Example 2 the Langmuir isotherm

This well-known equation, concerning the phenomenon of adsorption, is frequently arrived at by a kinetic method this depends on equating the condensation and evaporation rates for adsorbed molecules at the surface. However, since it applies to a condition of equilibrium, it might be expected that the same equation could be obtained by a purely statistical method. The advantage of the statistical derivation is that it shows much more clearly the precise conditions which are necessary if the Langmuir equation is to be obeyed, and thereby it shows the causes of deviations. 

The following is based on a derivation given by Everett,f with some simplifications of the notation. The necessary assumptions are (a) a gas molecule can only be adsorbed at a finite number of positions, called the sites , on the surface of the solid (b) the quantum states of adsorbed gas molecules are the same for all sites and independent of the presence of neighbouring molecules. The first of these is analogous to the assumption of a quasi-crystalline lattice in the case of a solution and its purpose is to give a countable number of configurations. Its justification depends, of course, on the fact that the surface is atomic in structure and may be expected to have potential energy wells , where adsorption takes place most readily. The assumption 

The second assumption makes possible the application of equation (14 10). It may be expected to break down either if the surface is appreciably heterogeneous or if there is appreciable attractive or repulsive interaction between the adsorbed molecules themselves. A third assui ption is also implicit in what follows, namely, that the gas does not dissociate on adsorption. However, this is not essential, and a form of the Langmuir isotherm can be obtained quite readily for the case where dissociation takes place. 

Let there be M sites on the given surface and a total of m gas molecules adsorbed. The ratio mIM is denoted 0, the fractional coverage mlM d. (14 26) 

Consider any one of the possible arrangements, such as is shown in Fig. 46, of the m molecules on the M sites and let s be the entropy per mole of the adsorbed molecules in this configuration. Let be the entropy per mole of the gas at unit pressure.. Therefore, if, from the 

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