156 J. W. Gibhs — Equilibrium of Heterogeneous Substances. 



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 compouents in connection 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 rn 

 v" m 

 v'" rn 



dp=i 



dt--\- 



m. 



tn. 



djJi^. (132) 



Now the value of the last of these determinants will be zero, when 

 the composition of one of the three phases is such as can be produced 

 by combining the other two. Hence, the pressure of three coexistent 

 phases will in general be a maximum or minimum for constant tem- 

 perature, and the temperature a maximum or minimvim for constant 

 pressure, when the above condition in regard to the composition of 

 the coexistent phases is satisfied. The series of simultaneous values 

 of t and p for which the condition is satisfied separates those simul- 

 taneous values of t and p for which three coexistent phases are not 

 possible, from those for which there are two triads of coexistent 

 phases. These propositions may be extended to higher values of ;i, 

 and illustrated by the boiling temperatures and pressures of saturated 

 solutions of ?^ — 2 different solids in solvents having two independ- 

 ently variable components. 



INTERNAL STABILITY OF HOM()(iENEOUS FLUIDS AS INDICATED BY 

 FUNDAMENTAL EQUATIONS. 



We will now consider the stability of a fluid enclosed in a rigid 

 envelop which is non-conducting to heat and impermeable to all the 

 components of the fluid. The fluid is supposed initially homogeneous 

 in the sense in which we have before used the word, i. e., uniform in 

 every respect throughout its whole extent. Let <Sj, S.^., , . . >S„ be 

 the ultiiiiate components of the fluid ; we may then consider every 

 body which can be formed out of the fluid to be composed of S^, S2, 

 . . . aS„, and that in only one way. Let m^, m^, . . . m„ denote 

 the quantities of these substances in any such body, and let f, ?/, v, 

 denote its energy, entropy, and volume. The fundamental equation 

 for compounds of iS^, ^.Sg, . . . S„, if completely determined, will give 

 us all possible sets of simultaneous values of these variables for homo- 

 geneous bodies. 



