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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the mathematical concept used in thermodynamics. For the engineering indicator, see <a href="Process_variable" title="Process variable">process variable</a>.</div>
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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.3em;border-bottom:1px solid #aaa;"><a href="Thermodynamics" title="Thermodynamics">Thermodynamics</a></th></tr><tr><td class="sidebar-image" style="display:block;margin:0.3em 0 0.4em;"><div class="sidebar-caption">The classical <a href="Carnot_heat_engine" title="Carnot heat engine">Carnot heat engine</a></div></td></tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;;color: var(--color-base)"><div class="sidebar-list-title-c">Branches</div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist">
<ul><li><a href="Thermodynamics" title="Thermodynamics">Classical</a></li>
<li><a href="Statistical_mechanics" title="Statistical mechanics">Statistical</a></li>
<li><a href="Chemical_thermodynamics" title="Chemical thermodynamics">Chemical</a></li>
<li><a href="Quantum_thermodynamics" title="Quantum thermodynamics">Quantum thermodynamics</a></li></ul>
</div>
<ul><li><a href="Equilibrium_thermodynamics" title="Equilibrium thermodynamics">Equilibrium</a> / <a href="Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">Non-equilibrium</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Laws_of_thermodynamics" title="Laws of thermodynamics">Laws</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist">
<ul><li><a href="Zeroth_law_of_thermodynamics" title="Zeroth law of thermodynamics">Zeroth</a></li>
<li><a href="First_law_of_thermodynamics" title="First law of thermodynamics">First</a></li>
<li><a href="Second_law_of_thermodynamics" title="Second law of thermodynamics">Second</a></li>
<li><a href="Third_law_of_thermodynamics" title="Third law of thermodynamics">Third</a></li></ul>
</div></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Thermodynamic_system" title="Thermodynamic system">Systems</a></div></div><div class="sidebar-list-content mw-collapsible-content">
<ul><li><a href="Closed_system" title="Closed system">Closed system</a></li>
<li><a href="Thermodynamic_system#Open_system" title="Thermodynamic system">Open system</a></li>
<li><a href="Isolated_system" title="Isolated system">Isolated system</a></li></ul>
<table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;">
<a href="Thermodynamic_state" title="Thermodynamic state">State</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Equation_of_state" title="Equation of state">Equation of state</a></li>
<li><a href="Ideal_gas" title="Ideal gas">Ideal gas</a></li>
<li><a href="Real_gas" title="Real gas">Real gas</a></li>
<li><a href="State_of_matter" title="State of matter">State of matter</a></li>
<li><a href="Phase_(matter)" title="Phase (matter)">Phase (matter)</a></li>
<li><a href="Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Equilibrium</a></li>
<li><a href="Control_volume" title="Control volume">Control volume</a></li>
<li><a href="Thermodynamic_instruments" title="Thermodynamic instruments">Instruments</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;">
<a href="Thermodynamic_process" title="Thermodynamic process">Processes</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Isobaric_process" title="Isobaric process">Isobaric</a></li>
<li><a href="Isochoric_process" title="Isochoric process">Isochoric</a></li>
<li><a href="Isothermal_process" title="Isothermal process">Isothermal</a></li>
<li><a href="Adiabatic_process" title="Adiabatic process">Adiabatic</a></li>
<li><a href="Isentropic_process" title="Isentropic process">Isentropic</a></li>
<li><a href="Isenthalpic_process" title="Isenthalpic process">Isenthalpic</a></li>
<li><a href="Quasistatic_process" title="Quasistatic process">Quasistatic</a></li>
<li><a href="Polytropic_process" title="Polytropic process">Polytropic</a></li>
<li><a href="Free_expansion" class="mw-redirect" title="Free expansion">Free expansion</a></li>
<li><a href="Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">Reversibility</a></li>
<li><a href="Irreversible_process" title="Irreversible process">Irreversibility</a></li>
<li><a href="Endoreversible_thermodynamics" title="Endoreversible thermodynamics">Endoreversibility</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;">
