SURFACES OF DISCONTINUITY 547 



p - ^^P \R^t dp, 



= '^P Km Tk. 



(18) 



where k is the surface density of the solute, not in Gibbs' sense 

 of an excess, but of the actual amount in the film. If, on the 

 other hand, the surface tension is increased by increasing con- 

 centration of the solute, V is negative or p' > p, and the solution 

 is less concentrated in the surface film than in the bulk of the 

 phase; there is "desorption." Actually in the approximate 

 form of Thomson's relation, viz. (17), a is differentiated with 

 respect to p, the equivalent of the volume concentration in the 

 surface; to make it the exact counterpart of the approximate 

 form of Gibbs' equation it should be 



p;_da_ 

 ^ ~ ~ Rtdp'' 



No doubt under the severe limitations imposed (which we have 

 just referred to) this change is justified, but it is well to notice 

 that in Thomson's actual result the concentration which is the 

 variable on which a depends is the surface concentration. In 

 Gibbs' adsorption law the variable is the chemical potential and 

 it matters not at all whether we refer to the potential at the 

 surface or in the bulk of the phase, since by the equations of 

 equilibrium they are equal; when we approximate we naturally 

 use the approximation for the potential in terms of the bulk 

 concentration. This indeed will serve as a cue to raise a small 

 point which, as the writer knows from experience, occasionally 

 causes some perplexity. The surface tension is of course 

 measured at the surface and we cannot help feeling that it should 

 be directly dependent on the concentration there. When one 

 sees the expression da/dc it is not altogether unpardonable to 

 feel somehow that in this differential coefficient a is the surface 

 tension at the surface of a hypothetical solution in which there 

 is no concentration at the surface. Any such idea must be 

 carefully avoided. Such a condition would of course be physi- 

 cally unrealizable, and the conception is entirely valueless. To 



