Température thermodynamique

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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 Concept historique de température</span> </button> <ul id="toc-Concept_historique_de_température-sublist" class="vector-toc-list"> <li id="toc-Thermodynamique_des_milieux_à_l'équilibre" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Thermodynamique_des_milieux_à_l'équilibre"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Thermodynamique des milieux à l'équilibre</span> </div> </a> <ul id="toc-Thermodynamique_des_milieux_à_l'équilibre-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Thermodynamique_hors_équilibre" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Thermodynamique_hors_équilibre"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Thermodynamique hors équilibre</span> </div> </a> <ul id="toc-Thermodynamique_hors_équilibre-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Remarques_sur_la_notion_de_température" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Remarques_sur_la_notion_de_température"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Remarques sur la notion de température</span> </div> </a> <button aria-controls="toc-Remarques_sur_la_notion_de_température-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>Afficher / masquer la sous-section Remarques sur la notion de température</span> </button> <ul id="toc-Remarques_sur_la_notion_de_température-sublist" class="vector-toc-list"> <li id="toc-Zéro_absolu" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Zéro_absolu"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Zéro absolu</span> </div> </a> <ul id="toc-Zéro_absolu-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Échelle_absolue_de_température" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Échelle_absolue_de_température"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Échelle absolue de température</span> </div> </a> <ul id="toc-Échelle_absolue_de_température-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Température_de_rayonnement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Température_de_rayonnement"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Température de rayonnement</span> </div> </a> <ul id="toc-Température_de_rayonnement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Milieu_à_plusieurs_températures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Milieu_à_plusieurs_températures"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Milieu à plusieurs températures</span> </div> </a> <ul id="toc-Milieu_à_plusieurs_températures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Milieu_dans_lequel_la_température_n'est_pas_un_paramètre_pertinent" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Milieu_dans_lequel_la_température_n'est_pas_un_paramètre_pertinent"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Milieu dans lequel la température n'est pas un paramètre pertinent</span> </div> </a> <ul id="toc-Milieu_dans_lequel_la_température_n'est_pas_un_paramètre_pertinent-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Température_négative" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Température_négative"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Température négative</span> </div> </a> <ul id="toc-Température_négative-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notes_et_références" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_et_références"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Notes et références</span> </div> </a> <button aria-controls="toc-Notes_et_références-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>Afficher / masquer la sous-section Notes et références</span> </button> <ul id="toc-Notes_et_références-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Références" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Références"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Références</span> </div> </a> <ul id="toc-Références-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Voir_aussi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voir_aussi"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Voir aussi</span> </div> </a> <ul id="toc-Voir_aussi-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="Sommaire" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table des matières" > <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="Basculer la table des matières" > <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 mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Basculer la table des matières</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Température thermodynamique</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Aller à un article dans une autre langue. Disponible en 43 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-43" 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">43 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%AF%D8%B1%D8%AC%D8%A9_%D8%AD%D8%B1%D8%A7%D8%B1%D8%A9_%D9%85%D8%B7%D9%84%D9%82%D8%A9" title="درجة حرارة مطلقة – arabe" lang="ar" hreflang="ar" data-title="درجة حرارة مطلقة" data-language-autonym="العربية" data-language-local-name="arabe" 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%90%D0%B1%D1%81%D0%B0%D0%BB%D1%8E%D1%82%D0%BD%D0%B0%D1%8F_%D1%82%D1%8D%D1%80%D0%BC%D0%B0%D0%B4%D1%8B%D0%BD%D0%B0%D0%BC%D1%96%D1%87%D0%BD%D0%B0%D1%8F_%D1%82%D1%8D%D0%BC%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D1%83%D1%80%D0%B0" title="Абсалютная тэрмадынамічная тэмпература – biélorusse" lang="be" hreflang="be" data-title="Абсалютная тэрмадынамічная тэмпература" data-language-autonym="Беларуская" data-language-local-name="biélorusse" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%90%D0%B1%D1%81%D0%B0%D0%BB%D1%8E%D1%82%D0%BD%D0%B0%D1%8F_%D1%82%D1%8D%D1%80%D0%BC%D0%B0%D0%B4%D1%8B%D0%BD%D0%B0%D0%BC%D1%96%D1%87%D0%BD%D0%B0%D1%8F_%D1%82%D1%8D%D0%BC%D0%BF%D1%8D%D1%80%D0%B0%D1%82%D1%83%D1%80%D0%B0" title="Абсалютная тэрмадынамічная тэмпэратура – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Абсалютная тэрмадынамічная тэмпэратура" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D1%87%D0%BD%D0%B0_%D1%82%D0%B5%D0%BC%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D1%83%D1%80%D0%B0" title="Термодинамична температура – bulgare" lang="bg" hreflang="bg" data-title="Термодинамична температура" data-language-autonym="Български" data-language-local-name="bulgare" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A6%B0%E0%A6%AE_%E0%A6%A4%E0%A6%BE%E0%A6%AA%E0%A6%AE%E0%A6%BE%E0%A6%A4%E0%A7%8D%E0%A6%B0%E0%A6%BE" title="পরম তাপমাত্রা – bengali" lang="bn" hreflang="bn" data-title="পরম তাপমাত্রা" 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/Temperatura_termodin%C3%A0mica" title="Temperatura termodinàmica – catalan" lang="ca" hreflang="ca" data-title="Temperatura termodinàmica" 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/Termodynamick%C3%A1_teplota" title="Termodynamická teplota – tchèque" lang="cs" hreflang="cs" data-title="Termodynamická