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Display information for equation id:math.2577.18 on revision:2577

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Spezifische Wärme von Festkörpern Eq: math.2577.18

Hash: 24cfe4846e59c4c7f08ed42ebc144357

TeX (original user input):

\begin{align}

& T<<{{\Theta }_{D}} \\

& \xi >>1 \\

& \Rightarrow \Psi \left( \xi  \right):=\frac{1}{{{\xi }^{3}}}\int_{0}^{\xi }{{}}dx\frac{{{x}^{3}}}{{{e}^{x}}-1}\approx \frac{1}{{{\xi }^{3}}}\int_{0}^{\infty }{{}}dx\frac{{{x}^{3}}}{{{e}^{x}}-1}=\frac{1}{{{\xi }^{3}}}\frac{{{\pi }^{4}}}{15} \\

& U=9\frac{{{\pi }^{4}}}{15}NkT{{\left( \frac{T}{{{\Theta }_{D}}} \right)}^{3}} \\

& \Rightarrow {{C}_{V}}=\frac{\partial U}{\partial T}=\frac{36}{15}{{\pi }^{4}}Nk{{\left( \frac{T}{{{\Theta }_{D}}} \right)}^{3}} \\

& {{c}_{v}}=\frac{12}{5}{{\pi }^{4}}R{{\left( \frac{T}{{{\Theta }_{D}}} \right)}^{3}} \\

\end{align}

TeX (checked):

{\begin{aligned}&T<<{{\Theta }_{D}}\\&\xi >>1\\&\Rightarrow \Psi \left(\xi \right):={\frac {1}{{\xi }^{3}}}\int _{0}^{\xi }{}dx{\frac {{x}^{3}}{{{e}^{x}}-1}}\approx {\frac {1}{{\xi }^{3}}}\int _{0}^{\infty }{}dx{\frac {{x}^{3}}{{{e}^{x}}-1}}={\frac {1}{{\xi }^{3}}}{\frac {{\pi }^{4}}{15}}\\&U=9{\frac {{\pi }^{4}}{15}}NkT{{\left({\frac {T}{{\Theta }_{D}}}\right)}^{3}}\\&\Rightarrow {{C}_{V}}={\frac {\partial U}{\partial T}}={\frac {36}{15}}{{\pi }^{4}}Nk{{\left({\frac {T}{{\Theta }_{D}}}\right)}^{3}}\\&{{c}_{v}}={\frac {12}{5}}{{\pi }^{4}}R{{\left({\frac {T}{{\Theta }_{D}}}\right)}^{3}}\\\end{aligned}}

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MathML (5.535 KB / 654 B) :

\begin{aligned} T < < Θ_{D} \\ ξ > > 1 \\ \Rightarrow Ψ (ξ) : = \frac{1}{ξ^{3}} \int_{0}^{ξ} d x \frac{x^{3}}{e^{x} - 1} \approx \frac{1}{ξ^{3}} \int_{0}^{\infty} d x \frac{x^{3}}{e^{x} - 1} = \frac{1}{ξ^{3}} \frac{π^{4}}{15} \\ U = 9 \frac{π^{4}}{15} N k T {(\frac{T}{Θ_{D}})}^{3} \\ \Rightarrow C_{V} = \frac{\partial U}{\partial T} = \frac{36}{15} π^{4} N k {(\frac{T}{Θ_{D}})}^{3} \\ c_{v} = \frac{12}{5} π^{4} R {(\frac{T}{Θ_{D}})}^{3} \end{aligned}

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Identifiers

$T$

$Θ_{D}$

$ξ$

$Ψ$

$ξ$

$ξ$

$ξ$

$x$

$x$

$e$

$x$

$ξ$

$x$

$x$

$e$

$x$

$ξ$

$π$

$U$

$π$

$N$

$k$

$T$

$T$

$Θ_{D}$

$C_{V}$

$U$

$T$

$π$

$N$

$k$

$T$

$Θ_{D}$

$c_{v}$

$π$

$R$

$T$

$Θ_{D}$

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