Enthalpy

Enthalpy ($H$) is defined as:

$H=U+pV$

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

$dH=dU+pdV+Vdp$

From the Second law of thermodynamics one obtains

$dH=TdS-pdV+pdV+Vdp$

thus we arrive at

$dH=TdS+Vdp$

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

$dH={(\frac{\partial H}{\partial S})}_{p}dS+{(\frac{\partial H}{\partial p})}_{S}dp$