teplota" data-language-autonym="Čeština" data-language-local-name="tchèque" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%C4%83%D0%BB%D0%BB%D0%B0_%D1%82%D0%B5%D0%BC%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D1%83%D1%80%D0%B0" title="Термодинамикăлла температура – tchouvache" lang="cv" hreflang="cv" data-title="Термодинамикăлла температура" data-language-autonym="Чӑвашла" data-language-local-name="tchouvache" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Termodynamisk_temperatur" title="Termodynamisk temperatur – danois" lang="da" hreflang="da" data-title="Termodynamisk temperatur" data-language-autonym="Dansk" data-language-local-name="danois" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Thermodynamische_Temperatur" title="Thermodynamische Temperatur – allemand" lang="de" hreflang="de" data-title="Thermodynamische Temperatur" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Thermodynamic_temperature" title="Thermodynamic temperature – anglais" lang="en" hreflang="en" data-title="Thermodynamic temperature" 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/Temperatura_absoluta" title="Temperatura absoluta – espagnol" lang="es" hreflang="es" data-title="Temperatura absoluta" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Absoluutne_temperatuur" title="Absoluutne temperatuur – estonien" lang="et" hreflang="et" data-title="Absoluutne temperatuur" data-language-autonym="Eesti" data-language-local-name="estonien" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D9%85%D8%A7%DB%8C_%D8%AA%D8%B1%D9%85%D9%88%D8%AF%DB%8C%D9%86%D8%A7%D9%85%DB%8C%DA%A9%DB%8C" title="دمای ترمودینامیکی – persan" lang="fa" hreflang="fa" data-title="دمای ترمودینامیکی" data-language-autonym="فارسی" data-language-local-name="persan" class="interlanguage-link-target"><span>فارسی</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%A4%BE%E0%A4%97%E0%A4%A4%E0%A4%BF%E0%A4%95_%E0%A4%A4%E0%A4%BE%E0%A4%AA%E0%A4%95%E0%A5%8D%E0%A4%B0%E0%A4%AE" title="ऊष्मागतिक तापक्रम – hindi" lang="hi" hreflang="hi" data-title="ऊष्मागतिक तापक्रम" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Termodinamikai_h%C5%91m%C3%A9rs%C3%A9klet" title="Termodinamikai hőmérséklet – hongrois" lang="hu" hreflang="hu" data-title="Termodinamikai hőmérséklet" data-language-autonym="Magyar" data-language-local-name="hongrois" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Suhu_termodinamika" title="Suhu termodinamika – indonésien" lang="id" hreflang="id" data-title="Suhu termodinamika" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésien" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Temperatura_assoluta" title="Temperatura assoluta – italien" lang="it" hreflang="it" data-title="Temperatura assoluta" data-language-autonym="Italiano" data-language-local-name="italien" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%86%B1%E5%8A%9B%E5%AD%A6%E6%B8%A9%E5%BA%A6" title="熱力学温度 – japonais" lang="ja" hreflang="ja" data-title="熱力学温度" data-language-autonym="日本語" data-language-local-name="japonais" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%97%E1%83%94%E1%83%A0%E1%83%9B%E1%83%9D%E1%83%93%E1%83%98%E1%83%9C%E1%83%90%E1%83%9B%E1%83%98%E1%83%99%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%A2%E1%83%94%E1%83%9B%E1%83%9E%E1%83%94%E1%83%A0%E1%83%90%E1%83%A2%E1%83%A3%E1%83%A0%E1%83%90" title="თერმოდინამიკური ტემპერატურა – géorgien" lang="ka" hreflang="ka" data-title="თერმოდინამიკური ტემპერატურა" data-language-autonym="ქართული" data-language-local-name="géorgien" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Absolyut_temperatura" title="Absolyut temperatura – karakalpak" lang="kaa" hreflang="kaa" data-title="Absolyut temperatura" data-language-autonym="Qaraqalpaqsha" data-language-local-name="karakalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D1%82%D0%B5%D0%BC%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D1%83%D1%80%D0%B0" title="Термодинамикалық температура – kazakh" lang="kk" hreflang="kk" data-title="Термодинамикалық температура" data-language-autonym="Қазақша" data-language-local-name="kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%97%B4%EC%97%AD%ED%95%99_%EC%98%A8%EB%8F%84" title="열역학 온도 – coréen" lang="ko" hreflang="ko" data-title="열역학 온도" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A2%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%D0%B8%D0%B9%D0%BD_%D1%82%D0%B5%D0%BC%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D1%83%D1%80" title="Термодинамикийн температур – mongol" lang="mn" hreflang="mn" data-title="Термодинамикийн температур" data-language-autonym="Монгол" data-language-local-name="mongol" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Suhu_termodinamik" title="Suhu termodinamik – malais" lang="ms" hreflang="ms" data-title="Suhu termodinamik" data-language-autonym="Bahasa Melayu" data-language-local-name="malais" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Absolute_temperatuur" title="Absolute temperatuur – néerlandais" lang="nl" hreflang="nl" data-title="Absolute temperatuur" data-language-autonym="Nederlands" data-language-local-name="néerlandais" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Termodynamisk_temperatur" title="Termodynamisk temperatur – norvégien nynorsk" lang="nn" hreflang="nn" data-title="Termodynamisk temperatur" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Termodynamisk_temperatur" title="Termodynamisk temperatur – norvégien bokmål" lang="nb" hreflang="nb" data-title="Termodynamisk temperatur" data-language-autonym="Norsk bokmål" data-language-local-name="norvégien bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Kelviin" title="Kelviin – oromo" lang="om" hreflang="om" data-title="Kelviin" data-language-autonym="Oromoo" data-language-local-name="oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pt badge-Q70893996 mw-list-item" title=""><a href="https://pt.wikipedia.org/wiki/Temperatura_termodin%C3%A2mica" title="Temperatura termodinâmica – portugais" lang="pt" hreflang="pt" data-title="Temperatura termodinâmica" data-language-autonym="Português" data-language-local-name="portugais" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%82%D0%B5%D0%BC%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D1%83%D1%80%D0%B0" title="Термодинамическая температура – russe" lang="ru" hreflang="ru" data-title="Термодинамическая температура" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Absolute_temperature" title="Absolute temperature – Simple English" lang="en-simple" hreflang="en-simple" data-title="Absolute temperature" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Termodynamick%C3%A1_teplota" title="Termodynamická teplota – slovaque" lang="sk" hreflang="sk" data-title="Termodynamická teplota" 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/Absolutna_temperatura" title="Absolutna temperatura – slovène" lang="sl" hreflang="sl" data-title="Absolutna temperatura" data-language-autonym="Slovenščina" data-language-local-name="slovène" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Absol%C3%BDut_temperatura" title="Absolýut temperatura – turkmène" lang="tk" hreflang="tk" data-title="Absolýut