108 BUTLER AUT. D 



which subsist between the different component substances," 

 (that is, the variations of the n + /i + 2 quantities, viz., 

 n -\- h potentials, and temperature and pressure, are subject to 

 r -^ h relations). 



We may illustrate the phase rule by reference to systems 

 containing a single component (w = 1). If there is only one 

 phase, |5 = 2, i.e., the temperature and the pressure may be 

 varied independently. If there are two phases, e.g., liquid and 

 vapor, only one independent variation of phase is possible, so 

 that the temperature and the pressure cannot be varied inde- 

 pendently of each other. A variation of the temperature 

 involves a necessary variation of the pressure, if the two phases 

 are to remain in equilibrium. If there are three phases of the 

 substance, ^^ = 0, i.e., it is impossible to vary either the tem- 

 perature or the pressure while the three phases remain. The 

 conditions under which three phases of the same substance 

 can coexist are thus invariant. Gibbs remarks that "it seems 

 not improbable that in the case of sulphur and some other sub- 

 stances there is more than one triad of coexistent phases" (a 

 prediction which has been verified in numerous cases), "but it is 

 entirely improbable that there are four coexistent phases of any 

 simple substance." 



14. The Relation between Variations of Temperature and 

 Pressure in a Univariant System* According to (96), a system 

 of r = w + 1 coexistent phases has one degree of freedom. The 

 pressure and the temperature cannot therefore be varied inde- 

 pendently and there must be a relation between a variation of 

 the temperature and the consequent change of pressure. 



We will first consider a system of one component in two phases, 

 e.g., liquid and vapor. The variations of each phase must 

 be in accordance with (56), so that we may write 



v' dp' = rj' dt' + m' dfi' ,1 .Q_s 



v"dp" = v"dt" + m"diJL".j ^ ^ 



If the two phases are to remain in equilibrium, 



dp' = dp", dt' = dt", dp' = dtx". 



* Gibbs, I, 97-98. 



