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

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

From the Second law of thermodynamics one obtains

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

thus we arrive at

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

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

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