일반관계식 문제풀이

 

 

 

 

편미분 관계식 관련 문제

 

질문

https://kin.naver.com/qna/detail.naver?d1id=6&dirId=6010103&docId=430454153 

5.6 번 증명 어떻게 해야하는지 모르겠어요 ㅠ

 

풀이

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

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

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

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

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

 

$${\left(\frac{\partial s}{\partial T}\right)}_P=\frac{1}{T}{\left(\frac{\partial h-v\partial P}{\partial T}\right)}_P=\frac{1}{T}{\left(\frac{\partial h}{\partial T}\right)}_P=\frac{c_P}{T}$$

$${\left(\frac{\partial s}{\partial T}\right)}_v=\frac{1}{T}{\left(\frac{\partial u+P\partial v}{\partial T}\right)}_v=\frac{1}{T}{\left(\frac{\partial u}{\partial T}\right)}_v=\frac{c_v}{T}$$

 

 

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

$$=v\frac{\frac{c_P}{T}}{\frac{c_v}{T}}{\left(\frac{\partial P}{\partial v}\right)}_T=-\frac{\frac{c_P}{T}}{\frac{c_v}{T}}\frac{1}{\frac{-1}{v}{\left(\frac{\partial v}{\partial P}\right)}_T}$$

$$-\frac{c_P}{c_v}\frac{1}{\kappa }$$

 

$$(\kappa =-\frac{1}{v}{\left(\frac{\partial v}{\partial P}\right)}_T={\beta }_T)$$