<a href="Thermodynamic_cycle" title="Thermodynamic cycle">Cycles</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Heat_engine" title="Heat engine">Heat engines</a></li>
<li><a href="Heat_pump_and_refrigeration_cycle" title="Heat pump and refrigeration cycle">Heat pumps</a></li>
<li><a href="Thermal_efficiency" title="Thermal efficiency">Thermal efficiency</a></li></ul></td>
</tr></tbody></table></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="List_of_thermodynamic_properties" title="List of thermodynamic properties">System properties</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div style="font-size:90%;padding-bottom:0.2em;border-bottom:1px solid #aaa;">Note: <a href="Conjugate_variables_(thermodynamics)" title="Conjugate variables (thermodynamics)">Conjugate variables</a> in <i>italics</i></div>
<table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;margin-top:0.4em;"><tbody><tr><td class="sidebar-content" style="padding-bottom:0.7em;">
<ul><li><a href="Thermodynamic_diagrams" title="Thermodynamic diagrams">Property diagrams</a></li>
<li><a href="Intensive_and_extensive_properties" title="Intensive and extensive properties">Intensive and extensive properties</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;">
</th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;;padding-bottom:0.4em;">
<div class="hlist">
<ul><li><a href="Work_(thermodynamics)" title="Work (thermodynamics)">Work</a></li>
<li><a href="Heat" title="Heat">Heat</a></li></ul>
</div></td>
</tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;">
<a href="State_function" title="State function">Functions of state</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;">
<ul><li><a href="Thermodynamic_temperature" title="Thermodynamic temperature">Temperature</a> / <i><a href="Entropy" title="Entropy">Entropy</a></i> (<a href="Introduction_to_entropy" title="Introduction to entropy">introduction</a>)</li>
<li><a href="Pressure" title="Pressure">Pressure</a> / <i><a href="Volume_(thermodynamics)" title="Volume (thermodynamics)">Volume</a></i></li>
<li><a href="Chemical_potential" title="Chemical potential">Chemical potential</a> / <i><a href="Particle_number" title="Particle number">Particle number</a></i></li>
<li><a href="Vapor_quality" title="Vapor quality">Vapor quality</a></li>
<li><a href="Reduced_properties" title="Reduced properties">Reduced properties</a></li></ul></td>
</tr></tbody></table></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Material_properties_(thermodynamics)" title="Material properties (thermodynamics)">Material properties</a></div></div><div class="sidebar-list-content mw-collapsible-content">
<ul><li><a href="Thermodynamic_databases_for_pure_substances" title="Thermodynamic databases for pure substances">Property databases</a></li></ul>
<div style="font-size:90%;margin-top:0.4em;border-top:1px solid #aaa;">
<table>
<tbody><tr><td style="vertical-align:middle; text-align:right"><a href="Heat_capacity" title="Heat capacity">Specific heat capacity</a> </td>
<td style="vertical-align:middle; text-align:left"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=}">
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<td><table><tbody><tr><td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">
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<tr><td style="vertical-align:middle; text-align:right"><a href="Compressibility" title="Compressibility">Compressibility</a> </td>
<td style="vertical-align:middle; text-align:left"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =-}">
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<td><table><tbody><tr><td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">
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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>
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</math></span><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span></td><td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>V</mi>
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<annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation>
</semantics>
</math></span><img src="./0cecdd9d069fa84159940068fc11a91b6b3b9ee4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}" loading="lazy"></span></td></tr><tr><td style="border-top:solid 1px black;"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span></td><td style="border-top:solid 1px black;"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \partial p}</annotation>
