6 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Integrating, 



/'7> C ' '/ -1/ 



where li', ia the integration constant whose value depends on the value 

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

 tion i 1). we obtain for ,d\, the free energy, per gram-molecule, of a 

 simple Bubstance in any given Btate, 



9t = --B^ln» x - Tf T %dT+% T-\- Kx4 N, (5) 



./ if 7 



and .Vis, like M } 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 tlic most general case, a Bystem composed of any num- 

 ber, //>! , of different molecular constituents,* and any amount. //, . //', , 

 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 ?i.,, n'.,, etc. gram-molecules of each 

 constituent. 



If&i, ~l'i . etc. represent the free energies per gram-molecule of the 

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



be gi\ en by the equation, 



A = (n, »i + n\ Zl\ + ...)- (»., 3, + n' a 3', + . . . I . 



If we substitute for 11, , ZT, . etc, their values as in equation i 5 ), the N 

 terms will all vanish and the equation will become, 



A ■=■ — R V < «j In i\ + n\ In r\ + ... — w 2 In r., — it'.. In i/ a — • • • ) 



— T f J ?h r " + Hl C ' vx + ' * ' " ~ " n - c„, — n'n c' VJ — . . . T 



t'T 



+ ^(«i?t?i + n\ $',+ ...- i,,% } , - ,,', 1,' 2 -...)+ (». ©, 



+ n\ Zl\ + . . ■- n, IX, — i/., (K's -...). 



■ 



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

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

 BtitUentS, SI are two different molecular species in the same phase. 



