How Does the Grand Canonical Ensemble Determine Site Atom Occupancy?

[j93]

Homework Statement

Grand Canonical ensemble problem- A surface of N sites that can have 0,1,2 atoms. It costs no energy to adsorb an atom. Grand canonical problem therefore in contact with particle reservoir. Assume \mu chem potentiol and temp T.
What is probability for site to be empty,1,or 2 atoms.
average number of atoms in gas.

Homework Equations

The Attempt at a Solution

I believe the answer is that they have equal probability because there is no energy cost to adsorb. ie 1/3 for all 3.

for second question I believe the partition function is $$Z = \Sigma e^{mu/kT}e^{E/kT}$$ where E = 0 for all 3 different states ie $$Z= 3^Ne^{mu/KT}$$. I don't think this is right though

 

[j93]

I think Z the partition function is
1$$(1 + e^{\frac{\mu N}{kT}}+ e^{\frac{2\mu N}{kT}})$$
or
$$(1+e^{\frac{\mu }{kT}}+ e^{\frac{2\mu }{kT}})^N$$