SURFACE CHEMISTRY 45 



For very high or very low surface concentrations, Gibbs' equation 

 takes simple limiting forms. At very low concentrations the molecules in 

 the adsorbed films are so far apart that they exert no appreciable forces on 

 one another. Under such conditions o, the surface concentration, will be 

 proportional to the volume concentration which in turn is proportional to 

 p. If we solve Eq. (i) on the assumption that o is proportional to p, we 

 find that it leads to the following equation of state 



F = o-kT , (6) 



which is the 2-dimensional analogue of Eq. (3), the equation of state of 

 an ideal gas, and it may therefore be called an equation of state of an 

 ideal 2-dimensional gas. 



The effect of forces acting between the adsorbed molecules is to modify 

 this equation. A 2-dimensional analogue of Van der Waals' equation 

 can be written in the form 



■r^ <^kT „ . . 



(1— ff/rii) ^^^ 



where a and Oi are constants. 



When the gas phase or the liquid phase contains a relatively high con- 

 centration, the concentration of the molecules in the adsorbed film tends 

 to increase to a limiting value Oi corresponding to a complete monomo- 

 lecular adsorbed film such as those that we observed in the study of oil 

 films on water. Although such films show surface elasticity, that is, they 

 are compressible, the forces required to compress the liquid and solid 

 films are so much greater than those required to compress the gaseous 

 film, that the surface compressibility can be neglected as the first approxi- 

 mation, so that we can consider Oi to be approximately constant, that is, 

 independent of p and F. Making this substitution in Eq. (i) and integrat- 

 ing, we obtain the following equation which should be applicable for con- 

 centrated surface films 



F = (7ikTln(p/po). (8) 



These two limiting equations are completely in accord with the general 

 relationships found by J. Traube (23). In Eq. (8) po is an integration 

 constant whose value cannot be found from Gibbs' equation. 



From the Boltzmann equation we can estimate the energy change A 

 involved in bringing a molecule from the vapor or liquid phase to the sur- 

 face phase. We thus have 



(j/p = const exp (A/kT) . (9) 



Traube found that with molecules of aliphatic compounds having 

 different lengths of hydrocarbon chains, the ratio F/p for dilute solutions 



