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편미분 관계식 관련 문제

 

질문

https://kin.naver.com/qna/detail.naver?d1id=6&dirId=6010103&docId=430454153 

5.6 번 증명 어떻게 해야하는지 모르겠어요 ㅠ

 

풀이

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

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

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

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

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

 

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

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

 

 

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

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

\[-\frac{c_P}{c_v}\frac{1}{\kappa}\]

 

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

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