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질문

열역학 정압비열 정적비열 관계식에 대해서 질문드립니다. 

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열역학 정압비열 정적비열 관계식에 대해서 질문드립니다.

의 관계식에서 p와V의 역할을 바꿔서와 같은 형태로 어떻게 바꿀수 있을까요?열역학 잘아시는분들 도와주세요.

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답변

 

\[ds =\left ( \frac{\partial s}{\partial T}\right )_{v}dT+\left ( \frac{\partial s}{\partial v}\right )_{T}dv\]

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

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

\[=\frac{c_{v}}{T} dT+\left ( \frac{\partial s}{\partial v}\right )_{T}dv\]

 

 

\[ds =\left ( \frac{\partial s}{\partial T}\right )_{P}dT+\left ( \frac{\partial s}{\partial P}\right )_{T}dP\]

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

\[ =\frac{1}{T} \left (\frac{\partial h}{\partial T}\right )_{P}dT+\left ( \frac{\partial s}{\partial P}\right )_{T}dP\]

\[=\frac{c_{P}}{T} dT+\left ( \frac{\partial s}{\partial P}\right )_{T}dP\]

 

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\[ds=\frac{c_{v}}{T} dT+\left ( \frac{\partial s}{\partial v}\right )_{T}dv\]

\[=\frac{c_{P}}{T} dT+\left ( \frac{\partial s}{\partial P}\right )_{T}dP\]

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

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

 

\[dT=\left ( \frac{\partial T}{\partial v}\right )_{P}dv+\left ( \frac{\partial T}{\partial P}\right )_{v}dP\]

 

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

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

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

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

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

 

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

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

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

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

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

 

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