6 PROCEEDINGS OP THE AMERICAN ACADEMY. 



Integrating, 





where fL^j is the integration constant whose value depends on the value 

 chosen for JJ, as the lower limit of integration. Substituting in equa- 

 tion (4), we obtain for '^i, the free energy, per gram-molecule, of a 

 simple substance in any given state, 



^1 



r.T\nv,- TT'-^dT+^T+m. + N, (5) 



where 



-^x 







and ^is, like 3f, a quantity -which will vanish in any expression for the 

 difference in free energy at constant temperature. 



From this we may obtain an equation for the change of free energy in 

 any process. In the most general case, a system composed of any num- 

 ber, ?/?! , of different molecular constituents,* and any amount, ni, n'l, 

 etc. gram-molecules of each constituent, undergoes any change, phys- 

 ical or chemical, arriving at a condition in which it is composed of 

 m^ molecular constituents, and rio , w'o , etc. gram-molecules of each 

 constituent. 



If 3i, ^'i , etc. represent the free energies per gram-molecule of the 

 various simple molecular constituents, a total change in free energy will 

 be given by the equation, 



A = (n, % + n\ Sl'i + . . . ) _ (n„ %. + n'. Sl'o + ...). 



If we substitute for ^i , '^'i , etc., their values as in equation (5), the N 

 terms will all vanish and the equation will become, 



A = — 11 T(7ii In Vi + n'l ]n v\ + . . . — rio In Vo — n'^ In v'.. — . . . ) 





^'"- c^, + n\ c' -\- . . . —n.c„ — n\ c' — 



dT 



T 



+ n'l m'l + . . . - n. 5Ho — n\^ W ^ -...). 



* Molecular constituent is used to mean a single molecular species in a single 

 phase. Thus two phases of tlie same molecular species are regarded as two con- 

 stituents, as are two different molecular species in the same phase. 



