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## Homework Statement

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

Can someone explain to me how this work out this formula

## Homework Equations

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

## 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]

## Homework Statement

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]