280 PROCEEDINGS OF THE AMERICAN ACADEMY. 



This equation states that the relative lowering of the activity of a 

 solvent by the addition of a small quantity of a solute is equal to the 

 number of mols of solute divided by the number of mols of solvent. 



This statement comprises in itself practically all the laws of dilute 

 solutions. Raoult's law is a special but only approximate form of 

 equation XIX, for equation XIX is true of every solution when infi- 

 nitely dilute, but Raoult's law is not true even at infinite dilution, 

 except when the vapor of the solvent is a perfect gas. 



If the solute, Xi, is dissolved, not in a pure solvent, but in a mixture 

 of X2, X3, etc., then for the perfect dilute solution we find in place of 

 equation XIX, 



N2d\n $, + Nsdlni3+ ■ • ■ =- d]S\. XX* 



Some Applications of the Preceding Equations. 



Equations I-XX can be combined in a very great variety of ways' 

 to give important results. A few examples, however, will suffice to 

 show the manner in which these equations may be employed. 



First, as a simple example, we may derive the formula for the lower- 

 ing of the freezing point of a perfect solution. According to equation 

 XIX, the activity of a pure liquid is always lowered by the addition of 

 a solute. If therefore a liquid and solid are together at the freezing 

 point and a solute is added to the liquid, the activity of the latter will 

 become lower than that of the solid, and the solid will melt. On the 

 other hand, if we start again with liquid and solid at the freezing point 

 and lower the temperature, we see from equation VIII that the activity 

 of the solid will decrease faster than that of the liquid and the liquid 

 will disappear. It is obvious, therefore, that by adding a solute to a 

 freezing mixture and at the same time lowering the temperature by a 

 suitable amount, the equilibrium between solid and liquid can be main- 

 tained. The necessary condition for the maintenance of equilibrium 

 is that the activity 1^2 of the solvent X2 in the liquid state remain equal 

 to the activity ^'2 of X2 in the solid state. Hence, 



d\ni'., = d\n.^2' 



Now, assuming that the solid does not dissolve any of the solute, the 

 change in activity of the solid Xo is due merely to change of tempera- 

 ture, and thus from equation VIII, 



din ^'2 = .jrp.2 ^ dT. 



