Enthalpy

Definition and physical meaning

For a thermodynamic system with internal energy $U$, pressure $p$, and volume $V$, the enthalpy is the state function

Physically, the $pV$ term accounts for the mechanical “flow work” needed to create space for the system in its surroundings . This makes $H$ a natural energy-like quantity for processes at approximately constant ambient pressure, and for open systems where material crosses a system boundary .

Differential form and natural variables

Using the first law of thermodynamics for a simple compressible system (only $p\,dV$ mechanical work and possible particle exchange), one obtains

where $S$ is the thermodynamic entropy , $T$ the temperature , $N$ the particle number , and $\mu$ the chemical potential . (For mixtures, replace $\mu\,dN$ by $\sum_i \mu_i\,dN_i$.)

This shows that $H$ is naturally expressed as $H(S,p,N)$ (for a single-component simple compressible system), with

• $T = \left(\partial H/\partial S\right)_{p,N}$,
• $V = \left(\partial H/\partial p\right)_{S,N}$,
• $\mu = \left(\partial H/\partial N\right)_{S,p}$.

Constant-pressure heating identity

With the usual $p\,dV$ work sign convention (so that $dU = \delta Q - p\,dV + \mu\,dN$), the enthalpy change simplifies to

For a closed system with fixed $N$ at constant pressure ($dp=0$), this reduces to

i.e. the heat absorbed at constant pressure equals the enthalpy change.

Relation to Legendre transforms

If the system admits a fundamental relation $U(S,V,N)$, then $H$ is obtained by a Legendre transform that trades the extensive variable $V$ for its conjugate intensive variable $p$ (defined by $p = -(\partial U/\partial V)_{S,N}$).