280 MOREY 



ART. G 



30. The Order of p-t Curves around an Invariant Point. 

 In the general consideration of phase equihbria it is convenient 

 to proceed from a consideration of the invariant points to the 

 various univarlant equihbria which proceed therefrom, and to 

 consider the sequence of the p-t curves around the invariant 

 point. Such a course is often of great value in determining the 

 stable phases in an investigation of complex systems. The 

 order* of the p-t curves may be deduced from the theorem that 

 whenever a linear relation exists between n of the n -f 1 phases 

 in a univariant equilibrium, the p-t curves of all the univariant 

 systems containing these phases coincide. But these curves 

 extend in both directions from the invariant point ; in one direc- 

 tion the equilibrium under consideration will be stable, in the 

 other, metastable, and to tell the actual position of any curve, 

 or to distinguish between the stable and metastable portions of 

 any one curve, a knowledge of the entropy and volume changes 

 is necessary. However, it will be shown that two adjoining 

 curves, i.e., curves that are not separated by either the stable 

 or metastable portions of other curves, e.g., the p-t curves of 

 the univariant ternary equilibria, P' + P" + P'" + P^^ and 

 pi _|_ pii _|_ pni _|_ pv ^ ^^jj coincide in their stable portions, that 

 is, are stable in the same direction from the invariant point, 

 when the phases P^^ and P^ lie on opposite sides of the straight 

 hne P'P"P"', and vice versa. With the aid of these theorems 

 and general considerations to be discussed later the actual 

 position of the p-t curves may be fixed within certain limits. 



The above theorem may be proved as follows. From the 

 definition of the chemical potential n, if the ^i of a substance 

 in a given phase is greater than the n of the same substance in 

 another phase, the two phases are not in equilibrium with 

 respect to that substance and it will tend to pass from the 

 phase in which its chemical potential is the greater into that 

 phase in which its chemical potential is the less. At the triple 

 point, ice + water + vapor in the one-component system, 



* By "the order of the p-i curves" is meant the sequence in which we 

 shall cut the curves as we circle around the invariant point, with the 

 stipulation that reversing the direction of rotation reverses the sequence 

 but not the order. 



