편미분 관계식 관련 문제

 

 

 

 

편미분 관계식 관련 문제

 

질문

파란색 동그라미 친 부분 설명부탁드립니다..

 

https://kin.naver.com/qna/detail.naver?d1id=11&dirId=1114&docId=416180068 

 

 

풀이

편미분 관계식을 활용한 문제입니다.

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

편미분 관계식 partial differential relations

 

$$du={\left(\frac{\partial u}{\partial v}\right)}_{T}dv+{\left(\frac{\partial u}{\partial T}\right)}_{v}dT$$

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

$$du={\left(\frac{\partial u}{\partial v}\right)}_T[{\left(\frac{\partial v}{\partial T}\right)}_{P}dT+{\left(\frac{\partial v}{\partial P}\right)}_{T}dP]+{\left(\frac{\partial u}{\partial T}\right)}_{v}dT$$

$$=[{\left(\frac{\partial u}{\partial v}\right)}_T{\left(\frac{\partial v}{\partial T}\right)}_P+{\left(\frac{\partial u}{\partial T}\right)}_v]dT+{\left(\frac{\partial v}{\partial P}\right)}_T{\left(\frac{\partial u}{\partial v}\right)}_{T}dP$$

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

 

$$\therefore {\left(\frac{\partial u}{\partial v}\right)}_T{\left(\frac{\partial v}{\partial T}\right)}_P+{\left(\frac{\partial u}{\partial T}\right)}_v={\left(\frac{\partial u}{\partial T}\right)}_P$$