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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Two_varieties_of_thermal_equilibrium" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Two_varieties_of_thermal_equilibrium"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Two varieties of thermal equilibrium</span> </div> </a> <button aria-controls="toc-Two_varieties_of_thermal_equilibrium-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon 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<span>Internal thermal equilibrium of an isolated body</span> </div> </a> <ul id="toc-Internal_thermal_equilibrium_of_an_isolated_body-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Thermal_contact" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Thermal_contact"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Thermal contact</span> </div> </a> <ul id="toc-Thermal_contact-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bodies_prepared_with_separately_uniform_temperatures,_then_put_into_purely_thermal_communication_with_each_other" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bodies_prepared_with_separately_uniform_temperatures,_then_put_into_purely_thermal_communication_with_each_other"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Bodies prepared with separately uniform temperatures, then put into purely thermal communication with each other</span> </div> </a> <ul id="toc-Bodies_prepared_with_separately_uniform_temperatures,_then_put_into_purely_thermal_communication_with_each_other-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Change_of_internal_state_of_an_isolated_system" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Change_of_internal_state_of_an_isolated_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Change of internal state of an isolated system</span> </div> </a> <button aria-controls="toc-Change_of_internal_state_of_an_isolated_system-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 Change of internal state of an isolated system subsection</span> </button> <ul id="toc-Change_of_internal_state_of_an_isolated_system-sublist" class="vector-toc-list"> <li id="toc-In_a_gravitational_field" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_a_gravitational_field"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>In a gravitational field</span> </div> </a> <ul id="toc-In_a_gravitational_field-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Distinctions_between_thermal_and_thermodynamic_equilibria" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Distinctions_between_thermal_and_thermodynamic_equilibria"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Distinctions between thermal and thermodynamic equilibria</span> </div> </a> <ul id="toc-Distinctions_between_thermal_and_thermodynamic_equilibria-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Thermal_equilibrium_of_planets" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Thermal_equilibrium_of_planets"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Thermal equilibrium of planets</span> </div> </a> <ul id="toc-Thermal_equilibrium_of_planets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citation_references" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Citation_references"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Citation references</span> </div> </a> <ul id="toc-Citation_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" > <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 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href="https://bn.wikipedia.org/wiki/%E0%A6%A4%E0%A6%BE%E0%A6%AA%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%B8%E0%A6%BE%E0%A6%AE%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%AC%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BE" title="তাপীয় সাম্যাবস্থা – Bangla" lang="bn" hreflang="bn" data-title="তাপীয় সাম্যাবস্থা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Equilibri_t%C3%A8rmic" title="Equilibri tèrmic – Catalan" lang="ca" hreflang="ca" data-title="Equilibri tèrmic" 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/Tepeln%C3%A1_rovnov%C3%A1ha" title="Tepelná rovnováha – Czech" lang="cs" hreflang="cs" data-title="Tepelná rovnováha" 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/Termisk_ligev%C3%A6gt" title="Termisk ligevægt – Danish" lang="da" hreflang="da" data-title="Termisk ligevægt" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Equilibrio_t%C3%A9rmico" title="Equilibrio térmico – Spanish" lang="es" hreflang="es" data-title="Equilibrio térmico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%B9%D8%A7%D8%AF%D9%84_%DA%AF%D8%B1%D9%85%D8%A7%DB%8C%DB%8C" title="تعادل گرمایی – Persian" lang="fa" hreflang="fa" data-title="تعادل گرمایی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89quilibre_thermique" title="Équilibre thermique – French" lang="fr" hreflang="fr" data-title="Équilibre thermique" 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/Cothroma%C3%ADocht_theirmeach" title="Cothromaíocht