EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 63 



position and state of this part. (By homogeneous is meant that the 

 part in question is uniform throughout, not only in chemical com- 

 position, but also in physical state.) If we consider the amount and 

 kind of matter in this homogeneous mass as fixed, its energy e is a 

 function of its entropy rj, and its volume v, and the differentials of 

 these quantities are subject to the relation 



de tdripdv, (11) 



t denoting the (absolute) temperature of the mass, and p its pressure. 

 For t dtj is the heat received, and p dv the work done, by the mass 

 during its change of state. But if we consider the matter in the 

 mass as variable, and write m^ m 2 , . . . m n for the quantities of the 

 various substances S lt S 2 , ... S n of which the mass is composed, e will 

 evidently be a function of rj, v, tn lt ra 2 , . . . m n , and we shall have for 

 the complete value of the differential of 



de = tdt] p dv + fadm^ + fJL 2 dm 2 . . . + p n dm n , (12) 



fjL lt /z 2 , ... fJL n denoting the differential coefficients of e taken with 

 respect to m^ w 2 , . . . m H . 



The substances S l} 8* . . . S n , of which we consider the mass com- 

 posed, must of course be such that the values of the differentials 

 doll, dm 2 ,...dm n shall be independent, and shall express every 

 possible variation in the composition of the homogeneous mass con- 

 sidered, including those produced by the absorption of substances 

 different from any initially present. It may therefore be necessary 

 to have terms in the equation relating to component substances 

 which do not initially occur in the homogeneous mass considered, 

 provided, of course, that these substances, or their components, are 

 to be found in some part of the whole given mass. 



If the conditions mentioned are satisfied, the choice of the sub- 

 stances which we are to regard as the components of the mass con- 

 sidered, may be determined entirely by convenience, and independently 

 of any theory in regard to the internal constitution of the mass. The 

 number of components will sometimes be greater, and sometimes 

 less, than the number of chemical elements present. For example, 

 in considering the equilibrium in a vessel containing water and free 

 hydrogen and oxygen, we should be obliged to recognize three com- 

 ponents in the gaseous part. But in considering the equilibrium of 

 dilute sulphuric acid with the vapor which it yields, we should have 

 only two components to consider in the liquid mass, sulphuric acid 

 (anhydrous, or of any particular degree of concentration) and (addi- 

 tional) water. If, however, we are considering sulphuric acid in a 

 state of maximum concentration in connection with substances which 

 might possibly afford water to the acid, it must be noticed that the 

 condition of the independence of the differentials will require that we 



