Editing Enthalpy (section)

==Definition==
The enthalpy {{mvar|H}} of a thermodynamic system is defined as the sum of its internal energy and the product of its pressure and volume:<ref name=GoldBk-enth/>
<math display="block">
 H = U + p V,
</math>
where {{mvar|U}} is the [[internal energy]], {{mvar|p}} is [[pressure]], and {{mvar|V}} is the [[Volume (thermodynamics)|volume]] of the system;  {{mvar|p{{tsp}}V}} is sometimes referred to as the pressure energy {{mvar|Ɛ}}{{sub|p}}.<ref>{{Cite web |date=2013-10-02 |title=1st Law of Thermodynamics |url=https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Thermodynamics/The_Four_Laws_of_Thermodynamics/First_Law_of_Thermodynamics |access-date=2023-10-17 |website=Chemistry LibreTexts |language=en}}</ref>

Enthalpy is an [[extensive property]]; it is proportional to the size of the system (for homogeneous systems). As [[intensive properties]], the ''[[specific enthalpy]]'' {{nobr|{{math|''h'' {{=}} ''H''/''m''}}}} is referenced to a unit of [[mass]] {{mvar|m}} of the system, and the ''molar enthalpy'' {{nobr|{{math|''H{{sub|m}}'' {{=}} ''H''/''n''}},}} where {{mvar|n}} is the number of [[Mole (unit)|moles]]. For inhomogeneous systems the enthalpy is the sum of the enthalpies of the component subsystems:
<math display="block">
 H = \sum_k H_k,
</math>
where
: {{mvar|H}} is the total enthalpy of all the subsystems,
: {{mvar|k}} refers to the various subsystems,
: {{mvar|H{{sub|k}}}} refers to the enthalpy of each subsystem.

A closed system may lie in thermodynamic equilibrium in a static [[gravitational field]], so that its pressure {{mvar|p}} varies continuously with [[altitude]], while, because of the equilibrium requirement, its temperature {{mvar|T}} is invariant with altitude. (Correspondingly, the system's [[gravitational potential energy]] density also varies with altitude.) Then the enthalpy summation becomes an [[integral]]:
<math display="block">
 H = \int \rho h \,\mathrm{d}V,
</math>
where
: {{mvar|ρ}} ("[[rho]]") is [[density]] (mass per unit volume),
: {{mvar|h}} is the specific enthalpy (enthalpy per unit mass),
: {{math|''ρh''}} represents the [[Energy density|enthalpy density]] (enthalpy per unit volume),
: {{math|d''V''}} denotes an [[infinitesimal]]ly small element of volume within the system, for example, the volume of an infinitesimally thin horizontal layer.
The integral therefore represents the sum of the enthalpies of all the elements of the volume.

The enthalpy of a closed homogeneous system is its energy function {{nobr|{{math|''H''(''S'', ''p'')}},}} with its entropy {{math|''S''[''p'']}} and its pressure {{mvar|p}} as [[Thermodynamic potential#Natural variables|natural state variables]] which provide a differential relation for {{math|d''H''}} of the simplest form, derived as follows. We start from the [[first law of thermodynamics]] for closed systems for an infinitesimal process:
<math display="block">
 \mathrm{d}U = \delta Q - \delta W,
</math>
where
: {{mvar|δQ}} is a small amount of heat added to the system,
: {{mvar|δW}} is a small amount of work performed by the system.

In a homogeneous system in which only [[Reversible process (thermodynamics)|reversible]] processes or pure heat transfer are considered, the [[second law of thermodynamics]] gives {{nobr|{{math|''δQ'' {{=}} ''T''{{tsp}}d''S''}},}} with {{mvar|T}} the [[absolute temperature]] and {{math|d''S''}} the infinitesimal change in [[entropy]] {{mvar|S}} of the system. Furthermore, if only {{mvar|pV}} work is done, {{nobr|{{math|''δW'' {{=}} ''p''{{tsp}}d''V''}}.}} As a result,
<math display="block">
 \mathrm{d}U = T\,\mathrm{d}S - p\,\mathrm{d}V.
</math>

Adding {{math|d(''pV'')}} to both sides of this expression gives
<math display="block">
 \mathrm{d}U + \mathrm{d}(pV) = T\,\mathrm{d}S - p\,\mathrm{d}V + \mathrm{d}(p\,V),
</math>
or
<math display="block">
 \mathrm{d}(U + pV) = T\,\mathrm{d}S + V\,\mathrm{d}p.
</math>
So
<math display="block">
 \mathrm{d}H(S, p) = T\,\mathrm{d}S + V\,\mathrm{d}p,
</math>
and the coefficients of the natural variable differentials {{math|d''S''}} and {{math|d''p''}} are just the single variables {{mvar|T}} and {{mvar|V}}.