THERMODYNAMIC AL SYSTEM OF GIBBS 125 



by a membrane which is permeable to the solvent, but not to the 

 solute. The difference of pressure on the two sides of the mem- 

 brane is the osmotic pressure of the solution. Let the potentials 

 of S\ and >S2 in the solution at the temperature t and the pressure 

 p' be Hi and ^2', and the potential of *Si in the solvent at the 

 same temperature and pressure y" be /i/'. For equilibrium 

 it is necessary that ^t/ = ni". All variations in the state of the 

 solution must satisfy (56), so that for constant temperature 



dp' = y dni' + ^ dM2'. (137) 



So long as the solution remains in osmotic equilibrium with the 

 solvent in its original state, din' = 0, so that 



Wo' 



rfp' = -7 ■ duL2'. (138) 



V 



By (136)= 



W , , At /W\ 

 ../ • aM2 = ,r (,.) • d[ ^, I, 



hence, integrating (138), we obtain 



At TYli 



Since — • 777^, is the pressure, as calculated by (122), of 



m^ IMi^^'^ gram molecules of a perfect gas in the volume v' and 

 at temperature t, this equation expresses van't Hoff's law of 

 osmotic pressure.! 



(6) Lowering of the Freezing Point. Consider the equilibrium 

 of the solution with a mass of the solid solvent. Applying (56) 



* Strictly, -7- • dix-^ = —7 • r^ — j—r - d —j -{ ;-•——• dp, but the 



V V dinh/v') V V dp 



last term vanishes at infinite dilution. 



t Z. physikal. Chem., 1, 481 (1887). M. Planck also gave a derivation 



of this law, Z. physikal. Chem., 6, 187 (1890). 



