How Do You Calculate Free Energy for Gas Atoms Adsorbed on a Metal Surface?

[TroyElliott]

The problem goes as follows... "

One model for the adsorption of gas atoms on a metal surface approximates the surface to be a corregated muffin-tin potential. A gas atom can lower its energy by sitting in one of the potential minima which are the adhesion sites on the surface each with a binding energy ∆. Ignore all other interactions. Show that the free energy of exactly N ≫ 1 atoms adsorbed to a metal surface with M > N adhesion sites is

F = −N∆ + MkBT[(1 − y) ln(1 − y) + y ln y],

where y = N/M."

My attempt at this problem goes as follows. This is a two level energy system, say with energy of 0 defined at the non-binding parts of the muffin-tin and $-\Delta$ in the binding wells. We can then write our partition function as follows,

$$Z = \binom{M}{0}+\binom{M}{1}e^{\beta \Delta}+\binom{M}{2}e^{\beta 2\Delta}+...+\binom{M}{N}e^{\beta N\Delta},$$

where $\binom{M}{N}$ is defined as $\frac{M!}{N!(M-N)!}.$

This can be written as a series given by $$Z = \sum_{k=0}^{N}\binom{M}{k}e^{k\beta \Delta}.$$

When can be slightly further simplified to $$\ln{Z} = \ln{(M!)}+\ln{(\sum_{k=0}^{N}\frac{e^{k\beta \Delta}}{k!(M-k)!})}.$$

I know that the free energy is given by $F = -k_{b}T\ln{(Z)}$, but I am not sure how to simplify the sum in order to get the correct free energy. Any ideas on where to go from here would be very much appreciated.

Thanks!

 

[robphy]

Since N ≫ 1, you can use the Stirling approximation.