﻿Molecular Thermodynamics. 235 



very small. This is essential to the rigour of the demon- 

 stration that the Raoult-van't Hoff laws are still the limiting 

 laws o£ dilute solution when the solvent is complex. 



Now G / is, at constant temperature and pressure, a func- 

 tion of c i, c 02 , . . . . , as well as of e l9 c 2 , Owing to the 



chemical equilibrium controlling coi, r 2, • • • • , liowever, these 

 quantities have, in the pure solvent, values depending only 

 on temperature and pressure, and the departures from these 

 limiting values, caused by the presence of solutes, will clearly 

 decrease with the concentrations of those solutes. Thus as C 

 diminishes, the ranges of variation to be considered of the 

 variables cbi, C02, «... are progressively limited, the same 

 being obvious in the case of c u c 2 , 



But clearly any finite, continuous, ditf'erentiable function 

 of several variables must behave as a linear function if the 

 range of variation considered of every variable is sufficiently 

 limited *. 



Thus 



LtG' = 2c iZoi + 2cA 5 ( 9 ^ 



C-yO 



i. e. 



Lt(M G')=Woi+2W,, • • • • C 10 ^ 



where Z 01 , , ? 1? , depend only on temperature and 



pressure, being, in fact, limiting values of Gr 01 ', , 



G/ , respectively. 



But clearly 7 i» > ^> , can be transferred to, and 



included in the " linear " terms, — in O1 , , fa, re- 

 spectively — whereupon the residual " general " terms will 

 satisfy the requirement that G', G s ', G 01 ', etc., should all 

 vanish as C becomes very small. We shall assume that 

 in (8) this adjustment has already been carried out. 



Returning now to (7) and comparing with (8), we see that 



C>OO iU o 00 L OU 0l J 



= It ["icoiC^oi-R^g"^!] I = 0M ( saY ); ' ( U ) 



since, in the pure solvent, c ol , and 777 assume values 



dependent only on temperature and pressure. 



* Merely the obvious property of tangency in re-dimensions. The 

 theorem quoted by Planck in treating simple solvents ['Thermo- 

 dynamics' (Trans. Ogg), 1917, p. 225] appears to be the particular case 

 of this, when the ranges of variation of the variables are all located (as 

 here in the case of c 1} c 2 , . . . .) close to zero. 



