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Display information for equation id:math.2565.25 on revision:2565

* Page found: Das Photonengas im Strahlungshohlraum (eq math.2565.25)

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Das Photonengas im Strahlungshohlraum Eq: math.2569.25

Hash: ab3df8d1c8c7b790a36593aa2db9b6cb

TeX (original user input):

\begin{align}

& p=kT{{\left( \frac{\partial }{\partial V}\ln Z \right)}_{T}}=-kT\sum\limits_{\nu }^{{}}{{}}\frac{\frac{h}{kT}\left( \frac{\partial \nu }{\partial V} \right){{e}^{-\frac{h\nu }{kT}}}}{\left( 1-{{e}^{-\frac{h\nu }{kT}}} \right)}=kT\sum\limits_{\nu }^{{}}{{}}\frac{\frac{h}{kT}\frac{\nu }{3V}{{e}^{-\frac{h\nu }{kT}}}}{\left( 1-{{e}^{-\frac{h\nu }{kT}}} \right)}=\frac{1}{3V}\sum\limits_{\nu }^{{}}{{}}\frac{h\nu {{e}^{-\frac{h\nu }{kT}}}}{\left( 1-{{e}^{-\frac{h\nu }{kT}}} \right)}= \\

& \Rightarrow p=\frac{1}{3V}\sum\limits_{\nu }^{{}}{{}}h\nu \left\langle {{N}_{\nu }} \right\rangle  \\

& p=\frac{1}{3}\frac{U}{V} \\

\end{align}

TeX (checked):

{\begin{aligned}&p=kT{{\left({\frac {\partial }{\partial V}}\ln Z\right)}_{T}}=-kT\sum \limits _{\nu }^{}{}{\frac {{\frac {h}{kT}}\left({\frac {\partial \nu }{\partial V}}\right){{e}^{-{\frac {h\nu }{kT}}}}}{\left(1-{{e}^{-{\frac {h\nu }{kT}}}}\right)}}=kT\sum \limits _{\nu }^{}{}{\frac {{\frac {h}{kT}}{\frac {\nu }{3V}}{{e}^{-{\frac {h\nu }{kT}}}}}{\left(1-{{e}^{-{\frac {h\nu }{kT}}}}\right)}}={\frac {1}{3V}}\sum \limits _{\nu }^{}{}{\frac {h\nu {{e}^{-{\frac {h\nu }{kT}}}}}{\left(1-{{e}^{-{\frac {h\nu }{kT}}}}\right)}}=\\&\Rightarrow p={\frac {1}{3V}}\sum \limits _{\nu }^{}{}h\nu \left\langle {{N}_{\nu }}\right\rangle \\&p={\frac {1}{3}}{\frac {U}{V}}\\\end{aligned}}

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\begin{aligned} p = k T {(\frac{\partial}{\partial V} \ln Z)}_{T} = - k T \sum_{ν}^{} \frac{\frac{h}{k T} (\frac{\partial ν}{\partial V}) e^{- \frac{h ν}{k T}}}{(1 - e^{- \frac{h ν}{k T}})} = k T \sum_{ν}^{} \frac{\frac{h}{k T} \frac{ν}{3 V} e^{- \frac{h ν}{k T}}}{(1 - e^{- \frac{h ν}{k T}})} = \frac{1}{3 V} \sum_{ν}^{} \frac{h ν e^{- \frac{h ν}{k T}}}{(1 - e^{- \frac{h ν}{k T}})} = \\ \Rightarrow p = \frac{1}{3 V} \sum_{ν}^{} h ν ⟨ N_{ν} ⟩ \\ p = \frac{1}{3} \frac{U}{V} \end{aligned}

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Identifiers

$p$

$k$

$T$

$V$

$Z$

$T$

$k$

$T$

$ν$

$h$

$k$

$T$

$ν$

$V$

$e$

$h$

$ν$

$k$

$T$

$e$

$h$

$ν$

$k$

$T$

$k$

$T$

$ν$

$h$

$k$

$T$

$ν$

$V$

$e$

$h$

$ν$

$k$

$T$

$e$

$h$

$ν$

$k$

$T$

$V$

$ν$

$h$

$ν$

$e$

$h$

$ν$

$k$

$T$

$e$

$h$

$ν$

$k$

$T$

$p$

$V$

$ν$

$h$

$ν$

$N_{ν}$

$p$

$U$

$V$

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