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Now let's derive property relations from the thermodynamic potentials.
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Properties such as temperature, pressure, volume, and entropy are related with each other
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and their relations can be derived from thermodynamic potential equations shown before.
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From dU equals TdS - PdV equation, dU over dS at constant V is T, and dU over dV at constant S is - P.
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Here, mathematical manipulation comes in. All point functions are exact functions,
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thus do not depend on the order of differentiation.
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So differentiate of U with regard to S first then V is the same with the reverse order,
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with regard to V first, then S. It is just the second order differentiation of U with regard to both S and V.
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Apply this theorem to above equations. Then, differentiation of (dU over dS) with regard to V is (dT over dV)
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at constant S. The diffentiation of (dU over dV) with regard to S is (- dP over dS) at constant V.
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We finally obtained this property relation. - dP over dS at constant V is dT over dV at constant S.
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The property relations are called Maxwell relations. They are summarized as follows.
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The first one from dU is shown here. The relation from dH is differentiation of T with respect to P
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equals differentiation of V with respect to S. So (dV over dS) at constant p is (dT over dP) at constant S.
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The relation from dF is thus (dP over dT) at constant V equals (dS over dV) at constant V.
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Likewise, we can derive the final relation from dG.
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- (dV over dT) at constant P equals to (dS over dP) at constant T.
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These 4 equations are the essential property relations or Maxwell relations.