Enthalpy

We want to define a state function $H$ such that $\d H|_p = \db Q$ . It is useful to have some exact analog for heat since $Q$ is inexact. We write

\begin{align*} \d U &= \d Q - p\d V = \db Q - \d(pV) + V\d p \\ \d(U + pV) &= \d U + p\d V + V\d p = \db Q + V\d p \\ H &:= U + pV = \text{enthalpy} \\ \d H|_p &= \db Q. \end{align*}

Differential enthalpy $\d H|_p$ represents both heat and the work required to maintain a constant pressure.