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<div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Real-gas relations</span> </div> </a> <button aria-controls="toc-Real-gas_relations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Real-gas relations subsection</span> </button> <ul id="toc-Real-gas_relations-sublist" class="vector-toc-list"> <li id="toc-Thermodynamic_expressions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Thermodynamic_expressions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Thermodynamic expressions</span> </div> </a> <ul id="toc-Thermodynamic_expressions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Adiabatic_process" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Adiabatic_process"> <div 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class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon 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class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%B0%D0%BA%D0%B0%D0%B7%D1%87%D1%8B%D0%BA_%D0%B0%D0%B4%D1%8B%D1%8F%D0%B1%D0%B0%D1%82%D1%8B" title="Паказчык адыябаты – Belarusian" lang="be" hreflang="be" data-title="Паказчык адыябаты" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Coeficient_de_dilataci%C3%B3_adiab%C3%A0tica" title="Coeficient de dilatació adiabàtica – Catalan" lang="ca" hreflang="ca" data-title="Coeficient de dilatació adiabàtica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Poissonova_konstanta" title="Poissonova konstanta – Czech" lang="cs" hreflang="cs" data-title="Poissonova konstanta" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Adiabateksponenten" title="Adiabateksponenten – Danish" lang="da" hreflang="da" data-title="Adiabateksponenten" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/W%C3%A4rmekapazit%C3%A4tsverh%C3%A4ltnis" title="Wärmekapazitätsverhältnis – German" lang="de" hreflang="de" data-title="Wärmekapazitätsverhältnis" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Coeficiente_de_dilataci%C3%B3n_adiab%C3%A1tica" title="Coeficiente de dilatación adiabática – Spanish" lang="es" hreflang="es" data-title="Coeficiente de dilatación adiabática" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Indice_adiabatique" title="Indice adiabatique – French" lang="fr" hreflang="fr" data-title="Indice adiabatique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/C%C3%B3imheas_saintoilleadh_teasa" title="Cóimheas saintoilleadh teasa – Irish" lang="ga" hreflang="ga" data-title="Cóimheas saintoilleadh teasa" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B9%84%EC%97%B4%EB%B9%84" title="비열비 – Korean" lang="ko" hreflang="ko" data-title="비열비" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Coefficiente_di_dilatazione_adiabatica" title="Coefficiente di dilatazione adiabatica – Italian" lang="it" hreflang="it" data-title="Coefficiente di dilatazione adiabatica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%99%D7%97%D7%A1_%D7%A7%D7%99%D7%91%D7%95%D7%9C%D7%99_%D7%94%D7%97%D7%95%D7%9D" title="יחס קיבולי החום – Hebrew" lang="he" hreflang="he" data-title="יחס קיבולי החום" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Adiabatikus_kitev%C5%91" title="Adiabatikus kitevő – Hungarian" lang="hu" hreflang="hu" data-title="Adiabatikus kitevő" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B0%D1%81%D0%BE%D0%BD%D0%BE%D0%B2%D0%B0_%D0%BA%D0%BE%D0%BD%D1%81%D1%82%D0%B0%D0%BD%D1%82%D0%B0" title="Поасонова константа – Macedonian" lang="mk" hreflang="mk" data-title="Поасонова константа" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Gamma_(thermodynamica)" title="Gamma (thermodynamica) – Dutch" lang="nl" hreflang="nl" data-title="Gamma (thermodynamica)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%AF%94%E7%86%B1%E6%AF%94" title="比熱比 – Japanese" lang="ja" hreflang="ja" data-title="比熱比" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Adiabateksponenten" title="Adiabateksponenten – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Adiabateksponenten" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wyk%C5%82adnik_adiabaty" title="Wykładnik adiabaty – Polish" lang="pl" hreflang="pl" data-title="Wykładnik adiabaty" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Coeficiente_de_expans%C3%A3o_adiab%C3%A1tica" title="Coeficiente de expansão adiabática – Portuguese" lang="pt" hreflang="pt" data-title="Coeficiente de expansão adiabática" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Coeficient_de_transformare_adiabatic%C4%83" title="Coeficient de transformare adiabatică – Romanian" lang="ro" hreflang="ro" data-title="Coeficient de transformare adiabatică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BA%D0%B0%D0%B7%D0%B0%D1%82%D0%B5%D0%BB%D1%8C_%D0%B0%D0%B4%D0%B8%D0%B0%D0%B1%D0%B0%D1%82%D1%8B" title="Показатель адиабаты – Russian" lang="ru" hreflang="ru" data-title="Показатель адиабаты" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Poissonova_kon%C5%A1tanta_(termodynamika)" title="Poissonova konštanta (termodynamika) – Slovak" lang="sk" hreflang="sk" data-title="Poissonova konštanta (termodynamika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Adiabatni_eksponent" title="Adiabatni eksponent – Slovenian" lang="sl" hreflang="sl" data-title="Adiabatni eksponent" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Is%C4%B1_s%C4%B1%C4%9Fas%C4%B1_oran%C4%B1" title="Isı sığası oranı – Turkish" lang="tr" hreflang="tr" data-title="Isı sığası oranı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a 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class="sidebar-image" style="display:block;margin:0.3em 0 0.4em;"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/Carnot_heat_engine" title="Carnot heat engine"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/220px-Carnot_heat_engine_2.svg.png" decoding="async" width="220" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/330px-Carnot_heat_engine_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/440px-Carnot_heat_engine_2.svg.png 2x" data-file-width="840" data-file-height="370" /></a></span><div class="sidebar-caption">The classical <a href="/wiki/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;text-align:center;;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="/wiki/Thermodynamics" title="Thermodynamics">Classical</a></li> <li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical</a></li> <li><a href="/wiki/Chemical_thermodynamics" title="Chemical thermodynamics">Chemical</a></li> <li><a href="/wiki/Quantum_thermodynamics" title="Quantum thermodynamics">Quantum thermodynamics</a></li></ul> </div> <ul><li><a href="/wiki/Equilibrium_thermodynamics" title="Equilibrium thermodynamics">Equilibrium</a> / <a href="/wiki/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;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/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="/wiki/Zeroth_law_of_thermodynamics" title="Zeroth law of thermodynamics">Zeroth</a></li> <li><a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">First</a></li> <li><a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">Second</a></li> <li><a href="/wiki/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;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Thermodynamic_system" title="Thermodynamic system">Systems</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Closed_system" title="Closed system">Closed system</a></li> <li><a href="/wiki/Thermodynamic_system#Open_system" title="Thermodynamic system">Open system</a></li> <li><a href="/wiki/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="/wiki/Thermodynamic_state" title="Thermodynamic state">State</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Equation_of_state" title="Equation of state">Equation of state</a></li> <li><a href="/wiki/Ideal_gas" title="Ideal gas">Ideal gas</a></li> <li><a href="/wiki/Real_gas" title="Real gas">Real gas</a></li> <li><a href="/wiki/State_of_matter" title="State of matter">State of matter</a></li> <li><a href="/wiki/Phase_(matter)" title="Phase (matter)">Phase (matter)</a></li> <li><a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Equilibrium</a></li> <li><a