theirmeach – Irish" lang="ga" hreflang="ga" data-title="Cothromaíocht theirmeach" 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/%EC%97%B4%ED%8F%89%ED%98%95" title="열평형 – Korean" lang="ko" hreflang="ko" data-title="열평형" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%8A%E0%A4%B7%E0%A5%8D%E0%A4%AE%E0%A5%80%E0%A4%AF_%E0%A4%B8%E0%A4%82%E0%A4%A4%E0%A5%81%E0%A4%B2%E0%A4%A8" title="ऊष्मीय संतुलन – Hindi" lang="hi" hreflang="hi" data-title="ऊष्मीय संतुलन" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Toplinska_ravnote%C5%BEa" title="Toplinska ravnoteža – Croatian" lang="hr" hreflang="hr" data-title="Toplinska ravnoteža" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Varmajafnv%C3%A6gi" title="Varmajafnvægi – Icelandic" lang="is" hreflang="is" data-title="Varmajafnvægi" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Aequilibrium_thermale" title="Aequilibrium thermale – Latin" lang="la" hreflang="la" data-title="Aequilibrium thermale" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Termisk_likevekt" title="Termisk likevekt – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Termisk likevekt" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Issiqlik_muvozanati" title="Issiqlik muvozanati – Uzbek" lang="uz" hreflang="uz" data-title="Issiqlik muvozanati" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%AD%E0%B7%8F%E0%B6%B4_%E0%B7%83%E0%B6%B8%E0%B6%AD%E0%B7%94%E0%B6%BD%E0%B7%92%E0%B6%AD%E0%B6%AD%E0%B7%8F%E0%B7%80%E0%B6%BA" title="තාප සමතුලිතතාවය – Sinhala" lang="si" hreflang="si" data-title="තාප සමතුලිතතාවය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Tepeln%C3%A1_rovnov%C3%A1ha" title="Tepelná rovnováha – Slovak" lang="sk" hreflang="sk" data-title="Tepelná 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href="https://sh.wikipedia.org/wiki/Toplotna_ravnote%C5%BEa" title="Toplotna ravnoteža – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Toplotna ravnoteža" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AF%86%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%9A%E0%AF%8D_%E0%AE%9A%E0%AE%AE%E0%AE%A8%E0%AE%BF%E0%AE%B2%E0%AF%88" title="வெப்பச் சமநிலை – Tamil" lang="ta" hreflang="ta" data-title="வெப்பச் சமநிலை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Is%C4%B1l_denge" title="Isıl denge – Turkish" lang="tr" hreflang="tr" data-title="Isıl denge" data-language-autonym="Türkçe" 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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">Not to be confused with <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Thermodynamic equilibrium</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Thermal_equilibrium_in_closed_system.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Thermal_equilibrium_in_closed_system.png/220px-Thermal_equilibrium_in_closed_system.png" decoding="async" width="220" height="134" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Thermal_equilibrium_in_closed_system.png/330px-Thermal_equilibrium_in_closed_system.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Thermal_equilibrium_in_closed_system.png/440px-Thermal_equilibrium_in_closed_system.png 2x" data-file-width="765" data-file-height="466" /></a><figcaption>Development of a thermal equilibrium in a <a href="/wiki/Closed_system" title="Closed system">closed system</a> over time through a <a href="/wiki/Heat_flow" class="mw-redirect" title="Heat flow">heat flow</a> that levels out temperature differences</figcaption></figure> <p>Two <a href="/wiki/Physical_system" title="Physical system">physical systems</a> are in <b>thermal equilibrium</b> if there is no net flow of thermal energy between them when they are connected by a path permeable to <a href="/wiki/Heat" title="Heat">heat</a>. Thermal equilibrium obeys the <a href="/wiki/Zeroth_law_of_thermodynamics" title="Zeroth law of thermodynamics">zeroth law of thermodynamics</a>. A system is said to be in thermal equilibrium with itself if the temperature within the system is spatially uniform and temporally constant. </p><p>Systems in <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">thermodynamic equilibrium</a> are always in thermal equilibrium, but the converse is not always true. If the connection between the systems allows transfer of energy as 'change in <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a>' but does not allow transfer of matter or transfer of energy as <a href="/wiki/Work_(physics)" title="Work (physics)">work</a>, the two systems may reach thermal equilibrium without reaching thermodynamic equilibrium. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Two_varieties_of_thermal_equilibrium">Two varieties of thermal equilibrium</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermal_equilibrium&action=edit&section=1" title="Edit section: Two varieties of thermal equilibrium"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Relation_of_thermal_equilibrium_between_two_thermally_connected_bodies">Relation of thermal equilibrium between two thermally connected bodies</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermal_equilibrium&action=edit&section=2" title="Edit section: Relation of thermal equilibrium between two thermally connected bodies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The relation of thermal equilibrium is an instance of equilibrium between two bodies, which means that it refers to transfer through a selectively permeable partition of matter or work; it is called a diathermal connection. According to Lieb and Yngvason, the essential meaning of the relation of thermal equilibrium includes that it is reflexive and symmetric. It is not included in the essential meaning whether it is or is not transitive. After discussing the semantics of the definition, they postulate a substantial physical axiom, that they call the "zeroth law of thermodynamics", that thermal equilibrium is a transitive relation. They comment that the equivalence classes of systems so established are called isotherms.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Internal_thermal_equilibrium_of_an_isolated_body">Internal thermal equilibrium of an isolated body</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermal_equilibrium&action=edit&section=3" title="Edit section: Internal thermal equilibrium of an isolated body"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Thermal equilibrium of a body in itself refers to the body when it is isolated. The background is that no heat enters or leaves it, and that it is allowed unlimited time to settle under its own intrinsic characteristics. When it is completely settled, so that macroscopic change is no longer detectable, it is in its own thermal equilibrium. It is not implied that it is necessarily in other kinds of internal equilibrium. For example, it is possible that a body might reach internal thermal equilibrium but not be in internal chemical equilibrium; glass is an example.<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> </p><p>One may imagine an isolated system, initially not in its own state of internal thermal equilibrium. It could be subjected to a fictive thermodynamic operation of partition into two subsystems separated by nothing, no wall. One could then consider the possibility of transfers of energy as heat between the two subsystems. A long time after the fictive partition operation, the two subsystems will reach a practically stationary state, and so be in the relation of thermal equilibrium with each other. Such an adventure could be conducted in indefinitely many ways, with different fictive partitions. All of them will result in subsystems that could be shown to be in thermal equilibrium with each other, testing subsystems from different partitions. For this reason, an isolated system, initially not its own state of internal thermal equilibrium, but left for a long time, practically always will reach a final state which may be regarded as one of internal thermal equilibrium. Such a final state is one of spatial uniformity or homogeneity of temperature.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The existence of such states is a basic postulate of classical thermodynamics.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> This postulate is sometimes, but not often, called the minus first law of thermodynamics.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> A notable exception exists for isolated quantum systems which are <a href="/wiki/Many-body_localization" title="Many-body localization">many-body localized</a> and which <i>never</i> reach internal thermal equilibrium. </p> <div class="mw-heading mw-heading2"><h2 id="Thermal_contact">Thermal contact</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermal_equilibrium&action=edit&section=4" title="Edit section: Thermal contact"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Heat_transfer" title="Heat transfer">Heat can flow</a> into or out of a <a href="/wiki/Closed_system" title="Closed system">closed system</a> by way of <a href="/wiki/Conduction_(heat)" class="mw-redirect" title="Conduction (heat)">thermal conduction</a> or of <a href="/wiki/Thermal_radiation" title="Thermal radiation">thermal radiation</a> to or from a thermal reservoir, and when this process is effecting net transfer of heat, the system is not in thermal equilibrium. While the transfer of energy as heat continues, the system's temperature can be changing. </p> <div class="mw-heading mw-heading2"><h2 id="Bodies_prepared_with_separately_uniform_temperatures,_then_put_into_purely_thermal_communication_with_each_other"><span id="Bodies_prepared_with_separately_uniform_temperatures.2C_then_put_into_purely_thermal_communication_with_each_other"></span>Bodies prepared with separately uniform temperatures, then put into purely thermal communication with each other</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermal_equilibrium&action=edit&section=5" title="Edit section: Bodies prepared with separately