Grand Canonical Partition Function and Adsorption Statistics

[Arjani]

Homework Statement

Consider a two dimensional surface on a three dimensional crystal. This surface has M positions that can adsorb particles, each of which can bind one particle only and an adsorption does not affect the adsorption on nearby sites. An adsorbed particle has energy ε and an empty site has energy 0.

(Question A and B come here, but I could answer those.)

The surface is now considered to be in diffusive and thermal equilibrium with a gas of temperature T and chemical potential μ, so the energy E and the number of adsorbed particles can now vary.

c) Calculate the grand canonical partitition function \mathcal{Z_1} (T, \mu) of one adsorption position and then the grand canonical partition function for the entire surface \mathcal{Z_M} (T, \mu).

d) Calculate the chance P(T,\mu) that one adsorption position is taken.

Homework Equations

\mathcal{Z} = \sum e^{\beta(\mu N_i - E_i)}

The Attempt at a Solution

So for \mathcal{Z_1} (T, \mu), N = 1 and E_i = 0 or E_i = \epsilon and so

\mathcal{Z_1} = \sum e^{\beta(\mu - E_i)} = e^{\beta(\mu - \epsilon)} + e^{\beta(\mu - 0)} = e^{\beta \mu}(e^{-\beta \epsilon} + 1)

This is my solution. However, I read that you have to take N = 0 for E_i = 0 and N = 1 for E_i = \epsilon, resulting in \mathcal{Z_1} = e^0 + e^{\beta(\mu - \epsilon)}, so I'm confused. What is correct here?

As for \mathcal{Z_M}, I'm not sure how to go about that. Can you do something like this?

\mathcal{Z_M} = \sum_{i=0}^{M} e^{\beta N_i} \sum_{i=0}^{\epsilon} e^{- \beta E_i}

d) This is simply P = \frac{e^{\beta(\mu - \epsilon)}}{\mathcal{Z_1}}?

 

[TSny]

Homework Equations

\mathcal{Z} = \sum e^{\beta(\mu N_i - E_i)}

Think about what the summation index is here.

The Attempt at a Solution

So for \mathcal{Z_1} (T, \mu), N = 1 and E_i = 0 or E_i = \epsilon and so

\mathcal{Z_1} = \sum e^{\beta(\mu - E_i)} = e^{\beta(\mu - \epsilon)} + e^{\beta(\mu - 0)} = e^{\beta \mu}(e^{-\beta \epsilon} + 1)

This is my solution. However, I read that you have to take N = 0 for E_i = 0 and N = 1 for E_i = \epsilon, resulting in \mathcal{Z_1} = e^0 + e^{\beta(\mu - \epsilon)}, so I'm confused. What is correct here?

The summation is over all allowed values of $N$, namely $N = 0$ for no particle absorbed at the site and $N = 1$ for one particle absorbed at the site. You didn't quite handle the $N_i=0$ case correctly in the expression \mathcal{Z} = \sum e^{\beta(\mu N_i - E_i)}. You should get the result that you stated from your reading.