## Grand Partition Function Question

1. The problem statement, all variables and given/known data
a) Suppose particles can be adsorbed onto a surface such that each adsorption site can be occupied by up to 6 atoms each in single-particle quantum state $\psi_i$ with an adsorption energy $\epsilon_i$. Write down the grand partition function for one site.

b) If $\frac{\epsilon_1 - \mu}{k_{B}T} = 0.7$ show that your expression for the grand partition function is very close to that of a Bose system where any number of particles may occupy the site.

c) Find the probability that there are 6 particles on the site.

2. Relevant equations

3. The attempt at a solution

My attempt so far at part a) (I'm not even sure if this is right, lol):

$$Z = \sum_{i=1}^{6} e^{- \frac{\epsilon_i - \mu}{k_B T}}$$

I don't really know how to get started on b. I did think that it might follow the reasoning that as u increase the number of particles which can be absorbed at a given site, the exponential part of Z becomes smaller and smaller, and so it approaches the geometric series that u get when you apply a similar analysis to a system of bose particles... Is that idea anywhere near right?

For part c) I thought that:
$$P_k = \frac{e^{\frac{\epsilion_{6} - \mu}{k_B T} }}{1 - e^{- \frac{\epsilon_1 - \mu}{k_B T}}}$$

Since we know from the previous part that the grand partition function can be approximated by the partition function of a bose system. Is any of this right?
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