# Grand Canonical ensemble

1. Mar 4, 2008

### j93

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. 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 dont think this is right though

Last edited: Mar 4, 2008
2. Mar 4, 2008

### j93

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

Last edited: Mar 4, 2008