124 BUTLER 



ART. D 



does not hold in every case in which the amount of the component 

 is very small, provided that the proper value of the molecular 

 weight in the solution is employed. The difficulty arises here 

 that there is no independent method by which the molecular 

 weights in solution can be determined. The general validity of 

 (135) is based on the fact that it has been found to hold in a very 

 large number of cases in which M-/^'' is given the value to be 

 expected for simple molecules according to the chemical formula. 

 The cumulative effect of this evidence is so strong that in doubt- 

 ful cases the value of the molecular weight in solution may be 

 determined from (135) itself. 



In deducing the limiting law of the variation of the potential 

 of a solute with its concentration we have considered a solute 

 having an appreciable vapor pressure. But there is no reason 

 to suppose that the behavior of involatile solutes is different 

 in this respect and we may regard (135) as generally applicable 

 to all components, the quantities of which cannot be negative 

 and which are present in very small amounts, provided that the 

 proper values of the molecular weights are used. 



IS. Equilibria Involving Dilute Solutions. In the last chapter 

 of the first volume of the Collected Works (Gibbs I, Chap. IX) 

 is printed a fragmentary manuscript of a proposed supplement 

 to The Equilibrium of Heterogeneous Substances, in which Gibbs 

 shows that the laws of dilute solutions obtained by van't Hoff 

 from his law of osmotic pressure can be derived by making use of 

 equation (135) for the potential of a solute. It will be of interest 

 to give these demonstrations as examples of the application of the 

 method of Gibbs to specific cases. We will consider a dilute 

 solution formed by dissolving a small quantity, m2 grams, of a 

 solute aS'2, in Wi grams of a solvent Si. The molecular weight 

 of the solute in the solution is ilf2^^\ We will assume that the 

 potential of S2 in the solution is given by (135), so that under 

 these conditions, at constant temperature and pressure 



At v_ 



^M2 = ^) • ± • d(^y (136) 



(a) Osmotic Pressure. Suppose that this solution is separated 

 from a quantity of the pure solvent at the same temperature 



