Grand partition function - find occupation energies

[squidlollipop]

Homework Statement

N sites
3 possible situations: empty with energy = 0, occupied by A with energy = E1, occupied by B with energy = E2.
fugacities: for A = 10^-5, for B = 10^-7
T = 37 C
1) if no B find E1 such that 90% of sites occupied by A
2) with B find E2 such that 10% of sites occupied by A

Homework Equations

grand partition function - fermion

Z = [product over k] (gsubk + gsubk*exp[-(esubk - u)/kT])

where gsubk is degeneracy of kth state, esubk is energy of kth state, u is chemical potential k is bolzman constant, T is temperature

The Attempt at a Solution

one can find pretty much any information from the grand partition function so i started there with the above equation. For part 1) since 90% are occupied by A i think gsubk would be (.9*N) and gsubk for the remaining empty would be (.1*N). Do i even need to consider the empty states since the energy is zero?

Z = (.9N + .9N*exp[-E1/kT]* 10^-5)(.2N) ?

the first term from A the second from empty state?

And what about with B? I don't have the exact %s for empty and occupied by B, only that 10% is occupied by A.

Thank you in advance for any help!

 

[mark726]

Thank you for your post. It seems like you are on the right track with using the grand partition function to solve this problem. However, there are a few things to consider in your approach:

1) You are correct in assuming that the degeneracy of the empty states does not need to be considered since their energy is zero. This means that the first term in your expression for Z should be just N, not .9N as you have written.

2) The second term in your expression should be (.9N*exp[-E1/kT]* 10^-5), not (.9N*exp[-E1/kT]* 10^-5)(.2N). This is because the second term corresponds to the states occupied by A, not the empty states.

3) To find E1, you can set the expression for Z equal to (.9N + .1N), since we want 90% of the sites to be occupied by A. This will give you an equation to solve for E1.

4) For part 2), you will need to consider the contribution from B to the grand partition function. This can be done by adding another term to the expression for Z, similar to the one you have for A.

I hope this helps. Let me know if you have any further questions. Good luck with your calculations!