</semantics>
</math></span><img src="./ebc4a48eb2412f08b54fe438b5139c88f9cfa372.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.487ex; height:2.509ex;" alt="{\displaystyle \partial p}" loading="lazy"></span></td></tr></tbody></table></td></tr>
<tr><td style="vertical-align:middle; text-align:right"><a href="Thermal_expansion" title="Thermal expansion">Thermal expansion</a> </td>
<td style="vertical-align:middle; text-align:left"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
<mo>=</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha =}</annotation>
</semantics>
</math></span><img src="./a92d4583d351f08c1c70985f0c843b2fff1b01e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.941ex; height:1.676ex;" alt="{\displaystyle \alpha =}" loading="lazy"></span></td>
<td><table><tbody><tr><td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>
</semantics>
</math></span><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span></td><td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation>
</semantics>
</math></span><img src="./0cecdd9d069fa84159940068fc11a91b6b3b9ee4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}" loading="lazy"></span></td></tr><tr><td style="border-top:solid 1px black;"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span></td><td style="border-top:solid 1px black;"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial T}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>T</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \partial T}</annotation>
</semantics>
</math></span><img src="./504aa558fff3d00d10b03cadb1085cb0b7bdc631.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="{\displaystyle \partial T}" loading="lazy"></span></td></tr></tbody></table></td></tr>
</tbody></table></div></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Thermodynamic_equations" title="Thermodynamic equations">Equations</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist">
<ul><li><a href="Carnot's_theorem_(thermodynamics)" title="Carnot's theorem (thermodynamics)">Carnot's theorem</a></li>
<li><a href="Clausius_theorem" title="Clausius theorem">Clausius theorem</a></li>
<li><a href="Fundamental_thermodynamic_relation" title="Fundamental thermodynamic relation">Fundamental relation</a></li>
<li><a href="Ideal_gas_law" title="Ideal gas law">Ideal gas law</a></li></ul>
</div>
<ul><li><a href="Maxwell_relations" title="Maxwell relations">Maxwell relations</a></li>
<li><a href="Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager reciprocal relations</a></li>
<li><a href="Bridgman's_thermodynamic_equations" title="Bridgman's thermodynamic equations">Bridgman's equations</a></li>
<li><i><a href="Table_of_thermodynamic_equations" title="Table of thermodynamic equations">Table of thermodynamic equations</a></i></li></ul></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Thermodynamic_potential" title="Thermodynamic potential">Potentials</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist">
<ul><li><a href="Thermodynamic_free_energy" title="Thermodynamic free energy">Free energy</a></li>
<li><a href="Free_entropy" title="Free entropy">Free entropy</a></li></ul>
</div>
<div class="plainlist"><ul><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="Internal_energy" title="Internal energy">Internal energy</a><br><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(S,V)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
<mo stretchy="false">(</mo>
<mi>S</mi>
<mo>,</mo>
<mi>V</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U(S,V)}</annotation>
</semantics>
</math></span><img src="./921f33f9c6551562ec836007b035c2de6323d2d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.912ex; height:2.843ex;" alt="{\displaystyle U(S,V)}" loading="lazy"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="Enthalpy" title="Enthalpy">Enthalpy</a><br><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(S,p)=U+pV}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
<mo stretchy="false">(</mo>
<mi>S</mi>
<mo>,</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>U</mi>
<mo>+</mo>
<mi>p</mi>
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H(S,p)=U+pV}</annotation>
</semantics>
</math></span><img src="./6407d78e5f39d07f70e2414a92e08e2e068519f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.254ex; height:2.843ex;" alt="{\displaystyle H(S,p)=U+pV}" loading="lazy"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a><br><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(T,V)=U-TS}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo stretchy="false">(</mo>
