Binary Adsorption on a Flat Surface

In the derivation of the i theory, was defined to derive the grand canonical ensemble. For convenience, the quantities for adsorbate 1 in relation to its E may be defined by 

Following the same prescription as before and noting that now molecules of type 2 may also form teeth in the particle in the box description, the energy N,l for the adsorbate number 1 is 

In this and subsequent equations, there are identical equations for adsorbate 2 with the indexes 1 and 2 switched. Added to this is the energy of interaction, j between the adsorbed molecules. Since this is a big box approximation, the energy between the molecules will be a weighted average, or for adsorbate 1 this is 

From the above considerations, the grand canonical ensemble may be written as 

Following the usual procedure and taking the partial differential with respect to of the In max term S one obtains, 

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