Enthalpy

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Definition of the enthalpy, H

$\left.H\right.=U+pV$

where $U$ is the internal energy, $p$ is the pressure, $V$ is the volume and (-pV) is a conjugate pair. The differential of this function is

$\left.dH\right.=dU+pdV+Vdp$

$\left.dH\right.=TdS -pdV +pdV+Vdp$

thus we arrive at

$\left.dH\right.=TdS +Vdp$

For H(S,p) we have the following total differential

$dH=\left(\frac{\partial H}{\partial S}\right)_p dS + \left(\frac{\partial H}{\partial p}\right)_S dp$