<mi>T</mi>
<mo>,</mo>
<mi>V</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>U</mi>
<mo>−<!-- − --></mo>
<mi>T</mi>
<mi>S</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A(T,V)=U-TS}</annotation>
</semantics>
</math></span><img src="./5e93692f031ba6484d82731c54db83a69daed3f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.867ex; height:2.843ex;" alt="{\displaystyle A(T,V)=U-TS}" loading="lazy"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a><br><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(T,p)=H-TS}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>G</mi>
<mo stretchy="false">(</mo>
<mi>T</mi>
<mo>,</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>H</mi>
<mo>−<!-- − --></mo>
<mi>T</mi>
<mi>S</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle G(T,p)=H-TS}</annotation>
</semantics>
</math></span><img src="./8dd7a8f0b8ae04963da133e3b202432e1b6caed4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.614ex; height:2.843ex;" alt="{\displaystyle G(T,p)=H-TS}" loading="lazy"></span></li></ul></div></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>History</li><li>Culture</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;">
History</th></tr><tr><td class="sidebar-content">
<div class="hlist">
<ul><li><a href="History_of_thermodynamics" title="History of thermodynamics">General</a></li>
<li><a href="History_of_entropy" title="History of entropy">Entropy</a></li>
<li><a href="Gas_laws" title="Gas laws">Gas laws</a></li></ul>
</div>
<ul><li><a href="History_of_perpetual_motion_machines" title="History of perpetual motion machines">"Perpetual motion" machines</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;">
<a href="Philosophy_of_thermal_and_statistical_physics" class="mw-redirect" title="Philosophy of thermal and statistical physics">Philosophy</a></th></tr><tr><td class="sidebar-content">
<div class="hlist">
<ul><li><a href="Entropy_(arrow_of_time)" class="mw-redirect" title="Entropy (arrow of time)">Entropy and time</a></li>
<li><a href="Entropy_and_life" title="Entropy and life">Entropy and life</a></li>
<li><a href="Brownian_ratchet" title="Brownian ratchet">Brownian ratchet</a></li>
<li><a href="Maxwell's_demon" title="Maxwell's demon">Maxwell's demon</a></li>
<li><a href="Heat_death_paradox" title="Heat death paradox">Heat death paradox</a></li>
<li><a href="Loschmidt's_paradox" title="Loschmidt's paradox">Loschmidt's paradox</a></li>
<li><a href="Synergetics_(Haken)" title="Synergetics (Haken)">Synergetics</a></li></ul>
</div></td>
</tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;">
Theories</th></tr><tr><td class="sidebar-content">
<div class="hlist">
<ul><li><a href="Caloric_theory" title="Caloric theory">Caloric theory</a></li></ul>
</div>
<ul><li><a href="Vis_viva" title="Vis viva"><i>Vis viva</i> <span style="font-size: 85%;">("living force")</span></a></li>
<li><a href="Mechanical_equivalent_of_heat" title="Mechanical equivalent of heat">Mechanical equivalent of heat</a></li>
<li><a href="Power_(physics)" title="Power (physics)">Motive power</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;">
<a href="List_of_important_publications_in_physics" class="mw-redirect" title="List of important publications in physics">Key publications</a></th></tr><tr><td class="sidebar-content">
<ul><li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="An_Inquiry_Concerning_the_Source_of_the_Heat_Which_Is_Excited_by_Friction" title="An Inquiry Concerning the Source of the Heat Which Is Excited by Friction">An Inquiry Concerning the<br>Source ... Friction</a></i></div></li>
<li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="On_the_Equilibrium_of_Heterogeneous_Substances" title="On the Equilibrium of Heterogeneous Substances">On the Equilibrium of<br>Heterogeneous Substances</a></i></div></li>
<li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="Reflections_on_the_Motive_Power_of_Fire" title="Reflections on the Motive Power of Fire">Reflections on the<br>Motive Power of Fire</a></i></div></li></ul></td>
</tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;">
Timelines</th></tr><tr><td class="sidebar-content">
<div class="hlist">
<ul><li><a href="Timeline_of_thermodynamics" title="Timeline of thermodynamics">Thermodynamics</a></li>
