Grand Partition Function Question

[ppyadof]

Homework Statement

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.

Homework Equations

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?

 

[Jhero]

Thank you for your interesting question. I would like to provide some guidance on your attempt at solving the problem.

For part a), your expression for the grand partition function is correct. However, it would be helpful to define the terms in your equation for better understanding. For example, \epsilon_i could represent the adsorption energy of a single particle in state \psi_i, and \mu could be the chemical potential. This way, your equation becomes more clear and easier to work with.

For part b), your intuition is correct. As the number of particles that can be adsorbed at a site increases, the exponential term in the grand partition function becomes smaller and the expression becomes closer to that of a Bose system. You can show this by taking the limit as the number of particles approaches infinity, which would result in the Bose-Einstein distribution.

For part c), your expression for the probability of having 6 particles on the site is not quite right. The correct expression should be:

P_6 = \frac{e^{-6 \frac{\epsilon_1 - \mu}{k_B T}}}{Z}

Where Z is the grand partition function you defined in part a). This probability represents the likelihood of having all 6 particles in the same state \psi_1, which is the lowest energy state. As you can see, as the chemical potential \mu approaches the energy of the lowest state \epsilon_1, the probability of having all 6 particles in that state increases.

I hope this helps you in your understanding of the problem. Keep up the good work in your studies!