href="/wiki/Control_volume" title="Control volume">Control volume</a></li> <li><a href="/wiki/Thermodynamic_instruments" title="Thermodynamic instruments">Instruments</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_process" title="Thermodynamic process">Processes</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Isobaric_process" title="Isobaric process">Isobaric</a></li> <li><a href="/wiki/Isochoric_process" title="Isochoric process">Isochoric</a></li> <li><a href="/wiki/Isothermal_process" title="Isothermal process">Isothermal</a></li> <li><a href="/wiki/Adiabatic_process" title="Adiabatic process">Adiabatic</a></li> <li><a href="/wiki/Isentropic_process" title="Isentropic process">Isentropic</a></li> <li><a href="/wiki/Isenthalpic_process" title="Isenthalpic process">Isenthalpic</a></li> <li><a href="/wiki/Quasistatic_process" title="Quasistatic process">Quasistatic</a></li> <li><a href="/wiki/Polytropic_process" title="Polytropic process">Polytropic</a></li> <li><a href="/wiki/Free_expansion" class="mw-redirect" title="Free expansion">Free expansion</a></li> <li><a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">Reversibility</a></li> <li><a href="/wiki/Irreversible_process" title="Irreversible process">Irreversibility</a></li> <li><a href="/wiki/Endoreversible_thermodynamics" title="Endoreversible thermodynamics">Endoreversibility</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_cycle" title="Thermodynamic cycle">Cycles</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Heat_engine" title="Heat engine">Heat engines</a></li> <li><a href="/wiki/Heat_pump_and_refrigeration_cycle" title="Heat pump and refrigeration cycle">Heat pumps</a></li> <li><a href="/wiki/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;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/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="/wiki/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="/wiki/Thermodynamic_diagrams" title="Thermodynamic diagrams">Property diagrams</a></li> <li><a href="/wiki/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;"> <a href="/wiki/Process_function" title="Process function">Process functions</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;;padding-bottom:0.4em;"> <div class="hlist"> <ul><li><a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">Work</a></li> <li><a href="/wiki/Heat" title="Heat">Heat</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/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="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">Temperature</a> / <i><a href="/wiki/Entropy" title="Entropy">Entropy</a></i> (<a href="/wiki/Introduction_to_entropy" title="Introduction to entropy">introduction</a>)</li> <li><a href="/wiki/Pressure" title="Pressure">Pressure</a> / <i><a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">Volume</a></i></li> <li><a href="/wiki/Chemical_potential" title="Chemical potential">Chemical potential</a> / <i><a href="/wiki/Particle_number" title="Particle number">Particle number</a></i></li> <li><a href="/wiki/Vapor_quality" title="Vapor quality">Vapor quality</a></li> <li><a href="/wiki/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;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Material_properties_(thermodynamics)" title="Material properties (thermodynamics)">Material properties</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/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;text-align:center;"> <table> <tbody><tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Heat_capacity" title="Heat capacity">Specific heat capacity</a> </td> <td style="vertical-align:middle; text-align:left"><span class="mwe-math-element"><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=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/891d40a9b18752b04065caee655d008b3ec11428" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.46ex; height:1.676ex;" alt="{\displaystyle c=}"></span></td> <td><table><tbody><tr><td><span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span></td><td><span class="mwe-math-element"><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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c609f4d3c5692ea4495479ef47594dc67f9fa464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.817ex; height:2.176ex;" alt="{\displaystyle \partial S}"></span></td></tr><tr><td style="border-top:solid 1px black;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span></td><td style="border-top:solid 1px black;"><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/504aa558fff3d00d10b03cadb1085cb0b7bdc631" 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}"></span></td></tr></tbody></table></td></tr> <tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Compressibility" title="Compressibility">Compressibility</a> </td> <td style="vertical-align:middle; text-align:left"><span class="mwe-math-element"><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 =-}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =-}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b01c042bf1456bd4d2a8caed1f4912820a7ecbb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.239ex; height:2.509ex;" alt="{\displaystyle \beta =-}"></span></td> <td><table><tbody><tr><td><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" 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}"></span></td><td><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" 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}"></span></td></tr><tr><td style="border-top:solid 1px black;"><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" 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}"></span></td><td style="border-top:solid 1px black;"><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc4a48eb2412f08b54fe438b5139c88f9cfa372" 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}"></span></td></tr></tbody></table></td></tr> <tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Thermal_expansion" title="Thermal expansion">Thermal expansion</a> </td> <td style="vertical-align:middle; text-align:left"><span class="mwe-math-element"><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</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a92d4583d351f08c1c70985f0c843b2fff1b01e7" 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 =}"></span></td> <td><table><tbody><tr><td><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" 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}"></span></td><td><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" 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}"></span></td></tr><tr><td style="border-top:solid 1px black;"><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" 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}"></span></td><td style="border-top:solid 1px black;"><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/504aa558fff3d00d10b03cadb1085cb0b7bdc631" 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}"></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;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Thermodynamic_equations" title="Thermodynamic equations">Equations</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Carnot%27s_theorem_(thermodynamics)" title="Carnot's theorem (thermodynamics)">Carnot's theorem</a></li> <li><a href="/wiki/Clausius_theorem" title="Clausius theorem">Clausius theorem</a></li> <li><a href="/wiki/Fundamental_thermodynamic_relation" title="Fundamental thermodynamic relation">Fundamental relation</a></li> <li><a href="/wiki/Ideal_gas_law" title="Ideal gas law">Ideal gas law</a></li></ul> </div> <ul><li><a href="/wiki/Maxwell_relations" title="Maxwell relations">Maxwell relations</a></li> <li><a href="/wiki/Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager reciprocal relations</a></li> <li><a href="/wiki/Bridgman%27s_thermodynamic_equations" title="Bridgman's thermodynamic equations">Bridgman's equations</a></li> <li><i><a