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Property relations : the Maxwell relations

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

Thermodynamic properties such as temperature, pressure, volume and entropy are related with each other.

Their mutual relations are called property relations or Maxwell relations, and the equations showing property relations are derived from the differential form of thermodynamic potentials. Since thermodynamic potentials are point functions, they are path-independent. In mathematical terminology, these functions are exact functions. Therefore, thermodynamic potential functions do not depend on the order of differentiation. Applying this property yields four property relations.

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Thermodynamics in Energy Engineering

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