열역학 체팽창계수 등온압축계수 일반관계식 문제풀이
질문
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풀이
\[dv=\left ( \frac{\partial v}{dP} \right )_{T}dP+\left ( \frac{\partial v}{dT} \right )_{P}dT\]
\[\frac{dv}{v}=\frac{1}{v}\left ( \frac{\partial v}{dP} \right )_{T}dP+\frac{1}{v}\left ( \frac{\partial v}{dT} \right )_{P}dT\]
\[\frac{dv}{v}=d\left ( ln \left ( v\right ) \right )\]
\[\frac{1}{v}\left ( \frac{\partial v}{dP} \right )_{T}dP+\frac{1}{v}\left ( \frac{\partial v}{dT} \right )_{P}dT=-\kappa dP+\beta dT\]
\[d\left ( ln \left ( v\right ) \right )=-\kappa dP+\beta dT\]
\[d\left ( ln \left ( v\right ) \right )\] 는 완전미분이므로
[열역학 Thermodynamics/12. 일반관계식 Thermodynamic Relations ] - 편미분 관계식 partial differential relations
\[-\left ( \frac{ \partial \kappa }{\partial T} \right )_{P}=\left ( \frac{ \partial \beta }{\partial P} \right )_{T}\]