href="/wiki/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;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">Potentials</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Thermodynamic_free_energy" title="Thermodynamic free energy">Free energy</a></li> <li><a href="/wiki/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="/wiki/Internal_energy" title="Internal energy">Internal energy</a><br /><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/921f33f9c6551562ec836007b035c2de6323d2d6" 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)}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Enthalpy" title="Enthalpy">Enthalpy</a><br /><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/6407d78e5f39d07f70e2414a92e08e2e068519f3" 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}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a><br /><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e93692f031ba6484d82731c54db83a69daed3f0" 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}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a><br /><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd7a8f0b8ae04963da133e3b202432e1b6caed4" 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}"></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;text-align:center;;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="/wiki/History_of_thermodynamics" title="History of thermodynamics">General</a></li> <li><a href="/wiki/History_of_entropy" title="History of entropy">Entropy</a></li> <li><a href="/wiki/Gas_laws" title="Gas laws">Gas laws</a></li></ul> </div> <ul><li><a href="/wiki/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="/wiki/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="/wiki/Entropy_(arrow_of_time)" class="mw-redirect" title="Entropy (arrow of time)">Entropy and time</a></li> <li><a href="/wiki/Entropy_and_life" title="Entropy and life">Entropy and life</a></li> <li><a href="/wiki/Brownian_ratchet" title="Brownian ratchet">Brownian ratchet</a></li> <li><a href="/wiki/Maxwell%27s_demon" title="Maxwell's demon">Maxwell's demon</a></li> <li><a href="/wiki/Heat_death_paradox" title="Heat death paradox">Heat death paradox</a></li> <li><a href="/wiki/Loschmidt%27s_paradox" title="Loschmidt's paradox">Loschmidt's paradox</a></li> <li><a href="/wiki/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="/wiki/Caloric_theory" title="Caloric theory">Caloric theory</a></li></ul> </div> <ul><li><a href="/wiki/Vis_viva" title="Vis viva"><i>Vis viva</i> <span style="font-size:85%;">("living force")</span></a></li> <li><a href="/wiki/Mechanical_equivalent_of_heat" title="Mechanical equivalent of heat">Mechanical equivalent of heat</a></li> <li><a href="/wiki/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="/wiki/List_of_important_publications_in_physics" 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="/wiki/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="/wiki/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="/wiki/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="/wiki/Timeline_of_thermodynamics" title="Timeline of thermodynamics">Thermodynamics</a></li> <li><a href="/wiki/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="/wiki/Maxwell%27s_thermodynamic_surface" title="Maxwell's thermodynamic surface">Maxwell's thermodynamic surface</a></li> <li><a href="/wiki/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;text-align:center;;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="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Bernoulli</a></li> <li><a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Boltzmann</a></li> <li><a href="/wiki/Percy_Williams_Bridgman" title="Percy Williams Bridgman">Bridgman</a></li> <li><a href="/wiki/Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">Carathéodory</a></li> <li><a href="/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Carnot</a></li> <li><a href="/wiki/Beno%C3%AEt_Paul_%C3%89mile_Clapeyron" class="mw-redirect" title="Benoît Paul Émile Clapeyron">Clapeyron</a></li> <li><a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Clausius</a></li> <li><a href="/wiki/Th%C3%A9ophile_de_Donder" title="Théophile de Donder">de Donder</a></li> <li><a href="/wiki/Pierre_Duhem" title="Pierre Duhem">Duhem</a></li> <li><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li> <li><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">von Helmholtz</a></li> <li><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule</a></li> <li><a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li> <li><a href="/wiki/Gilbert_N._Lewis" title="Gilbert N. Lewis">Lewis</a></li> <li><a href="/wiki/Fran%C3%A7ois_Massieu" title="François Massieu">Massieu</a></li> <li><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li> <li><a href="/wiki/Julius_von_Mayer" title="Julius von Mayer">von Mayer</a></li> <li><a href="/wiki/Walther_Nernst" title="Walther Nernst">Nernst</a></li> <li><a href="/wiki/Lars_Onsager" title="Lars Onsager">Onsager</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/William_John_Macquorn_Rankine" class="mw-redirect" title="William John Macquorn Rankine">Rankine</a></li> <li><a href="/wiki/John_Smeaton" title="John Smeaton">Smeaton</a></li> <li><a href="/wiki/Georg_Ernst_Stahl" title="Georg Ernst Stahl">Stahl</a></li> <li><a href="/wiki/Peter_Tait_(physicist)" class="mw-redirect" title="Peter Tait (physicist)">Tait</a></li> <li><a href="/wiki/Benjamin_Thompson" title="Benjamin Thompson">Thompson</a></li> <li><a href="/wiki/Johannes_Diderik_van_der_Waals" title="Johannes Diderik van der Waals">van der Waals</a></li> <li><a href="/wiki/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;text-align:center;;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="/wiki/Nucleation" title="Nucleation">Nucleation</a></li> <li><a href="/wiki/Self-assembly" title="Self-assembly">Self-assembly</a></li> <li><a href="/wiki/Self-organization" title="Self-organization">Self-organization</a></li> <li><a href="/wiki/Order_and_disorder" title="Order and disorder">Order and disorder</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below"> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Thermodynamics" title="Category:Thermodynamics">Category</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Thermodynamics_sidebar" title="Template:Thermodynamics sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Thermodynamics_sidebar" title="Template talk:Thermodynamics sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Thermodynamics_sidebar" title="Special:EditPage/Template:Thermodynamics sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <table class="wikitable sortable mw-collapsible floatright"> <caption>Heat capacity ratio for various gases<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </caption> <tbody><tr> <th>Gas </th> <th style="background:#e0e0e0; text-align:center;">Temp. [°C] </th> <th style="background:#e0e0e0; text-align:center;"><span class="texhtml">γ</span> </th></tr> <tr> <td rowspan="7"><a href="/wiki/Hydrogen" title="Hydrogen">H<sub>2</sub></a> </td> <td>−181</td> <td>1.597 </td></tr> <tr> <td>−76</td> <td>1.453 </td></tr> <tr> <td>20</td> <td>1.410 </td></tr> <tr> <td>100</td> <td>1.404 </td></tr> <tr> <td>400</td> <td>1.387 </td></tr> <tr> <td>1000</td> <td>1.358 </td></tr> <tr> <td>2000</td> <td>1.318 </td></tr> <tr> <td><a href="/wiki/Helium" title="Helium">He</a> </td> <td>20</td> <td>1.66 </td></tr> <tr> <td rowspan="2"><a href="/wiki/Argon" title="Argon">Ar</a> </td> <td>−180</td> <td>1.760 </td></tr> <tr> <td>20</td> <td>1.670 </td></tr> <tr> <td rowspan="6"><a href="/wiki/Oxygen" title="Oxygen">O<sub>2</sub></a> </td> <td>−181</td> <td>1.450 </td></tr> <tr> <td>−76</td> <td>1.415 </td></tr> <tr> <td>20</td> <td>1.400 </td></tr> <tr> <td>100</td> <td>1.399 </td></tr> <tr> <td>200</td> <td>1.397 </td></tr> <tr> <td>400</td> <td>1.394 </td></tr> <tr> <td><a href="/wiki/Nitrogen" title="Nitrogen">N<sub>2</sub></a> </td> <td>−181</td> <td>1.470 </td></tr> <tr> <td><a href="/wiki/Chlorine" title="Chlorine">Cl<sub>2</sub></a> </td> <td>20</td> <td>1.340 </td></tr> <tr> <td><a