130 BUTLER 



ART. D 



volume for a gram of the component Si. In the same way we 

 may determine the partial energies, entropies and volumes for a 

 gram of the other components. Similarly, since x = e + pr, 

 we have 



Xi = €i + pvi. (155) 



At a given temperature and pressure, the quantities e, -q, v, x 

 are all homogeneous functions of the first degree with respect 

 to Ml, . . . lUn. Therefore, by (52), 



e = mill + rrhh • ■ • + Wne„, (156) 



and, by (54), 



rriidli + nhdh . • . + w„c?e„ = 0, (157) 



and similar equations may be obtained for rj, v and x-* 



The variations of the potentials with pressure and temperature 

 are easily found in terms of these quantities. Thus, by (39), 



\dp/t. m ^* 



so that, differentiating this equation with respect to mi, we have 

 9 /ar\ dv d / d^\ dv 



/af\ ^ ^ or — (—\ 



\dp/ drrii °^ dp \dmi/ 



drrii \dp/ drrii dp \dmi/ drrii 



i.e., expressing the invariant quantities in full, 



\dp/t,m \dmi/ 1. p. m„ etc. 



Similarly, by (39), 



\(ll / p, m 



* The partial molar values of these quantities are obtained by multi- 

 plying the values per gram given here by the molecular weight. Practi- 

 cal methods of evaluating the partial molar quantities have been worked 

 out by G. N. Lewis and collaborators (G. N. Lewis and M. Randall, 

 Thermodynamics and the Free Energy of Chemical Substances, 1923). 



