Legendre transform from internal energy to enthalpy

Suppose the macroscopic energy of a single-component system is written in the energy representation as $U=U(S,V,N)$, where $U$ is the internal energy and $S,V,N$ are the natural extensive variables. The conjugate intensive variables are temperature $T$, pressure $p$, and chemical potential $\mu$.

Definition (enthalpy as a Legendre transform in $V$).
Define $p$ in the energy representation by

The enthalpy is the Legendre transform of $U$ that replaces $V$ by $p$:

At the minimizing volume $V^\star(S,p,N)$, the stationarity condition reproduces the defining relation $p=-(\partial U/\partial V)_{S,N}$, and then $H=U+pV$ evaluated at $V^\star$.

Differentials and interpretation.
Starting from the fundamental differential

the transformed potential satisfies

Thus $H$ is the natural potential when entropy (or heat input) and pressure are controlled, because changes in $p$ appear linearly with coefficient $V$. Physically, the additional term $pV$ accounts for the mechanical work needed to “make room” for the system against an external pressure reservoir.

Link to ensembles.
In the isothermal–isobaric ensemble , the combination $U+pV$ appears in the Boltzmann weight before taking the temperature transform, and this structure is reflected in the isothermal–isobaric partition function . Completing the additional Legendre transform in $S$ leads to the Gibbs free energy construction .