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Display information for equation id:math.2473.16 on revision:2473

* Page found: Thermodynamische Potenziale (eq math.2473.16)

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Thermodynamische Potenziale Eq: math.2473.16

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TeX (original user input):

\begin{align}
& \frac{{{\partial }^{2}}U}{\partial S\partial V}=\frac{{{\partial }^{2}}U}{\partial V\partial S}\Leftrightarrow -{{\left( \frac{\partial p}{\partial S} \right)}_{V,N}}={{\left( \frac{\partial T}{\partial V} \right)}_{S,N}} \\
& \frac{{{\partial }^{2}}U}{\partial S\partial N}=\frac{{{\partial }^{2}}U}{\partial N\partial S}\Leftrightarrow {{\left( \frac{\partial \mu }{\partial S} \right)}_{V,N}}={{\left( \frac{\partial T}{\partial N} \right)}_{S,V}} \\
& \frac{{{\partial }^{2}}U}{\partial V\partial N}=\frac{{{\partial }^{2}}U}{\partial N\partial V}\Leftrightarrow {{\left( \frac{\partial \mu }{\partial V} \right)}_{S,N}}=-{{\left( \frac{\partial p}{\partial N} \right)}_{S,V}} \\
& -\frac{{{\partial }^{2}}F}{\partial T\partial V}=-\frac{{{\partial }^{2}}F}{\partial V\partial T}\Leftrightarrow {{\left( \frac{\partial p}{\partial T} \right)}_{V,N}}={{\left( \frac{\partial S}{\partial V} \right)}_{T,N}} \\
\end{align}

TeX (checked):

{\begin{aligned}&{\frac {{{\partial }^{2}}U}{\partial S\partial V}}={\frac {{{\partial }^{2}}U}{\partial V\partial S}}\Leftrightarrow -{{\left({\frac {\partial p}{\partial S}}\right)}_{V,N}}={{\left({\frac {\partial T}{\partial V}}\right)}_{S,N}}\\&{\frac {{{\partial }^{2}}U}{\partial S\partial N}}={\frac {{{\partial }^{2}}U}{\partial N\partial S}}\Leftrightarrow {{\left({\frac {\partial \mu }{\partial S}}\right)}_{V,N}}={{\left({\frac {\partial T}{\partial N}}\right)}_{S,V}}\\&{\frac {{{\partial }^{2}}U}{\partial V\partial N}}={\frac {{{\partial }^{2}}U}{\partial N\partial V}}\Leftrightarrow {{\left({\frac {\partial \mu }{\partial V}}\right)}_{S,N}}=-{{\left({\frac {\partial p}{\partial N}}\right)}_{S,V}}\\&-{\frac {{{\partial }^{2}}F}{\partial T\partial V}}=-{\frac {{{\partial }^{2}}F}{\partial V\partial T}}\Leftrightarrow {{\left({\frac {\partial p}{\partial T}}\right)}_{V,N}}={{\left({\frac {\partial S}{\partial V}}\right)}_{T,N}}\\\end{aligned}}

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\begin{aligned} \frac{\partial^{2} U}{\partial S \partial V} = \frac{\partial^{2} U}{\partial V \partial S} \Leftrightarrow - {(\frac{\partial p}{\partial S})}_{V, N} = {(\frac{\partial T}{\partial V})}_{S, N} \\ \frac{\partial^{2} U}{\partial S \partial N} = \frac{\partial^{2} U}{\partial N \partial S} \Leftrightarrow {(\frac{\partial μ}{\partial S})}_{V, N} = {(\frac{\partial T}{\partial N})}_{S, V} \\ \frac{\partial^{2} U}{\partial V \partial N} = \frac{\partial^{2} U}{\partial N \partial V} \Leftrightarrow {(\frac{\partial μ}{\partial V})}_{S, N} = - {(\frac{\partial p}{\partial N})}_{S, V} \\ - \frac{\partial^{2} F}{\partial T \partial V} = - \frac{\partial^{2} F}{\partial V \partial T} \Leftrightarrow {(\frac{\partial p}{\partial T})}_{V, N} = {(\frac{\partial S}{\partial V})}_{T, N} \end{aligned}

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$U$

$S$

$V$

$U$

$V$

$S$

$p$

$S$

$V$

$N$

$T$

$V$

$S$

$N$

$U$

$S$

$N$

$U$

$N$

$S$

$μ$

$S$

$V$

$N$

$T$

$N$

$S$

$V$

$U$

$V$

$N$

$U$

$N$

$V$

$μ$

$V$

$S$

$N$

$p$

$N$

$S$

$V$

$F$

$T$

$V$

$F$

$V$

$T$

$p$

$T$

$V$

$N$

$S$

$V$

$T$

$N$

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