temperatura" data-language-autonym="Türkmençe" data-language-local-name="turkmène" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Mutlak_s%C4%B1cakl%C4%B1k" title="Mutlak sıcaklık – turc" lang="tr" hreflang="tr" data-title="Mutlak sıcaklık" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D1%96%D1%87%D0%BD%D0%B0_%D1%82%D0%B5%D0%BC%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D1%83%D1%80%D0%B0" title="Термодинамічна температура – ukrainien" lang="uk" hreflang="uk" data-title="Термодинамічна температура" 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%AD%D8%B1%DA%A9%DB%8C%D8%A7%D8%AA%DB%8C_%D8%AF%D8%B1%D8%AC%DB%81_%D8%AD%D8%B1%D8%A7%D8%B1%D8%AA_%DB%8C%D8%A7_%D9%85%D8%B7%D9%84%D9%82_%D8%AF%D8%B1%D8%AC%DB%81_%D8%AD%D8%B1%D8%A7%D8%B1%D8%AA" title="حرحرکیاتی درجہ حرارت یا مطلق درجہ حرارت – ourdou" lang="ur" hreflang="ur" data-title="حرحرکیاتی درجہ حرارت یا مطلق درجہ حرارت" 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/Termodinamik_harorat" title="Termodinamik harorat – ouzbek" lang="uz" hreflang="uz" data-title="Termodinamik harorat" 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-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Thermodynamische_temperateure" 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data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Un article de Wikipédia, l'encyclopédie libre.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="fr" dir="ltr"><div class="infobox_v3 infobox infobox--frwiki noarchive"> <div class="entete" style=""> <div>Température</div> </div> <table><caption class="hidden" style="">Données clés</caption> <tbody><tr> <th scope="row"><a href="/wiki/Unit%C3%A9s_de_base_du_Syst%C3%A8me_international" title="Unités de base du Système international">Unités SI</a></th> <td> <a href="/wiki/Kelvin" title="Kelvin">kelvin</a> (K)</td> </tr> <tr> <th scope="row"><a href="/wiki/Dimension_(physique)" title="Dimension (physique)">Dimension</a></th> <td> <span class="nowrap"><a class="mw-selflink selflink">Θ</a></span></td> </tr> <tr> <th scope="row">Nature</th> <td> Grandeur <a href="/wiki/Scalaire_(physique)" title="Scalaire (physique)">scalaire</a> <a href="/wiki/Grandeur_intensive" class="mw-redirect" title="Grandeur intensive">intensive</a></td> </tr> <tr> <th scope="row">Symbole usuel</th> <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> </tr> <tr> <th scope="row">Lien à d'autres grandeurs</th> <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 \mathrm {d} S={\frac {\delta Q}{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} S={\frac {\delta Q}{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4434f0305e8d6aeab2abee298bfd3b068ceef5f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.613ex; height:5.343ex;" alt="{\displaystyle \mathrm {d} S={\frac {\delta Q}{T}}}" /></span><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite_crochet">[</span>a<span class="cite_crochet">]</span></a></sup><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 T=\left({\frac {\partial U}{\partial S}}\right)_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=\left({\frac {\partial U}{\partial S}}\right)_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a127685bd48eef4d294443c7b6fedaca0b37efd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.589ex; height:6.176ex;" alt="{\displaystyle T=\left({\frac {\partial U}{\partial S}}\right)_{V}}" /></span><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite_crochet">[</span>b<span class="cite_crochet">]</span></a></sup><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 T=(k\beta )^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>β</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=(k\beta )^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ba2bc42cc9c99a294468b8e2c0fd93a9e481a60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.42ex; height:3.176ex;" alt="{\displaystyle T=(k\beta )^{-1}}" /></span></td> </tr> </tbody></table><p class="navbar bordered noprint" style=""><span class="plainlinks navigation-not-searchable"><a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Temp%C3%A9rature_thermodynamique&action=edit">modifier</a></span> <span typeof="mw:File"><a href="/wiki/Mod%C3%A8le:Infobox_Grandeur_physique" title="Consultez la documentation du modèle"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Info_Simple.svg/12px-Info_Simple.svg.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Info_Simple.svg/18px-Info_Simple.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/Info_Simple.svg/24px-Info_Simple.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></p></div> <p>La <b>température thermodynamique</b> est une formalisation de la notion expérimentale de <a href="/wiki/Temp%C3%A9rature" title="Température">température</a> et constitue l’une des <a href="/wiki/Grandeur_physique" title="Grandeur physique">grandeurs</a> principales de la <a href="/wiki/Thermodynamique" title="Thermodynamique">thermodynamique</a>. Elle est intrinsèquement liée à l'<a href="/wiki/Entropie_(thermodynamique)" title="Entropie (thermodynamique)">entropie</a>. </p><p>Usuellement notée <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>, la température thermodynamique se mesure en <a href="/wiki/Kelvin" title="Kelvin">kelvins</a> (symbole K). Encore souvent qualifiée de « <a href="/wiki/Temp%C3%A9rature_absolue" title="Température absolue">température absolue</a> », elle constitue une mesure absolue parce qu’elle traduit directement le phénomène physique fondamental qui la sous-tend : l’<a href="/wiki/%C3%89nergie_thermique" title="Énergie thermique">agitation</a> des <a href="/wiki/Particule_%C3%A9l%C3%A9mentaire" title="Particule élémentaire">particules</a> constituant la <a href="/wiki/Mati%C3%A8re" title="Matière">matière</a> (translation, vibration, rotation, <a href="/wiki/Niveau_d%27%C3%A9nergie" title="Niveau d'énergie">niveaux d'énergie</a> électronique). Son point origine, ou <a href="/wiki/Z%C3%A9ro_absolu" title="Zéro absolu">zéro absolu</a>, correspond par définition à l’<a href="/wiki/%C3%89tat_de_la_mati%C3%A8re" title="État de la matière">état de la matière</a> où ces particules ont une entropie minimale, ce qui correspond généralement à une énergie minimale<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Concept_historique_de_température"><span id="Concept_historique_de_temp.C3.A9rature"></span>Concept historique de température</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&veaction=edit&section=1" title="Modifier la section : Concept historique de température" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&action=edit&section=1" title="Modifier le code source de la section : Concept historique de température"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La <a href="/wiki/Temp%C3%A9rature" title="Température">température</a> est une notion qui a évolué avec la connaissance de la matière. On peut distinguer diverses étapes du concept. </p> <div class="mw-heading mw-heading3"><h3 id="Thermodynamique_des_milieux_à_l'équilibre"><span id="Thermodynamique_des_milieux_.C3.A0_l.27.C3.A9quilibre"></span>Thermodynamique des milieux à l'équilibre</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&veaction=edit&section=2" title="Modifier