Using Maxwell's relations, derive the first and second energy equations. 

Maxwell's relations are derived from the thermodynamic potentials and are fundamental in thermodynamics and statistical mechanics. To derive the first and second energy equations, we utilize the thermodynamic potentials and the differential forms of internal energy \( U \), enthalpy \( H \), Helmholtz free energy \( F __________ _____ _____ ___ ______ _____ _______ __________ ______ ________ _______.
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