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Display information for equation id:math.2573.28 on revision:2573
* Page found: Spezifische Wärme zweiatomiger idealer Gase (eq math.2573.28)
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TeX (original user input):
\begin{align}
& {{Z}_{r}}\approx 1+3\exp \left( -\frac{2{{\Theta }_{r}}}{T} \right) \\
& \ln {{Z}_{r}}\approx 3\exp \left( -\frac{2{{\Theta }_{r}}}{T} \right) \\
& \Rightarrow {{U}_{r}}\approx Nk{{T}^{2}}\frac{\partial }{\partial T}\left( 3\exp \left( -\frac{2{{\Theta }_{r}}}{T} \right) \right)=6Nk{{\Theta }_{r}}\left( \exp \left( -\frac{2{{\Theta }_{r}}}{T} \right) \right) \\
& \Rightarrow {{c}_{vr}}=12R{{\left( \frac{{{\Theta }_{r}}}{T} \right)}^{2}}\exp \left( -\frac{2{{\Theta }_{r}}}{T} \right) \\
\end{align}
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<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd></mtd><mtd><msub><mi>Z</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><mo>≈</mo><mn>1</mn><mo>+</mo><mn>3</mn><mi>exp</mi><mo>⁡</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><msub><mi mathvariant="normal">Θ</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi>ln</mi><mo>⁡</mo><msub><mi>Z</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><mo>≈</mo><mn>3</mn><mi>exp</mi><mo>⁡</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><msub><mi mathvariant="normal">Θ</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><msub><mi>U</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><mo>≈</mo><mi>N</mi><mi>k</mi><msup><mi>T</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>T</mi></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mn>3</mn><mi>exp</mi><mo>⁡</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><msub><mi mathvariant="normal">Θ</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mn>6</mn><mi>N</mi><mi>k</mi><msub><mi mathvariant="normal">Θ</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>exp</mi><mo>⁡</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><msub><mi mathvariant="normal">Θ</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><msub><mi>c</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>v</mi><mi>r</mi></mrow></mrow></msub><mo>=</mo><mn>1</mn><mn>2</mn><mi>R</mi><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msub><mi mathvariant="normal">Θ</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi>exp</mi><mo>⁡</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><msub><mi mathvariant="normal">Θ</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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