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Grand partition function

  • Thread starter Arjani
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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 [itex]\mathcal{Z_1} (T, \mu)[/itex] of one adsorption position and then the grand canonical partition function for the entire surface [itex]\mathcal{Z_M} (T, \mu)[/itex].

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

Homework Equations



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

The Attempt at a Solution



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

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

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

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

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

d) This is simply [itex]P = \frac{e^{\beta(\mu - \epsilon)}}{\mathcal{Z_1}}[/itex]?
 
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Answers and Replies

  • #2
TSny
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Homework Equations



[itex]\mathcal{Z} = \sum e^{\beta(\mu N_i - E_i)}[/itex]
Think about what the summation index is here.

The Attempt at a Solution



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

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

This is my solution. However, I read that you have to take [itex]N = 0[/itex] for [itex]E_i = 0[/itex] and [itex]N = 1[/itex] for [itex]E_i = \epsilon[/itex], resulting in [itex]\mathcal{Z_1} = e^0 + e^{\beta(\mu - \epsilon)}[/itex], 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 [itex]\mathcal{Z} = \sum e^{\beta(\mu N_i - E_i)}[/itex]. You should get the result that you stated from your reading.
 

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