42 Criterion for determining the Critical Point of a Gas. 



the mean of the volumes is ,, . These results agree very 



remarkably with those (5) calculated from Clausius. 



Reconsidering the question theoretically, and in the light of 

 the curves of Andrews and James Thomson, we are led to the 

 following criterion for the critical point — viz. from equation 



(1) at T,= const., we must have ^ = ° (*• e - a P oint of inflec- 

 tion), and ^ = (i. e. the tangent parallel to the axis of v). 



n c i v \ o 



These give much more easily than before, and at once, the 

 values 



v=Sd+2p, 



T 2 = 



(40 



27E(« + /Sy 



In short, provided the true theoretical relation between p, t*, 

 and t for a gas has been found, the critical point may be de- 

 termined by equating the volume of the liquid to that of the 

 gas, both being at the same temperature and pressure. 



The above numerical results have been deduced from Clau- 

 sius's formula for carbonic acid, verified by Andrews's table. 

 The formula which is nearest to that of Clausius is given by 

 Van der Waals (see Phil. Mag. June 1880, p. 398), and is 



__ RT a 



where 



- R _ 1-00646 

 T (=273°) 



a=-00874, 

 6 = -0023. 



Applying the theory to this equation, we get T— 273°= 29 0, 25, 

 ^> = 61*2, and v = j^ ; the discrepancy between these two last 

 results and Andrews's observed ones being very considerable. 

 With regard to the position of the horizontal line (p con- 

 stant) for a given isothermal, Clausius has assigned as the cri- 

 terion by which to determine it, that, the straight line joined 

 by the curved portion of the isothermal between its ends, con- 

 stitutes a reversible cycle, which is such that the whole work 

 done in it is zero. If ot be the unknown pressure along this 

 straight line, and v i} r 3 the least and greatest roots of the cubic 