uniform temperatures, then put into purely thermal communication with each other"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If bodies are prepared with separately microscopically stationary states, and are then put into purely thermal connection with each other, by conductive or radiative pathways, they will be in thermal equilibrium with each other just when the connection is followed by no change in either body. But if initially they are not in a relation of thermal equilibrium, heat will flow from the hotter to the colder, by whatever pathway, conductive or radiative, is available, and this flow will continue until thermal equilibrium is reached and then they will have the same temperature. </p><p>One form of thermal equilibrium is radiative exchange equilibrium.<sup id="cite_ref-Prevost_1791_7-0" class="reference"><a href="#cite_note-Prevost_1791-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Planck_1914_40_8-0" class="reference"><a href="#cite_note-Planck_1914_40-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Two bodies, each with its own uniform temperature, in solely radiative connection, no matter how far apart, or what partially obstructive, reflective, or refractive, obstacles lie in their path of radiative exchange, not moving relative to one another, will exchange thermal radiation, in net the hotter transferring energy to the cooler, and will exchange equal and opposite amounts just when they are at the same temperature. In this situation, <a href="/wiki/Kirchhoff%27s_law_of_thermal_radiation" title="Kirchhoff's law of thermal radiation">Kirchhoff's law of equality of radiative emissivity and absorptivity</a> and the <a href="/wiki/Helmholtz_reciprocity" title="Helmholtz reciprocity">Helmholtz reciprocity</a> principle are in play. </p> <div class="mw-heading mw-heading2"><h2 id="Change_of_internal_state_of_an_isolated_system">Change of internal state of an isolated system</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermal_equilibrium&action=edit&section=6" title="Edit section: Change of internal state of an isolated system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If an initially <a href="/wiki/Isolated_system" title="Isolated system">isolated physical system</a>, without internal walls that establish <a href="/wiki/Adiabatic_wall" title="Adiabatic wall">adiabatically isolated</a> subsystems, is left long enough, it will usually reach a state of thermal equilibrium in itself, in which its temperature will be <a href="/wiki/Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniform</a> throughout, but not necessarily a state of thermodynamic equilibrium, if there is some structural barrier that can prevent some possible processes in the system from reaching equilibrium; glass is an example. Classical thermodynamics in general considers idealized systems that have reached internal equilibrium, and idealized transfers of matter and <a href="/wiki/Energy_transfer" class="mw-redirect" title="Energy transfer">energy</a> between them. </p><p>An isolated physical system may be <a href="/wiki/Inhomogeneous" class="mw-redirect" title="Inhomogeneous">inhomogeneous</a>, or may be composed of several subsystems separated from each other by walls. If an initially inhomogeneous physical system, without internal walls, is isolated by a thermodynamic operation, it will in general over time change its internal state. Or if it is composed of several subsystems separated from each other by walls, it may change its state after a thermodynamic operation that changes its walls. Such changes may include change of temperature or spatial distribution of temperature, by changing the state of constituent materials. A rod of iron, initially prepared to be hot at one end and cold at the other, when isolated, will change so that its temperature becomes uniform all along its length; during the process, the rod is not in thermal equilibrium until its temperature is uniform. In a system prepared as a block of ice floating in a bath of hot water, and then isolated, the ice can melt; during the melting, the system is not in thermal equilibrium; but eventually, its temperature will become uniform; the block of ice will not re-form. A system prepared as a mixture of petrol vapour and air can be ignited by a spark and produce carbon dioxide and water; if this happens in an isolated system, it will increase the temperature of the system, and during the increase, the system is not in thermal equilibrium; but eventually, the system will settle to a uniform temperature. </p><p>Such changes in isolated systems are irreversible in the sense that while such a change will occur spontaneously whenever the system is prepared in the same way, the reverse change will practically never occur spontaneously within the isolated system; this is a large part of the content of the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a>. Truly perfectly isolated systems do not occur in nature, and always are artificially