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Display information for equation id:math.2573.25 on revision:2573
* Page found: Spezifische Wärme zweiatomiger idealer Gase (eq math.2573.25)
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TeX (original user input):
\begin{align}
& {{Z}_{r}}\approx \int_{0}^{\infty }{{}}dl(2l+1)\exp \left( -\frac{{{\Theta }_{r}}}{T}l(l+1) \right) \\
& \frac{{{\Theta }_{r}}}{T}l(l+1):=x \\
& \Rightarrow {{Z}_{r}}\approx \frac{T}{{{\Theta }_{r}}}\int_{0}^{\infty }{{}}dx{{e}^{-x}}=\frac{T}{{{\Theta }_{r}}} \\
& \Rightarrow {{U}_{r}}=-N\frac{\partial }{\partial \beta }\ln {{Z}_{r}}=Nk{{T}^{2}}\frac{\partial }{\partial T}\ln {{Z}_{r}}\approx NkT \\
& \Rightarrow {{c}_{vr}}={{N}_{A}}k=R \\
\end{align}
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MathML (3.859 KB / 619 B) :
<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><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mi>d</mi><mi>l</mi><mo stretchy="false">(</mo><mn>2</mn><mi>l</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</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"><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><mi>l</mi><mo stretchy="false">(</mo><mi>l</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><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><mi>l</mi><mo stretchy="false">(</mo><mi>l</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>:</mi><mo>=</mo><mi>x</mi></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><msub><mi>Z</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><mo>≈</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow><mrow data-mjx-texclass="ORD"><msub><mi mathvariant="normal">Θ</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub></mrow></mfrac></mrow><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mi>d</mi><mi>x</mi><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>x</mi></mrow></mrow></msup><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow><mrow data-mjx-texclass="ORD"><msub><mi mathvariant="normal">Θ</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub></mrow></mfrac></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><mo>−</mo><mi>N</mi><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>β</mi></mrow></mrow></mfrac></mrow><mi>ln</mi><mo>⁡</mo><msub><mi>Z</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><mi>ln</mi><mo>⁡</mo><msub><mi>Z</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><mo>≈</mo><mi>N</mi><mi>k</mi><mi>T</mi></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><msub><mi>N</mi><mrow data-mjx-texclass="ORD"><mi>A</mi></mrow></msub><mi>k</mi><mo>=</mo><mi>R</mi></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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