168 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



different kinds of gas, and by V as before the total volume, the 

 increase of entropy may be written in the form 



^(ra^j) log F S 1 (m 1 a 1 log vj. 



And if we denote by r l} r z , etc., the numbers of the molecules of the 

 several different kinds of gas, we shall have 



rj = Cm^ , r 2 = (7ra 2 a 2 , etc., 

 where G denotes a constant. Hence 



v l : V : : m^ : ^(m^) : : r x : 2^ ; 

 and the increase of entropy may be written 



2^*1 log Siri-Sifo log rj 

 ~~C~ 



The Phases of Dissipated Energy of an Ideal Gas-mixture with 

 Components which are Chemically Related. 



We will now pass to the consideration of the phases of dissipated 

 energy (see page 140) of an ideal gas-mixture, in which the number 

 of the proximate components exceeds that of the ultimate. 



Let us first suppose that an ideal gas-mixture has for proximate 

 components the gases G I} 6r 2 , and 6r 3 , the units of which are denoted 

 by ,, o, ($o, and that in ultimate analysis 



tr A * ' O ' t/ 



'" " . ."; . , 3 = X 1 1 +X 2 2 , . (299) 



\! and X 2 denoting positive constants, such that X 1 + X 2 = 1. . The 

 phases which we are to consider are those for which the energy of 

 the gas-mixture is a minimum for constant entropy and volume and 

 constant quantities of G 1 and 6r 2 , as determined in ultimate analysis. 

 For such phases, by (86), . ... 



fa Sm 1 + fa Sm z + JUL B Sm 3 ^ (300) 



for such values of the variations as do not affect the quantities of 

 G 1 and 6r 2 as determined in ultimate analysis. Values of Sm 1} <5m 2 , 

 (Sm 3 proportional to X 1? X 2 , 1, and only such, are evidently consistent 

 with this restriction : therefore 



X 1 /z 1 +X 2 ^ 2 = ^ 3 . (301) 



If we substitute in this equation values of fa, fi 2) /* 3 taken from 

 (276), we obtain, after arranging the terms and dividing by t, 



^ 1 m i i x 1 m <> l m s A , ri /o^n\ 



\ a i^-^+^2 l ^~-^^S-^ *=A+Blogt-j, (302) 

 where 



(303) 



(304) 

 (305) 