prepared. </p> <div class="mw-heading mw-heading3"><h3 id="In_a_gravitational_field">In a gravitational field</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermal_equilibrium&action=edit&section=7" title="Edit section: In a gravitational field"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One may consider a system contained in a very tall adiabatically isolating vessel with rigid walls initially containing a thermally heterogeneous distribution of material, left for a long time under the influence of a steady gravitational field, along its tall dimension, due to an outside body such as the earth. It will settle to a state of uniform temperature throughout, though not of uniform pressure or density, and perhaps containing several phases. It is then in internal thermal equilibrium and even in thermodynamic equilibrium. This means that all local parts of the system are in mutual radiative exchange equilibrium. This means that the temperature of the system is spatially uniform.<sup id="cite_ref-Planck_1914_40_8-1" class="reference"><a href="#cite_note-Planck_1914_40-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> This is so in all cases, including those of non-uniform external force fields. For an externally imposed gravitational field, this may be proved in macroscopic thermodynamic terms, by the calculus of variations, using the method of Lagrange multipliers.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Considerations of kinetic theory or statistical mechanics also support this statement.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Distinctions_between_thermal_and_thermodynamic_equilibria">Distinctions between thermal and thermodynamic equilibria</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermal_equilibrium&action=edit&section=8" title="Edit section: Distinctions between thermal and thermodynamic equilibria"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is an important distinction between thermal and <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">thermodynamic equilibrium</a>. According to Münster (1970), in states of thermodynamic equilibrium, the state variables of a system do not change at a measurable rate. Moreover, "The proviso 'at a measurable rate' implies that we can consider an equilibrium only with respect to specified processes and defined experimental conditions." Also, a state of thermodynamic equilibrium can be described by fewer macroscopic variables than any other state of a given body of matter. A single isolated body can start in a state which is not one of thermodynamic equilibrium, and can change till thermodynamic equilibrium is reached. Thermal equilibrium is a relation between two bodies or closed systems, in which transfers are allowed only of energy and take place through a partition permeable to heat, and in which the transfers have proceeded till the states of the bodies cease to change.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>An explicit distinction between 'thermal equilibrium' and 'thermodynamic equilibrium' is made by C.J. Adkins. He allows that two systems might be allowed to exchange heat but be constrained from exchanging work; they will naturally exchange heat till they have equal temperatures, and reach thermal equilibrium, but in general, will not be in thermodynamic equilibrium. They can reach thermodynamic equilibrium when they are allowed also to exchange work.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>Another explicit distinction between 'thermal equilibrium' and 'thermodynamic equilibrium' is made by B. C. Eu. He considers two systems in thermal contact, one a thermometer, the other a system in which several irreversible processes are occurring. He considers the case in which, over the time scale of interest, it happens that both the thermometer reading and the irreversible processes are steady. Then there is thermal equilibrium without thermodynamic equilibrium. Eu proposes consequently that the zeroth law of thermodynamics can be considered to apply even when thermodynamic equilibrium is not present; also he proposes that if changes are occurring so fast that a steady temperature cannot be defined, then "it is no longer possible to describe the process by means of a thermodynamic formalism. In other words, thermodynamics has no meaning for such a process."