href="/wiki/Neon" title="Neon">Ne</a> </td> <td>19</td> <td>1.640 </td></tr> <tr> <td><a href="/wiki/Xenon" title="Xenon">Xe</a> </td> <td>19</td> <td>1.660 </td></tr> <tr> <td><a href="/wiki/Krypton" title="Krypton">Kr</a> </td> <td>19</td> <td>1.680 </td></tr> <tr> <td><a href="/wiki/Mercury_(element)" title="Mercury (element)">Hg</a> </td> <td>360</td> <td>1.670 </td></tr> <tr> <td rowspan="3"><a href="/wiki/Water_vapor" title="Water vapor">H<sub>2</sub>O</a> </td> <td>20</td> <td>1.330 </td></tr> <tr> <td>100</td> <td>1.324 </td></tr> <tr> <td>200</td> <td>1.310 </td></tr> <tr> <td rowspan="5"><a href="/wiki/Carbon_dioxide" title="Carbon dioxide">CO<sub>2</sub></a> </td> <td>0</td> <td>1.310 </td></tr> <tr> <td>20</td> <td>1.300 </td></tr> <tr> <td>100</td> <td>1.281 </td></tr> <tr> <td>400</td> <td>1.235 </td></tr> <tr> <td>1000</td> <td>1.195 </td></tr> <tr> <td><a href="/wiki/Carbon_monoxide" title="Carbon monoxide">CO</a> </td> <td>20</td> <td>1.400 </td></tr> <tr> <td><a href="/wiki/Nitric_oxide" title="Nitric oxide">NO</a> </td> <td>20</td> <td>1.400 </td></tr> <tr> <td><a href="/wiki/Nitrous_oxide" title="Nitrous oxide">N<sub>2</sub>O</a> </td> <td>20</td> <td>1.310 </td></tr> <tr> <td rowspan="3"><a href="/wiki/Methane" title="Methane">CH<sub>4</sub></a> </td> <td>−115</td> <td>1.410 </td></tr> <tr> <td>−74</td> <td>1.350 </td></tr> <tr> <td>20</td> <td>1.320 </td></tr> <tr> <td><a href="/wiki/Ammonia" title="Ammonia">NH<sub>3</sub></a> </td> <td>15</td> <td>1.310 </td></tr> <tr> <td><a href="/wiki/Sulfur_dioxide" title="Sulfur dioxide">SO<sub>2</sub></a> </td> <td>15</td> <td>1.290 </td></tr> <tr> <td><a href="/wiki/Ethane" title="Ethane">C<sub>2</sub>H<sub>6</sub></a> </td> <td>15</td> <td>1.220 </td></tr> <tr> <td><a href="/wiki/Propane" title="Propane">C<sub>3</sub>H<sub>8</sub></a> </td> <td>16</td> <td>1.130 </td></tr> <tr> <td rowspan="6">Dry <a href="/wiki/Air" class="mw-redirect" title="Air">air</a> </td> <td>-15</td> <td>1.404 </td></tr> <tr> <td>0</td> <td>1.403 </td></tr> <tr> <td>20</td> <td>1.400 </td></tr> <tr> <td>200</td> <td>1.398 </td></tr> <tr> <td>400</td> <td>1.393 </td></tr> <tr> <td>1000</td> <td>1.365 </td></tr></tbody></table> <p>In <a href="/wiki/Thermal_physics" title="Thermal physics">thermal physics</a> and <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a>, the <b>heat capacity ratio</b>, also known as the <b>adiabatic index</b>, the <b>ratio of specific heats</b>, or <b>Laplace's coefficient</b>, is the ratio of the <a href="/wiki/Heat_capacity" title="Heat capacity">heat capacity</a> at constant pressure (<span class="texhtml mvar" style="font-style:italic;">C<sub>P</sub></span>) to heat capacity at constant volume (<span class="texhtml mvar" style="font-style:italic;">C<sub>V</sub></span>). It is sometimes also known as the <i><a href="/wiki/Isentropic" class="mw-redirect" title="Isentropic">isentropic</a> expansion factor</i> and is denoted by <span class="texhtml">γ</span> (<a href="/wiki/Gamma_(letter)" class="mw-redirect" title="Gamma (letter)">gamma</a>) for an ideal gas<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> or <span class="texhtml mvar" style="font-style:italic;">κ</span> (<a href="/wiki/Kappa" title="Kappa">kappa</a>), the isentropic exponent for a real gas. The symbol <span class="texhtml">γ</span> is used by aerospace and chemical engineers. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ={\frac {C_{P}}{C_{V}}}={\frac {{\bar {C}}_{P}}{{\bar {C}}_{V}}}={\frac {c_{P}}{c_{V}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\frac {C_{P}}{C_{V}}}={\frac {{\bar {C}}_{P}}{{\bar {C}}_{V}}}={\frac {c_{P}}{c_{V}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ffcaf70b28897898a347d4997740f2ff0c1b43d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.636ex; height:6.509ex;" alt="{\displaystyle \gamma ={\frac {C_{P}}{C_{V}}}={\frac {{\bar {C}}_{P}}{{\bar {C}}_{V}}}={\frac {c_{P}}{c_{V}}},}"></span> where <span class="texhtml mvar" style="font-style:italic;">C</span> is the heat capacity, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f27ad0814e382ab26260a9183960cf7b00e71e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.676ex;" alt="{\displaystyle {\bar {C}}}"></span> the <a href="/wiki/Molar_heat_capacity" title="Molar heat capacity">molar heat capacity</a> (heat capacity per mole), and <span class="texhtml mvar" style="font-style:italic;">c</span> the <a href="/wiki/Specific_heat_capacity" title="Specific heat capacity">specific heat capacity</a> (heat capacity per unit mass) of a gas. The suffixes <span class="texhtml mvar" style="font-style:italic;">P</span> and <span class="texhtml mvar" style="font-style:italic;">V</span> refer to constant-pressure and constant-volume conditions respectively. </p><p>The heat capacity ratio is important for its applications in <a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">thermodynamical reversible processes</a>, especially involving <a href="/wiki/Ideal_gases" class="mw-redirect" title="Ideal gases">ideal gases</a>; the <a href="/wiki/Speed_of_sound" title="Speed of sound">speed of sound</a> depends on this factor. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Thought_experiment">Thought experiment</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heat_capacity_ratio&action=edit&section=1" title="Edit section: Thought experiment"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To understand this relation, consider the following <a href="/wiki/Thought_experiment" title="Thought experiment">thought experiment</a>. A closed <a href="/wiki/Pneumatic_cylinder" title="Pneumatic cylinder">pneumatic cylinder</a> contains air. The <a href="/wiki/Piston" title="Piston">piston</a> is locked. The pressure inside is equal to atmospheric pressure. This cylinder is heated to a certain target temperature. Since the piston cannot move, the volume is constant. The temperature and pressure will rise. When the target temperature is reached, the heating is stopped. The amount of energy added equals <span class="texhtml"><i>C<sub>V</sub> </i>Δ<i>T</i></span>, with <span class="texhtml">Δ<i>T</i></span> representing the change in temperature. </p><p>The piston is now freed and moves outwards, stopping as the pressure inside the chamber reaches atmospheric pressure. We assume the expansion occurs without exchange of heat (<a href="/wiki/Adiabatic_process" title="Adiabatic process">adiabatic expansion</a>). Doing this <a href="/wiki/Mechanical_work" class="mw-redirect" title="Mechanical work">work</a>, air inside the cylinder will cool to below the target temperature. </p><p>To return to the target temperature (still with a free piston), the air must be heated, but is no longer under constant volume, since the piston is free to move as the gas is reheated. This extra heat amounts to about 40% more than the previous amount added. In this example, the amount of heat added with a locked piston is proportional to <span class="texhtml mvar" style="font-style:italic;">C<sub>V</sub></span>, whereas the total amount of heat added is proportional to <span class="texhtml mvar" style="font-style:italic;">C<sub>P</sub></span>. Therefore, the heat capacity ratio in this example is 1.4. </p><p>Another way of understanding the difference between <span class="texhtml mvar" style="font-style:italic;">C<sub>P</sub></span> and <span class="texhtml mvar" style="font-style:italic;">C<sub>V</sub></span> is that <span class="texhtml mvar" style="font-style:italic;">C<sub>P</sub></span> applies if work is done to the system, which causes a change in volume (such as by moving a piston so as to compress the contents of a cylinder), or if work is done by the system, which changes its temperature (such as heating the gas in a cylinder to cause a piston to move). <span class="texhtml mvar" style="font-style:italic;">C<sub>V</sub></span> applies only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\,\mathrm {d} V=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\,\mathrm {d} V=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed36bbf78bda5b4ab79dbf6c77c30cbc39d8d689" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.473ex; height:2.176ex;" alt="{\displaystyle P\,\mathrm {d} V=0}"></span>, that is, no work is done. Consider the difference between adding heat to the gas with a locked piston and adding heat with a piston free to move, so that pressure remains constant. </p><p>In the second case, the gas will both heat and expand, causing the piston to do mechanical work on the atmosphere. The heat that is added to the gas goes only partly into heating the gas, while the rest is transformed into the mechanical work performed by the piston. </p><p>In the first, constant-volume case (locked piston), there is no external motion, and thus no mechanical work is done on the atmosphere; <span class="texhtml mvar" style="font-style:italic;">C<sub>V</sub></span> is used. