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Display information for equation id:math.2509.62 on revision:2509
* Page found: Reale Gase (eq math.2509.62)
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TeX (original user input):
\begin{align}
& ds={{\left( \frac{\partial s}{\partial T} \right)}_{v}}dT+{{\left( \frac{\partial s}{\partial v} \right)}_{T}}dv \\
& {{\left( \frac{\partial s}{\partial T} \right)}_{v}}=\frac{{{c}_{v}}}{T} \\
& {{\left( \frac{\partial s}{\partial v} \right)}_{T}}={{\left( \frac{\partial p}{\partial T} \right)}_{v}}=\frac{R}{v-b} \\
& ds={{\left( \frac{\partial s}{\partial T} \right)}_{v}}dT+{{\left( \frac{\partial s}{\partial v} \right)}_{T}}dv=\frac{{{c}_{v}}}{T}dT+\frac{R}{v-b}dv \\
& \Rightarrow s\left( T,v \right)=\int_{{{T}_{0}}}^{T}{{}}dT\acute{\ }\frac{{{c}_{v}}(T\acute{\ })}{T\acute{\ }}+R\ln \left( v-b \right)+const. \\
\end{align}
TeX (checked):
{\begin{aligned}&ds={{\left({\frac {\partial s}{\partial T}}\right)}_{v}}dT+{{\left({\frac {\partial s}{\partial v}}\right)}_{T}}dv\\&{{\left({\frac {\partial s}{\partial T}}\right)}_{v}}={\frac {{c}_{v}}{T}}\\&{{\left({\frac {\partial s}{\partial v}}\right)}_{T}}={{\left({\frac {\partial p}{\partial T}}\right)}_{v}}={\frac {R}{v-b}}\\&ds={{\left({\frac {\partial s}{\partial T}}\right)}_{v}}dT+{{\left({\frac {\partial s}{\partial v}}\right)}_{T}}dv={\frac {{c}_{v}}{T}}dT+{\frac {R}{v-b}}dv\\&\Rightarrow s\left(T,v\right)=\int _{{T}_{0}}^{T}{}dT{\acute {\ }}{\frac {{{c}_{v}}(T{\acute {\ }})}{T{\acute {\ }}}}+R\ln \left(v-b\right)+const.\\\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>d</mi><mi>s</mi><mo>=</mo><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>∂</mi><mi>s</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>T</mi></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mi>v</mi></mrow></msub><mi>d</mi><mi>T</mi><mo>+</mo><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow 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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>∂</mi><mi>s</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>v</mi></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></msub><mo>=</mo><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>∂</mi><mi>p</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>T</mi></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mi>v</mi></mrow></msub><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>R</mi></mrow><mrow data-mjx-texclass="ORD"><mrow 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data-mjx-texclass="ORD"><mi>T</mi></mrow></msub><mi>d</mi><mi>v</mi><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msub><mi>c</mi><mrow data-mjx-texclass="ORD"><mi>v</mi></mrow></msub></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></mfrac></mrow><mi>d</mi><mi>T</mi><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>R</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>v</mi><mo>−</mo><mi>b</mi></mrow></mrow></mfrac></mrow><mi>d</mi><mi>v</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>,</mo><mi>v</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>T</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow><mrow 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data-mjx-texclass="OPEN">(</mo><mi>v</mi><mo>−</mo><mi>b</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mi>c</mi><mi>o</mi><mi>n</mi><mi>s</mi><mi>t</mi><mo>.</mo></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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