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Display information for equation id:math.2473.15 on revision:2473

* Page found: Thermodynamische Potenziale (eq math.2473.15)

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Thermodynamische Potenziale Eq: math.2473.15

Hash: decb810c7e249ebb41428d44b630daf0

TeX (original user input):

\begin{align}
& dJ\left( T,V,\mu  \right)=TdS-pdV+\mu dN-TdS-SdT-\mu dN-Nd\mu =-pdV-SdT-Nd\mu  \\
& {{\left( \frac{\partial J}{\partial T} \right)}_{V,\mu }}=-S \\
& {{\left( \frac{\partial J}{\partial V} \right)}_{T,\mu }}=-p \\
& {{\left( \frac{\partial J}{\partial \mu } \right)}_{T,V}}=-N \\
\end{align}

TeX (checked):

{\begin{aligned}&dJ\left(T,V,\mu \right)=TdS-pdV+\mu dN-TdS-SdT-\mu dN-Nd\mu =-pdV-SdT-Nd\mu \\&{{\left({\frac {\partial J}{\partial T}}\right)}_{V,\mu }}=-S\\&{{\left({\frac {\partial J}{\partial V}}\right)}_{T,\mu }}=-p\\&{{\left({\frac {\partial J}{\partial \mu }}\right)}_{T,V}}=-N\\\end{aligned}}

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MathML (2.746 KB / 440 B) :

\begin{aligned} d J (T, V, μ) = T d S - p d V + μ d N - T d S - S d T - μ d N - N d μ = - p d V - S d T - N d μ \\ {(\frac{\partial J}{\partial T})}_{V, μ} = - S \\ {(\frac{\partial J}{\partial V})}_{T, μ} = - p \\ {(\frac{\partial J}{\partial μ})}_{T, V} = - N \end{aligned}

<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>d</mi><mi>J</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>T</mi><mo>,</mo><mi>V</mi><mo>,</mo><mi>&#x03BC;</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mi>T</mi><mi>d</mi><mi>S</mi><mo>&#x2212;</mo><mi>p</mi><mi>d</mi><mi>V</mi><mo>+</mo><mi>&#x03BC;</mi><mi>d</mi><mi>N</mi><mo>&#x2212;</mo><mi>T</mi><mi>d</mi><mi>S</mi><mo>&#x2212;</mo><mi>S</mi><mi>d</mi><mi>T</mi><mo>&#x2212;</mo><mi>&#x03BC;</mi><mi>d</mi><mi>N</mi><mo>&#x2212;</mo><mi>N</mi><mi>d</mi><mi>&#x03BC;</mi><mo>=</mo><mo>&#x2212;</mo><mi>p</mi><mi>d</mi><mi>V</mi><mo>&#x2212;</mo><mi>S</mi><mi>d</mi><mi>T</mi><mo>&#x2212;</mo><mi>N</mi><mi>d</mi><mi>&#x03BC;</mi></mtd></mtr><mtr><mtd></mtd><mtd><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>J</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>T</mi></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>V</mi><mo>,</mo><mi>&#x03BC;</mi></mrow></mrow></msub><mo>=</mo><mo>&#x2212;</mo><mi>S</mi></mtd></mtr><mtr><mtd></mtd><mtd><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>J</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>V</mi></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>T</mi><mo>,</mo><mi>&#x03BC;</mi></mrow></mrow></msub><mo>=</mo><mo>&#x2212;</mo><mi>p</mi></mtd></mtr><mtr><mtd></mtd><mtd><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>J</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>&#x03BC;</mi></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>T</mi><mo>,</mo><mi>V</mi></mrow></mrow></msub><mo>=</mo><mo>&#x2212;</mo><mi>N</mi></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>

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Identifiers

$d$

$J$

$T$

$V$

$μ$

$T$

$d$

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$p$

$d$

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$μ$

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$N$

$T$

$d$

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$S$

$d$

$T$

$μ$

$d$

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$N$

$d$

$μ$

$p$

$d$

$V$

$S$

$d$

$T$

$N$

$d$

$μ$

$J$

$T$

$V$

$μ$

$S$

$J$

$V$

$T$

$μ$

$p$

$J$

$μ$

$T$

$V$

$N$

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