la section : Thermodynamique des milieux à l'équilibre" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&action=edit&section=2" title="Modifier le code source de la section : Thermodynamique des milieux à l'équilibre"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On s'intéresse ici aux <a href="/wiki/%C3%89quilibre_thermodynamique" title="Équilibre thermodynamique">états thermodynamiques d'équilibre</a> et aux possibilités (ou à l'impossibilité) de passer d'un état à un autre par une <a href="/wiki/Processus_quasi_statique" title="Processus quasi statique">transformation quasi statique</a>, c'est-à-dire s'effectuant par une suite d'états d'équilibre. Cette partie de la <a href="/wiki/Thermodynamique" title="Thermodynamique">thermodynamique</a> devrait en toute logique s'appeler « thermostatique », mais ce mot est défini comme <a href="/wiki/Adjectif" title="Adjectif">adjectif</a> dans un autre sens. </p><p>Au <abbr class="abbr" title="17ᵉ siècle"><span class="romain">XVII</span><sup style="font-size:72%">e</sup></abbr> – <abbr class="abbr" title="18ᵉ siècle"><span class="romain">XVIII</span><sup style="font-size:72%">e</sup></abbr> siècle, l'apparition du <a href="/wiki/Thermom%C3%A8tre" title="Thermomètre">thermomètre</a> permet de quantifier les notions d'état chaud ou froid d'un milieu, quantités accessibles à la perception corporelle. La température thermométrique est ce que mesure l'appareil : un effet, en l'occurrence la <a href="/wiki/Dilatation_thermique" title="Dilatation thermique">dilatation thermique</a> d'un liquide. Le lien avec la température se fait par le choix d'une échelle de référence, qui est le problème de la <a href="/wiki/Thermom%C3%A9trie" title="Thermométrie">thermométrie</a>. Mais cela ne renseigne en rien sur la nature de la température et ne donne que peu d'indications sur ses propriétés. </p><p>À la même époque, <a href="/wiki/Joseph_Black" title="Joseph Black">Joseph Black</a> établit le lien entre 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}"> <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> et chaleur <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" /></span> en introduisant la <a href="/wiki/Capacit%C3%A9_thermique" title="Capacité thermique">capacité thermique</a> <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_{V}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00c18988639f2937d3fe4dc00dfa0b0bd7e0bec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.158ex; height:2.509ex;" alt="{\displaystyle C_{V}}" /></span><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup>. Si on se place à volume constant : </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 \delta Q=C_{V}\,\mathrm {d} T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>Q</mi> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta Q=C_{V}\,\mathrm {d} T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4f1e5c7d0bd5db67e428011ab6439e0b6081d79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.459ex; height:2.676ex;" alt="{\displaystyle \delta Q=C_{V}\,\mathrm {d} T}" /></span></dd></dl> <p>Cette loi montre que la température est une quantité substituable à la chaleur et donc à l'<a href="/wiki/%C3%89nergie_interne" title="Énergie interne">énergie interne</a> en l'absence de <a href="/wiki/Travail_d%27une_force" title="Travail d'une force">travail</a>. </p><p>Les travaux sur les <a href="/wiki/Machine_thermique" title="Machine thermique">machines thermiques</a> au <abbr class="abbr" title="19ᵉ siècle"><span class="romain">XIX</span><sup style="font-size:72%">e</sup></abbr> siècle, en particulier ceux de <a href="/wiki/Sadi_Carnot_(physicien)" title="Sadi Carnot (physicien)">Carnot</a>, montrent l'impossibilité des cycles monothermes. Le <a href="/wiki/Deuxi%C3%A8me_principe_de_la_thermodynamique" title="Deuxième principe de la thermodynamique">deuxième principe de la thermodynamique</a> établi par <a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite_crochet">[</span>3<span class="cite_crochet">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite_crochet">[</span>4<span class="cite_crochet">]</span></a></sup> conduit à l'introduction de la notion d'<a href="/wiki/Entropie_(thermodynamique)" title="Entropie (thermodynamique)">entropie</a> S, liée à une variation de <a href="/wiki/Chaleur_(thermodynamique)" title="Chaleur (thermodynamique)">chaleur</a> <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 \delta Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf715eece146c816847a8c5d56eae97798453d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle \delta Q}" /></span> reçue par le système et à la température thermodynamique <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>. Pour une transformation réversible : </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 \mathrm {d} S={\frac {\delta Q}{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} S={\frac {\delta Q}{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4434f0305e8d6aeab2abee298bfd3b068ceef5f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.613ex; height:5.343ex;" alt="{\displaystyle \mathrm {d} S={\frac {\delta Q}{T}}}" /></span></dd></dl> <p>Cette définition permet d'annuler les <a href="/wiki/Transfert_thermique" title="Transfert thermique">échanges de chaleur</a> entre deux milieux de température égale mais ne suffit pas pour définir une température thermodynamique de manière unique. </p> <style data-mw-deduplicate="TemplateStyles:r222902168">.mw-parser-output div.NavFrame{background-color:var(--background-color-base,#fff);border:1px var(--border-color-base,#a2a9b1);border-style:solid;margin-bottom:1em;clear:both}.mw-parser-output div.NavFrame div.NavHead{min-height:1.6em;font-weight:bold;background-color:#efefef;color:var(--color-emphasized,#000);text-align:center}.mw-parser-output div.NavContent{display:block;margin:0;text-align:left}body:not(.mw-mf) .mw-parser-output div.NavFrame{padding:2px;text-align:center;border-collapse:collapse;font-size:95%}body:not(.mw-mf) .mw-parser-output div.NavPic{padding:2px;float:left}body:not(.mw-mf) .mw-parser-output div.NavEnd{height:0;clear:both}@media screen{html.skin-theme-clientpref-night .mw-parser-output div.NavFrame div.NavHead{background-color:#27292d}html.skin-theme-clientpref-night .mw-parser-output table div.NavFrame a{color:#88a3e8}html.skin-theme-clientpref-night .mw-parser-output table div.NavFrame a.new{color:#fd7865}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output div.NavFrame div.NavHead{background-color:#27292d}html.skin-theme-clientpref-os .mw-parser-output table div.NavFrame a{color:#88a3e8}html.skin-theme-clientpref-os .mw-parser-output table div.NavFrame a.new{color:#fd7865}}</style><div class="NavFrame" style="" title="[afficher]/[masquer]"> <div class="NavHead" style="">Cas d'un gaz parfait à volume constant.</div> <div class="NavContent" style=""> <p>Il suffit de deux paramètres pour définir l'<a href="/wiki/Fonction_d%27%C3%A9tat" title="Fonction d'état">état thermodynamique</a> d'un <a href="/wiki/Gaz_parfait" title="Gaz parfait">gaz parfait</a>, par exemple l'<a href="/wiki/%C3%89nergie_interne" title="Énergie interne">énergie interne</a> <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" /></span> et le volume <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> : </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 S=S\!