Editing Enthalpy (section)

==Other expressions==
The above expression of {{math|d''H''}} in terms of entropy and pressure may be unfamiliar to some readers. There are also expressions in terms of more directly measurable variables such as temperature and pressure:<ref name=Guggenheim>
{{cite book
 |last=Guggenheim |first=E. A.
 |year=1959
 |title=Thermodynamics
 |publisher=North-Holland Publishing Company
 |place=Amsterdam, NL
}}</ref>{{rp|style=AMA|p= 88}}<ref>
{{cite book
 |first1=M. J. |last1=Moran
 |first2=H. N. |last2=Shapiro
 |year=2006
 |title=Fundamentals of Engineering Thermodynamics |edition=5th
 |publisher=John Wiley & Sons
 |page=[https://archive.org/details/fundamentalsengi00mora_077/page/n523 511]
 |isbn=9780470030370
 |url=https://archive.org/details/fundamentalsengi00mora_077 |url-access=limited
}}
</ref>
<math display="block">
 \mathrm{d}H = C_\mathsf{p}\,\mathrm{d}T + V(1 - \alpha T)\,\mathrm{d}p,
</math>
where {{math|''C{{sub|p}}''}} is the [[heat capacity]] at ''constant [[pressure]]'', and {{mvar|α}} is the [[Coefficient of thermal expansion|coefficient of (cubic) thermal expansion]]:
<math display="block">
 \alpha = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_p.
</math>

With this expression one can, in principle, determine the enthalpy if {{mvar|C{{sub|p}}}} and {{mvar|V}} are known as functions of {{mvar|p}} and {{mvar|T}}. However the expression is more complicated than <math>\mathrm{d}H = T\,\mathrm{d}S + V\,\mathrm{d}p</math> because {{mvar|T}} is not a natural variable for the enthalpy {{mvar|H}}.

At constant pressure, <math>\mathrm{d}P = 0</math> so that <math>\mathrm{d}H = C_\mathsf{p}\,\mathrm{d}T.</math> For an [[ideal gas]], <math>\mathrm{d}H</math> reduces to this form even if the process involves a pressure change, because {{nobr|{{math|''αT'' {{=}} 1}}.}}<ref group="note"><math>\alpha T = \frac{T}{V} \left(\frac{\partial(\frac{nRT}{P})}{\partial T}\right)_p = \frac{nRT}{PV} = 1.</math></ref>

In a more general form, the first law describes the internal energy with additional terms involving the [[chemical potential]] and the number of particles of various types. The differential statement for {{math|d''H''}} then becomes
<math display="block">
 \mathrm{d}H = T\,\mathrm{d}S + V\,\mathrm{d}p + \sum_i \mu_i\,\mathrm{d}N_i,
</math>
where {{mvar|μ{{sub|i}}}} is the chemical potential per particle for a type&nbsp;{{mvar|i}} particle, and {{mvar|N{{sub|i}}}} is the number of such particles. The last term can also be written as {{math|''μ{{sub|i}}''&thinsp;d''n{{sub|i}}''}} (with {{math|d''n''{{sub|''i'' 0}}}} the number of moles of component {{mvar|i}} added to the system and, in this case, {{mvar|μ{{sub|i}}}} the molar chemical potential) or as {{math|''μ{{sub|i}}''{{tsp}}d''m{{sub|i}}''}} (with {{math|d''m{{sub|i}}''}} the mass of component {{mvar|i}} added to the system and, in this case, {{mvar|μ{{sub|i}}}} the specific chemical potential).

===Characteristic functions and natural state variables===

The enthalpy {{nobr|{{math|''H''(''S''[''p''], ''p'', {{mset|''N{{sub|i}}''}})}}}} expresses the thermodynamics of a system in the ''energy representation''. As a [[State function|function of state]], its arguments include one intensive and several extensive [[state variable]]s. The state variables {{nobr|{{math|''S''[''p'']}},}} {{mvar|p}}, and {{math|{{mset|''N{{sub|i}}''}}}} are said to be the [[Thermodynamic potential#Natural variables|''natural state variables'']] in this representation. They are suitable for describing processes in which they are determined by factors in the surroundings. For example, when a virtual parcel of atmospheric air moves to a different altitude, the pressure surrounding it changes, and the process is often so rapid that there is too little time for heat transfer. This is the basis of the so-called [[adiabatic approximation]] that is used in [[meteorology]].<ref>
{{cite book
 |last1=Iribarne |first1=J. V.
 |last2=Godson   |first2=W. L.
 |year=1981
 |title=Atmospheric Thermodynamics |edition=2nd
 |publisher=Kluwer Academic Publishers
 |place=Dordrecht, NL
 |isbn=90-277-1297-2
 |pages=235–236
}}
</ref>

Conjugate with the enthalpy, with these arguments, the other characteristic function of state of a thermodynamic system is its entropy, as a function {{nobr|{{math|''S''[''p''](''H'', ''p'', {{mset|N{{sub|i}}}})}}}} of the same list of variables of state, except that the entropy {{math|''S''[''p'']}} is replaced in the list by the enthalpy {{mvar|H}}. It expresses the ''entropy representation''. The state variables {{mvar|H}}, {{mvar|p}}, and {{math|{{mset|''N{{sub|i}}''}}}} are said to be the ''natural state variables'' in this representation. They are suitable for describing processes in which they are experimentally controlled. For example, {{mvar|H}} and {{mvar|p}} can be controlled by allowing heat transfer, and by varying only the external pressure on the piston that sets the volume of the system.<ref>
{{cite book
 |last=Tschoegl |first=N. W.
 |year=2000
 |title=Fundamentals of Equilibrium and Steady-State Thermodynamics
 |publisher=Elsevier
 |place=Amsterdam, NL
 |isbn=0-444-50426-5
 |page=17
}}
</ref><ref>
{{cite book
 |last=Callen |first=H. B.
 |orig-date=1960 |year=1985
 |title=Thermodynamics and an Introduction to Thermostatistics |edition=1st (1960), 2nd (1985)
 |publisher=John Wiley & Sons
 |place=New York, NY
 |isbn=0-471-86256-8
 |at=Chapter&nbsp;5
}}
</ref><ref>
{{cite book
 |last=Münster |first=A.
 |year=1970
 |title=Classical Thermodynamics
 |translator-first=E. S. |translator-last=Halberstadt
 |publisher=Wiley–Interscience
 |place=London, UK
 |isbn=0-471-62430-6
 |page=6
}}
</ref>