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. </p> <div class="mw-heading mw-heading2"><h2 id="Ideal-gas_relations">Ideal-gas relations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heat_capacity_ratio&action=edit&section=2" title="Edit section: Ideal-gas relations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For an ideal gas, the molar heat capacity is at most a function of temperature, since the <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a> is solely a function of temperature for a <a href="/wiki/Closed_system" title="Closed system">closed system</a>, i.e., <span class="mwe-math-element"><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=U(n,T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=U(n,T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ff4ba3771f82ae89bec9c7252f098cdcf085d18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.538ex; height:2.843ex;" alt="{\displaystyle U=U(n,T)}"></span>, where <span class="texhtml mvar" style="font-style:italic;">n</span> is the <a href="/wiki/Amount_of_substance" title="Amount of substance">amount of substance</a> in moles. In thermodynamic terms, this is a consequence of the fact that the <a href="/wiki/Internal_pressure" title="Internal pressure">internal pressure</a> of an ideal gas vanishes. </p><p><a href="/wiki/Mayer%27s_relation" title="Mayer's relation">Mayer's relation</a> allows us to deduce the value of <span class="texhtml mvar" style="font-style:italic;">C<sub>V</sub></span> from the more easily measured (and more commonly tabulated) value of <span class="texhtml mvar" style="font-style:italic;">C<sub>P</sub></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{V}=C_{P}-nR.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <mi>n</mi> <mi>R</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{V}=C_{P}-nR.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c735a9f48139834b75a82b71b5aef95cd6df2c2c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.031ex; height:2.509ex;" alt="{\displaystyle C_{V}=C_{P}-nR.}"></span> </p><p>This relation may be used to show the heat capacities may be expressed in terms of the heat capacity ratio (<span class="texhtml">γ</span>) and the <a href="/wiki/Gas_constant" title="Gas constant">gas constant</a> (<span class="texhtml mvar" style="font-style:italic;">R</span>): <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{P}={\frac {\gamma nR}{\gamma -1}}\quad {\text{and}}\quad C_{V}={\frac {nR}{\gamma -1}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mi>n</mi> <mi>R</mi> </mrow> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>R</mi> </mrow> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{P}={\frac {\gamma nR}{\gamma -1}}\quad {\text{and}}\quad C_{V}={\frac {nR}{\gamma -1}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa5cdf30ecda8d81af8c4e18333bd8f7db81315" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.725ex; height:5.843ex;" alt="{\displaystyle C_{P}={\frac {\gamma nR}{\gamma -1}}\quad {\text{and}}\quad C_{V}={\frac {nR}{\gamma -1}},}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Relation_with_degrees_of_freedom">Relation with degrees of freedom</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heat_capacity_ratio&action=edit&section=3" title="Edit section: Relation with degrees of freedom"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The classical <a href="/wiki/Equipartition_theorem" title="Equipartition theorem">equipartition theorem</a> predicts that the heat capacity ratio (<span class="texhtml">γ</span>) for an ideal gas can be related to the thermally accessible <a href="/wiki/Degrees_of_freedom_(physics_and_chemistry)" title="Degrees of freedom (physics and chemistry)">degrees of freedom</a> (<span class="texhtml mvar" style="font-style:italic;">f</span>) of a molecule by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =1+{\frac {2}{f}},\quad {\text{or}}\quad f={\frac {2}{\gamma -1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>f</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>or</mtext> </mrow> <mspace width="1em" /> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =1+{\frac {2}{f}},\quad {\text{or}}\quad f={\frac {2}{\gamma -1}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1acdc03d5c04cadf460f11509b9a86acf77e380" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.357ex; height:5.676ex;" alt="{\displaystyle \gamma =1+{\frac {2}{f}},\quad {\text{or}}\quad f={\frac {2}{\gamma -1}}.}"></span> </p><p>Thus we observe that for a <a href="/wiki/Monatomic" class="mw-redirect" title="Monatomic">monatomic</a> gas, with 3 translational degrees of freedom per atom: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ={\frac {5}{3}}=1.6666\ldots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mn>1.6666</mn> <mo>…</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\frac {5}{3}}=1.6666\ldots ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e648d9195b1a7a368b1d49748b4934d7843cd3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.061ex; height:5.176ex;" alt="{\displaystyle \gamma ={\frac {5}{3}}=1.6666\ldots ,}"></span> </p><p>As an example of this behavior, at 273 K (0 °C) the noble gases He, Ne, and Ar all have nearly the same value of <span class="texhtml">γ</span>, equal to 1.664. </p><p>For a <a href="/wiki/Diatomic" class="mw-redirect" title="Diatomic">diatomic</a> gas, often 5 degrees of freedom are assumed to contribute at room temperature since each molecule has 3 translational and 2 <a href="/wiki/Rotational_degrees_of_freedom" class="mw-redirect" title="Rotational degrees of freedom">rotational degrees of freedom</a>, and the single vibrational degree of freedom is often not included since vibrations are often not thermally active except at high temperatures, as predicted by <a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">quantum statistical mechanics</a>. Thus we have <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ={\frac {7}{5}}=1.4.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>5</mn> </mfrac> </mrow> <mo>=</mo> <mn>1.4.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\frac {7}{5}}=1.4.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ce0bfa71dbf43f87bd64f6294436c39c114e26" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.076ex; height:5.176ex;" alt="{\displaystyle \gamma ={\frac {7}{5}}=1.4.}"></span> </p><p>For example, terrestrial air is primarily made up of <a href="/wiki/Diatomic" class="mw-redirect" title="Diatomic">diatomic</a> gases (around 78% <a href="/wiki/Nitrogen" title="Nitrogen">nitrogen</a>, N<sub>2</sub>, and 21% <a href="/wiki/Oxygen" title="Oxygen">oxygen</a>, O<sub>2</sub>), and at standard conditions it can be considered to be an ideal gas. