# Statistical mechanics. Partition function.

1. Mar 15, 2014

### LagrangeEuler

1. The problem statement, all variables and given/known data
If $Z$ is homogeneous function with property
$Z(\alpha T,\alpha^{-\frac{3}{\nu}}V,N)$
and you calculate Z(T,V,N). Could you calculate directly $Z(\alpha T,\alpha^{-\frac{3}{\nu}}V,N)$.

2. Relevant equations
$Z(T,V,N)=\frac{1}{h^{3N}N!}(2\pi m k T)^{\frac{3N}{2}}\int...\int_{V} e^{-\frac{U}{kT}}$

3. The attempt at a solution
When I have for exaple
$Z(\alpha T,\alpha^{-\frac{3}{\nu}}V,N)=\frac{1}{h^{3N}N!}(2\pi m k \alpha T)^{\frac{3N}{2}}\int...\int_{\alpha^{-\frac{3}{\nu}}V} e^{-\frac{U}{k\alpha T}}$
I'm not sure how to integrate this $e^{-\frac{U}{k\alpha T}}$ in exponent.
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2. Mar 19, 2014

### LagrangeEuler

From this if I understand well
$\int...\int_{\alpha^{-\frac{3}{\nu}V}}e^{-\frac{U}{k\alpha T}}$
need to be equal
$\int...\int_{\alpha^{-\frac{3}{\nu}V}}e^{-\frac{U}{k\alpha T}}=\alpha^{-\frac{3N}{\nu}}...$