Measukement and Calculation 37 



of the freezing-point, elevation of the boiling-point, decrease 

 of the vapor tension of solutions, and osmotic pressure are 

 all related phenomena, and may be obtained one from the 

 other for any given solution.' Three of the most satis- 

 factory methods for determining osmotic pressure in this 

 way will be briefly described here: 



1. The freezing-point method : The freezing-point of a 

 solution is always lower than that of the pure solvent. This 

 depression of the freezing-point is proportional to the num- 

 ber of solute particles present, and therefore to the osmotic 

 pressure. 



The depression of the freezing-point can best be deter- 

 mined by means of Beckmann's apparatus," which may be 

 found described in any of the texts on physical chemistry. 

 A determination of the freezing-point is first made for 

 distilled water ; this is followed by a determination for the 

 solution to be tested, care being taken not to disturb the 

 adjustment of the thermometer between the determinations. 

 The difference between the two observations will be the 

 required depression, which may be denoted by A^. The rela- 

 tion between this quantity and the osmotic pressure is ex- 

 pressed, for aqueous solutions, by the following equation: 



P/= 9173.2 A/ ,» 



wherein Pf is the osmotic pressure at the freezing-point 

 of the solution, measured in millimeters of mercury. 



The osmotic pressure at any desiriKl temperature other 

 than the f roezing-point, say T in the absolute scale, may be 

 obtained by applying the principle of Gay-Lussac, which 



> J. H. van't Hoff, " Die Eollo des osmotisohon Druckos in der Analogie 

 zwisohon LOsuurou und Gasou," Zeitschr.f. physik. Chciu,, Vol. I (1887), pp. 481-508. 



SB. Beokmann, " Ueber dio Methode der Molekulargowiohtsbestimmung duroh 

 Gefrierpunktaornlodriffung," ibid.. Vol. 11 (1888), pp. 638-45. 



a NBBNST-PAIiMffin, Tluorctical Chemistry (London, 1895), p. 132. The pressure 

 here is reduced from atmospheres to millimeters of mercury. 



