Gaseous monolayers two-dimensional perfect gas

It should be clear that monolayer molecules may either merely rest on the surface without appreciably penetrating it or be almost wholly immersed in the top surface molecules of the underlying material. Adsorption on solids belongs to the second type where localized monolayers are formed (see Section 8.4). 

For the first type, the Gibbs monolayers are often described as two-dimensional gaseous (or gas-like) monolayers formed on the surface of dilute solutions, having a low surface excess, and the surface pressure, n, is regarded as the two-dimensional osmotic pressure of this solution. When a two-dimensional gaseous monolayer is present on a dilute solution, the surface tension decreases linearly with the increase in concentration of the added surfactant, at constant temperature 

If we combine the (d]4dc)r = -m expression and Equation (432) with the Gibbs adsorption isotherm (Equation (229)), we have 

However, there are conceptual differences between the surface equation of state and the adsorption isotherm, so that the surface equation of state is only concerned with the lateral motions of the monolayer molecules and their lateral cohesive and adhesive interactions with the solvent molecules present in the monolayer, whereas an adsorption isotherm is also concerned with the interactions normal to the surface, between the monolayer molecules (as adsorbate) and solvent molecules (as adsorbent). 

Since the surface excess is the inverse of the area available per mole of solute, A ie. in the monolayer, [r = 1M4o1J, then from Equation (433) we have 

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