Because V is a state function, ((V/T) P /P) T = ( (V/P) T /T) P .
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Because V is a state function, (∂(∂V/∂T)P/∂P)T = (∂ (∂V/∂P)T /∂T)P. Using this relationship, show that the isothermal compressibility and isobaric expansion coefficient are related by (∂β /∂P)T = −(∂κ/∂T)P.
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