156 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



will also be desirable to consider more rigorously and more in detail 

 the equilibrium of such a gas-mixture with solids and liquids, with 

 respect to the above rule. 



By differentiation and comparison with (98) we obtain 



flj-ci-<h ci 



= e ai t a i e 

 v 



g a - c g - 03 .2 



Z = e "a t * e 

 v 



(275) 



etc. 



Equations (^75) indicate that the relation between the temperature, 

 the density of any component, and the potential for that component, is 

 not affected by the presence of the other components. They may 

 also be written 



etc. 



Eliminating fa, // 2 , etc. from (273) and (274) by means of (275) and 

 (276), we obtain 



(277) 



v 



ri = 2j_ ( m x H 1 + m^ log 1 4- m^ log - - ). (278) 



\ m 1 / 



Equation (277) expresses the familiar principle that the pressure in a 

 gas-mixture is equal to the sum of the pressures which the component 

 gases would possess if existing separately with the same volume at 

 the same temperature. Equation (278) expresses a similar principle 

 in regard to the entropy of the gas-mixture. 



From (276) and (277) we may easily obtain the fundamental equa- 

 tion between \fs, t, v, m 1} m 2 , etc. For by substituting in (94) the 

 values of p, yu 1 , jm 2 , etc. taken from these equations, we obtain 



ii (c 1 -H 1 -c 1 \ogt+a l log^ 1 ) V (279) 



If we regard the proportion of the various components as constant, 

 this equation may be simplified by writing 



m for 2 1 m 1 , 



cm for S 1 (c 1 m 1 ), 



wm for Z 1 (a 1 'm/ 1 ), 



Em for 



and Hm am log m for 



