In my book it talks about deriving the Maxwell relations and how they can be obtained by considering the differential changes of the enthalpy. By doing this from $$H=U+PV$$ we get $$dH=TdS+VdP$$ and from equation $$U=-PV+H$$ we get $$dU=-PdV+TdS$$ My question is why is it for $dH=TdS+VdP$ its $VdP$ and not $PdV$ and similarly for $U=-PV+H$ why is it $-PdV$ and not $-VdP$?


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The starting point is the fundamental differential $$dU = T dS - P dV\tag{1}$$ This is a fundamental result in thermodynamics and is obtained by combining the first and second law on a closed system undergong reversible process.

For enthalpy start with $H=U+PV$ and take its differential: $$dH = dU + P dV + V dP$$ Applying (1) we obtain $$\boxed{dH = T dS + V dP}$$


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