150 J. W. Gibbs — Equilibrmrn of Heterogeneous Substances. 



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 sup- 

 posed 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 neces- 

 sary 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 poten- 

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

 equilibrium) are simply those which by (121) would necessary sub- 

 sist between the same potentials in any homogeneous mass containing 

 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 t, for one unit of the 

 body, as appears from the equation 



1;=: /.nn (122) 



to which (96) reduces in this case. Therefore, when n = \, the fun- 

 damental equation between the quantities in the set (102) (see page 

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

 from the other by simple substitution. But, with this single excep- 

 tion, an eqiiation 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. 



Also in the case of a body of variable composition, when all the 

 quantities of the components except one vanish, the potential for 

 that one will be equal to the value of t, for one unit of the body. 

 We may make this occur for any given composition of the body by 



