[문제]
https://kin.naver.com/qna/detail.naver?d1id=11&dirId=1114&docId=434996363
Cengel 9th ed. 12-21 Problem
[풀이]
\[ k=\frac{c_p}{c_v}\]
\[ \frac{k}{k-1}=\frac{\frac{c_p}{c_v}}{\frac{c_p}{c_v}-1}=\frac{c_p}{c_p-c_v} \]
\[s=s\left(T, P\right) \]
\[ds=\left( \frac{\partial s}{\partial T} \right)_{P}dT + \left( \frac{\partial s}{\partial P} \right)_{T}dP \]
\[=\frac{1}{T}\left( \frac{\partial h - v\partial P}{\partial T} \right)_{P}dT+ \left( \frac{\partial s}{\partial P} \right)_{T}dP \]
\[ =\frac{1}{T}\left( \frac{\partial h}{\partial T} \right)_{P}dT+ \left( \frac{\partial s}{\partial P} \right)_{T}dP \]
\[ =\frac{c_P}{T}dT+ \left( \frac{\partial s}{\partial P} \right)_{T}dP \]
\[ \left( \frac{\partial s}{\partial T} \right)_{P} = \frac{c_P}{T} \]
\[ c_P = T\left( \frac{\partial s}{\partial T} \right)_{P} \]
\[ c_P-c_v = T \left(\frac{\partial P}{\partial T} \right)_v \left(\frac{\partial v}{\partial T} \right)_P\]
[열역학 Thermodynamics/12. 일반관계식 Thermodynamic Relations ] - Mayer 관계식 Mayer's relation
https://syssurr.tistory.com/17
\[ \frac{k}{k-1}=\frac{c_p}{c_p-c_v} = \frac{T\left( \frac{\partial s}{\partial T} \right)_{P}}{T \left(\frac{\partial P}{\partial T} \right)_v \left(\frac{\partial v}{\partial T} \right)_P} \]
\[ = \left( \frac{\partial s}{\partial T} \right)_{P} \left(\frac{\partial T}{\partial P} \right)_v \left(\frac{\partial T}{\partial v} \right)_P \]
\[ = \left[-\left( \frac{\partial s}{\partial P} \right)_{T} \left( \frac{\partial P}{\partial T} \right)_{s}\right] \left(\frac{\partial T}{\partial P} \right)_v \left[ -\left(\frac{\partial P}{\partial s} \right)_T\right]\]
\[ = \left( \frac{\partial P}{\partial T} \right)_{s} \left(\frac{\partial T}{\partial P} \right)_v \left( \frac{\partial s}{\partial P} \right)_{T} \left(\frac{\partial P}{\partial s} \right)_T \]
\[ = \left( \frac{\partial P}{\partial T} \right)_{s} \left(\frac{\partial T}{\partial P} \right)_v \]
\[ \left( \frac{\partial P}{\partial T} \right)_{s} = \frac{k}{k-1}\left(\frac{\partial P}{\partial T} \right)_v\]
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