<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Thermal_equilibrium_of_planets">Thermal equilibrium of planets</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermal_equilibrium&action=edit&section=9" title="Edit section: Thermal equilibrium of planets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Planetary_equilibrium_temperature" title="Planetary equilibrium temperature">Planetary equilibrium temperature</a></div> <p>A planet is in thermal equilibrium when the incident energy reaching it (typically the <a href="/wiki/Solar_irradiance" title="Solar irradiance">solar irradiance</a> from its parent star) is equal to the <a href="/wiki/Infrared" title="Infrared">infrared</a> energy radiated away to space. </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=Thermal_equilibrium&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Thermal_center" title="Thermal center">Thermal center</a></li> <li><a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Thermodynamic equilibrium</a></li> <li><a href="/wiki/Radiative_equilibrium" title="Radiative equilibrium">Radiative equilibrium</a></li> <li><a href="/wiki/Thermal_oscillator" title="Thermal oscillator">Thermal oscillator</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermal_equilibrium&action=edit&section=11" title="Edit section: Citations"><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 reflist-columns references-column-width" style="column-width: 40em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink">'<i><a href="#cite_ref-1">^</a><b></b></i></span><i><b> <span class="reference-text">Lieb, E.H., Yngvason, J. (1999). The physics and mathematics of the second law of thermodynamics, </span></b></i><b>Physics Reports<i>, </i></b><i>314</i>..a': 1–96, p. 55–56. </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Adkins, C.J. (1968/1983), pp. 249–251.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="/wiki/Max_Planck" title="Max Planck">Planck, M.</a>, (1897/1903), p. 3.</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"><a href="/wiki/L%C3%A1szl%C3%B3_Tisza" title="László Tisza">Tisza, L.</a> (1966), p. 108.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Bailyn, M. (1994), p. 20.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><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="CITEREFMarslandBrownValente2015" class="citation journal cs1">Marsland, Robert; Brown, Harvey R.; Valente, Giovanni (2015). "Time and irreversibility in axiomatic thermodynamics". <i>American Journal of Physics</i>. <b>83</b> (7): 628–634. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015AmJPh..83..628M">2015AmJPh..83..628M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.4914528">10.1119/1.4914528</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/11311%2F1043322">11311/1043322</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:117173742">117173742</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=Time+and+irreversibility+in+axiomatic+thermodynamics&rft.volume=83&rft.issue=7&rft.pages=628-634&rft.date=2015&rft_id=info%3Ahdl%2F11311%2F1043322&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A117173742%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1119%2F1.4914528&rft_id=info%3Abibcode%2F2015AmJPh..83..628M&rft.aulast=Marsland&rft.aufirst=Robert&rft.au=Brown%2C+Harvey+R.&rft.au=Valente%2C+Giovanni&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermal+equilibrium" class="Z3988"></span></span> </li> <li id="cite_note-Prevost_1791-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Prevost_1791_7-0">^</a></b></span> <span class="reference-text"><a href="/wiki/Pierre_Prevost_(physicist)" title="Pierre Prevost (physicist)">Prevost, P.</a> (1791). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7ZLOAAAAMAAJ&pg=PA314">Mémoire sur l'equilibre du feu. <i>Journal de Physique</i> (Paris), vol. 38 pp. 314-322.</a></span> </li> <li id="cite_note-Planck_1914_40-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Planck_1914_40_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Planck_1914_40_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/Max_Planck" title="Max Planck">Planck, M.</a> (1914), p. 40.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Gibbs, J.W. (1876/1878), pp. 144-150.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="/wiki/Dirk_ter_Haar" title="Dirk ter Haar">ter Haar, D.</a>, <a href="/wiki/Harald_Wergeland" title="Harald Wergeland">Wergeland, H.</a> (1966), pp. 127–130.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Münster, A. (1970), pp. 309–310.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Bailyn, M. (1994), pp. 254-256.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVerkleyGerkema2004" class="citation journal cs1">Verkley, W. T. M.; Gerkema, T. (2004). <a rel="nofollow" class="external text" href="https://doi.org/10.1175%2F1520-0469%282004%29061%3C0931%3AOMEP%3E2.0.CO%3B2">"On Maximum Entropy Profiles"</a>. <i>Journal of the Atmospheric Sciences</i>. <b>61</b> (8): 931–936. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004JAtS...61..931V">2004JAtS...61..931V</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1175%2F1520-0469%282004%29061%3C0931%3AOMEP%3E2.0.CO%3B2">10.1175/1520-0469(2004)061<0931:OMEP>2.0.CO;2</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1520-0469">1520-0469</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Atmospheric+Sciences&rft.atitle=On+Maximum+Entropy+Profiles&rft.volume=61&rft.issue=8&rft.pages=931-936&rft.date=2004&rft.issn=1520-0469&rft_id=info%3Adoi%2F10.1175%2F1520-0469%282004%29061%3C0931%3AOMEP%3E2.0.CO%3B2&rft_id=info%3Abibcode%2F2004JAtS...61..931V&rft.aulast=Verkley&rft.aufirst=W.