100 EQUILIBKIUM OF HETEROGENEOUS SUBSTANCES. 



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 minimum 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 n } 

 and illustrated by the boiling temperatures and pressures of saturated 

 solutions of n 2 different solids in solvents having two independently 

 variable components. 



Internal Stability of Homogeneous 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 S v 8 2> ...S n be the 

 ultimate components of the fluid ; we may then consider every body 

 which can be formed out of the fluid to be composed of 8 lt $ 2 , . . . S n , 

 and that in only one way. Let m p m 2 , . . . m n denote the quantities of 

 these substances in any such body, and let e, 77, v, denote its energy, 

 entropy, and volume. The fundamental equation for compounds of 

 8 V S 2 , ... S n , if completely determined, will give us all possible sets of 

 simultaneous values of these variables for homogeneous bodies. 



Now, if it is possible to assign such values to the constants T, P, 

 M v M 2 , ...M n that the value of the expression 



e - Tr\ + Pv - M 1 m 1 - M 2 m 2 . . , - M n m n (133) 



shall be zero for the given fluid, arid shall be positive for every other 

 phase of the same components, i.e., for every homogeneous body* 

 not identical in nature and state with the given fluid (but composed 

 entirely of S lf $ 2 , . . . S n ), the condition of the given fluid will be 

 stable. 



For, in any condition whatever of the given mass, whether or not 

 homogeneous, or fluid, if the value of the expression (133) is not 



* A vacuum is throughout this discussion to be regarded as a limiting case of an 

 extremely rarified body. We may thus avoid the necessity of the specific mention of 

 a vacuum in propositions of this kind. 



