160 PROCEEDINGS OF THE AMERICAN ACADEMY. 



its tendency to escape is measured by that pressure. If n gram mole- 

 cules of another gas B are now introduced, while the external pressure 

 remains constant, the partial pressure of A is diminished by the fraction 



n 

 of its original value, and its escaping tendency will be diminished 



in the same ratio. Therefore gaseous solutions also obey equation (12), 

 and in this case the explanation is obviously the one which has been 

 given here for solutions in general. 



If now we imagine the external pressure to be produced by the gas A 

 outside, while within the pressure is borne by A and B, and if we imag- 

 ine the outside and inside connected by a membrane permeable to A 

 alone, then the gas A having a lower partial pressure inside will pass in 

 from the outside, and equilibrium will not be established until the partial 

 pressure of A is the same inside and outside ; that is, until the total 

 pressure inside is greater than that outside by the partial pressure of B. 

 This is an exact analogy to the osmotic pressure in solutions, and in fact 

 the same explanation can be given for both.* 



Let us consider a solution and the pure solvent at the same external 

 pressure originally, connected by a semipermeable membrane. The 

 tendency to pass from solution to solvent is less tlian the tendency to 

 pass from the solvent to solution, as we have seen. The solvent will 

 therefore flow into the solution until, in some way, this tendency is bal- 

 anced. Let us suppose that it is balanced by an increase in the external 

 pressure on the solution. Then, since the function ^ of the solvent has 

 been decreased by the thermal pressure of the solute, according to equa- 

 tion (14) it must be restored to its original value by an external pressure 

 equal to that thermal pressure. In other words, the osmotic pressure of 

 a dilute solution must be equal to the partial thermal pressure of the 

 solute and, n RT 



n 



V 



which is the law of van 't Iloff. This may be shown mathematically as 

 follows : — 



By generalizing equation (13) we obtain, as the equation of the 

 solution, when the external pressure upon the solution is variable, 



P-\-dP = (S + dj3+dfS'-a-da. (15) 



* Since this paper was written an exactly similar statement of this analogy 

 between the osmotic pressure and tlie pressure which a mixture of gases would 

 show under the above conditions has been given by Ikeda, Zeit. Phys. Chem., 

 XXXII. 280 (1900). 



