THERMODYNAMIC AL SYSTEM OF GIBBS 95 



if iJLi = Hi" = Hi", etc. But if ni" were greater than hi, hi'", 

 etc., there would be variations of the state of the system (if 

 Hi" is positive, those for which 8mi" is positive) which satisfy 

 (71), but for which 



Hi8mi' + Hi'^mi" + Hi"5mi"' + etc. > 0. 



But since the quantities Snii, 8mi", etc., may be both positive 

 and negative, there are similar variations in which all these 

 quantities have the opposite sign and for which 



Hi8mi' + Hi'^rrii" + Hi"^mi"' + etc. < 0. 



The same considerations apply to the other sets of terms of the 

 types thy], p8v, H^8m2, etc., so that we may conclude that if (64) 

 holds for all possible variations which satisfy (65), (66) and (67), 

 the equalities (68), (69) and (70) must be satisfied. 



Equations (68) and (69) express the conditions of thermal and 

 mechanical equilibrium, viz., that the temperature and pressure 

 must be constant throughout the system. Equations (70), 

 which state that the value of h for every component must be 

 constant throughout the system, are "the conditions character- 

 istic of chemical equilibrium." Gibbs calls the quantities 

 Hi, H2, etc., the potentials of the substances Si, Si, etc., and ex- 

 presses the conditions (70) in the following statement: "The 

 potential for each component substance must he constant throughout 

 the whole mass." 



We will now consider the case in which one or more of the 

 substances Si, S2,-.. Sn are only possible components of some 

 parts of the system. Let S2 be a possible component of that 

 part of the system distinguished by ("). Then 8mi" cannot 

 have a negative value, so that equation (64) does not require 

 that H2" shall be equal to the value of H2 for those parts of the 

 system of which S2 is an actual component, but only that it 

 shall not be less than that value. For if H2" were greater than 

 Ma'i Hi"', etc., the sum of the terms 



fii'Snh' + iJ,2"8nh" + iX2"'8m2"' + etc. 



would be positive if 8m2" were positive, but since 8m2" cannot 

 be negative, this expression can never have a negative value. 

 The condition of equilibrium (64) is therefore satisfied. 



