일반관계식 문제 - dh 관계식 유도
질문
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
\[\left(\frac{\partial h}{\partial P}\right )_{T} = T\left ( \frac{\partial s}{\partial P}\right )_{T}+v\]