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

$${\left(\frac{\partial h}{\partial T}\right)}_P={\left(\frac{\partial h}{\partial s}\right)}_P{\left(\frac{\partial s}{\partial T}\right)}_P=T{\left(\frac{\partial s}{\partial T}\right)}_P$$

$$\therefore {\left(\frac{\partial h}{\partial T}\right)}_P=T{\left(\frac{\partial s}{\partial T}\right)}_P$$

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$$h=h(P,T)$$

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

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

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

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

(1),(2)에서 P, T가 서로 independent 하므로

$$\therefore {\left(\frac{\partial h}{\partial P}\right)}_T=T{\left(\frac{\partial s}{\partial P}\right)}_T+v$$

 

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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

$${\left(\frac{\partial h}{\partial P}\right)}_T=T{\left(\frac{\partial s}{\partial P}\right)}_T+v$$