Surface Concentration, and Formation of Liquid Films. ( M 



surface being kept constant is —pdv. If these two rever- 

 sible operations be performed successively, the final result 

 is independent of the order in which they are performed, and 

 by the second law of thermodynamics the same must be the 

 case of the work done in reaching it. Consequently 



rds 



and therefore 



s ~ \ p + £ ds ). dr = ~ pdv + ( r + [{ v dv ) Chi 



(h _ _ dp 



dv ds ' 



(i) 



(1) shows that the surface tension will vary with the volume 

 of the solution (i, e., with the concentration) only when the 

 osmotic pressure depends on the surface. In order that 

 this should be the case, we must suppose that in the thin 

 surface-film which is the seat of the capillary forces, the con- 

 centration of the solute is different from what it is in the 

 interior of the solution. In the case of an excess of con- 

 centration in the surface, any increase in the area will result 

 in drawing away from the interior a certain amount of solute 

 which was previously operative in producing osmotic pressure; 

 with a defect the opposite will be the case. The actual con- 

 centration in the surface-film is indeterminate, in the absence 

 of any knowledge as to how it varies with the distance away 

 from the surface, but the whole excess of solute associated with 

 the surface may readily be calculated from equation (1). Let 

 a be the " surface excess," i. e. the number of grain-molecules 

 of the solute associated with each sq. cm. of the surface which 

 are drawn out from the interior and made ineffective on the 

 osmotic pressure. If N is the number of gram-molecules 

 of the solute originally dissolved, the concentration in the 

 interior, on which variable alone both the surface tension 

 and the osmotic pressure depend, wiil be 



-^ (2) 



Changing to the concentration as the variable by (2), we 

 have 



dr _ dr dc _ _ c dr 

 dv dc dv v dc 



and dp _ dp dc _ _a dp 



ds dc ds v dc 



Phil. Mag. S. 6. Vol. 13. No. 73. Jan. 1907 H 



