EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



99 



Concerning Cases in which the Number of Coexistent Phases is 



less than 



When n > 1, if the quantities of all the components S 1 ,S 2 ,...S n 

 are proportional in two coexistent phases, the two equations of the 

 form of (127) and (128) relating to these phases will be sufficient 

 for the elimination of the variations of all the potentials. In fact, 

 the condition of the coexistence of the two phases together with the 

 condition of the equality of the n 1 ratios of m/, m 2 ', . . . m n ' with 

 the n 1 ratios of m/', ra 2 ", . . . m n " is sufficient to determine p as a 

 function of t if the fundamental equation is known for each of the 

 phases. The differential equation in this case may be expressed in 

 the form of (130), m' and w" denoting either the quantities of any 

 one of the components or the total quantities of matter in the bodies 

 to which they relate. Equation (131) will also hold true in this case 

 if the total quantity of matter in each of the bodies is unity. But 

 this case differs from the preceding in that the matter which absorbs 

 the heat Q in passing from one state to another, and to which the 

 other letters in the formula relate, although the same in quantity, 

 is not in general the same in kind at different temperatures and 

 pressures. Yet the case will often occur that one of the phases is 

 essentially invariable in composition, especially when it is a crystalline 

 body, and in this case the matter to which the letters in (131) relate 

 will not vary with the temperature and pressure. 



When 7i = 2, two coexistent phases are capable, when the tem- 

 perature is constant, of a single variation in phase. But as (130) 

 will hold true in this case when m 1 ':wi a '::m l ":m|", it follows that 

 for constant temperature the pressure is in general a maximum or 

 a minimum when the composition of the two phases is identical. 

 In like manner, the temperature of the two coexistent phases is in 

 general a maximum or a minimum, for constant pressure, when the 

 composition of the two phases is identical. Hence, the series of 

 simultaneous values of t and p for which the composition of two 

 coexistent phases is identical separates those simultaneous values of 

 t and p for which no coexistent phases are possible from those for 

 which there are two pair of coexistent phases. This may be applied 

 to a liquid having two independently variable components in con- 

 nection with the vapor which it yields, or in connection with any 

 solid which may be formed in it. 



When n = 3, we have for three coexistent phases three equations 

 of the form of (127), from which we may obtain the following, 



v 



m 



v m m 2 

 v'" m/" m 2 ' 



ra x 



ra^' 77i, 

 m/" m 



1 3 



// 



// 



2 m 3 



/// /// 



O lti/f> 



dp* (132) 