<li><a href="Timeline_of_heat_engine_technology" title="Timeline of heat engine technology">Heat engines</a></li></ul>
</div></td>
</tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;">
<div class="hlist"><ul><li>Art</li><li>Education</li></ul></div></th></tr><tr><td class="sidebar-content">
<ul><li><a href="Maxwell's_thermodynamic_surface" title="Maxwell's thermodynamic surface">Maxwell's thermodynamic surface</a></li>
<li><a href="Entropy_(energy_dispersal)" title="Entropy (energy dispersal)">Entropy as energy dispersal</a></li></ul></td>
</tr></tbody></table></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;;color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist">
<ul><li><a href="Daniel_Bernoulli" title="Daniel Bernoulli">Bernoulli</a></li>
<li><a href="Ludwig_Boltzmann" title="Ludwig Boltzmann">Boltzmann</a></li>
<li><a href="Percy_Williams_Bridgman" title="Percy Williams Bridgman">Bridgman</a></li>
<li><a href="Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">Carathéodory</a></li>
<li><a href="Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Carnot</a></li>
<li><a href="Beno%C3%AEt_Paul_%C3%89mile_Clapeyron" class="mw-redirect" title="Benoît Paul Émile Clapeyron">Clapeyron</a></li>
<li><a href="Rudolf_Clausius" title="Rudolf Clausius">Clausius</a></li>
<li><a href="Th%C3%A9ophile_de_Donder" class="mw-redirect" title="Théophile de Donder">de Donder</a></li>
<li><a href="Pierre_Duhem" title="Pierre Duhem">Duhem</a></li>
<li><a href="Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li>
<li><a href="Hermann_von_Helmholtz" title="Hermann von Helmholtz">von Helmholtz</a></li>
<li><a href="James_Prescott_Joule" title="James Prescott Joule">Joule</a></li>
<li><a href="Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li>
<li><a href="Gilbert_N._Lewis" title="Gilbert N. Lewis">Lewis</a></li>
<li><a href="Fran%C3%A7ois_Massieu" title="François Massieu">Massieu</a></li>
<li><a href="James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li>
<li><a href="Julius_von_Mayer" title="Julius von Mayer">von Mayer</a></li>
<li><a href="Walther_Nernst" title="Walther Nernst">Nernst</a></li>
<li><a href="Lars_Onsager" title="Lars Onsager">Onsager</a></li>
<li><a href="Max_Planck" title="Max Planck">Planck</a></li>
<li><a href="William_John_Macquorn_Rankine" class="mw-redirect" title="William John Macquorn Rankine">Rankine</a></li>
<li><a href="John_Smeaton" title="John Smeaton">Smeaton</a></li>
<li><a href="Georg_Ernst_Stahl" title="Georg Ernst Stahl">Stahl</a></li>
<li><a href="Peter_Tait_(physicist)" class="mw-redirect" title="Peter Tait (physicist)">Tait</a></li>
<li><a href="Benjamin_Thompson" title="Benjamin Thompson">Thompson</a></li>
<li><a href="Johannes_Diderik_van_der_Waals" title="Johannes Diderik van der Waals">van der Waals</a></li>
<li><a href="John_James_Waterston" title="John James Waterston">Waterston</a></li></ul>
</div></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;;color: var(--color-base)"><div class="sidebar-list-title-c">Other</div></div><div class="sidebar-list-content mw-collapsible-content">
<ul><li><a href="Nucleation" title="Nucleation">Nucleation</a></li>
<li><a href="Self-assembly" title="Self-assembly">Self-assembly</a></li>
<li><a href="Self-organization" title="Self-organization">Self-organization</a></li></ul></div></div></td>
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<div role="note" class="hatnote navigation-not-searchable">See also: <a href="List_of_thermodynamic_properties" title="List of thermodynamic properties">List of thermodynamic properties</a></div>
<p>In <a href="Thermodynamics" title="Thermodynamics">thermodynamics</a>, a <a href="Quantity" title="Quantity">quantity</a> that is well defined so as to describe the path of a process through the <a href="Equilibrium_state" class="mw-redirect" title="Equilibrium state">equilibrium state</a> space of a <a href="Thermodynamic_system" title="Thermodynamic system">thermodynamic system</a> is termed a <b>process function</b>,<sup id="cite_ref-Sychev1991_1-0" class="reference"><a href="#cite_note-Sychev1991-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> or, alternatively, a <b>process quantity</b>, or a <b>path function</b>. As an example, <a href="Mechanical_work" class="mw-redirect" title="Mechanical work">mechanical work</a> and <a href="Heat" title="Heat">heat</a> are process functions because they describe quantitatively the transition between equilibrium states of a thermodynamic system.
