362 KEYES ART. J 



Comparing the two entropy expressions gives for the final 

 entropy equation 



f dt V — Bimi 



Tji = mi / Ci* — + aiWi log 



"^"^^^ '^^ + m./7. (67) 



(v - Bimi) ai 



The f function ei + piWi — ^771 may now be formed by sub- 

 stituting the energy and entropy, with the result 



f 1 = mi / ci* dt + miEi + aimit + miBipi 



— mii / ^* T ~" ^1^1^ log — — miHit, (68) 



and for a mixture, employing the rule of Gibbs, 



f = 2 f 1 ^ 2 *"'^ / ci* (^f + ^ mi^Ji + 2j ^1^1^ 

 + 2j ^1-^iPi — ^ mit j 





ttii 



— /, miOii log — — 2j 'f^iHit. (69) 



The equations for ^ui, m, ... and Ci, C2, ... can be readily 

 obtained from the last equation by differentiation, i.e., 



/•' r dt 



Ml = / ci* d^ + ^1 + pifii + ait - t ci* J 



ait 

 - ait log — - Hit, (70) 



Pi 



mici = mici* + (^ _ 5^^^)^^^^^,' (71) 



\_dt \ dt J J 

 using (Vila) and neglecting higher terms in the reciprocal of 



