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https://m.kin.naver.com/mobile/qna/detail.naver?d1Id=11&dirId=1115&docId=407575849 

https://kin.naver.com/qna/detail.naver?d1Id=11&dirId=1115&docId=407575849 

 

 

풀이

\[dh = Tds + vdP\]

[열역학 Thermodynamics/7. 엔트로피 Entropy] - T ds 관계식 Tds equation

\[h=h(s,P)\]

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

[열역학 Thermodynamics/12. 일반관계식 Thermodynamic Relations ] - dh 일반관계식 dh general relation

s, P 가 서로 independent 하므로

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

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

(by chain rule)

[열역학 Thermodynamics/12. 일반관계식 Thermodynamic Relations ] - 편미분 관계식 partial differential relations

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

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


\[h=h(P,T)\]

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

위 결과에서

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

이므로

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

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

 

\[dh = Tds + vdP\]

\[s=s(P,T)\]

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

ds 식을 위 dh 식(dh=Tds+vdP)에 대입하면

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

\[dh=\left \{ T \left ( \frac{\partial s}{\partial P}\right )_{T}+v \right \}dP+T\left ( \frac{\partial s}{\partial T}\right )_{P}dT...(2)\]

(1),(2)에서 P, T가 서로 independent 하므로

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

 


보다 간단하게는

\[dh = Tds + vdP\]

양 변을 P가 constant 일때 T로 편미분을 취하면

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

\[\left ( \frac{\partial P}{\partial T}\right )_{P}=0\]

이므로

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

\[dh = Tds + vdP\]

양 변을 T가 constant 일때 P로 편미분을 취하면

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

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

이므로

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

 

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