64 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



consider the acid in the state of maximum concentration as one of 

 the components. The quantity of this component will then be cap- 

 able of variation both in the positive and in the negative sense, while 

 the quantity of the other component can increase but cannot decrease 

 below the value 0. 



For brevity's sake, we may call a substance S a an actual component 

 of any homogeneous mass, to denote that the quantity m a of that 

 substance in the given mass may be either increased or diminished 

 (although we may have so chosen the other component substances 

 that m a = 0); and we may call a substance $ & a possible component 

 to denote that it may be combined with, but cannot be subtracted 

 from the homogeneous mass in question. In this case, as we have 

 seen in the above example, we must so choose the component sub- 

 stances that m b = 0. 



The units by which we measure the substances of which we regard 

 the given mass as composed may each be chosen independently. To 

 fix our ideas for the purpose of a general discussion, we may suppose 

 all substances measured by weight or mass. Yet in special cases, it 

 may be more convenient to adopt chemical equivalents as the units 

 of the component substances. 



It may be observed that it is not necessary for the validity of 

 equation (12) that the variations of nature and state of the mass to 

 which the equation refers should be such as do not disturb its homo- 

 geneity, provided that in all parts of the mass the variations of 

 nature and state are infinitely small. For, if this last condition be 

 not violated, an equation like (12) is certainly valid for all the infin- 

 itesimal parts of the (initially) homogeneous mass ; i.e., if we write 

 De, Dq, etc., for the energy, entropy, etc., of any infinitesimal part, 



dDe = t dDrj p dDv + fa dDm l + // 2 dDm 2 ... + /ut n dDm n , (13) 



whence we may derive equation (12) by integrating for the whole 

 initially homogeneous mass. 



We will now suppose that the whole mass is divided into parts so 

 that each part is homogeneous, and consider such variations in the 

 energy of the system as are due to variations in the composition and 

 state of the several parts remaining (at least approximately) homoge- 

 neous, and together occupying the whole space within the envelop. 

 We will at first suppose the case to be such that the component sub- 

 stances are the same for each of the parts, each of the substances 

 $1, $ 2 , . . . S n being an actual component of each part. If we distinguish 

 the letters referring to the different parts by accents, the variation in 

 the energy of the system may be expressed by Se' + <$e" + etc., and the 

 general condition of equilibrium requires that 



" + etc. ^0 (14) 



