{title2=$H=U+pV$}
{wiki}

Enthalpy is the <state function> obtained by adding <pressure> times <volume (thermodynamics)> to <internal energy>:
$$
H=U+pV.
$$
Here $U$ denotes <internal energy>, also written $E$ in some references. Like <internal energy>, <enthalpy> is measured in <joules> and is an <extensive variable>. Its definition packages the <internal energy> change and expansion <work> into one useful quantity for fixed-<pressure> calculations. It does not define a body's stored <heat>. https://goldbook.iupac.org/terms/view/E02141/plain[IUPAC Gold Book, enthalpy]{ref}

For a fixed amount and composition of a <simple compressible system>, the relation between neighboring equilibrium states is
$$
dH=T\,dS+V\,dp.
$$
Here $T$ is absolute <temperature> and $S$ is <entropy>. Thus <entropy> and <pressure> are the natural variables of this <thermodynamic potential>, with <partial derivatives> $T=(\partial H/\partial S)_p$ and $V=(\partial H/\partial p)_S$. Fixed composition matters: changing composition adds further terms. The formula follows in one line from the equilibrium <first law of thermodynamics> identity:
$$
dH=dU+p\,dV+V\,dp=(T\,dS-p\,dV)+p\,dV+V\,dp=T\,dS+V\,dp.
$$
The cancellation of $p\,dV$ is the reason <enthalpy> replaces <volume (thermodynamics)> with <pressure> as a natural variable. This identity concerns equilibrium states; it does not identify $T\,dS$ with actual irreversible <heat> transfer. https://davidtong.org/teaching/statistical-physics/statmechhtml/S4#S4.SS4.SSS1[David Tong, Statistical Physics, section 4.4.1: enthalpy]{ref}

For a <closed system> undergoing a <quasistatic process> at constant <pressure>, require fixed amount and composition, only <pressure>-<volume (thermodynamics)> <work>, no <friction>, and unchanged bulk <kinetic energy> and <potential energy>. Then the actual <first law of thermodynamics> balance gives
$$
\delta Q=dU+p\,dV=dH-V\,dp,\qquad
dp=0\ \Longrightarrow\ Q_p=\Delta H.
$$
The symbol $Q_p$ means transferred <heat> for this fixed-<pressure> process. If the piston example in <heat> additionally meets these conditions, its $20\,\mathrm{kJ}$ of incoming <heat> equals $\Delta H$, while $\Delta U=15\,\mathrm{kJ}$ and $p\Delta V=W=5\,\mathrm{kJ}$. The example shows exactly what <enthalpy> adds to <internal energy>: the expansion <work> required at the imposed <pressure>.

Within a single <thermodynamic phase>, <heat capacity at constant pressure> gives
$$
C_p=\left(\frac{\partial H}{\partial T}\right)_p,\qquad
\Delta H=\int_{T_1}^{T_2}C_p(T)\,dT
$$
with the amount and composition fixed. Constant $C_p$ gives $\Delta H=C_p\Delta T$; dividing by <mass> gives <specific heat capacity at constant pressure>. At equilibrium <melting>, <enthalpy> can instead change through <latent heat> while <temperature> stays fixed. The <sensible heat> example therefore adds two single-<thermodynamic phase> <enthalpy> changes and one <latent heat> contribution.

The usefulness of <enthalpy> lies in matching the imposed constraint: a fixed-<pressure> experiment naturally reports $\Delta H$, while fixed-<volume (thermodynamics)> heating with no other <work> reports $\Delta U$. Neither equality is universal. Additional <work> or changing <pressure> alters the relation between supplied <heat> and $\Delta H$; irreversible <entropy> production also prevents identifying actual $\delta Q$ with $T\,dS$. The separate <Joule-Thomson coefficient> branch uses <enthalpy> to describe throttling under its own assumptions.


 Enthalpy (source code)