\left(U,V\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>S</mi> <mspace width="negativethinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mi>U</mi> <mo>,</mo> <mi>V</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=S\!\left(U,V\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a00efa6f87c31be71c8c64f0dcc9b449e3ab86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.51ex; height:2.843ex;" alt="{\displaystyle S=S\!\left(U,V\right)}" /></span></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Sch%C3%A9ma_d%27un_syst%C3%A8me_thermodynamique_ferm%C3%A9_divariant.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Sch%C3%A9ma_d%27un_syst%C3%A8me_thermodynamique_ferm%C3%A9_divariant.png/220px-Sch%C3%A9ma_d%27un_syst%C3%A8me_thermodynamique_ferm%C3%A9_divariant.png" decoding="async" width="220" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Sch%C3%A9ma_d%27un_syst%C3%A8me_thermodynamique_ferm%C3%A9_divariant.png/330px-Sch%C3%A9ma_d%27un_syst%C3%A8me_thermodynamique_ferm%C3%A9_divariant.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Sch%C3%A9ma_d%27un_syst%C3%A8me_thermodynamique_ferm%C3%A9_divariant.png/440px-Sch%C3%A9ma_d%27un_syst%C3%A8me_thermodynamique_ferm%C3%A9_divariant.png 2x" data-file-width="786" data-file-height="345" /></a><figcaption>Schéma d'un système fermé formé de deux sous-systèmes A et B, séparés par une <a href="/wiki/Syst%C3%A8me_thermodynamique" title="Système thermodynamique">paroi diatherme</a>.</figcaption></figure> <p>Considérons l'interaction entre deux <a href="/wiki/Syst%C3%A8me_isol%C3%A9" title="Système isolé">systèmes isolés</a> de l'extérieur, séparés par une <a href="/wiki/Syst%C3%A8me_thermodynamique" title="Système thermodynamique">paroi diatherme</a> qui ne laisse passer aucune particule mais permet les échanges de chaleur (voir figure). Si, initialement, les deux systèmes ne sont pas à l'équilibre, un échange thermique a lieu entre A et B jusqu'à ce que l'équilibre soit atteint. Les volumes de A et B restant invariants, la variation de l'entropie totale, somme des variations des entropies de A et B, ne dépend que des variations d'énergies internes de A et B : </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 \mathrm {d} S=\left({\frac {\partial S_{\rm {A}}}{\partial U_{\rm {A}}}}\right)_{V_{\rm {A}}}\mathrm {d} U_{\rm {A}}+\left({\frac {\partial S_{\rm {B}}}{\partial U_{\rm {B}}}}\right)_{V_{\rm {B}}}\mathrm {d} U_{\rm {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} S=\left({\frac {\partial S_{\rm {A}}}{\partial U_{\rm {A}}}}\right)_{V_{\rm {A}}}\mathrm {d} U_{\rm {A}}+\left({\frac {\partial S_{\rm {B}}}{\partial U_{\rm {B}}}}\right)_{V_{\rm {B}}}\mathrm {d} U_{\rm {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cd222f05cd2fb7c1300fcdcdde3319836bb8cb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.193ex; height:6.676ex;" alt="{\displaystyle \mathrm {d} S=\left({\frac {\partial S_{\rm {A}}}{\partial U_{\rm {A}}}}\right)_{V_{\rm {A}}}\mathrm {d} U_{\rm {A}}+\left({\frac {\partial S_{\rm {B}}}{\partial U_{\rm {B}}}}\right)_{V_{\rm {B}}}\mathrm {d} U_{\rm {B}}}" /></span></dd></dl> <p>Puisque le système est isolé, la variation de l'énergie interne totale est nulle ; il en résulte que : </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 \mathrm {d} U_{\rm {A}}=-\mathrm {d} U_{\rm {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} U_{\rm {A}}=-\mathrm {d} U_{\rm {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1302d83fc80560260b89610c37009d702b4922ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.527ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} U_{\rm {A}}=-\mathrm {d} U_{\rm {B}}}" /></span></dd></dl> <p>d'où : </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 \mathrm {d} S=\left[\left({\frac {\partial S_{\rm {A}}}{\partial U_{\rm {A}}}}\right)-\left({\frac {\partial S_{\rm {B}}}{\partial U_{\rm {B}}}}\right)\right]\mathrm {d} U_{\rm {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} S=\left[\left({\frac {\partial S_{\rm {A}}}{\partial U_{\rm {A}}}}\right)-\left({\frac {\partial S_{\rm {B}}}{\partial U_{\rm {B}}}}\right)\right]\mathrm {d} U_{\rm {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46a9e3b85e4dc70b0c2cda2daa44948228fa3c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.104ex; height:6.176ex;" alt="{\displaystyle \mathrm {d} S=\left[\left({\frac {\partial S_{\rm {A}}}{\partial U_{\rm {A}}}}\right)-\left({\frac {\partial S_{\rm {B}}}{\partial U_{\rm {B}}}}\right)\right]\mathrm {d} U_{\rm {A}}}" /></span></dd></dl> <p>De plus, comme le système est à l'équilibre, son <a href="/wiki/Entropie_(thermodynamique)" title="Entropie (thermodynamique)">entropie</a> est maximale : toutes les dérivées partielles de l'entropie sont nulles et : </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 \left({\frac {\partial S_{\rm {A}}}{\partial U_{\rm {A}}}}\right)_{V_{\rm {A}}}=\left({\frac {\partial S_{\rm {B}}}{\partial U_{\rm {B}}}}\right)_{V_{\rm {B}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial S_{\rm {A}}}{\partial U_{\rm {A}}}}\right)_{V_{\rm {A}}}=\left({\frac {\partial S_{\rm {B}}}{\partial U_{\rm {B}}}}\right)_{V_{\rm {B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87fdcd1e4e8ecff010b788a60cb2bb07541c7ce9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.94ex; height:6.676ex;" alt="{\displaystyle \left({\frac {\partial S_{\rm {A}}}{\partial U_{\rm {A}}}}\right)_{V_{\rm {A}}}=\left({\frac {\partial S_{\rm {B}}}{\partial U_{\rm {B}}}}\right)_{V_{\rm {B}}}}" /></span></dd></dl> <p>En l'absence de <a href="/wiki/Travail_d%27une_force" title="Travail d'une force">travail</a> du système considéré on peut écrire : </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 \delta Q=\mathrm {d} U=T\,\mathrm {d} S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>U</mi> <mo>=</mo> <mi>T</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta Q=\mathrm {d} U=T\,\mathrm {d} S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe2856c65fdeeba9e8764cda00cdeef4b27bcc2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.974ex; height:2.676ex;" alt="{\displaystyle \delta Q=\mathrm {d} U=T\,\mathrm {d} S}" /></span></dd></dl> <p>donc : </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 \left({\frac {\partial U}{\partial S}}\right)_{V}=T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial U}{\partial S}}\right)_{V}=T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/928dca417dee6801864d68ffedd4f56d7b95181f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.589ex; height:6.176ex;" alt="{\displaystyle \left({\frac {\partial U}{\partial S}}\right)_{V}=T}" /></span></dd></dl> <p>et par suite <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_{\rm {A}}=T_{\rm {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\rm {A}}=T_{\rm {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b91db671c9a53a9e553c4ecf5b4f962cfd71a18f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.674ex; height:2.509ex;" alt="{\displaystyle T_{\rm {A}}=T_{\rm {B}}}" /></span> </p><p>Ce résultat pourrait être obtenu pour toute fonction monotone <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 f(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71e79e53bc5161b01ee450c15644a4e9417fb0ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.724ex; height:2.843ex;" alt="{\displaystyle f(T)}" /></span> telle que : </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 \left({\frac {\partial U}{\partial S}}\right)_{V}=f(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial U}{\partial S}}\right)_{V}=f(T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d577f7ae1318f16e7123d45a62fb0803b35fd4cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.677ex; height:6.176ex;" alt="{\displaystyle \left({\frac {\partial U}{\partial S}}\right)_{V}=f(T)}" /></span></dd></dl> <p>Le choix <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 f(T)=T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(T)=T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/885e42063a0d23331053c5797cb05ef64f8bd18d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.459ex; height:2.843ex;" alt="{\displaystyle f(T)=T}" /></span> correspond au choix d'une <a href="/wiki/%C3%89chelle_(mesure)" title="Échelle (mesure)">échelle</a> de température qui est à justifier. </p> </div> <div class="NavEnd"> </div> </div> <p>Divers travaux, en particulier ceux dus à <a href="/wiki/Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">Carathéodory</a><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite_crochet">[</span>5<span class="cite_crochet">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite_crochet">[</span>6<span class="cite_crochet">]</span></a></sup>, permettent de trouver l'expression liant la température à l'entropie à partir de critères mathématiques. En effet, on souhaite faire de <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 \mathrm {d} S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72ba425d7d7a0f229457dea3c0be4a47ea303cc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.792ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} S}" /></span> une <a href="/wiki/Forme_diff%C3%A9rentielle_exacte" title="Forme différentielle exacte">différentielle exacte</a> afin que la variation de l'entropie d'un état à un autre soit indépendante du chemin suivi, ce qui n'est pas le cas de <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 \delta Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf715eece146c816847a8c5d56eae97798453d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle \delta Q}" /></span>. La quantité qui apparait naturellement est <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/T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee06bfe8f48b840ea1c11f78977a90f661f2375e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.961ex; height:2.843ex;" alt="{\displaystyle 1/T}" /></span>, quantité que l'on retrouve comme potentiel dans les échanges hors équilibre (voir ci-dessous). La dernière étape consiste alors à définir une échelle de température. On se base pour cela sur les propriétés d'un gaz parfait<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite_crochet">[</span>c<span class="cite_crochet">]</span></a></sup>, ce qui assure la cohérence de l'approche. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r222902168" /><div class="NavFrame" style="" title="[afficher]/[masquer]"> <div class="NavHead" style="">Température et entropie.</div> <div class="NavContent" style=""> <p><b>Recherche d'un facteur intégrant</b> </p><p>On cherche une quantité (facteur intégrant) <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 w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" /></span>, fonction des <a href="/wiki/Variable_d%27%C3%A9tat" title="Variable d'état">variables d'état</a>, qui permet de passer de la quantité élémentaire de chaleur <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 \delta Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf715eece146c816847a8c5d56eae97798453d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle \delta Q}" /></span> à une <a href="/wiki/Forme_diff%C3%A9rentielle_exacte" title="Forme différentielle exacte">différentielle exacte</a> <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 \mathrm {d} S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72ba425d7d7a0f229457dea3c0be4a47ea303cc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.792ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} S}" /></span>, <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> étant alors défini comme entropie du système et sur laquelle on pourra appuyer un raisonnement comme celui de l'encadré précédent<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite_crochet">[</span>d<span class="cite_crochet">]</span></a></sup> : </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 w(U,V,p)\cdot \delta Q=\mathrm {d} S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo>,</mo> <mi>V</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>δ</mi> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w(U,V,p)\cdot \delta Q=\mathrm {d} S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d2a8efcbb378cad8259dccdfa7813043a2d12d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.737ex; height:2.843ex;" alt="{\displaystyle w(U,V,p)\cdot \delta Q=\mathrm {d} S}" /></span></dd></dl> <p>On 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 \mathrm {d} S=w\,\mathrm {d} U+w\,p\,\mathrm {d} V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>=</mo> <mi>w</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>U</mi> <mo>+</mo> <mi>w</mi> <mspace width="thinmathspace"></mspace> <mi>p</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} S=w\,\mathrm {d} U+w\,p\,\mathrm {d} V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8614fd0203b689877591b754af128bf394ed2a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.544ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} S=w\,\mathrm {d} U+w\,p\,\mathrm {d} V}" /></span></dd></dl> <p>Le critère utilisé est le <a href="/wiki/Th%C3%A9or%C3%A8me_de_Schwarz" title="Théorème de Schwarz">théorème de Schwarz</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 {\frac {\partial ^{2}S}{\partial U\partial V}}={\frac {\partial ^{2}S}{\partial V\partial U}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}S}{\partial U\partial V}}={\frac {\partial ^{2}S}{\partial V\partial U}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a10a524538c990322abc6361067a1dfda65debb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.183ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial ^{2}S}{\partial U\partial V}}={\frac {\partial ^{2}S}{\partial V\partial U}}}" /></span></dd></dl> <p>Ici : </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 {\frac {\partial ^{2}S}{\partial U\partial V}}={\frac {\partial }{\partial U}}\left({\frac {\partial S}{\partial V}}\right)_{U}={\frac {\partial }{\partial U}}\left(w\,p\right)=w\,\left({\frac {\partial