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). </p><p>For a linear triatomic molecule such as CO<sub style="font-size: 80%;vertical-align: -0.35em">2</sub>, there are only 5 degrees of freedom (3 translations and 2 rotations), assuming vibrational modes are not excited. However, as mass increases and the frequency of vibrational modes decreases, vibrational degrees of freedom start to enter into the equation at far lower temperatures than is typically the case for diatomic molecules. For example, it requires a far larger temperature to excite the single vibrational mode for <style data-mw-deduplicate="TemplateStyles:r1123817410">.mw-parser-output .template-chem2-su{display:inline-block;font-size:80%;line-height:1;vertical-align:-0.35em}.mw-parser-output .template-chem2-su>span{display:block;text-align:left}.mw-parser-output sub.template-chem2-sub{font-size:80%;vertical-align:-0.35em}.mw-parser-output sup.template-chem2-sup{font-size:80%;vertical-align:0.65em}</style><span class="chemf nowrap">H<sub class="template-chem2-sub">2</sub></span>, for which one quantum of vibration is a fairly large amount of energy, than for the bending or stretching vibrations of CO<sub style="font-size: 80%;vertical-align: -0.35em">2</sub>. </p><p>For a non-linear <a href="/wiki/Triatomic_molecule" title="Triatomic molecule">triatomic</a> gas, such as water vapor, which has 3 translational and 3 rotational degrees of freedom, this model predicts <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ={\frac {8}{6}}=1.3333\ldots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>6</mn> </mfrac> </mrow> <mo>=</mo> <mn>1.3333</mn> <mo>…</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\frac {8}{6}}=1.3333\ldots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2424b05a0220a3ae6cd373f8b6b5971b5d0b643" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.061ex; height:5.176ex;" alt="{\displaystyle \gamma ={\frac {8}{6}}=1.3333\ldots .}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Real-gas_relations">Real-gas relations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heat_capacity_ratio&action=edit&section=4" title="Edit section: Real-gas relations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As noted above, as temperature increases, higher-energy vibrational states become accessible to molecular gases, thus increasing the number of degrees of freedom and lowering <span class="texhtml">γ</span>. Conversely, as the temperature is lowered, rotational degrees of freedom may become unequally partitioned as well. As a result, both <span class="texhtml mvar" style="font-style:italic;">C<sub>P</sub></span> and <span class="texhtml mvar" style="font-style:italic;">C<sub>V</sub></span> increase with increasing temperature. </p><p>Despite this, if the density is fairly low and <a href="/wiki/Intermolecular_forces" class="mw-redirect" title="Intermolecular forces">intermolecular forces</a> are negligible, the two heat capacities may still continue to differ from each other by a fixed constant (as above, <span class="texhtml"><i>C<sub>P</sub></i> = <i>C<sub>V</sub></i> + <i>nR</i></span>), which reflects the relatively constant <span class="texhtml mvar" style="font-style:italic;">PV</span> difference in work done during expansion for constant pressure vs. constant volume conditions. Thus, the ratio of the two values, <span class="texhtml">γ</span>, decreases with increasing temperature. </p><p>However, when the gas density is sufficiently high and intermolecular forces are important, thermodynamic expressions may sometimes be used to accurately describe the relationship between the two heat capacities, as explained below. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other <a href="/wiki/Chemical_reactions" class="mw-redirect" title="Chemical reactions">chemical reactions</a>, in which case thermodynamic expressions arising from simple <a href="/wiki/Equations_of_state" class="mw-redirect" title="Equations of state">equations of state</a> may not be adequate. </p> <div class="mw-heading mw-heading3"><h3 id="Thermodynamic_expressions">Thermodynamic expressions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heat_capacity_ratio&action=edit&section=5" title="Edit section: Thermodynamic expressions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Values based on approximations (particularly <span class="texhtml"><i>C<sub>P</sub></i> − <i>C<sub>V</sub></i> = <i>nR</i></span>) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. An experimental value should be used rather than one based on this approximation, where possible. A rigorous value for the ratio <span class="texhtml mvar" style="font-style:italic;"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">C<sub>P</sub></span><span class="sr-only">/</span><span class="den">C<sub>V</sub></span></span>⁠</span></span> can also be calculated by determining <span class="texhtml mvar" style="font-style:italic;">C<sub>V</sub></span> from the residual properties expressed as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{P}-C_{V}=-T{\frac {\left({\frac {\partial V}{\partial T}}\right)_{P}^{2}}{\left({\frac {\partial V}{\partial P}}\right)_{T}}}=-T{\frac {\left({\frac {\partial P}{\partial T}}\right)_{V}^{2}}{\left({\frac {\partial P}{\partial V}}\right)_{T}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{P}-C_{V}=-T{\frac {\left({\frac {\partial V}{\partial T}}\right)_{P}^{2}}{\left({\frac {\partial V}{\partial P}}\right)_{T}}}=-T{\frac {\left({\frac {\partial P}{\partial T}}\right)_{V}^{2}}{\left({\frac {\partial P}{\partial V}}\right)_{T}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22ca15b0c7da096c18c8b640046274b35304361c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:39.079ex; height:10.676ex;" alt="{\displaystyle C_{P}-C_{V}=-T{\frac {\left({\frac {\partial V}{\partial T}}\right)_{P}^{2}}{\left({\frac {\partial V}{\partial P}}\right)_{T}}}=-T{\frac {\left({\frac {\partial P}{\partial T}}\right)_{V}^{2}}{\left({\frac {\partial P}{\partial V}}\right)_{T}}}.}"></span> </p><p>Values for <span class="texhtml mvar" style="font-style:italic;">C<sub>P</sub></span> are readily available and recorded, but values for <span class="texhtml mvar" style="font-style:italic;">C<sub>V</sub></span> need to be determined via relations such as these. See <a href="/wiki/Relations_between_specific_heats" class="mw-redirect" title="Relations between specific heats">relations between specific heats</a> for the derivation of the thermodynamic relations between the heat capacities. </p><p>The above definition is the approach used to develop rigorous expressions from equations of state (such as <a href="/wiki/Peng%E2%80%93Robinson_equation_of_state" class="mw-redirect" title="Peng–Robinson equation of state">Peng–Robinson</a>), which match experimental values so closely that there is little need to develop a database of ratios or <span class="texhtml mvar" style="font-style:italic;">C<sub>V</sub></span> values. Values can also be determined through <a href="/wiki/Finite_difference_method" title="Finite difference method">finite-difference approximation</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Adiabatic_process">Adiabatic process</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heat_capacity_ratio&action=edit&section=6" title="Edit section: Adiabatic process"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Adiabatic_process" title="Adiabatic process">adiabatic process</a> and <a href="/wiki/Polytropic_process" title="Polytropic process">polytropic process</a></div> <p>This ratio gives the important relation for an <a href="/wiki/Isentropic" class="mw-redirect" title="Isentropic">isentropic</a> (<a href="/wiki/Quasistatic_process" title="Quasistatic process">quasistatic</a>, <a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">reversible</a>, <a href="/wiki/Adiabatic_process" title="Adiabatic process">adiabatic process</a>) process of a simple compressible calorically-perfect <a href="/wiki/Ideal_gas" title="Ideal gas">ideal gas</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PV^{\gamma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PV^{\gamma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7e276ce55b886ed772a4ba9030d93d20651078d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.787ex; height:2.343ex;" alt="{\displaystyle PV^{\gamma }}"></span> is constant</dd></dl> <p>Using the ideal gas law, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PV=nRT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>V</mi> <mo>=</mo> <mi>n</mi> <mi>R</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PV=nRT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/934032db2ac1f12624f85a90eeba651dcf4af377" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.426ex; height:2.176ex;" alt="{\displaystyle