176 SCIENCE PROGRESS. 



each of these properties and vapour-pressure. Since at the 

 freezing-point a solution has the same vapour-pressure as 

 the solid solvent, solutions in the same solvent having the 

 same freezing-point must have the same vapour-pressure. 

 But solutions which at a given temperature have the same 

 vapour-pressure have been shown to have the same osmotic 

 pressure, consequently solutions having the same freezing- 

 point must have the same osmotic pressure. From the 

 fact that Raoult's determinations have shown that solutions 

 having the same freezing-point have the same molecular 

 concentration, we have another mode of proving that Avo- 

 gadro's hypothesis is applicable to dilute solutions, or that 

 isotonic solutions contain per unit volume the same number 

 of dissolved molecules. 



In order to ascertain how freezing-point observations 

 accord with the general equation PV = '08 19 T, a quantita- 

 tive connection has to be established between the lowering 

 of the vapour-pressure and the lowering of the freezing- 

 point. Such a connection was deduced as early as 1870, by 

 Guldberg, who started with the well-known equations 

 expressing the heats of vaporisation of the solid and the 

 liquid solvents in terms of the pressure, temperature, and 

 volume of the vapour to which each gives rise. Assuming 

 the vapour to obey gaseous laws, Guldberg's relationship 

 may be formulated as 



log e ///=MW/2 x (i/T— i/T) (2). 



Here/ is the vapour-pressure of the liquid solvent, and 

 p' that of the solid solvent, and therefore of the solution, at 

 T', the absolute freezing-point of the solution. T is 

 the absolute freezing-point of the pure solvent. W is 

 the difference between the heats of vaporisation of 1 gr. of 

 solid solvent and 1 gr. of liquid solvent, and is thus the 

 heat of fusion of 1 gr. of solid solvent. If we now substitute 

 for \og e p/p', its value at T" as given by the vapour-pressure 

 equation (6) (vol. i., p. 414), we obtain on writing AT for T-T' 



P = -0819 /0WAT/2T* (3). 



* In this equation W/2T is assumed to be the same at the freezing- 

 points of the solvent and the solution. Arrhenius has shown that if the 



