96 PHENOMENA, ATOMS, AND MOLECULES 



part of the surface of the water. The spreading force F (in dynes per cm.) 

 exerted by the film is measured by the mechanical force applied to the 

 piston. On one side of' the piston is a surface of water which exerts a 

 force yw tending to cause the film to expand and on the other side is the 

 expanded film whose upper surface exerts a force Yb while the lower 

 surface exerts a force which we may represent by y^. Thus for equilibrium 

 we have 



Yir— F = Y« + Yi 



If the molecules in the film did not have any active groups, the surface 

 tension y^ would be equal to the normal interfacial energy Ybtt- But the 

 active groups in the interface because of their thermal agitation will tend 

 to act hke a two-dimensional gas. When these active groups or heads are 

 far enough apart they will exert a force F^ following the gas law 



F, = iikT or F,a = kT (52) 



where a is the area per molecule. 



When the molecules are packed as closely as in the ordinary expanded 

 film, we should, by the analogy with the b term in the van der Waals 

 equation, write 



F,{a — a,)=kT (53) 



where Oo is the area per molecule for a highly compressed film. 



Furthermore the analogy with the equation of state for gases would 

 suggest that when the film is compressed sufficiently, attractive forces 

 between the active groups would come into play and that these might be 

 largely responsible for the small spreading forces observed with some con- 

 tracted films. In any case, however, we may place 



Yi= Y«"- —^^ (54) 



and thus from Equation (51) we get 



F=:y,y — yu — yR^y + Ff. (55) 



For very low compressive forces we may neglect the attractive forces 

 between the heads and thus by combining Equations (55) and (53) we get 



(F-Fo){a-a,)=kT (56) 



as our equation of state for expanded films. Here F© is used as an abbrevia- 

 tion for the three y terms in (55) so that 



Fo = Y'v— Y« — Y^-- (57) 



