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cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Afficher / masquer la sous-section Température d'équilibre thermique</span> </button> <ul id="toc-Température_d'équilibre_thermique-sublist" class="vector-toc-list"> <li id="toc-Définition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Définition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Définition</span> </div> </a> <ul id="toc-Définition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Calcul_simplifié" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Calcul_simplifié"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Calcul simplifié</span> </div> </a> <ul id="toc-Calcul_simplifié-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Voir_aussi" class="vector-toc-list-item vector-toc-level-1 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Disponible en 28 langues." > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-28" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">28 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%88%D8%A7%D8%B2%D9%86_%D8%AD%D8%B1%D8%A7%D8%B1%D9%8A" title="توازن حراري – arabe" lang="ar" hreflang="ar" data-title="توازن حراري" data-language-autonym="العربية" data-language-local-name="arabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a 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 – bengali" lang="bn" hreflang="bn" data-title="তাপীয় সাম্যাবস্থা" data-language-autonym="বাংলা" data-language-local-name="bengali" 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-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%BE%D8%A7%D9%88%D8%B3%DB%95%D9%86%DA%AF%DB%8C%DB%8C_%DA%AF%DB%95%D8%B1%D9%85%DB%8C" title="ھاوسەنگیی گەرمی – sorani" lang="ckb" hreflang="ckb" data-title="ھاوسەنگیی گەرمی" data-language-autonym="کوردی" data-language-local-name="sorani" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs badge-Q70893996 mw-list-item" title=""><a href="https://cs.wikipedia.org/wiki/Tepeln%C3%A1_rovnov%C3%A1ha" title="Tepelná rovnováha – tchèque" lang="cs" hreflang="cs" data-title="Tepelná rovnováha" data-language-autonym="Čeština" data-language-local-name="tchèque" 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 – danois" lang="da" hreflang="da" data-title="Termisk ligevægt" data-language-autonym="Dansk" data-language-local-name="danois" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Thermal_equilibrium" title="Thermal equilibrium – anglais" lang="en" hreflang="en" data-title="Thermal equilibrium" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</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 – espagnol" lang="es" hreflang="es" data-title="Equilibrio térmico" data-language-autonym="Español" data-language-local-name="espagnol" 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 – persan" lang="fa" hreflang="fa" data-title="تعادل گرمایی" data-language-autonym="فارسی" data-language-local-name="persan" class="interlanguage-link-target"><span>فارسی</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 – irlandais" lang="ga" hreflang="ga" data-title="Cothromaíocht theirmeach" data-language-autonym="Gaeilge" data-language-local-name="irlandais" class="interlanguage-link-target"><span>Gaeilge</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 – croate" lang="hr" hreflang="hr" data-title="Toplinska ravnoteža" data-language-autonym="Hrvatski" data-language-local-name="croate" 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 – islandais" lang="is" hreflang="is" data-title="Varmajafnvægi" data-language-autonym="Íslenska" data-language-local-name="islandais" class="interlanguage-link-target"><span>Íslenska</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 – coréen" lang="ko" hreflang="ko" data-title="열평형" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</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 – norvégien nynorsk" lang="nn" hreflang="nn" data-title="Termisk likevekt" data-language-autonym="Norsk nynorsk" data-language-local-name="norvégien nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Echilibru_termic" title="Echilibru termic – roumain" lang="ro" hreflang="ro" data-title="Echilibru termic" data-language-autonym="Română" data-language-local-name="roumain" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Toplotna_ravnote%C5%BEa" title="Toplotna ravnoteža – serbo-croate" lang="sh" hreflang="sh" data-title="Toplotna ravnoteža" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croate" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</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 – cingalais" lang="si" hreflang="si" data-title="තාප සමතුලිතතාවය" data-language-autonym="සිංහල" data-language-local-name="cingalais" 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 – slovaque" lang="sk" hreflang="sk" data-title="Tepelná