PV=nRT}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{1-\gamma }T^{\gamma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </msup> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{1-\gamma }T^{\gamma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08838420003c7dfdd355fcdce6b1f27af0a8faab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.892ex; height:2.676ex;" alt="{\displaystyle P^{1-\gamma }T^{\gamma }}"></span> is constant</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TV^{\gamma -1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle TV^{\gamma -1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0a5e44aad94a8b77668e6652878884d7ae9974f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.779ex; height:2.676ex;" alt="{\displaystyle TV^{\gamma -1}}"></span> is constant</dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">P</span> is the pressure of the gas, <span class="texhtml mvar" style="font-style:italic;">V</span> is the volume, and <span class="texhtml mvar" style="font-style:italic;">T</span> is the <a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">thermodynamic temperature</a>. </p><p>In gas dynamics we are interested in the local relations between pressure, density and temperature, rather than considering a fixed quantity of gas. By considering the density <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =M/V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =M/V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ed457a6aafb262f754ae8801d87eb036b8c5fe9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.692ex; height:2.843ex;" alt="{\displaystyle \rho =M/V}"></span> as the inverse of the volume for a unit mass, we can take <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =1/V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =1/V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a61bc7155024455af4923f9b190d01d816707be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.413ex; height:2.843ex;" alt="{\displaystyle \rho =1/V}"></span> in these relations. Since for constant entropy, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>, we have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\propto \rho ^{\gamma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>∝</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\propto \rho ^{\gamma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aea1e866ea8075352208c669fea9c1093e9a476" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.171ex; height:2.843ex;" alt="{\displaystyle P\propto \rho ^{\gamma }}"></span>, or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln P=\gamma \ln \rho +\mathrm {constant} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>P</mi> <mo>=</mo> <mi>γ</mi> <mi>ln</mi> <mo>⁡</mo> <mi>ρ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln P=\gamma \ln \rho +\mathrm {constant} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/520715c774ffbfa6bec4e48741fe648fdc95e0eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.856ex; height:2.676ex;" alt="{\displaystyle \ln P=\gamma \ln \rho +\mathrm {constant} }"></span>, it follows that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =\left.{\frac {\partial \ln P}{\partial \ln \rho }}\right|_{S}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =\left.{\frac {\partial \ln P}{\partial \ln \rho }}\right|_{S}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/209f544141d3ce25f2f5cac16b9bcf653f7b71fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.56ex; height:6.176ex;" alt="{\displaystyle \gamma =\left.{\frac {\partial \ln P}{\partial \ln \rho }}\right|_{S}.}"></span> </p><p>For an imperfect or non-ideal gas, <a href="/wiki/Subramanyan_Chandrasekhar" class="mw-redirect" title="Subramanyan Chandrasekhar">Chandrasekhar</a><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> defined three different adiabatic indices so that the adiabatic relations can be written in the same form as above; these are used in the theory of <a href="/wiki/Stellar_structure" title="Stellar structure">stellar structure</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Gamma _{1}&=\left.{\frac {\partial \ln P}{\partial \ln \rho }}\right|_{S},\\[2pt]{\frac {\Gamma _{2}-1}{\Gamma _{2}}}&=\left.{\frac {\partial \ln T}{\partial \ln P}}\right|_{S},\\[2pt]\Gamma _{3}-1&=\left.{\frac {\partial \ln T}{\partial \ln \rho }}\right|_{S}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Gamma _{1}&=\left.{\frac {\partial \ln P}{\partial \ln \rho }}\right|_{S},\\[2pt]{\frac {\Gamma _{2}-1}{\Gamma _{2}}}&=\left.{\frac {\partial \ln T}{\partial \ln P}}\right|_{S},\\[2pt]\Gamma _{3}-1&=\left.{\frac {\partial \ln T}{\partial \ln \rho }}\right|_{S}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7646c0aafdd5e68c49c35bd36abaf2bbefc2149f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.999ex; margin-bottom: -0.172ex; width:20.395ex; height:19.509ex;" alt="{\displaystyle {\begin{aligned}\Gamma _{1}&=\left.{\frac {\partial \ln P}{\partial \ln \rho }}\right|_{S},\\[2pt]{\frac {\Gamma _{2}-1}{\Gamma _{2}}}&=\left.{\frac {\partial \ln T}{\partial \ln P}}\right|_{S},\\[2pt]\Gamma _{3}-1&=\left.{\frac {\partial \ln T}{\partial \ln \rho }}\right|_{S}.\end{aligned}}}"></span> </p><p>All of these are equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> in the case of an ideal gas. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heat_capacity_ratio&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Relations_between_heat_capacities" title="Relations between heat capacities">Relations between heat capacities</a></li> <li><a href="/wiki/Heat_capacity" title="Heat capacity">Heat capacity</a></li> <li><a href="/wiki/Specific_heat_capacity" title="Specific heat capacity">Specific heat capacity</a></li> <li><a href="/wiki/Speed_of_sound" title="Speed of sound">Speed of sound</a></li> <li><a href="/wiki/Thermodynamic_equations" title="Thermodynamic equations">Thermodynamic equations</a></li> <li><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a></li> <li><a href="/wiki/Volumetric_heat_capacity" title="Volumetric heat capacity">Volumetric heat capacity</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heat_capacity_ratio&action=edit&section=8" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">γ first appeared in an article by the French mathematician, engineer, and physicist <a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Siméon Denis Poisson</a>: <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFPoisson1808" class="citation journal cs1 cs1-prop-foreign-lang-source">Poisson (1808). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=mdp.39015074785588&view=1up&seq=327">"Mémoire sur la théorie du son"</a> [Memoir on the theory of sound]. <i>Journal de l'École Polytechnique</i> (in French). <b>7</b> (14): <span class="nowrap">319–</span>392.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+de+l%27%C3%89cole+Polytechnique&rft.atitle=M%C3%A9moire+sur+la+th%C3%A9orie+du+son&rft.volume=7&rft.issue=14&rft.pages=%3Cspan+class%3D%22nowrap%22%3E319-%3C%2Fspan%3E392&rft.date=1808&rft.au=Poisson&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dmdp.39015074785588%26view%3D1up%26seq%3D327&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat+capacity+ratio" class="Z3988"></span> On p. 332, Poisson defines γ merely as a small deviation from equilibrium which causes small variations of the equilibrium value of the density ρ.</li></ul> In Poisson's article of 1823 – <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoisson1823" class="citation journal cs1 cs1-prop-foreign-lang-source">Poisson (1823). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=iau.31858046217836&view=1up&seq=11">"Sur la vitesse du son"</a> [On the speed of sound]. <i>Annales de chimie et de physique</i>. 2nd series (in French). <b>23</b>: <span class="nowrap">5–</span>16.