일반관계식 문제 - Mayer 관계식 이용(Cengel 9th ed. 12-21 solution)

[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

 

\[ \frac{k}{k-1}=\frac{c_p}{c_p-c_v} = \frac{T\left( \frac{\partial s}{\partial T} \right)_{P}}{T \left(\frac{\partial P}{\partial T} \right)_v  \left(\frac{\partial v}{\partial T} \right)_P} \]

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

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

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

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

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