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Display information for equation id:math.2585.51 on revision:2585
* Page found: Paramagnetismus (eq math.2585.51)
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Hash: 9d6a5967ea5b3b5be9bce7b6e453ee29
TeX (original user input):
\begin{align}
& T\to \infty \\
& \Rightarrow S\left( T \right)=kN\ln 2 \\
& \\
& T->0 \\
& \Rightarrow S(T)\to kN\left[ \ln 2+\ln \frac{{{e}^{x}}}{2}-x\left( 1-2{{e}^{-2x}} \right) \right]=2kNx{{e}^{-2x}}\to 0 \\
& x:=\frac{\mu B}{2kT}\to \infty \\
\end{align}
TeX (checked):
{\begin{aligned}&T\to \infty \\&\Rightarrow S\left(T\right)=kN\ln 2\\&\\&T->0\\&\Rightarrow S(T)\to kN\left[\ln 2+\ln {\frac {{e}^{x}}{2}}-x\left(1-2{{e}^{-2x}}\right)\right]=2kNx{{e}^{-2x}}\to 0\\&x:={\frac {\mu B}{2kT}}\to \infty \\\end{aligned}}
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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>T</mi><mo accent="false">→</mo><mi mathvariant="normal">∞</mi></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><mi>S</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>T</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mi>k</mi><mi>N</mi><mi>ln</mi><mo>⁡</mo><mn>2</mn></mtd></mtr><mtr><mtd></mtd><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mi>T</mi><mo>−</mo><mo>></mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><mi>S</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo accent="false">→</mo><mi>k</mi><mi>N</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mi>ln</mi><mo>⁡</mo><mn>2</mn><mo>+</mo><mi>ln</mi><mo>⁡</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mi>x</mi></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow><mo>−</mo><mi>x</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mn>1</mn><mo>−</mo><mn>2</mn><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>2</mn><mi>x</mi></mrow></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mn>2</mn><mi>k</mi><mi>N</mi><mi>x</mi><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>2</mn><mi>x</mi></mrow></mrow></msup><mo accent="false">→</mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd><mtd><mi>x</mi><mi>:</mi><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>μ</mi><mi>B</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>k</mi><mi>T</mi></mrow></mrow></mfrac></mrow><mo accent="false">→</mo><mi mathvariant="normal">∞</mi></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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