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]

$$k=\frac{c_p}{c_v}$$

$$\frac{k}{k-1}=\frac{\frac{c_p}{c_v}}{\frac{c_p}{c_v}-1}=\frac{c_p}{c_p-c_v}$$

 

$$s=s(T,P)$$

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

$$=\frac{1}{T}{\left(\frac{\partial h-v\partial P}{\partial T}\right)}_{P}dT+{\left(\frac{\partial s}{\partial P}\right)}_{T}dP$$

$$=\frac{1}{T}{\left(\frac{\partial h}{\partial T}\right)}_{P}dT+{\left(\frac{\partial s}{\partial P}\right)}_{T}dP$$

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

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

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

 

$$c_P-c_v=T{\left(\frac{\partial P}{\partial T}\right)}_v{\left(\frac{\partial v}{\partial T}\right)}_P$$

[열역학 Thermodynamics/12. 일반관계식 Thermodynamic Relations ] - Mayer 관계식 Mayer's relation

https://syssurr.tistory.com/17

 

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

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

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

$$={\left(\frac{\partial P}{\partial T}\right)}_s{\left(\frac{\partial T}{\partial P}\right)}_v{\left(\frac{\partial s}{\partial P}\right)}_T{\left(\frac{\partial P}{\partial s}\right)}_T$$

$$={\left(\frac{\partial P}{\partial T}\right)}_s{\left(\frac{\partial T}{\partial P}\right)}_v$$

$${\left(\frac{\partial P}{\partial T}\right)}_s=\frac{k}{k-1}{\left(\frac{\partial P}{\partial T}\right)}_v$$