94 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



etc., denote the potentials for these substances in the homogeneous 



mass, 



+ etc. (121) 



To show this, we will suppose the mass considered to be very large. 

 Then, the first member of (121) denotes the increase of the energy of 

 the mass produced by the addition of the matter represented by the 

 first member of (120), and the second member of (121) denotes the 

 increase of energy of the same mass produced by the addition of 

 the matter represented by the second member of (120), the entropy 

 and volume of the mass remaining in each case unchanged. Therefore, 

 as the two members of (120) represent the same matter in kind and 

 quantity, the two members of (121) must be equal. 



But it must be understood that equation (120) is intended to 

 denote equivalence of the substances represented in the mass con- 

 sidered, and not merely chemical identity ; in other words, it is 

 supposed that there are no passive resistances to change in the mass 

 considered which prevent the substances represented by one member 

 of (120) from passing into those represented by the other. For 

 example, in respect to a mixture of vapor of water and free hydrogen 

 and oxygen (at ordinary temperatures), we may not write 



but water is to be treated as an independent substance, and no 

 necessary relation will subsist between the potential for water and 

 the potentials for hydrogen and oxygen. 



The reader will observe that the relations expressed by equations 

 (43) and (51) (which are essentially relations between the potentials 

 for actual components in different parts of a mass in a state of 

 equilibrium) are simply those which by (121) would necessarily 

 subsist between the same potentials in any homogeneous mass con- 

 taining as variable components all the substances to which the 

 potentials relate. 



In the case of a body of invariable composition, the potential for 

 the single component is equal to the value of f for one unit of the 

 body, as appears from the equation 



=/xm, : . (122) 



to which (96) reduces in this case. Therefore, when 7i = l, the funda- 

 mental equation between the quantities in the set (102) (see page 88) 

 and that between the quantities in (103) may be derived either from 

 the other by simple substitution. But, with this single exception, an 

 equation between the quantities in one of the sets (99)-(103) cannot 

 be derived from the equation between the quantities in another of 

 these sets without differentiation. 



