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Display information for equation id:math.2555.54 on revision:2555

* Page found: Das ideale Fermigas (eq math.2555.54)

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Das ideale Fermigas Eq: math.2555.54

Hash: 2c96c136ad5dad51037053f3e4cc8e6e

TeX (original user input):

\begin{align}

& \bar{N}=\frac{\left( 2s+1 \right)}{2}\left( \frac{V}{{{h}^{3}}} \right)4\pi {{\left( 2mkT \right)}^{\frac{3}{2}}}\int_{0}^{\infty }{{}}dy\frac{{{y}^{\frac{1}{2}}}}{\left( {{e}^{y-\eta }}+1 \right)}=\frac{\left( 2s+1 \right)}{2}\left( \frac{V}{{{h}^{3}}} \right)4\pi {{\left( 2mkT \right)}^{\frac{3}{2}}}\left[ \frac{2}{3}{{\left( \frac{\mu }{kT} \right)}^{\frac{3}{2}}}+\frac{{{\pi }^{2}}}{12}{{\left( \frac{\mu }{kT} \right)}^{-\frac{1}{2}}} \right] \\

& \Rightarrow \bar{N}=\frac{2}{3}\frac{\left( 2s+1 \right)}{2}\left( \frac{V}{{{h}^{3}}} \right)4\pi {{\left( 2m\mu  \right)}^{\frac{3}{2}}}\left[ 1+\frac{{{\pi }^{2}}}{8}{{\left( \frac{kT}{\mu } \right)}^{2}} \right] \\

\end{align}

TeX (checked):

{\begin{aligned}&{\bar {N}}={\frac {\left(2s+1\right)}{2}}\left({\frac {V}{{h}^{3}}}\right)4\pi {{\left(2mkT\right)}^{\frac {3}{2}}}\int _{0}^{\infty }{}dy{\frac {{y}^{\frac {1}{2}}}{\left({{e}^{y-\eta }}+1\right)}}={\frac {\left(2s+1\right)}{2}}\left({\frac {V}{{h}^{3}}}\right)4\pi {{\left(2mkT\right)}^{\frac {3}{2}}}\left[{\frac {2}{3}}{{\left({\frac {\mu }{kT}}\right)}^{\frac {3}{2}}}+{\frac {{\pi }^{2}}{12}}{{\left({\frac {\mu }{kT}}\right)}^{-{\frac {1}{2}}}}\right]\\&\Rightarrow {\bar {N}}={\frac {2}{3}}{\frac {\left(2s+1\right)}{2}}\left({\frac {V}{{h}^{3}}}\right)4\pi {{\left(2m\mu \right)}^{\frac {3}{2}}}\left[1+{\frac {{\pi }^{2}}{8}}{{\left({\frac {kT}{\mu }}\right)}^{2}}\right]\\\end{aligned}}

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\begin{aligned} \bar{N} = \frac{(2 s + 1)}{2} (\frac{V}{h^{3}}) 4 π {(2 m k T)}^{\frac{3}{2}} \int_{0}^{\infty} d y \frac{y^{\frac{1}{2}}}{(e^{y - η} + 1)} = \frac{(2 s + 1)}{2} (\frac{V}{h^{3}}) 4 π {(2 m k T)}^{\frac{3}{2}} [\frac{2}{3} {(\frac{μ}{k T})}^{\frac{3}{2}} + \frac{π^{2}}{12} {(\frac{μ}{k T})}^{- \frac{1}{2}}] \\ \Rightarrow \bar{N} = \frac{2}{3} \frac{(2 s + 1)}{2} (\frac{V}{h^{3}}) 4 π {(2 m μ)}^{\frac{3}{2}} [1 + \frac{π^{2}}{8} {(\frac{k T}{μ})}^{2}] \end{aligned}

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Identifiers

$\bar{N}$

$s$

$V$

$h$

$π$

$m$

$k$

$T$

$y$

$y$

$e$

$y$

$η$

$s$

$V$

$h$

$π$

$m$

$k$

$T$

$μ$

$k$

$T$

$π$

$μ$

$k$

$T$

$\bar{N}$

$s$

$V$

$h$

$π$

$m$

$μ$

$π$

$k$

$T$

$μ$

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