Gibbs monolayers thermodynamics of adsorption

On the other hand, if the solute is partially or completely immiscible with the subphase solvent, the solute molecules form a layer on the surface of the sub-phase (or in the interface between the two immiscible phases) which is different from the Gibbs mono-layer, and if the layer is monomolecular, it is called a Langmuir insoluble monolayer (see Section 5.6). 

The solute and solvent molecules present in any solution have different intensities of attractive force fields, and also have different molecular volumes and shapes. A concentration difference between the surface region and the bulk solution occurs because the molecules that have the greater fields of force tend to pass into the interior, and those with the smaller force fields remain at the surface. The Gibbs surface layer of a solution is more concentrated in the constituents that have smaller attractive force fields, and thus whose intrinsic surface free energy is smaller than the interior. As we stated in Section 3.3, this concentration difference of one constituent of a solution at the surface is termed adsorption. In qualitative terms, if the solution has a smaller surface tension than its pure solvent, the solute is concentrated in the surface layer indicating a positive adsorption according to 

Gibbs equation. The adsorption process involves the transport of molecules from the bulk solution to the interface, where they form a specially oriented monomolecular layer according to the nature of the two phases. When a Gibbs monolayer forms, it does not necessarily mean that the molecules are touching each other in this monolayer. Instead, if the anchoring from the sub-phase molecules is weak, the molecules may move freely in the two-dimensional interfacial area. Thus, the physically measurable monolayer area is sometimes much larger than the close-packed area where all the molecules touch each other. When any monolayer is fairly well populated with adsorbed molecules, it exerts a lateral spreading (film) pressure, it, which is equal to the depression of the surface tension (see Section 5.5.2). 

For a two-component system, the total number of moles of component (1) can be written (from Equation (194) in section 3.2.5) as 

From the Gibbs convention where Vs = 0 is assumed, we know that V=Va+ V, where V is the total volume of the system (or solution). If we substitute [V = V - V ] into Equation (413), we have 

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