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Display information for equation id:math.2492.56 on revision:2492
* Page found: Gleichgewichtsbedingungen (eq math.2492.56)
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\begin{align}
& {{\left( \frac{\partial g\acute{\ }\left( T,p\acute{\ } \right)}{\partial p\acute{\ }} \right)}_{T}}\left[ {{\left( \frac{\partial P}{\partial r} \right)}_{T}}-\frac{2}{{{r}^{2}}}\sigma \right]={{\left( \frac{\partial g\acute{\ }\acute{\ }}{\partial p\acute{\ }\acute{\ }} \right)}_{T}}{{\left( \frac{\partial P}{\partial r} \right)}_{T}} \\
& \left( \frac{\partial g\acute{\ }\acute{\ }}{\partial p\acute{\ }\acute{\ }} \right)=v\acute{\ }\acute{\ } \\
& \left( \frac{\partial g\acute{\ }\left( T,p\acute{\ } \right)}{\partial p\acute{\ }} \right)=v\acute{\ } \\
& \Rightarrow {{\left( \frac{\partial P}{\partial r} \right)}_{T}}=-\frac{2}{{{r}^{2}}}\sigma \frac{v\acute{\ }}{v\acute{\ }\acute{\ }-v\acute{\ }}\approx -\frac{2}{{{r}^{2}}}\sigma \frac{v\acute{\ }}{v\acute{\ }\acute{\ }}=idGas=-\frac{2\sigma v\acute{\ }}{R{{\operatorname{Tr}}^{2}}}P \\
& \Rightarrow \ln \frac{P}{{{P}_{\infty }}}=\frac{2\sigma v\acute{\ }}{R\operatorname{Tr}} \\
& P={{P}_{\infty }}(T)\exp \frac{2\sigma v\acute{\ }}{R\operatorname{Tr}} \\
\end{align}
TeX (checked):
{\begin{aligned}&{{\left({\frac {\partial g{\acute {\ }}\left(T,p{\acute {\ }}\right)}{\partial p{\acute {\ }}}}\right)}_{T}}\left[{{\left({\frac {\partial P}{\partial r}}\right)}_{T}}-{\frac {2}{{r}^{2}}}\sigma \right]={{\left({\frac {\partial g{\acute {\ }}{\acute {\ }}}{\partial p{\acute {\ }}{\acute {\ }}}}\right)}_{T}}{{\left({\frac {\partial P}{\partial r}}\right)}_{T}}\\&\left({\frac {\partial g{\acute {\ }}{\acute {\ }}}{\partial p{\acute {\ }}{\acute {\ }}}}\right)=v{\acute {\ }}{\acute {\ }}\\&\left({\frac {\partial g{\acute {\ }}\left(T,p{\acute {\ }}\right)}{\partial p{\acute {\ }}}}\right)=v{\acute {\ }}\\&\Rightarrow {{\left({\frac {\partial P}{\partial r}}\right)}_{T}}=-{\frac {2}{{r}^{2}}}\sigma {\frac {v{\acute {\ }}}{v{\acute {\ }}{\acute {\ }}-v{\acute {\ }}}}\approx -{\frac {2}{{r}^{2}}}\sigma {\frac {v{\acute {\ }}}{v{\acute {\ }}{\acute {\ }}}}=idGas=-{\frac {2\sigma v{\acute {\ }}}{R{{\operatorname {Tr} }^{2}}}}P\\&\Rightarrow \ln {\frac {P}{{P}_{\infty }}}={\frac {2\sigma v{\acute {\ }}}{R\operatorname {Tr} }}\\&P={{P}_{\infty }}(T)\exp {\frac {2\sigma v{\acute {\ }}}{R\operatorname {Tr} }}\\\end{aligned}}
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