126 BUTLER ART. D 



to the two phases, we have, for a variation of the solution, at 

 constant pressure, 



= n'dt + mi' dm' + nh'dni', (140) 



and for a variation of the soUd phase, at constant pressure, 



= r,"dt + m/'d/xi". (141) 



In order to preserve equiUbrium 



so that if mi = mi", i.e., if we take quantities of the soUd and 

 of the solution which contain equal amounts of *Si, 



W - v')dt = m'dfii'. (142) 



Now, by (136), 



Atv' /mA At , ,^ 



so that, integrating (142), we obtain 



At 

 W - V) ^^ = ^i^) • ^2', (143) 



where At is the change of temperature when the value of m^' 

 increases from zero to its value in the given solution. Thus 

 the lowering of the freezing point is 



At mi' At^ rrii 



- ^ - 7^7' • Mix-. = -Q- • <l^' ("*) 



where 



W - v") t 



Q = 



mi 



is the heat absorbed in the melting of unit weight of the solvent 

 into the solution.! 



* The term m^ — • dt, which vanishes when 7112' = 0, is neglected 

 at 



here. 



t van't Hoff, Z. physikal. Che?n., 1, 481, (1887). 