rovnováha" data-language-autonym="Slovenčina" data-language-local-name="slovaque" 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/Toplotno_ravnovesje" title="Toplotno ravnovesje – slovène" lang="sl" hreflang="sl" data-title="Toplotno ravnovesje" data-language-autonym="Slovenščina" data-language-local-name="slovène" class="interlanguage-link-target"><span>Slovenščina</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 – tamoul" lang="ta" hreflang="ta" data-title="வெப்பச் சமநிலை" data-language-autonym="தமிழ்" data-language-local-name="tamoul" 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 – turc" lang="tr" hreflang="tr" data-title="Isıl denge" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk badge-Q70893996 mw-list-item" title=""><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BF%D0%BB%D0%BE%D0%B2%D0%B0_%D1%80%D1%96%D0%B2%D0%BD%D0%BE%D0%B2%D0%B0%D0%B3%D0%B0" title="Теплова рівновага – ukrainien" lang="uk" hreflang="uk" data-title="Теплова рівновага" data-language-autonym="Українська" data-language-local-name="ukrainien" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AD%D8%B1%D8%A7%D8%B1%D8%AA%DB%8C_%D8%AA%D9%88%D8%A7%D8%B2%D9%86" title="حرارتی توازن – ourdou" lang="ur" hreflang="ur" data-title="حرارتی توازن" data-language-autonym="اردو" data-language-local-name="ourdou" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Issiqlik_muvozanati" title="Issiqlik muvozanati – ouzbek" lang="uz" hreflang="uz" data-title="Issiqlik muvozanati" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="ouzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%86%B1%E5%B9%B3%E8%A1%A1" title="熱平衡 – chinois" lang="zh" hreflang="zh" data-title="熱平衡" data-language-autonym="中文" data-language-local-name="chinois" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%86%B1%E5%B9%B3%E8%A1%A1" title="熱平衡 – cantonais" lang="yue" hreflang="yue" data-title="熱平衡" data-language-autonym="粵語" data-language-local-name="cantonais" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span 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metadata bandeau-article bandeau-niveau-ebauche"><div class="bandeau-cell bandeau-icone" style="display:table-cell;padding-right:0.5em"><span class="noviewer" typeof="mw:File"><a href="/wiki/Fichier:Noia_64_apps_energy.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Noia_64_apps_energy.png/35px-Noia_64_apps_energy.png" decoding="async" width="35" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Noia_64_apps_energy.png/53px-Noia_64_apps_energy.png 1.5x, //upload.wikimedia.org/wikipedia/commons/6/6f/Noia_64_apps_energy.png 2x" data-file-width="64" data-file-height="64" /></a></span></div><div class="bandeau-cell" style="display:table-cell;padding-right:0.5em"> <p><strong class="bandeau-titre">Cet article est une <a href="/wiki/Aide:%C3%89bauche" title="Aide:Ébauche">ébauche</a> concernant la <a href="/wiki/Thermodynamique" title="Thermodynamique">thermodynamique</a>.</strong> </p> <div class="nomobile"> Vous pouvez partager vos connaissances en l’améliorant (<b><a href="/wiki/Aide:Comment_modifier_une_page" title="Aide:Comment modifier une page">comment ?</a></b>) selon les recommandations des <a href="/wiki/Projet:Accueil" title="Projet:Accueil">projets correspondants</a>.</div> </div></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fichier: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/250px-Thermal_equilibrium_in_closed_system.png" decoding="async" width="250" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Thermal_equilibrium_in_closed_system.png/375px-Thermal_equilibrium_in_closed_system.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Thermal_equilibrium_in_closed_system.png/500px-Thermal_equilibrium_in_closed_system.png 2x" data-file-width="765" data-file-height="466" /></a><figcaption>Développement d'un équilibre thermique au cours du temps dans un <a href="/wiki/Syst%C3%A8me_isol%C3%A9" title="Système isolé">système isolé</a> composé de deux compartiments initialement à températures différentes et échangeant de la chaleur.</figcaption></figure> <p>En <a href="/wiki/Physique" title="Physique">physique</a>, et particulièrement en <a href="/wiki/Thermodynamique" title="Thermodynamique">thermodynamique</a>, l'<b>équilibre thermique</b> entre deux <a href="/wiki/Corps_(entit%C3%A9)" title="Corps (entité)">corps</a> de <a href="/wiki/Temp%C3%A9rature" title="Température">températures</a> différentes mis en contact est l'état atteint lorsque ces températures deviennent égales, l'échange d'<a href="/wiki/%C3%89nergie_thermique" title="Énergie thermique">énergie thermique</a> (<a href="/wiki/Chaleur_(thermodynamique)" title="Chaleur (thermodynamique)">chaleur</a>) entre ces deux corps étant alors nul. