The Polanyi Treatment

A still different approach to multilayer adsorption considers that there is a potential field at the surface of a solid into which adsorbate molecules fall. The adsorbed layer thus resembles the atmosphere of a planet—it is most compressed at the surface of the solid and decreases in density outward. The general idea is quite old, but was first formalized by Polanyi in about 1914—see Brunauer [34]. As illustrated in Fig. XVII-12, one can draw surfaces of equipo-tential that appear as lines in a cross-sectional view of the surface region. The space between each set of equipotential surfaces corresponds to a definite volume, and there will thus be a relationship between potential U and volume 0. 

If we consider the case of a gas in adsorption equilibrium with a surface, there must be no net free energy change on transporting a small amount from one region to the other. Therefore, since the potential represents the work done by the adsorption forces when adsorbate is brought up to a distance x from the surface, there must be a compensating compressional increase in the free energy of the adsorbate. Thus 

In the case of multilayer adsorption it seems reasonable to suppose that condensation to a liquid film occurs (as in curves T or of Fig. XVII-13). If one now assumes that the amount adsorbed can be attributed entirely to such a film, and that the liquid is negligibly compressible, the thickness x of the film is related to n by 

Where Pg is the pressure of vapor in equilibrium with the adsorbed film. The characteristic curve is now just PF In Pq/P versus x (or against / , if E is not known). Dubinin and co-workers (see Ref. 58) have made much use of a semiempirical relation between and t/  

Isotherms Based on an Assumed Variation of Potential with Distance 

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