ORIGIN OF THE SMALL BUBBLES OF FROTH. 213 



d(fiT)/dT = - da-jdY) 

 changing the variable this becomes, 



r = — dvjdfA 

 which is the expression given by Willard Gibbs. For dilute 

 solutions dfi = Bddc/c, where R is the gas constant, the 

 absolute temperature, and c the concentration in the bulk 

 of the solution, so at the absolute temperature 0. 



c d ( _ a ds 



R6 dc V dd 



similar units being used throughout. 



This expression contains an additional term to those 

 which appear in the formula given by other writers. 1 The 

 value of the extra term, 'though quite capable of experi- 

 mental determination, has not yet in any case been found, 

 its importance, therefore, in connection with the calculation 

 of the surface excess of solute, remains to be seen. 



For dilute solutions the heat change associated with 

 adsorption, under isothermal conditions at the absolute 

 temperature is, per unit area of surface, 



(ds/dO - dsJdO), 

 where s is the surface tension of the solvent and s c that of 

 the solution. 



Owing to the variation of the value of the additional 

 molecular restraint within the surface stratum, what is 

 here established, by the movement of the molecules of the 

 solute just considered, is a concentration gradient; travel- 

 ling from the interior of the solution, the concentration will 

 be constant up to a point at a depth beneath the surface 

 equal to the range of molecular influence; it will here begin 

 to increase, attaining a maximum value at the surface. On 

 the other hand, with a solution in which, with increasing 

 concentration, the surface tension becomes greater, the 



1 See Milner, Phil. Mag., Vol. xm, p. 100, 1907; 

 and Lewis, Phil. Mag., Vol. xv, p. 499, 1908, and Vol. xvn, p. 466, 1909. 