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Température_d'équilibre_thermique"><span id="Temp.C3.A9rature_d.27.C3.A9quilibre_thermique"></span>Température d'équilibre thermique</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%89quilibre_thermique&veaction=edit&section=1" title="Modifier la section : Température d'équilibre thermique" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%89quilibre_thermique&action=edit&section=1" title="Modifier le code source de la section : Température d'équilibre thermique"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Définition"><span id="D.C3.A9finition"></span>Définition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%89quilibre_thermique&veaction=edit&section=2" title="Modifier la section : Définition" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%89quilibre_thermique&action=edit&section=2" title="Modifier le code source de la section : Définition"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lorsque deux corps à des températures différentes sont mis en contact, il s'effectue entre eux un <a href="/wiki/Transfert_thermique" title="Transfert thermique">transfert thermique</a>, c'est-à-dire un échange de <a href="/wiki/Chaleur_(thermodynamique)" title="Chaleur (thermodynamique)">chaleur</a>. La température la plus élevée baisse, la plus basse augmente. Ce transfert s'effectue jusqu'à ce que les deux corps soient à la même température, la température d'équilibre. Le transfert thermique cesse alors. La température d'équilibre est intermédiaire entre les deux températures initiales. </p> <div class="mw-heading mw-heading3"><h3 id="Calcul_simplifié"><span id="Calcul_simplifi.C3.A9"></span>Calcul simplifié</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%89quilibre_thermique&veaction=edit&section=3" title="Modifier la section : Calcul simplifié" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%89quilibre_thermique&action=edit&section=3" title="Modifier le code source de la section : Calcul simplifié"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Supposons une évolution dans laquelle un <span class="nowrap">corps 1</span>, ayant une température élevée <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f304724948a3ef606c4a92459e22b87a954d993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{1}}"></span>, transmet de la <a href="/wiki/Chaleur_(thermodynamique)" title="Chaleur (thermodynamique)">chaleur</a> à un <span class="nowrap">corps 2</span>, ayant une température basse <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ba5f12fbb0ff766aec6e22148b429373608555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{2}}"></span>. À l'équilibre ils atteignent la température <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_{E}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{E}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceedee3f98b23ebe40bc74343ef0ffb9070f4f83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.845ex; height:2.509ex;" alt="{\displaystyle T_{E}}"></span>. Les deux corps échangent les chaleurs : </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 Q_{1}=m_{1}c_{p_{1}}(T_{E}-T_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{1}=m_{1}c_{p_{1}}(T_{E}-T_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2657899d16320af1f968441fc8ad17be28a76200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.89ex; height:3.009ex;" alt="{\displaystyle Q_{1}=m_{1}c_{p_{1}}(T_{E}-T_{1})}"></span></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 Q_{2}=m_{2}c_{p_{2}}(T_{E}-T_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{2}=m_{2}c_{p_{2}}(T_{E}-T_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dd0a0d77fac6b91f47a917993f21c2e3fb8dbc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.89ex; height:3.009ex;" alt="{\displaystyle Q_{2}=m_{2}c_{p_{2}}(T_{E}-T_{2})}"></span></dd></dl> <p>avec : </p> <ul><li><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_{p_{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{p_{1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/018a199669dbb3a2a451f826c6023372a7a1d070" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.898ex; height:2.343ex;" alt="{\displaystyle c_{p_{1}}}"></span> et <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_{p_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{p_{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf94535c286e869370a599ef959faa3014dc5ac4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.898ex; height:2.343ex;" alt="{\displaystyle c_{p_{2}}}"></span> les <a href="/wiki/Capacit%C3%A9_thermique_massique" title="Capacité thermique massique">capacités thermiques massiques</a> respectives des <span class="nowrap">corps 1</span> et <span class="nowrap">corps 2</span>, en <a href="/wiki/Joule_par_kilogramme_kelvin" title="Joule par kilogramme kelvin">joules par kilogramme kelvin</a> (<abbr class="abbr" title="joule par kilogramme