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annales+de+chimie+et+de+physique&rft.atitle=Sur+la+vitesse+du+son&rft.volume=23&rft.pages=%3Cspan+class%3D%22nowrap%22%3E5-%3C%2Fspan%3E16&rft.date=1823&rft.au=Poisson&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Diau.31858046217836%26view%3D1up%26seq%3D11&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat+capacity+ratio" class="Z3988"></span></li></ul> γ was expressed as a function of density D (p. 8) or of pressure P (p. 9).<br /> Meanwhile, in 1816 the French mathematician and physicist <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> had found that the speed of sound depends on the ratio of the specific heats. <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaplace1816" class="citation journal cs1 cs1-prop-foreign-lang-source">Laplace (1816). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=hvd.hx3dvc&view=1up&seq=244">"Sur la vitesse du son dans l'air et dans l'eau"</a> [On the speed of sound in air and in water]. <i>Annales de chimie et de physique</i>. 2nd series (in French). <b>3</b>: <span class="nowrap">238–</span>241.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annales+de+chimie+et+de+physique&rft.atitle=Sur+la+vitesse+du+son+dans+l%27air+et+dans+l%27eau&rft.volume=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E238-%3C%2Fspan%3E241&rft.date=1816&rft.au=Laplace&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dhvd.hx3dvc%26view%3D1up%26seq%3D244&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat+capacity+ratio" class="Z3988"></span></li></ul> However, he didn't denote the ratio as γ.<br /> In 1825, Laplace stated that the speed of sound is proportional to the square root of the ratio of the specific heats: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaplace1825" class="citation book cs1 cs1-prop-foreign-lang-source">Laplace, P.S. (1825). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=gri.ark:/13960/t9d532b05&view=1up&seq=137"><i>Traité de mecanique celeste</i></a> [<i>Treatise on celestial mechanics</i>] (in French). Vol. 5. Paris, France: Bachelier. pp. <span class="nowrap">127–</span>137.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Trait%C3%A9+de+mecanique+celeste&rft.place=Paris%2C+France&rft.pages=%3Cspan+class%3D%22nowrap%22%3E127-%3C%2Fspan%3E137&rft.pub=Bachelier&rft.date=1825&rft.aulast=Laplace&rft.aufirst=P.S.&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dgri.ark%3A%2F13960%2Ft9d532b05%26view%3D1up%26seq%3D137&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat+capacity+ratio" class="Z3988"></span> On p. 127, Laplace defines the symbols for the specific heats, and on p. 137 (at the bottom of the page), Laplace presents the equation for the speed of sound in a perfect gas.</li></ul> In 1851, the Scottish mechanical engineer <a href="/wiki/William_John_Macquorn_Rankine" class="mw-redirect" title="William John Macquorn Rankine">William Rankine</a> showed that the speed of sound is proportional to the square root of Poisson's γ: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRankine1851" class="citation journal cs1">Rankine, William John Macquorn (1851). <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=umn.31951000614090i&view=1up&seq=239">"On Laplace's <i>Theory of Sound</i>"</a>. <i>Philosophical Magazine</i>. 4th series. <b>1</b> (3): <span class="nowrap">225–</span>227.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Magazine&rft.atitle=On+Laplace%27s+Theory+of+Sound&rft.volume=1&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E225-%3C%2Fspan%3E227&rft.date=1851&rft.aulast=Rankine&rft.aufirst=William+John+Macquorn&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Dumn.31951000614090i%26view%3D1up%26seq%3D239&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat+capacity+ratio" class="Z3988"></span></li></ul> It follows that Poisson's γ is the ratio of the specific heats — although Rankine didn't state that explicitly. <ul><li>See also: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrehl2009" class="citation book cs1">Krehl, Peter O. K. (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PmuqCHDC3pwC&pg=PA276"><i>History of Shock Waves, Explosions and Impact: A Chronological and Biographical Reference</i></a>. Berlin and Heidelberg, Germany: Springer Verlag. p. 276. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783540304210" title="Special:BookSources/9783540304210"><bdi>9783540304210</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+Shock+Waves%2C+Explosions+and+Impact%3A+A+Chronological+and+Biographical+Reference&rft.place=Berlin+and+Heidelberg%2C+Germany&rft.pages=276&rft.pub=Springer+Verlag&rft.date=2009&rft.isbn=9783540304210&rft.aulast=Krehl&rft.aufirst=Peter+O.+K.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DPmuqCHDC3pwC%26pg%3DPA276&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat+capacity+ratio" class="Z3988"></span></li></ul> </span></li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heat_capacity_ratio&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhite1998" class="citation book cs1">White, Frank M. (October 1998). <i>Fluid Mechanics</i> (4th ed.). New York: <a href="/wiki/McGraw_Hill" class="mw-redirect" title="McGraw Hill">McGraw Hill</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-228192-7" title="Special:BookSources/978-0-07-228192-7"><bdi>978-0-07-228192-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fluid+Mechanics&rft.place=New+York&rft.edition=4th&rft.pub=McGraw+Hill&rft.date=1998-10&rft.isbn=978-0-07-228192-7&rft.aulast=White&rft.aufirst=Frank+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat+capacity+ratio" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLange1967" class="citation book cs1">Lange, Norbert A. (1967). <i>Lange's Handbook of Chemistry</i> (10th ed.). New York: <a href="/wiki/McGraw_Hill" class="mw-redirect" title="McGraw Hill">McGraw Hill</a>. p. 1524. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-036261-1" title="Special:BookSources/978-0-07-036261-1"><bdi>978-0-07-036261-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lange%27s+Handbook+of+Chemistry&rft.place=New+York&rft.pages=1524&rft.edition=10th&rft.pub=McGraw+Hill&rft.date=1967&rft.isbn=978-0-07-036261-1&rft.aulast=Lange&rft.aufirst=Norbert+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat+capacity+ratio" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChandrasekhar1939" class="citation book cs1">Chandrasekhar, S. (1939). <i>An Introduction to the Study of Stellar Structure</i>. Chicago: <a href="/wiki/University_of_Chicago_Press" title="University of Chicago Press">University of Chicago Press</a>. p. 56. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-60413-8" title="Special:BookSources/978-0-486-60413-8"><bdi>978-0-486-60413-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Study+of+Stellar+Structure&rft.place=Chicago&rft.pages=56&rft.pub=University+of+Chicago+Press&rft.date=1939&rft.isbn=978-0-486-60413-8&rft.aulast=Chandrasekhar&rft.aufirst=S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHeat+capacity+ratio" class="Z3988"></span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Heat_capacity_ratio&oldid=1239795511">https://en.wikipedia.org/w/index.php?title=Heat_capacity_ratio&oldid=1239795511</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Thermodynamic_properties" title="Category:Thermodynamic properties">Thermodynamic properties</a></li><li><a href="/wiki/Category:Physical_quantities" title="Category:Physical quantities">Physical quantities</a></li><li><a href="/wiki/Category:Ratios" title="Category:Ratios">Ratios</a></li><li><a href="/wiki/Category:Thought_experiments_in_physics" title="Category:Thought experiments in physics">Thought experiments in physics</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_French-language_sources_(fr)" title="Category:CS1 French-language sources (fr)">CS1 French-language sources (fr)</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 11 August 2024, at 16:04<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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