Anwendung auf das ideale Gas

Rekapitulation

$\displaystyle \beta$ $\displaystyle =\frac{\partial\ln\Omega}{\partial E}\left<x_{\alpha}\right>=\frac{1}{\beta }\frac{\partial\ln\Omega}{\partial\left<x_{\alpha}\right>}$ (3.316)
  $\displaystyle \left<p\right>=\frac{1}{\beta}\frac{\partial\ln\Omega}{\partial V}$ (3.317)
$\displaystyle \Omega$ $\displaystyle \varpropto V_{\chi}^{N}\left( E\right)$ (3.318)

also ist $ \ln\Omega=N\ln V+\ln\chi\left( E\right) +const.$

$\displaystyle \frac{\partial\ln\Omega}{\partial V}=\frac{N}{V}=\beta\left<p\right>$ (3.319)

also

  $\displaystyle \left<p\right>=kT\frac{N}{V}$ (3.320)
  $\displaystyle \Rightarrow pV=NkT                           ideale Gasgleichung$ (3.321)

weiter

$\displaystyle \beta$ $\displaystyle =\frac{\partial\ln\chi\left( E\right) }{\partial E}$ (3.322)
  $\displaystyle \Rightarrow E=E\left( T\right)                  f\ddot {u}r ein ideales Gas$ (3.323)

Othmar Marti
Experimentelle Physik
Universiät Ulm