating V. 



