열역학 맥스웰 관계식 문제 증명 (편미분 방정식)
질문
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풀이
\[dv=\left ( \frac{\partial v}{\partial P} \right )_{T}dP+\left ( \frac{\partial v}{\partial T} \right )_{P}dT\]
\[ds=\left ( \frac{\partial s}{\partial P} \right )_{v}dP+\left ( \frac{\partial s}{\partial v} \right )_{P}dv\]
\[ds=\left ( \frac{\partial s}{\partial P} \right )_{v}dP+\left ( \frac{\partial s}{\partial v} \right )_{P}\left [ \left ( \frac{\partial v}{\partial P} \right )_{T}dP+\left ( \frac{\partial v}{\partial T} \right )_{P}dT \right ]\]
\[ds=\left [ \left ( \frac{\partial s}{\partial P} \right )_{v}+\left ( \frac{\partial s}{\partial v} \right )_{P}\left ( \frac{\partial v}{\partial P} \right )_{T}\right ]dP+\left ( \frac{\partial s}{\partial v} \right )_{P}\left ( \frac{\partial v}{\partial T} \right )_{P}dT\]
\[ds=\left ( \frac{\partial s}{\partial P} \right )_{T}dP+\left ( \frac{\partial s}{\partial T} \right )_{P}dT\]
\[\left ( \frac{\partial s}{\partial P} \right )_{T}=\left ( \frac{\partial s}{\partial P} \right )_{v}+\left ( \frac{\partial s}{\partial v} \right )_{P}\left ( \frac{\partial v}{\partial P} \right )_{T}\]
\[\left ( \frac{\partial s}{\partial P} \right )_{T}\left ( \frac{\partial P}{\partial s} \right )_{T}=\left ( \frac{\partial s}{\partial P} \right )_{v}\left ( \frac{\partial P}{\partial s} \right )_{T}+\left ( \frac{\partial s}{\partial v} \right )_{P}\left ( \frac{\partial v}{\partial P} \right )_{T}\left ( \frac{\partial P}{\partial s} \right )_{T}\]
\[1=\left ( \frac{\partial s}{\partial P} \right )_{v}\left ( \frac{\partial P}{\partial s} \right )_{T}+\left ( \frac{\partial s}{\partial v} \right )_{P}\left ( \frac{\partial v}{\partial s} \right )_{T}\]
\[1=\left ( \frac{\partial s}{\partial P} \right )_{v}\left [ -\left ( \frac{\partial T}{\partial v} \right )_{P} \right ] +\left ( \frac{\partial s}{\partial v} \right )_{P}\left ( \frac{\partial T}{\partial P} \right )_{v}\]
\[\left ( \frac{\partial T}{\partial P} \right )_{v}\left ( \frac{\partial s}{\partial v} \right )_{P}-\left ( \frac{\partial T}{\partial v} \right )_{P}\left ( \frac{\partial s}{\partial P} \right )_{v}=1\]
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