282 PROCEEDINGS OF THE AMERICAN ACADEMY. 



the activity of the water is lowered by the same per cent as when 

 1 gram of sugar is added to 1000 grams of pure water. 



An interesting system is one composed of two phases, both of which 

 are mixtures of the same composition. An important example of such 

 a system is a constant boiling mixture and its saturated vapor. Here 

 A^2, ^"^^3) etc., which are the molecular fractions in the one phase, are 

 equal respectively to N'^, N's, etc., in the other phase. If the condi- 

 tions are changed by changing the temperature or pressure or by adding 

 a third substance Xi to one or both of the phases, then equilibrium can 

 only be maintained by keeping the activity of each component the same 

 in both phases ; thus we may write as usual, 



d\ni2 = <^ln ^'2, «^ In ^3 = d\n i's , 



etc. ; but since N2 = N'^, etc., we may write 



iVs^lncfs + A^s^lnfs + • • • = N'.dhxt'^. + A^'s^^ln^^'s +• • • 



Now the first member of this equation represents a change which may 

 be the resultant of the changes produced by change of temperature, 

 change of pressure, and the addition of dNi mols of the solute Xi. 

 Each of these changes is represented alone by equations XVI, XVII, or 

 XX. Therefore, 



( 



P.N ^^J^ 



( 



\ dP J T,N ^^^ 



liT 



dNi = - dNr. 



dNx jT.P 



We may therefore write the sum of these as follows : 



N^d\xxL^N^d\slk^^ • ■ ' ^^-^^^dT+^dP-dN^. 

 Likewise we find 



iV'2^1n^'2 + iV's^lnfs + • • • = ^' ~^f dT + -^dP - dN'„ 



where dN'i is the number of mols of the solute in one mol of the second 

 phase. Equating the second members of these two equations we have, 



