THEORIES OF DYEING 329 



This equation is a purely empirical one and has no 

 theoretical basis although it has been found to fit the facts 

 with a fair degree of accuracy ; moreover, it shows that many 

 types of equilibria of very different nature can be described 

 by similar formulae. Willard Gibbs, however, by thermo- 

 dynamical methods deduced another more comprehensive 

 formula to describe adsorption phenomena; if r is the adsorp- 

 tion coefficient, i.e. the mass of solute adsorbed per unit area 

 of surface, c the concentration of solute in the mass of solution, 

 and a the surface energy per unit of surface, then 



n r\ rr 



r= - 



R T ' d c • 



This formula contains the differential coefficient ol the 



function connecting surface energy and concentration (i.e., -r— ), 



and this will be positive if both change in the same sense, but 

 negative if they change in opposite senses. This, taken in con- 

 junction with the minus sign on the right-hand side of the 

 equation, shows that, if the surface energy increases with 

 increasing concentration, there will be a diminished concentra- 

 tion in the surface, i.e. a negative adsorption. On the other 

 hand, if, as is commonly the case, the surface energy decreases 

 with increasing concentration, there will be an increased con- 

 centration in the surface, i.e. ordinary positive adsorption will 

 occur. 



This equation was investigated experimentally by Lewis in 

 1908, and applied by him with some ingenuity and success to 

 the equilibrium involved in dyeing. He determined the altera- 

 tion of surface tension at the interface between two immiscible 

 liquids, this alteration being due to the surface adsorption by 

 one liquid of a solute contained in another. It was found that 

 in all cases the experimental value of r was many times — any- 

 thing up to 80 — the theoretical value. This discrepancy between 

 the theoretical and observed values of r has been confirmed by 

 other workers, and so may be taken as well established ; but 

 nevertheless it does not affect the value of the theory for com- 

 parative purposes. In applying the theory to dyes it is evident 

 that if the theory holds, the dyestuffs in solution should possess 

 the property of lowering the interfacial tension. This, of course, 

 cannot be determined in ordinary dyeing owing to the solid 



