394 Dr. R. D. Kleeman on Fundamental 



of the liquid by an infinite distance from one another, and 

 U\ is the internal energy of a molecule in the liquid, and 

 U 2 and u 2 are the corresponding quantities for the saturated 

 vapour. 



Another equation connecting p, T, t\ and v 2 can be obtained 

 from Clapeyron's equation. Since L the internal heat of 



evaporation is equal to — (Ui — T5 2 + u l — u 2 ) the equation 



may be written ma 



dp 

 TJ 1 — TJ 2 + u 1 —U2=ma{v 2 —v 1 )T-^. .-. . (6) 



If p, v 1 and v 2 be expressed in terms of T from equations (3), 

 (4), and (5), and substituted in equation (6) it should 

 identically vanish. 



Conditions to be satisfied by the Equation of State at 

 the Critical Point. 



The equation of state must satisfy a number of conditions 

 at the critical point. The conditions usually given are 



H> f v =0, g = 0, .... (7) 



where the first equation denotes the equation of state. 

 These conditions have been obtained from considerations of 

 the continuity of state. But a number of others can be 

 found which the equation of state must satisfy. 

 One of them is equation (6) which becomes 



Ui — U 2 U\ — u 2 d\J t du _ ^ T dp 

 v 2 — vi v 2 — v ± 



— iu dV du fmdp \ / x 



at the critical point. 



Another equation is obtained as follows. Let us write 

 Li = A 1 w 1 and L 2 =A 1 iyc, where 1^ is the energy necessary 

 to separate the molecules of a gram of liquid by an infinite 

 distance from one another, L 2 the corresponding energy 

 necessary for a gram of the saturated vapour, and A 1 and x 

 are appropriate functions of T. We have then L = L X — L 2 , 

 and Clapeyron's equation becomes 



A 1 (n-.y)=fe-n)(Tj- P ). 



At the critical point x = l, and the equation becomes 



A,=p-T$. Now L 1 = ^(U 1+Ul -u a ), 



a x ))i a 



where u a is the value of u when ^=x),the corresponding 



