HETEROGENEOUS EQUILIBRIUM 235 



the given energy and volume. The concept of phase, and the 

 derivation of the Phase Rule, result from the appUcation of 

 equation (1) [97] to the consideration of "the different homo- 

 geneous bodies which can be formed out of any set of component 

 substances." "It will be convenient to have a term which 

 shall refer solely to the composition and thermodynamic state of 

 any such body without regard to its quantity or form. We 

 may call such bodies as differ in composition or state different 

 phases of the matter considered, regarding all bodies which 

 differ only in quantity and form as different examples of the 

 same phase. Phases which can exist together, the dividing 

 surfaces being plane, in an equilibrium which does not depend 

 on passive resistances to change, we shall call coexistent. 



"If a homogeneous body has n independently variable com- 

 ponents, the phase of the body is evidently capable of n + 1 

 independent variations." This follows from the fact that there 

 are n + 2 independent variables, pressure, temperature, and 

 the n quantities yiii, H2, ... Mn connected by an equation of the 

 form of (1) [97]. "A system of r coexistent phases, each of 

 which has the same n independently variable components is 

 capable of n + 2 — r variations of phase," or degrees of freedom, 

 F. "For the temperature, the pressure, and the potentials for 

 the actual* components have the same values in the different 

 phases, and the variations in these quantities are by [97] subject 

 to as many conditions as there are different phases. Therefore, 

 the number of independent variations in the values of these 

 quantities, i.e., the number of independent variations of phase 

 of the system, will be n + 2 — r." 



"Hence, if r = w + 2, no variation in the phases (remaining 

 coexistent) is possible. It does not seem probable that r can 

 ever exceed n -\- 2. An example of w = 1 and r = 3 is seen in 

 the coexistent solid, liquid, and gaseous forms of any substance 

 of invariable composition. It seems not improbable that in 

 the case of sulphur and some other simple substances there is 

 more than one triad of coexistent phases; but it is entirely 



* The distinction between "actual" and "possible" components need 

 not be discussed in this place. See Gibbs, I, 66. 



