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## Homework Statement

I. Finding the partition function Z.

II. If the middle level (only) is degenerate, i.e. there are two states with the same energy, show that the partition function is:

[tex] Z = (1+exp(\frac{-\epsilon}{k_{B}T}))^{2}[/tex]

III. State the Helmholtz free energy F of the assembly in part II.

IV. Show the entropy of the assembly in part II is:

[tex] S = 2Nk_{B} ln(1 + exp^({\frac{-\epsilon}{k_{B}T}})) + \frac{2Nk_{B}(\frac{-\epsilon}{k_{B}T}) exp(\frac{-\epsilon}{k_{B}T})}{1+exp(\frac{-\epsilon}{k_{B}T})}[/tex]

## Homework Equations

Partition function for a system that can exist in energy levels [tex]\epsilon_{1},\epsilon_{2},..,\epsilon_{i},..[/tex] etc. is defined as:

[tex]Z=\sum_{i}exp(\frac{-\epsilon_{i}}{k_{B}T})[/tex]

## The Attempt at a Solution

Part I:

Part II: Don't know how to do this! :|

Part III:

Free energy: [tex]F=-Nk_{B}Tln(Z)[/tex]

Each atom replaced by 3 oscillators..

.. therefore: [tex]F=-3Nk_{B}Tln(Z)[/tex]

Define: [tex]Z = \frac{exp(-\frac{\theta}{2T})}{1-exp(-\frac{\theta}{T})}[/tex]

.. hence after substitution:

[tex]F=-3Nk_{B}Tln(\frac{exp(-\frac{\theta}{2T})}{1-exp(-\frac{\theta}{T})})}[/tex]

Which rearranges to:

[tex]Z = \frac{exp(-\frac{\theta}{2T})}{1-exp(-\frac{\theta}{T})}[/tex]

Part IV: Don't know how to do this! :|