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Display information for equation id:math.2556.34 on revision:2556

* Page found: Das ideale Bosegas (eq math.2556.34)

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Hash: 1d1c8bd313a2cd4b38601ba8e035f42f

TeX (original user input):

\begin{align}
  & \frac{N\acute{\ }}{V}\approx \left( 2s+1 \right)\frac{2\pi }{{{h}^{3}}}{{\left( 2mkT \right)}^{\frac{3}{2}}}\int_{0}^{\infty }{{}}dy\frac{{{y}^{\frac{1}{2}}}}{{{e}^{y}}-1}\approx \left( 2s+1 \right){{\left( \frac{2\pi mkT}{{{h}^{2}}} \right)}^{\frac{3}{2}}}\frac{2}{\sqrt{\pi }}\int_{0}^{\infty }{{}}dy{{e}^{-y}}{{y}^{\frac{1}{2}}} \\ 
 & \frac{2}{\sqrt{\pi }}\int_{0}^{\infty }{{}}dy{{e}^{-y}}{{y}^{\frac{1}{2}}}=1 \\ 
 & {{\left( \frac{2\pi mkT}{{{h}^{2}}} \right)}^{\frac{3}{2}}}={{\lambda }^{-3}} \\ 
\end{align}

TeX (checked):

{\begin{aligned}&{\frac {N{\acute {\ }}}{V}}\approx \left(2s+1\right){\frac {2\pi }{{h}^{3}}}{{\left(2mkT\right)}^{\frac {3}{2}}}\int _{0}^{\infty }{}dy{\frac {{y}^{\frac {1}{2}}}{{{e}^{y}}-1}}\approx \left(2s+1\right){{\left({\frac {2\pi mkT}{{h}^{2}}}\right)}^{\frac {3}{2}}}{\frac {2}{\sqrt {\pi }}}\int _{0}^{\infty }{}dy{{e}^{-y}}{{y}^{\frac {1}{2}}}\\&{\frac {2}{\sqrt {\pi }}}\int _{0}^{\infty }{}dy{{e}^{-y}}{{y}^{\frac {1}{2}}}=1\\&{{\left({\frac {2\pi mkT}{{h}^{2}}}\right)}^{\frac {3}{2}}}={{\lambda }^{-3}}\\\end{aligned}}

LaTeXML (experimentell; verwendet MathML) rendering

MathML (44.396 KB / 4.84 KB) :

N ´ V ( 2 s + 1 ) 2 π h 3 ( 2 m k T ) 3 2 0 𝑑 y y 1 2 e y - 1 ( 2 s + 1 ) ( 2 π m k T h 2 ) 3 2 2 π 0 𝑑 y e - y y 1 2 2 π 0 𝑑 y e - y y 1 2 = 1 ( 2 π m k T h 2 ) 3 2 = λ - 3 missing-subexpression 𝑁 ´ absent 𝑉 2 𝑠 1 2 𝜋 superscript 3 superscript 2 𝑚 𝑘 𝑇 3 2 superscript subscript 0 differential-d 𝑦 superscript 𝑦 1 2 superscript 𝑒 𝑦 1 2 𝑠 1 superscript 2 𝜋 𝑚 𝑘 𝑇 superscript 2 3 2 2 𝜋 superscript subscript 0 differential-d 𝑦 superscript 𝑒 𝑦 superscript 𝑦 1 2 missing-subexpression 2 𝜋 superscript subscript 0 differential-d 𝑦 superscript 𝑒 𝑦 superscript 𝑦 1 2 1 missing-subexpression superscript 2 𝜋 𝑚 𝑘 𝑇 superscript 2 3 2 superscript 𝜆 3 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle\frac{N\acute{\ }}{V% }\approx\left(2s+1\right)\frac{2\pi}{{{h}^{3}}}{{\left(2mkT\right)}^{\frac{3}{% 2}}}\int_{0}^{\infty}{{}}dy\frac{{{y}^{\frac{1}{2}}}}{{{e}^{y}}-1}\approx\left% (2s+1\right){{\left(\frac{2\pi mkT}{{{h}^{2}}}\right)}^{\frac{3}{2}}}\frac{2}{% \sqrt{\pi}}\int_{0}^{\infty}{{}}dy{{e}^{-y}}{{y}^{\frac{1}{2}}}\\ &\displaystyle\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}{{}}dy{{e}^{-y}}{{y}^{\frac% {1}{2}}}=1\\ &\displaystyle{{\left(\frac{2\pi mkT}{{{h}^{2}}}\right)}^{\frac{3}{2}}}={{% \lambda}^{-3}}\\ \end{aligned}}}

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MathML (experimentell; keine Bilder) rendering

MathML (5.313 KB / 635 B) :

N´V(2s+1)2πh3(2mkT)320dyy12ey1(2s+1)(2πmkTh2)322π0dyeyy122π0dyeyy12=1(2πmkTh2)32=λ3

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