104 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



It may easily be shown that the same must be true in the limiting 

 cases in which T=0 and P = 0. For negative values of P, (133) is 

 always capable of negative values, as its value for a vacuum is Pv. 



For any body of the temperature T and pressure P, the expression 

 (133) may by (91) be reduced to the form 



f Mjn^ - M 2 m 2 . . . - M n m n . (135) 



We have already seen (page 77) that an expression like (133), 

 when T, P, M v M 2 , . . . M n and v have any given finite values, 

 cannot have an infinite negative value as applied to any real body. 

 Hence, in determining whether (133) is capable of a negative value 

 for any phase of the components $ lt $ 2 , . . . S n , and if not, whether it is 

 capable of the value zero for any other phase than that of which the 

 stability is in question, we have only to consider the least value of 

 which it is capable for a constant value of v. Any body giving this 

 value must satisfy the condition that for constant volume 



de-Tdij M l dm l -Mtdmt...-M n dm n ^.O, (136) 



or, if we substitute the value of de taken from equation (86), using 

 subscript a ... g for the quantities relating to the actual components 

 of the body, and subscript h . . . k for those relating to the possible, 



tdr) + /uL a dm a ...+juL g dm g +jUL h dm h ...+/jL k dm k 



-Tdrj-M l dm l -M 2 dm 2 ...-M n dm n ^0. (137) 



That is, the temperature of the body must be equal to T, and the 

 potentials of its components must satisfy the same conditions as if it 

 were in contact and in equilibrium with a body having potentials 

 M lt M 2 , . . . M n . Therefore the same relations must subsist between 

 fj. a . . . fJL g) and M l ... M n as between the units of the corresponding 



substances, so that 



* 



m a jUL a ... + m g /uL g = m l M l ...+m n M n ; (138) 



and as we have by (93) 



e = trj-pv+jui a m a ... -\-fjigmg, (139) 



the expression (133) will reduce (for the body or bodies for which it 

 has the least value per unit of volume) to 



(P-p)v, (HO) 



the value of which will be positive, null, or negative, according as the 



value of 



P-p (141) 



is positive, null, or negative. 



Hence, the conditions in regard to the stability of a fluid of which 

 all the ultimate components are independently variable admit a very 

 simple expression. If the pressure of the fluid is greater than that 



