EQUILIBRIUM QF HETEROGENEOUS SUBSTANCES. 



65 



for all variations which do not conflict with the equations of condi- 

 tion. These equations must express that the entropy of the whole 



given mass does not vary, nor itejyojljnig^or the total quantities oT 



any of the substances $,, &,, ... S n . We will suppose that there are 

 no other equations of condition. It will then be necessary for 

 equilibrium that 



-p'W +yM/($m 1 / +/z 2 / (Sm 2 / ... +/z n '<$m n ' 



... +fJL n "Sm n " 



for any values of the variations for which 



f" + etc. = 0, 



^ + etc. = 0, ' 

 '" + etc. = 0, 



(15) 



(16) 

 (17) 



' -' 



Sm n f + Sm n " + Sm n '" + etc. = 0. \ 

 For this it is evidently necessary and sufficient that 



/ =3," =2,'" = etc. 



(19) 

 (20) 



(21) 



Equations (19) and (20) express the conditions of thermal and 

 mechanical equilibrium, viz., that the temperature and the pressure 

 must be constant throughout the whole mass. In equations (21) we 

 have the conditions characteristic of chemical equilibrium. If we 

 call a quantity JUL X) as defined by such an equation as (12), the potential 

 for the substance S x in the homogeneous mass considered, these con- 

 ditions may be expressed as follows : 



The potential for each component substance must be constant 

 throughout the whole mass. 



It will be remembered that we have supposed that there is no 

 restriction upon the freedom of motion or combination of the com- 

 ponent substances, and that each is an actual component of all parts 

 of the given mass. 



The state of the whole mass will be completely determined (if we 

 regard as immaterial the position and form of the various homoge- 

 neous parts of which it is composed), when the values are determined 



of the quantities of which the variations occur in (15). The number 

 G.I. E 



