252 MOREY 



ART. G 



which give the rate of change of total volume and of total 

 entropy, respectively, on addition of mj. Since 



V = {mi-\- mijv, —-= V - {\ - x) — 



drrii dx 



and, similarly, 



an dv 



- — = ^ — (1 — a;) — • 

 dm2 dx 



djJLi dfi2 



Substituting these values of — - and Trin (10), inserting this 



O^ 01/ 



value of diJi2 in (9) and rearranging, gives 



^y' _ ," _ (^' _ ^") ^£^ dp = [v - n" - {x' - x") ~\^ dt 



x' - x" dfji2 „ , , 



+ 1 T,^ndx". 12) 



1 — X dx 



This is a general equation* for the equilibrium between two 



dfJL2 



phases in a binary system. The term r-j, can in general be 



OJu 



evaluated only from experimental data; indeed, the whole of 

 chemical equilibrium is contained in the evaluation of this 

 term. Gibbs has indicated the form it takes for dilute solutions, 

 and has shown that it is necessarilyt positive for stable phases. 

 13. Derivation of an Equation Applying to the Solubility 

 (t-x) Curve. Equation (12) can be written in the form 



x' — x" du.2 



Av^' dp = AV^ dt + j^^ ^, dx", (13) 



* This equation can be derived in a number of different ways; the 

 introduction of equation (1) [97] is not necessary nor is it the most 

 convenient way. It is used here as being more in harmony with the 

 general mode of treatment. Cf. E. D. Williamson and G. W. Morey, 

 J. Am. Chem. Soc, 40, 49 (1918). 



dfX2 



t Gibbs, I, 112. The proof refers to - — but it is easily shown that 



al7l2 



if this is positive — — , must be positive also. 

 dx" 