+T.+M.&rft.au=Gerkema%2C+T.&rft_id=https%3A%2F%2Fdoi.org%2F10.1175%252F1520-0469%25282004%2529061%253C0931%253AOMEP%253E2.0.CO%253B2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AThermal+equilibrium" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Akmaev, R.A. (2008). On the energetics of maximum-entropy temperature profiles, <i>Q. J. R. Meteorol. Soc.</i>, <b>134</b>:187–197.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Maxwell, J.C. (1867).</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Boltzmann, L. (1896/1964), p. 143.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Chapman, S., Cowling, T.G. (1939/1970), Section 4.14, pp. 75–78.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><a href="/wiki/J._R._Partington" title="J. R. Partington">Partington, J.R.</a> (1949), pp. 275–278.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Coombes, C.A., Laue, H. (1985). A paradox concerning the temperature distribution of a gas in a gravitational field, <i>Am. J. Phys.</i>, <b>53</b>: 272–273.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Román, F.L., White, J.A., Velasco, S. (1995). Microcanonical single-particle distributions for an ideal gas in a gravitational field, <i>Eur. J. Phys.</i>, <b>16</b>: 83–90.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Velasco, S., Román, F.L., White, J.A. (1996). On a paradox concerning the temperature distribution of an ideal gas in a gravitational field, <i>Eur. J. 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The Thermodynamics of Irreversible Processes and Generalized Hydrodynamics</i>, Kluwer Academic Publishers, Dordrecht, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4020-0788-4" title="Special:BookSources/1-4020-0788-4">1-4020-0788-4</a>, page 13.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Citation_references">Citation references</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Thermal_equilibrium&action=edit&section=12" title="Edit section: Citation references"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Adkins, C.J. (1968/1983). <i>Equilibrium Thermodynamics</i>, third edition, McGraw-Hill, London, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-25445-0" title="Special:BookSources/0-521-25445-0">0-521-25445-0</a>.</li> <li>Bailyn, M. (1994). <i>A Survey of Thermodynamics</i>, American Institute of Physics Press, New York, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-88318-797-3" title="Special:BookSources/0-88318-797-3">0-88318-797-3</a>.</li> <li><a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Boltzmann, L.</a> (1896/1964). <i>Lectures on Gas Theory</i>, translated by S.G. Brush, University of California Press, Berkeley.</li> <li><a href="/wiki/Sydney_Chapman_(mathematician)" title="Sydney Chapman (mathematician)">Chapman, S.</a>, <a href="/wiki/Thomas_George_Cowling" class="mw-redirect" title="Thomas George Cowling">Cowling, T.G.</a> (1939/1970). <i>The Mathematical Theory of Non-uniform gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases</i>, third edition 1970, Cambridge University Press, London.</li> <li><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs, J.W.</a> (1876/1878). On the equilibrium of heterogeneous substances, <i>Trans. Conn. Acad.</i>, <b>3</b>: 108-248, 343-524, reprinted in <i>The Collected Works of J. Willard Gibbs, Ph.D, LL. D.</i>, edited by W.R. Longley, R.G. Van Name, Longmans, Green & Co., New York, 1928, volume 1, pp. 55–353.</li> <li><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell, J.C.</a> (1867). On the dynamical theory of gases, <i>Phil. Trans. Roy. Soc. London</i>, <b>157</b>: 49–88.</li> <li>Münster, A. (1970). <i>Classical Thermodynamics</i>, translated by E.S. Halberstadt, Wiley–Interscience, London.</li> <li><a href="/wiki/J._R._Partington" title="J. R. Partington">Partington, J.R.</a> (1949). <i>An Advanced Treatise on Physical Chemistry</i>, volume 1, <i>Fundamental Principles. The Properties of Gases</i>, Longmans, Green and Co., London.</li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck, M.</a>, (1897/1903). <a rel="nofollow" class="external text" href="https://archive.org/details/treatiseonthermo00planrich"><i>Treatise on Thermodynamics</i></a>, translated by A. Ogg, first English edition, <a href="/wiki/Longman" title="Longman">Longmans, Green and Co.</a>, London.</li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck, M.</a> (1914). <i>The Theory of Heat Radiation</i>, second edition translated by M. Masius, P. Blakiston's Son and Co., Philadelphia.</li> <li><a href="/wiki/Dirk_ter_Haar" title="Dirk ter Haar">ter Haar, D.</a>, <a href="/wiki/Harald_Wergeland" title="Harald Wergeland">Wergeland, H.</a> (1966). <i>Elements of Thermodynamics</i>, Addison-Wesley Publishing, Reading MA.</li> <li><a href="/wiki/L%C3%A1szl%C3%B3_Tisza" title="László Tisza">Tisza, L.</a> (1966). <i>Generalized Thermodynamics</i>, M.I.T. 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