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Display information for equation id:math.2573.25 on revision:2573

* Page found: Spezifische Wärme zweiatomiger idealer Gase (eq math.2573.25)

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Spezifische Wärme zweiatomiger idealer Gase Eq: math.2573.25

Hash: 966777eacd5d272a0dae4094aa64beb3

TeX (original user input):

\begin{align}

& {{Z}_{r}}\approx \int_{0}^{\infty }{{}}dl(2l+1)\exp \left( -\frac{{{\Theta }_{r}}}{T}l(l+1) \right) \\

& \frac{{{\Theta }_{r}}}{T}l(l+1):=x \\

& \Rightarrow {{Z}_{r}}\approx \frac{T}{{{\Theta }_{r}}}\int_{0}^{\infty }{{}}dx{{e}^{-x}}=\frac{T}{{{\Theta }_{r}}} \\

& \Rightarrow {{U}_{r}}=-N\frac{\partial }{\partial \beta }\ln {{Z}_{r}}=Nk{{T}^{2}}\frac{\partial }{\partial T}\ln {{Z}_{r}}\approx NkT \\

& \Rightarrow {{c}_{vr}}={{N}_{A}}k=R \\

\end{align}

TeX (checked):

{\begin{aligned}&{{Z}_{r}}\approx \int _{0}^{\infty }{}dl(2l+1)\exp \left(-{\frac {{\Theta }_{r}}{T}}l(l+1)\right)\\&{\frac {{\Theta }_{r}}{T}}l(l+1):=x\\&\Rightarrow {{Z}_{r}}\approx {\frac {T}{{\Theta }_{r}}}\int _{0}^{\infty }{}dx{{e}^{-x}}={\frac {T}{{\Theta }_{r}}}\\&\Rightarrow {{U}_{r}}=-N{\frac {\partial }{\partial \beta }}\ln {{Z}_{r}}=Nk{{T}^{2}}{\frac {\partial }{\partial T}}\ln {{Z}_{r}}\approx NkT\\&\Rightarrow {{c}_{vr}}={{N}_{A}}k=R\\\end{aligned}}

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MathML (3.859 KB / 619 B) :

\begin{aligned} Z_{r} \approx \int_{0}^{\infty} d l (2 l + 1) \exp (- \frac{Θ_{r}}{T} l (l + 1)) \\ \frac{Θ_{r}}{T} l (l + 1) : = x \\ \Rightarrow Z_{r} \approx \frac{T}{Θ_{r}} \int_{0}^{\infty} d x e^{- x} = \frac{T}{Θ_{r}} \\ \Rightarrow U_{r} = - N \frac{\partial}{\partial β} \ln Z_{r} = N k T^{2} \frac{\partial}{\partial T} \ln Z_{r} \approx N k T \\ \Rightarrow c_{v r} = N_{A} k = R \end{aligned}

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Identifiers

$Z_{r}$

$l$

$l$

$Θ_{r}$

$T$

$l$

$l$

$Θ_{r}$

$T$

$l$

$l$

$x$

$Z_{r}$

$T$

$Θ_{r}$

$x$

$e$

$x$

$T$

$Θ_{r}$

$U_{r}$

$N$

$β$

$Z_{r}$

$N$

$k$

$T$

$T$

$Z_{r}$

$N$

$k$

$T$

$c_{v r}$

$N_{A}$

$k$

$R$

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