Number of Adsorbate Configurations

The number of ways one can distribute adsorbates on a clean surface with surface sites can be calculated in the following way. For the first adsorbate, there are available free sites to choose between for the second adsorbate, there are sites remaining to choose between and so on. This process continues until the iV th adsorbate, for which there are (N -N +l) free sites left to choose between. The number of ways to place the adsorbates is then product of the individual possibilities  

However, the adsorbates we have placed on the surface are indistinguishable. This means that if we exchange an adsorbate that was put in site X with an adsorbate that was put in site Y, then the resulting configuration would be indistinguishable from the original state. Therefore, Equation (A.3.2.1) overcounts the number of physically distinguishable distributions by a factor equal to the number of ways we can order the adsorbates (in the chosen sites on the surface). The number of such ways can be described as the number of ways we can pick an adsorbate for site one, multiplied by the number of ways we can pick one for site two, etc., until all adsorbates have been placed in their sites. For the first pick, there are A adsorbates to choose between for the second choice, there are to choose between etc. 

So in total, there are A ways to order the adsorbates, and the total number of configurations, is therefore given by 

This expression is often referred to as a binomial coefficient since it is the same as the coefficient in front of the x th term in the polynomial expansion of the binomial 

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Spectroscopic investigations of alkenes (68,83,104,254,337,456,538) have suggested 7i-bonded or di-o-bonded species as possible reactive intermediates. However, the complexity of butadiene allows for a large number of adsorbate configurations (539,540), and future spectroscopic investigations are needed to explore the reaction mechanism of diene hydrogenation. 

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