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### Derivation of the Second Energy Equation Using Maxwell’s Relations We begin with Maxwell’s relations, which are derived from the fundamental thermodynamic equations. One of Maxwell’s relations is: \[ \left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V \] We also know from thermodynamics that the internal energy \( U \) is expressed as: \[ dU = TdS - PdV \] Taking the partial derivative of \( U \) with respect to pressure at constant temperature: \[ \left(\frac{\partial U}{\partial P}\right)_T = T \left(\frac{\partial S}{\partial P}\right)_T - P \left(\frac{\partial V}{\partial _____ ___ _________ __________ __________ ____ _______ ___ ___ ___ _________.
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