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Display information for equation id:math.2507.53 on revision:2507
* Page found: Das ideale Gas (eq math.2507.53)
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\begin{align}
& Z=\frac{1}{N!}\frac{1}{{{\hbar }^{3N}}}\int_{V}^{{}}{{}}{{d}^{3}}{{q}_{1}}...\int_{V}^{{}}{{}}{{d}^{3}}{{q}_{N}}\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}{{p}_{1}}...\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}{{p}_{N}}\exp \left( -\frac{\beta }{2m}\sum\limits_{i}^{{}}{{}}{{p}_{i}}^{2} \right)=\frac{1}{N!}{{\left[ V\Phi \left( \beta \right) \right]}^{N}} \\
& \Rightarrow \Psi =-\ln Z=\frac{F(T,V)}{kT}=-N\ln \left[ V\Phi \left( \beta \right) \right]+\ln N!\approx -N\ln \Phi \left( \beta \right)-N\ln \frac{V}{N}-N \\
\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><mi>Z</mi><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>N</mi><mi>!</mi></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><msup><mi data-mjx-alternate="1">ℏ</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>3</mn><mi>N</mi></mrow></mrow></msup></mrow></mfrac></mrow><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mi>V</mi></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover></mstyle><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><msub><mi>q</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo>.</mo><mo>.</mo><mo>.</mo><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mi>V</mi></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover></mstyle><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><msub><mi>q</mi><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msup><mi>R</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover></mstyle><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo>.</mo><mo>.</mo><mo>.</mo><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msup><mi>R</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover></mstyle><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msub><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"><mi>β</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>m</mi></mrow></mrow></mfrac></mrow><munderover><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><msup><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>N</mi><mi>!</mi></mrow></mrow></mfrac></mrow><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mi>V</mi><mi mathvariant="normal">Φ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>β</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></msup></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><mi mathvariant="normal">Ψ</mi><mo>=</mo><mo>−</mo><mi>ln</mi><mo>⁡</mo><mi>Z</mi><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>F</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>T</mi></mrow></mrow></mfrac></mrow><mo>=</mo><mo>−</mo><mi>N</mi><mi>ln</mi><mo>⁡</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mi>V</mi><mi mathvariant="normal">Φ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>β</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>+</mo><mi>ln</mi><mo>⁡</mo><mi>N</mi><mi>!</mi><mo>≈</mo><mo>−</mo><mi>N</mi><mi>ln</mi><mo>⁡</mo><mi mathvariant="normal">Φ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>β</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>−</mo><mi>N</mi><mi>ln</mi><mo>⁡</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>V</mi></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></mfrac></mrow><mo>−</mo><mi>N</mi></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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