</p><p>Path functions depend on the path taken to reach one state from another. Different routes give different quantities. Examples of path functions include <a href="Work_(thermodynamics)" title="Work (thermodynamics)">work</a>, <a href="Heat" title="Heat">heat</a> and <a href="Arc_length" title="Arc length">arc length</a>. In contrast to path functions, <a href="State_function" title="State function">state functions</a> are independent of the path taken. Thermodynamic <a href="State_function" title="State function">state variables</a> are point functions, differing from path functions. For a given state, considered as a point, there is a definite value for each state variable and state function.
</p><p>Infinitesimal changes in a process function <span class="texhtml mvar" style="font-style:italic;">X</span> are often indicated by <span class="texhtml mvar" style="font-style:italic;">δX</span> to distinguish them from infinitesimal changes in a state function <span class="texhtml mvar" style="font-style:italic;">Y</span> which is written <span class="texhtml mvar" style="font-style:italic;">dY</span>. The quantity <span class="texhtml mvar" style="font-style:italic;">dY</span> is an <a href="Exact_differential" title="Exact differential">exact differential</a>, while <span class="texhtml mvar" style="font-style:italic;">δX</span> is not, it is an <a href="Inexact_differential" title="Inexact differential">inexact differential</a>. Infinitesimal changes in a process function may be integrated, but the integral between two states depends on the particular path taken between the two states, whereas the integral of a state function is simply the difference of the state functions at the two points, independent of the path taken.
</p><p>In general, a process function <span class="texhtml mvar" style="font-style:italic;">X</span> may be either <a href="Holonomic_constraints" title="Holonomic constraints">holonomic</a> or non-holonomic. For a holonomic process function, an auxiliary state function (or integrating factor) <span class="texhtml mvar" style="font-style:italic;">λ</span> may be defined such that <span class="texhtml"><i>Y</i> = <i>λX</i></span> is a state function. For a non-holonomic process function, no such function may be defined. In other words, for a holonomic process function, <span class="texhtml mvar" style="font-style:italic;">λ</span> may be defined such that <span class="texhtml"><i>dY</i> = <i>λδX</i></span> is an exact differential. For example, thermodynamic work is a holonomic process function since the integrating factor <span class="texhtml"><i>λ</i> = <style data-mw-deduplicate="TemplateStyles:r1214402035">
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</style><span class="sfrac"><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>p</i></span></span></span></span> (where <span class="texhtml mvar" style="font-style:italic;">p</span> is pressure) will yield exact differential of the volume state function <span class="texhtml"><i>dV</i> = <span class="sfrac"><span class="tion"><span class="num"><i>δW</i></span><span class="sr-only">/</span><span class="den"><i>p</i></span></span></span></span>. The <a href="Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a> as stated by <a href="Carath%C3%A9odory" class="mw-redirect" title="Carathéodory">Carathéodory</a> essentially amounts to the statement that heat is a holonomic process function since the integrating factor <span class="texhtml"><i>λ</i> = <span class="sfrac"><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>T</i></span></span></span></span> (where <span class="texhtml mvar" style="font-style:italic;">T</span> is temperature) will yield the exact differential of an entropy state function <span class="texhtml mvar" style="font-style:italic;"><i>dS</i> = <span class="sfrac"><span class="tion"><span class="num"><i>δQ</i></span><span class="sr-only">/</span><span class="den"><i>T</i></span></span></span></span>.<sup id="cite_ref-Sychev1991_1-1" class="reference"><a href="#cite_note-Sychev1991-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-Sychev1991-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Sychev1991_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Sychev1991_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFSychev1991" class="citation book cs1">Sychev, V. V. (1991). <i>The Differential Equations of Thermodynamics</i>. Taylor & Francis. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1560321217</bdi>.</cite></span>
</li>
</ol></div>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Thermodynamics" title="Thermodynamics">Thermodynamics</a></li></ul>
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