kelvin">J kg<sup>−1</sup> K<sup>−1</sup></abbr>) ;</li> <li><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 m_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31aafa60e48d39ccce922404c0b80340b2cc777a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{1}}"></span> et <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 m_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ecebe334d5cadc3ffcf245eb02919034d7a2ec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{2}}"></span> les <a href="/wiki/Masse" title="Masse">masses</a> respectives des <span class="nowrap">corps 1</span> et <span class="nowrap">corps 2</span>, en <a href="/wiki/Kilogramme" title="Kilogramme">kilogrammes</a> (<abbr class="abbr" title="kilogramme">kg</abbr>) ;</li> <li><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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f304724948a3ef606c4a92459e22b87a954d993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{1}}"></span> et <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ba5f12fbb0ff766aec6e22148b429373608555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{2}}"></span> les températures initiales respectives des <span class="nowrap">corps 1</span> et <span class="nowrap">corps 2</span>, en <a href="/wiki/Kelvin" title="Kelvin">kelvins</a> (<abbr class="abbr" title="kelvin">K</abbr>) ;</li> <li><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_{E}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{E}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceedee3f98b23ebe40bc74343ef0ffb9070f4f83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.845ex; height:2.509ex;" alt="{\displaystyle T_{E}}"></span> la température d'équilibre thermique, en kelvins (<abbr class="abbr" title="kelvin">K</abbr>).</li></ul> <p>En l'absence de tout autre échange d'<a href="/wiki/%C3%89nergie" title="Énergie">énergie</a>, notamment de <a href="/wiki/Travail_d%27une_force" title="Travail d'une force">travail</a>, l'énergie dégagée par le <span class="nowrap">corps 1</span> est entièrement récupérée par le <span class="nowrap">corps 2</span> (selon le <a href="/wiki/Premier_principe_de_la_thermodynamique" title="Premier principe de la thermodynamique">premier principe de la thermodynamique</a>), ce qui se traduit par : </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 Q_{1}+Q_{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{1}+Q_{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28f6d1402108281adb6f222d9be4e58c758a3766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.886ex; height:2.509ex;" alt="{\displaystyle Q_{1}+Q_{2}=0}"></span></dd></dl> <p>De cette équation on tire : </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 T_{E}={\frac {m_{1}c_{p_{1}}T_{1}+m_{2}c_{p_{2}}T_{2}}{m_{1}c_{p_{1}}+m_{2}c_{p_{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{E}={\frac {m_{1}c_{p_{1}}T_{1}+m_{2}c_{p_{2}}T_{2}}{m_{1}c_{p_{1}}+m_{2}c_{p_{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15ffdbeb66dba52f15d5e1470b650852d34d581" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.428ex; height:6.343ex;" alt="{\displaystyle T_{E}={\frac {m_{1}c_{p_{1}}T_{1}+m_{2}c_{p_{2}}T_{2}}{m_{1}c_{p_{1}}+m_{2}c_{p_{2}}}}}"></span></dd></dl> <dl><dt>Remarques</dt></dl> <ul><li>Il est supposé que les capacités thermiques ne dépendent pas de la température, ce qui n'est approximativement vrai que sur de courtes plages de température.</li> <li>Cette relation reste valable si toutes les températures sont exprimées en <a href="/wiki/Degr%C3%A9_Celsius" title="Degré Celsius">degrés Celsius</a> (<abbr class="abbr" title="degré Celsius">°C</abbr>).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%89quilibre_thermique&veaction=edit&section=4" title="Modifier la section : Voir aussi" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%89quilibre_thermique&action=edit&section=4" title="Modifier le code source de la section : Voir aussi"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%C3%89quilibre_thermodynamique" title="Équilibre thermodynamique">Équilibre thermodynamique</a></li> <li><a href="/wiki/Principe_z%C3%A9ro_de_la_thermodynamique" title="Principe zéro de la thermodynamique">Principe zéro de la thermodynamique</a></li></ul> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:Physique" title="Portail de la physique"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/24px-Circle-icons-physics-logo.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/36px-Circle-icons-physics-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/48px-Circle-icons-physics-logo.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Physique" title="Portail:Physique">Portail de la physique</a></span> </span></li> </ul>    </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; 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