224 J. W. Gibbs — Equilibrium of Heterogeneous ISubstances. 

 Therefore, 



\dmjt,2], m i dp Y*^^ 

 \dm2/t, V, 1 



It will be observed that the first member of this equation relates 

 solely to the liquid or solid, and the second member solely to the 

 gas. Now we may suppose the same gaseous mass to be capable of 

 equilibrium with several diiferent liquids or solids, and the first mem- 

 ber of this equation must therefore have the same value for all such 

 liquids or solids ; which is quite inadmissible. In the simplest case, in 

 which the liquid or solid is identical in substance with the vapor 

 which it yields, it is evident that the expression in question denotes 

 the reciprocal of the density of the solid or liquid. Hence, when a 

 gas is in equilibrium with one of its components both in the solid 

 and liquid states (as when a moist gas is in equilibrium with ice and 

 water), it would be necessary that the solid and liquid should have 

 the same density. 



The foregoing considerations appear sufiicient to justify the defini- 

 tion of an ideal gas-mixture which we have chosen. It is of course 

 immaterial whether we regard the definition as expressed by equation 

 (273), or by (279), or by any other fundamental equation which can 

 be derived from these. 



The fundamental equations for an ideal gas-mixture corresponding 

 to (255), (265), and (271) may easily be derived from these equations 

 by using inversely the substitutions given on page 217. They are 



^,(c. m,) log '-^^£^=r,-\.2, {a,m, log^-^,^J, (291) 

 = V^^.{-.>n^^og^-^^-B,m,Y (292) 



- 2^ {c,m,+a, m,) t log t +^\ [ct, m, t log ^h^^^^). (293) 



The components to which the fundamental equations (273), (279), 

 (291) (292), 293) refer, may themselves be gas-mixtures. We may 

 for example apply the fundamental equations of a binary gas-mixture 



