230 J. W. Gibbs — Equilibrhim of Heterogeneous ^Substances. 



The Phases of Dissipated Energy of an Ideal Gas-ndxtare loith 

 Components v^hivh are Vhemically Related. 



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

 energy (see page 200) 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 6r,, 6^3, and 6^g, the units of which are 

 denoted by @^, @2, @3, and that in ultimate analysis 



@3 = A,®, +A2@2, (299) 



A, and A2 denoting positive constants, such that Aj + Ag = 1. The 

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

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

 constant quantities of G^ and 6r g, as determined in ultimate analysis. 

 For such phases, by (86), 



/^i 8m ^ 4- //g 6m. ^ + fx^ Sm^^O. (300) 



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

 (tj and 6rg as determined in ultimate analysis. Values of dm^, 

 6ino, (Si)ip^ proportional to A,, A,, — 1, and only such, are evidently 

 consistent with this restriction : therefore 



Aj Ml + Ao ^2 = 1^2- (301) 



If we substitute in this equation values of fi^, /^2? /'a taken from 

 (2*76), we obtain, after arranging the terms and dividing by t, 



^1 «i log V+ '^^ ""■' ^""^ "V ~ "'^ ^""^ "zT ^ -^+ Slog^ — ?, (302) 

 where 



^ = A, JTj + Aa^o — ^3-A,Cj-A2C2-f Cg-Ajffj-Aatta + ^s^ (•'^03) 



^czrAjCj-fA^eo-Cg, (304) 



G~\^E^-\-\„E^— E^. (305) 



If we denote by /^, and fi^ the volumes (determined under stand- 

 ard conditions of temperature and pressure) of the quantities of 

 the gases G ^^ and G^ which are contained in a unit of volume of the 

 gas 6^3, we shall have 



/A = ^'\and /i. = ^-|^, (300) 



and (302) will reduce to the form 



log "^^ T\ a ^ = - + — log ^ - — . (307) 



^\nj' A B. 'C 



m„ V 



3 "3 "s 



