(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

We had a lecture about partition function, canonical ensemble etc.

Can someone explain to me how this work out this formula

2. Relevant equations

we are supposed to find the mean energy and preasure of a gas with given partition function

3. The attempt at a solution

mean energy is given [tex] \overline{-}U=\sum_{r}E_{r}p_{r}[/tex]

I know also that Boltzman's probability distribution is described by

[tex] p_{r}= \frac{e^{-\beta E_{r}}}{\sum_{r}e^{-\beta E_{r}}}[/tex]

because the partition function is definied as [tex] z=\sum_{r}e^{-\beta E_{r}^}[/tex]

so rewriting now the Boltzman's probablility distribution I get

[tex] p_{r}= \frac{e^{-\beta E_{r}}}{z}[/tex]

1. The problem statement, all variables and given/known data

now going back to the mean energy I can write

[tex] \overline{-}U=\frac{1}{z}\sum_{r}E_{r}e^{-\beta E_{r}[/tex]

These are operations I do not understand. Could someone explain them step by step ?

[tex] \sum_{r}E_{r}e^{-\beta E_{r}}= -\frac{\partial}{\partial \beta}\sum_{r}e^{-\beta E_{r}}= -\frac{\partial}{\partial \beta}z[/tex]

and the final one

[tex] U= -\frac{1}{z}\frac{\partial z}{\partial \beta}=-\frac{\partial lnz}{\partial \beta}[/tex]

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# Homework Help: Mean energy and preassure inolving partition function

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