40 WILSON ART c 



Here 



a Rt 



P = - -, + 7' (1) 



V^ V — 



/dp\ _ 2a _ Rt 



\dv/t ~ V' ~ (v -by~^ ^^^ 



at the limit of stability. Eliminating t, the locus in the pv plane 



is* 



a 2ab . . 



p = -,--r' (3) 



v^ v^ 



We have also the equation 



\dvyt 



Qa 2Rt 



= - ~T + 7 ^3 = (4) 



to represent the inflections of the isothermals. Equations (1), 

 (2), (4) have a common solution, which must be also a solution 

 of (3), and this is the critical point. If (1) be regarded as a 

 cubic in v the critical point is that for which the cubic has three 

 equal roots. For this point 



the actual physical isothermal representing complete equilibrium, in 

 the detailed discussion of the thermodynamic surface including the 

 questions of stability (whether entire or limited) in Lectures XX-XXIII. 

 This general discussion completed, the lecturer returns to a considerable 

 development and illustration with the aid of the equation of van der 

 Waals. 



* The limit of stability is defined by {dp/dv)t = 0, i.e., when AC — 

 B' = 0. It may be observed that by this definition there may lie within 

 the limit of stability states with negative values of p, i.e., with tensions 

 instead of pressures. From (3) we have v = 2b when p = 0. Then 

 Rbt/a = 1/4. In terms of the critical values v/vc = 2/3, t/tc = 27/32. 

 Thus for temperatures below 27ic/32 = .Siitc the van der Waals' iso- 

 thermal dips down to negative values of p. Indeed as v decreases toward 

 b, p in (3) decreases toward —a/b^ = —27pc, and t toward zero. Al- 

 though all negative values of p represent instability in vapor phases, we 

 do know that under careful experimental conditions liquids can be made 

 to support very considerable tensions without going over into the vapor 

 phase, thus parts of these isothermals for negative p can be realized 

 qualitatively even if the quantitative relations are quite inadequately 

 represented by (1). 



