Application of the Grand Partition Function

As a simple example of the application of this method we again derive the Langmuir equation. 

The last equation above follows directly from the binomial theorem. 

Many modifications to the simple Langmuir model are amenable to analysis in this way. For example, consider adsorption on a set of M independent pairs of identical sites such that there is an interaction energy (2w) when both sites of a given pair are occupied. The expression for the grand partition function for such a system is 

The BET isotherm may also be easily derived via the grand partition function. As with the Langmuir model we consider a system of M independent sites with no interaction between adsorbed molecules. However, in the BET model each site in the first adsorbed layer is considered to provide a site for the second layer of adsorbed molecules and similarly for the third layer, 

Summing the terms in parentheses as geometric scries, we obtain 

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Statislical Thermodynamic Approach 77 Application of the Grand Partition Function ... 

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