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Display information for equation id:math.2547.69 on revision:2547

* Page found: Das ideale Fermigas (eq math.2547.69)

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Das ideale Fermigas Eq: math.2555.69

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\begin{align}

& U\approx \frac{2}{5}\left( \frac{V}{{{h}^{3}}} \right)4\pi \frac{\left( 2s+1 \right)}{2}{{\left( 2m \right)}^{\frac{3}{2}}}{{\left( {{E}_{F}} \right)}^{\frac{5}{2}}}{{\left[ 1-\frac{{{\pi }^{2}}}{12}{{\left( \frac{kT}{{{E}_{F}}} \right)}^{2}} \right]}^{\frac{5}{2}}}\left[ 1+\frac{5}{2}\frac{{{\pi }^{2}}}{4}{{\left( \frac{kT}{{{E}_{F}}} \right)}^{2}} \right] \\

& \frac{2}{5}\left( \frac{V}{{{h}^{3}}} \right)4\pi \frac{\left( 2s+1 \right)}{2}{{\left( 2m \right)}^{\frac{3}{2}}}{{\left( {{E}_{F}} \right)}^{\frac{5}{2}}}\approx \frac{3}{5}\bar{N}{{E}_{F}} \\

& {{\left[ 1-\frac{{{\pi }^{2}}}{12}{{\left( \frac{kT}{{{E}_{F}}} \right)}^{2}} \right]}^{\frac{5}{2}}}\left[ 1+\frac{5}{2}\frac{{{\pi }^{2}}}{4}{{\left( \frac{kT}{{{E}_{F}}} \right)}^{2}} \right]\approx 1+5\frac{{{\pi }^{2}}}{12}{{\left( \frac{kT}{{{E}_{F}}} \right)}^{2}} \\

& \Rightarrow U\approx \frac{3}{5}\bar{N}{{E}_{F}}\left[ 1+5\frac{{{\pi }^{2}}}{12}{{\left( \frac{kT}{{{E}_{F}}} \right)}^{2}} \right] \\

\end{align}

TeX (checked):

{\begin{aligned}&U\approx {\frac {2}{5}}\left({\frac {V}{{h}^{3}}}\right)4\pi {\frac {\left(2s+1\right)}{2}}{{\left(2m\right)}^{\frac {3}{2}}}{{\left({{E}_{F}}\right)}^{\frac {5}{2}}}{{\left[1-{\frac {{\pi }^{2}}{12}}{{\left({\frac {kT}{{E}_{F}}}\right)}^{2}}\right]}^{\frac {5}{2}}}\left[1+{\frac {5}{2}}{\frac {{\pi }^{2}}{4}}{{\left({\frac {kT}{{E}_{F}}}\right)}^{2}}\right]\\&{\frac {2}{5}}\left({\frac {V}{{h}^{3}}}\right)4\pi {\frac {\left(2s+1\right)}{2}}{{\left(2m\right)}^{\frac {3}{2}}}{{\left({{E}_{F}}\right)}^{\frac {5}{2}}}\approx {\frac {3}{5}}{\bar {N}}{{E}_{F}}\\&{{\left[1-{\frac {{\pi }^{2}}{12}}{{\left({\frac {kT}{{E}_{F}}}\right)}^{2}}\right]}^{\frac {5}{2}}}\left[1+{\frac {5}{2}}{\frac {{\pi }^{2}}{4}}{{\left({\frac {kT}{{E}_{F}}}\right)}^{2}}\right]\approx 1+5{\frac {{\pi }^{2}}{12}}{{\left({\frac {kT}{{E}_{F}}}\right)}^{2}}\\&\Rightarrow U\approx {\frac {3}{5}}{\bar {N}}{{E}_{F}}\left[1+5{\frac {{\pi }^{2}}{12}}{{\left({\frac {kT}{{E}_{F}}}\right)}^{2}}\right]\\\end{aligned}}

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\begin{aligned} U \approx \frac{2}{5} (\frac{V}{h^{3}}) 4 π \frac{(2 s + 1)}{2} {(2 m)}^{\frac{3}{2}} {(E_{F})}^{\frac{5}{2}} {[1 - \frac{π^{2}}{12} {(\frac{k T}{E_{F}})}^{2}]}^{\frac{5}{2}} [1 + \frac{5}{2} \frac{π^{2}}{4} {(\frac{k T}{E_{F}})}^{2}] \\ \frac{2}{5} (\frac{V}{h^{3}}) 4 π \frac{(2 s + 1)}{2} {(2 m)}^{\frac{3}{2}} {(E_{F})}^{\frac{5}{2}} \approx \frac{3}{5} \bar{N} E_{F} \\ {[1 - \frac{π^{2}}{12} {(\frac{k T}{E_{F}})}^{2}]}^{\frac{5}{2}} [1 + \frac{5}{2} \frac{π^{2}}{4} {(\frac{k T}{E_{F}})}^{2}] \approx 1 + 5 \frac{π^{2}}{12} {(\frac{k T}{E_{F}})}^{2} \\ \Rightarrow U \approx \frac{3}{5} \bar{N} E_{F} [1 + 5 \frac{π^{2}}{12} {(\frac{k T}{E_{F}})}^{2}] \end{aligned}

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Identifiers

$U$

$V$

$h$

$π$

$s$

$m$

$E_{F}$

$π$

$k$

$T$

$E_{F}$

$π$

$k$

$T$

$E_{F}$

$V$

$h$

$π$

$s$

$m$

$E_{F}$

$\bar{N}$

$E_{F}$

$π$

$k$

$T$

$E_{F}$

$π$

$k$

$T$

$E_{F}$

$π$

$k$

$T$

$E_{F}$

$U$

$\bar{N}$

$E_{F}$

$π$

$k$

$T$

$E_{F}$

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