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질문

열역학 일반관계식, 맥스웰 관계식 증명

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풀이

\[\left ( \frac{\partial s}{\partial v} \right )_{P}=\frac{1}{T}\left( \frac{\partial h-v\partial P}{\partial v} \right )_{P}\]

\[=\frac{1}{T}\left( \frac {\partial h}{\partial v}\right )_{P}=\frac{1}{T}\left( \frac {\partial h}{\partial T}\right )_{P}\left( \frac {\partial T}{\partial v}\right )_{P}\]

\[=\frac{c_{P}}{T}\frac{v}{v}\left( \frac {\partial T}{\partial v}\right )_{P}=\frac{c_{P}}{Tv\beta }\]

\[\left ( \beta =\frac{1}{v}\left ( \frac{\partial v}{\partial T} \right )_{P} =\alpha \right )\]

 

 

\[\left ( \frac{\partial s}{\partial P} \right )_{v}=\frac{1}{T}\left( \frac{\partial u+P\partial v}{\partial P} \right )_{v}\]

\[=\frac{1}{T}\left( \frac {\partial u}{\partial P}\right )_{v}=\frac{1}{T}\left( \frac {\partial u}{\partial T}\right )_{v}\left( \frac {\partial T}{\partial P}\right )_{v}\]

\[=\frac{c_{v}}{T}\left[ -\left( \frac {\partial T}{\partial v}\right )_{P}\left( \frac {\partial v}{\partial P}\right )_{T}\right ]\]

\[=\frac{c_{v}}{T}\frac{v}{1}\left( \frac {\partial T}{\partial v}\right )_{P}\frac{-1}{v}\left( \frac {\partial v}{\partial P}\right )_{T}\]

\[=\frac{c_{v} \kappa_{T}}{T\beta}\]

\[\left ( \kappa_{T} =\frac{-1}{v}\left ( \frac{\partial v}{\partial P} \right )_{T} =\beta_{T}\right )\]

이 문제는 아직 못 풀었네요

풀면 올릴게요

 

 

\[\left(\frac{\partial a}{\partial P} \right )_{v}=\left(\frac{\partial u-T\partial s-s\partial T}{\partial P} \right )_{v}\]

\[=\left(\frac{\partial u-\left( \partial u+P\partial v\right )-s\partial T}{\partial P} \right )_{v}=\left(\frac{-P\partial v-s\partial T}{\partial P} \right )_{v}\]

\[=-s\left(\frac{\partial T}{\partial P} \right )_{v}=s\left(\frac{\partial T}{\partial v} \right )_{P}\left(\frac{\partial v}{\partial P} \right )_{T}\]

\[=-s\frac{v}{1}\left(\frac{\partial T}{\partial v} \right )_{P}\frac{-1}{v}\left(\frac{\partial v}{\partial P} \right )_{T}=\frac{-s\kappa_{T}}{\beta}\]

 

 

\[\left(\frac{\partial a}{\partial v} \right )_{P}=\left(\frac{-P\partial v-s\partial T}{\partial v} \right )_{P}\]

\[=-P-s\left ( \frac{\partial T}{\partial v} \right )_{P}=-P-s\frac{v}{v}\left ( \frac{\partial T}{\partial v} \right )_{P}=-P-\frac{s}{v\beta}\]

 

\[\left ( \frac{\partial h}{\partial s} \right )_{v}=\left ( \frac{T\partial s+v\partial P}{\partial s} \right )_{v}=T+v\left ( \frac{\partial P}{\partial s} \right )_{v}\]

\[=T-v\left ( \frac{\partial P}{\partial u+P\partial v} \right )_{v}=T-v\left ( \frac{\partial P}{\partial u} \right )_{v}\]

\[=T-vT\left ( \frac{\partial P}{\partial u+P\partial v} \right )_{v}=T\left (1 -v\left ( \frac{\partial P}{\partial u} \right )_{v} \right )\]

\[=T\left (1 +v\left ( \frac{\partial P}{\partial T} \right )_{v}\left ( \frac{\partial T}{\partial u} \right )_{v} \right )=T\left (1 +\frac{v}{c_{v}}\left ( \frac{\partial P}{\partial T} \right )_{v}\right )\]

\[=T\left (1 -\frac{v}{c_{v}}\left ( \frac{\partial P}{\partial v} \right )_{T}\left ( \frac{\partial v}{\partial T} \right )_{P}\right )=T\left (1 +\frac{v\beta}{c_{v}\kappa_{T}}\right )\]

 

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