열역학 체팽창계수 등온압축계수 일반관계식 문제풀이

 

 

 

 

 

 

질문

 

kin.naver.com/qna/detail.nhn?d1Id=11&dirId=1114&docId=370539865

열역학 부피팽창률 등온압축률 질문

 

풀이

 

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

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

 

$$\frac{dv}{v}=d\left(ln\left(v\right)\right)$$

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

 

[열역학 Thermodynamics/12. 일반관계식 Thermodynamic Relations ] - 압축계수 팽창계수 compressibility & expansion coefficient

압축계수 팽창계수 compressibility & expansion coefficient

 

$$d\left(ln\left(v\right)\right)=-\kappa dP+\beta dT$$

 

$$d\left(ln\left(v\right)\right)$$ 는 완전미분이므로

 

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

편미분 관계식 partial differential relations

 

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