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Display information for equation id:math.2573.40 on revision:2573

* Page found: Spezifische Wärme zweiatomiger idealer Gase (eq math.2573.40)

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Spezifische Wärme zweiatomiger idealer Gase Eq: math.2573.40

Hash: ff64cd5f585ec5f1f71085616943005b

TeX (original user input):

\begin{align}

& {{Z}_{S}}=\sum\limits_{n=0}^{\infty }{{}}\exp \left( -\beta \left( \hbar \omega \left( n+\frac{1}{2} \right) \right) \right)=\exp \left( -\beta \hbar \omega \frac{1}{2} \right)\sum\limits_{n=0}^{\infty }{{}}{{\left[ \exp \left( -\beta \hbar \omega  \right) \right]}^{n}} \\

& =\exp \left( -\beta \hbar \omega \frac{1}{2} \right)\frac{1}{1-\left[ \exp \left( -\beta \hbar \omega  \right) \right]} \\

& {{U}_{S}}=-N\frac{\partial }{\partial \beta }\ln {{Z}_{S}}=N\left( \frac{1}{\left[ \exp \left( \beta \hbar \omega  \right)-1 \right]}+\frac{1}{2} \right)\hbar \omega =N\left( \left\langle n \right\rangle +\frac{1}{2} \right)\hbar \omega  \\

& {{c}_{Vs}}={{N}_{A}}\frac{\partial }{\partial T}\left( \frac{1}{\left[ \exp \left( \frac{\hbar \omega }{kT} \right)-1 \right]}+\frac{1}{2} \right)\hbar \omega ={{N}_{A}}k\frac{{{\left( \frac{{{\Theta }_{S}}}{T} \right)}^{2}}{{e}^{\left( \frac{{{\Theta }_{S}}}{T} \right)}}}{{{\left( {{e}^{\left( \frac{{{\Theta }_{S}}}{T} \right)}}-1 \right)}^{2}}} \\

& {{\Theta }_{S}}:=\frac{\hbar \omega }{k} \\

\end{align}

TeX (checked):

{\begin{aligned}&{{Z}_{S}}=\sum \limits _{n=0}^{\infty }{}\exp \left(-\beta \left(\hbar \omega \left(n+{\frac {1}{2}}\right)\right)\right)=\exp \left(-\beta \hbar \omega {\frac {1}{2}}\right)\sum \limits _{n=0}^{\infty }{}{{\left[\exp \left(-\beta \hbar \omega \right)\right]}^{n}}\\&=\exp \left(-\beta \hbar \omega {\frac {1}{2}}\right){\frac {1}{1-\left[\exp \left(-\beta \hbar \omega \right)\right]}}\\&{{U}_{S}}=-N{\frac {\partial }{\partial \beta }}\ln {{Z}_{S}}=N\left({\frac {1}{\left[\exp \left(\beta \hbar \omega \right)-1\right]}}+{\frac {1}{2}}\right)\hbar \omega =N\left(\left\langle n\right\rangle +{\frac {1}{2}}\right)\hbar \omega \\&{{c}_{Vs}}={{N}_{A}}{\frac {\partial }{\partial T}}\left({\frac {1}{\left[\exp \left({\frac {\hbar \omega }{kT}}\right)-1\right]}}+{\frac {1}{2}}\right)\hbar \omega ={{N}_{A}}k{\frac {{{\left({\frac {{\Theta }_{S}}{T}}\right)}^{2}}{{e}^{\left({\frac {{\Theta }_{S}}{T}}\right)}}}{{\left({{e}^{\left({\frac {{\Theta }_{S}}{T}}\right)}}-1\right)}^{2}}}\\&{{\Theta }_{S}}:={\frac {\hbar \omega }{k}}\\\end{aligned}}

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\begin{aligned} Z_{S} = \sum_{n = 0}^{\infty} \exp (- β (ℏ ω (n + \frac{1}{2}))) = \exp (- β ℏ ω \frac{1}{2}) \sum_{n = 0}^{\infty} {[\exp (- β ℏ ω)]}^{n} \\ = \exp (- β ℏ ω \frac{1}{2}) \frac{1}{1 - [\exp (- β ℏ ω)]} \\ U_{S} = - N \frac{\partial}{\partial β} \ln Z_{S} = N (\frac{1}{[\exp (β ℏ ω) - 1]} + \frac{1}{2}) ℏ ω = N (⟨ n ⟩ + \frac{1}{2}) ℏ ω \\ c_{V s} = N_{A} \frac{\partial}{\partial T} (\frac{1}{[\exp (\frac{ℏ ω}{k T}) - 1]} + \frac{1}{2}) ℏ ω = N_{A} k \frac{{(\frac{Θ_{S}}{T})}^{2} e^{(\frac{Θ_{S}}{T})}}{{(e^{(\frac{Θ_{S}}{T})} - 1)}^{2}} \\ Θ_{S} : = \frac{ℏ ω}{k} \end{aligned}

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Identifiers

$Z_{S}$

$n$

$β$

$ω$

$n$

$β$

$ω$

$n$

$β$

$ω$

$n$

$β$

$ω$

$β$

$ω$

$U_{S}$

$N$

$β$

$Z_{S}$

$N$

$β$

$ω$

$ω$

$N$

$n$

$ω$

$c_{V s}$

$N_{A}$

$T$

$ω$

$k$

$T$

$ω$

$N_{A}$

$k$

$Θ_{S}$

$T$

$e$

$Θ_{S}$

$T$

$e$

$Θ_{S}$

$T$

$Θ_{S}$

$ω$

$k$

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