p}{\partial U}}\right)_{V}+p\,\left({\frac {\partial w}{\partial U}}\right)_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> </mfrac> </mrow> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>w</mi> <mspace width="thinmathspace"></mspace> <mi>p</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>w</mi> <mspace width="thinmathspace"></mspace> <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>U</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>+</mo> <mi>p</mi> <mspace width="thinmathspace"></mspace> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>w</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}S}{\partial U\partial V}}={\frac {\partial }{\partial U}}\left({\frac {\partial S}{\partial V}}\right)_{U}={\frac {\partial }{\partial U}}\left(w\,p\right)=w\,\left({\frac {\partial p}{\partial U}}\right)_{V}+p\,\left({\frac {\partial w}{\partial U}}\right)_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea443bcead3f20b71b12dbe631de8f21c7a779e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:62.64ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial ^{2}S}{\partial U\partial V}}={\frac {\partial }{\partial U}}\left({\frac {\partial S}{\partial V}}\right)_{U}={\frac {\partial }{\partial U}}\left(w\,p\right)=w\,\left({\frac {\partial p}{\partial U}}\right)_{V}+p\,\left({\frac {\partial w}{\partial U}}\right)_{V}}" /></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 {\frac {\partial ^{2}S}{\partial V\partial U}}={\frac {\partial }{\partial V}}\left({\frac {\partial S}{\partial U}}\right)_{V}=\left({\frac {\partial w}{\partial V}}\right)_{U}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>w</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}S}{\partial V\partial U}}={\frac {\partial }{\partial V}}\left({\frac {\partial S}{\partial U}}\right)_{V}=\left({\frac {\partial w}{\partial V}}\right)_{U}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6492afa70001710a5b37e6d4bb0f2c3108a67ff2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.89ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial ^{2}S}{\partial V\partial U}}={\frac {\partial }{\partial V}}\left({\frac {\partial S}{\partial U}}\right)_{V}=\left({\frac {\partial w}{\partial V}}\right)_{U}}" /></span></dd></dl> <p>Parmi tous les facteurs intégrants possibles, on choisit une quantité <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(U,V,p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo>,</mo> <mi>V</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(U,V,p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b006285b3743ef129519df581c997ecf8f4fc3b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.253ex; height:2.843ex;" alt="{\displaystyle T(U,V,p)}" /></span> que l'on définit comme température : </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 \left({\frac {\partial w}{\partial U}}\right)_{V}={\frac {\mathrm {d} w}{\mathrm {d} T}}\left({\frac {\partial T}{\partial U}}\right)_{V}={\frac {1}{C_{V}}}{\frac {\mathrm {d} w}{\mathrm {d} T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>w</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>w</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> </mrow> </mfrac> </mrow> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>w</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial w}{\partial U}}\right)_{V}={\frac {\mathrm {d} w}{\mathrm {d} T}}\left({\frac {\partial T}{\partial U}}\right)_{V}={\frac {1}{C_{V}}}{\frac {\mathrm {d} w}{\mathrm {d} T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/903389f01ffdabcafecc0f3974ef80ae1f2969c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.484ex; height:6.176ex;" alt="{\displaystyle \left({\frac {\partial w}{\partial U}}\right)_{V}={\frac {\mathrm {d} w}{\mathrm {d} T}}\left({\frac {\partial T}{\partial U}}\right)_{V}={\frac {1}{C_{V}}}{\frac {\mathrm {d} w}{\mathrm {d} T}}}" /></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 \left({\frac {\partial w}{\partial V}}\right)_{U}={\frac {\mathrm {d} w}{\mathrm {d} T}}\left({\frac {\partial T}{\partial V}}\right)_{U}=\mu {\frac {\mathrm {d} w}{\mathrm {d} T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>w</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>w</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> </mrow> </mfrac> </mrow> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo>=</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>w</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial w}{\partial V}}\right)_{U}={\frac {\mathrm {d} w}{\mathrm {d} T}}\left({\frac {\partial T}{\partial V}}\right)_{U}=\mu {\frac {\mathrm {d} w}{\mathrm {d} T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be1f7e761717619eb5848e63ec14e14780073013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.895ex; height:6.176ex;" alt="{\displaystyle \left({\frac {\partial w}{\partial V}}\right)_{U}={\frac {\mathrm {d} w}{\mathrm {d} T}}\left({\frac {\partial T}{\partial V}}\right)_{U}=\mu {\frac {\mathrm {d} w}{\mathrm {d} T}}}" /></span></dd></dl> <p>On a défini les quantités : </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_{V}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00c18988639f2937d3fe4dc00dfa0b0bd7e0bec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.158ex; height:2.509ex;" alt="{\displaystyle C_{V}}" /></span> <a href="/wiki/Coefficients_calorim%C3%A9triques_et_thermo%C3%A9lastiques" title="Coefficients calorimétriques et thermoélastiques">capacité thermique isochore</a> ;</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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" /></span> <a href="/wiki/Coefficients_calorim%C3%A9triques_et_thermo%C3%A9lastiques" title="Coefficients calorimétriques et thermoélastiques">coefficient de Joule-Gay-Lussac</a> (ou coefficient de Joule-Thomson isoénergétique).</li></ul> <p><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 w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" /></span> est donc solution de l'équation : </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 {\frac {\mathrm {d} w}{w}}=\mathrm {d} \ln w={\frac {\left({\frac {\partial p}{\partial U}}\right)_{V}}{\mu -{\frac {p}{C_{V}}}}}\,\mathrm {d} T={\frac {\left({\frac {\partial p}{\partial T}}\right)_{V}}{\mu C_{V}-p}}\,\mathrm {d} T={\frac {1}{\lambda }}\left({\frac {\partial p}{\partial T}}\right)_{V}\,\mathrm {d} T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>w</mi> </mrow> <mi>w</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ln</mi> <mo>⁡</mo> <mi>w</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <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>U</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mrow> <mi>μ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <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>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mrow> <mi>μ</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>−</mo> <mi>p</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> </mrow> <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>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} w}{w}}=\mathrm {d} \ln w={\frac {\left({\frac {\partial p}{\partial U}}\right)_{V}}{\mu -{\frac {p}{C_{V}}}}}\,\mathrm {d} T={\frac {\left({\frac {\partial p}{\partial T}}\right)_{V}}{\mu C_{V}-p}}\,\mathrm {d} T={\frac {1}{\lambda }}\left({\frac {\partial p}{\partial T}}\right)_{V}\,\mathrm {d} T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3f4ef844f312faf720dea38dcbf9b5919950694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:60.389ex; height:9.676ex;" alt="{\displaystyle {\frac {\mathrm {d} w}{w}}=\mathrm {d} \ln w={\frac {\left({\frac {\partial p}{\partial U}}\right)_{V}}{\mu -{\frac {p}{C_{V}}}}}\,\mathrm {d} T={\frac {\left({\frac {\partial p}{\partial T}}\right)_{V}}{\mu C_{V}-p}}\,\mathrm {d} T={\frac {1}{\lambda }}\left({\frac {\partial p}{\partial T}}\right)_{V}\,\mathrm {d} T}" /></span></dd></dl> <p><b>Unicité</b> </p><p>On remarque que cette équation définit la température à une constante multiplicative près, en effet si l'on pose : </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 w(T)=\kappa w(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>κ</mi> <mi>w</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w(T)=\kappa w(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d63f53ee10099bfdb104e56ea79713fdca02af41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.111ex; height:2.843ex;" alt="{\displaystyle w(T)=\kappa w(\theta )}" /></span></dd></dl> <p><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 \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" /></span> ne change pas puisque : </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 C_{V}(\theta )\,\mu (\theta )=C_{V}(T)\,\mu (T)}"> <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 stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{V}(\theta )\,\mu (\theta )=C_{V}(T)\,\mu (T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d08342b8e361d1971dd06373f0f020d6d4746d81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.682ex; height:2.843ex;" alt="{\displaystyle C_{V}(\theta )\,\mu (\theta )=C_{V}(T)\,\mu (T)}" /></span></dd></dl> <p>et : </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 {\frac {\mathrm {d} \ln w(\theta )}{\mathrm {d} \theta }}={\frac {\mathrm {d} \ln w(T)}{\mathrm {d} T}}{\frac {\mathrm {d} T}{\mathrm {d} \theta }}={\frac {1}{\lambda }}\left({\frac {\partial p}{\partial T}}\right)_{V}{\frac {\mathrm {d} T}{\mathrm {d} \theta }}={\frac {1}{\lambda }}\left({\frac {\partial p}{\partial \theta }}\right)_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ln</mi> <mo>⁡</mo> <mi>w</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ln</mi> <mo>⁡</mo> <mi>w</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> </mrow> <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>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> </mrow> <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>θ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} \ln w(\theta )}{\mathrm {d} \theta }}={\frac {\mathrm {d} \ln w(T)}{\mathrm {d} T}}{\frac {\mathrm {d} T}{\mathrm {d} \theta }}={\frac {1}{\lambda }}\left({\frac {\partial p}{\partial T}}\right)_{V}{\frac {\mathrm {d} T}{\mathrm {d} \theta }}={\frac {1}{\lambda }}\left({\frac {\partial p}{\partial \theta }}\right)_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a6d0548621b1c1b1386832c1673a8c267af25b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:57.514ex; height:6.343ex;" alt="{\displaystyle {\frac {\mathrm {d} \ln w(\theta )}{\mathrm {d} \theta }}={\frac {\mathrm {d} \ln w(T)}{\mathrm {d} T}}{\frac {\mathrm {d} T}{\mathrm {d} \theta }}={\frac {1}{\lambda }}\left({\frac {\partial p}{\partial T}}\right)_{V}{\frac {\mathrm {d} T}{\mathrm {d} \theta }}={\frac {1}{\lambda }}\left({\frac {\partial p}{\partial \theta }}\right)_{V}}" /></span></dd></dl> <p>ce qui conduit à une équation identique pour <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 w(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31d89d3e07d4f33cb5518cd6f715fd0e4f54ccd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.564ex; height:2.843ex;" alt="{\displaystyle w(\theta )}" /></span>. </p><p><b>Échelle</b> </p><p>On a donc montré qu'il existe une variable d'état <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> telle que <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 {\frac {\delta Q}{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\delta Q}{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8961bbca4cefeb5f1a0b7614ff14b12d135af2f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.723ex; height:5.343ex;" alt="{\displaystyle {\frac {\delta Q}{T}}}" /></span> soit une différentielle exacte. Il reste à la préciser en choisissant une échelle de référence, laquelle est implicitement liée à une équation d'état. Pour des raisons évidentes on va prendre <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 \mu =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3753282c0ad2ea1e7d63f39425efd13c37da3169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="{\displaystyle \mu =0}" /></span>. L'équation sur <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 w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" /></span> se simplifie en : </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 \mathrm {d} \ln w=-{\frac {1}{p}}\left({\frac {\partial p}{\partial T}}\right)_{V}\,\mathrm {d} T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ln</mi> <mo>⁡</mo> <mi>w</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> <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>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \ln w=-{\frac {1}{p}}\left({\frac {\partial p}{\partial T}}\right)_{V}\,\mathrm {d} T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb35efc326bd2d70a9ad3c2ffbb19d5f49f57a05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.606ex; height:6.176ex;" alt="{\displaystyle \mathrm {d} \ln w=-{\frac {1}{p}}\left({\frac {\partial p}{\partial T}}\right)_{V}\,\mathrm {d} T}" /></span></dd></dl> <p>Prenons le cas d'une transformation isochore d'un état 1 vers un état 2, l'expression de <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 w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" /></span> devient : </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 \int _{1}^{2}\mathrm {d} \ln \left({\frac {1}{T}}\right)=-\ln {\frac {T_{2}}{T_{1}}}=-\int _{1}^{2}{\frac {1}{p}}\left({\frac {\partial p}{\partial T}}\right)_{V}\mathrm {d} T=-\ln {\frac {p_{2}}{p_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> <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>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> <mo>=</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{1}^{2}\mathrm {d} \ln \left({\frac {1}{T}}\right)=-\ln {\frac {T_{2}}{T_{1}}}=-\int _{1}^{2}{\frac {1}{p}}\left({\frac {\partial p}{\partial T}}\right)_{V}\mathrm {d} T=-\ln {\frac {p_{2}}{p_{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e61208697acc7fee5e9a831e233555c99fd2a53d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:57.807ex; height:6.343ex;" alt="{\displaystyle \int _{1}^{2}\mathrm {d} \ln \left({\frac {1}{T}}\right)=-\ln {\frac {T_{2}}{T_{1}}}=-\int _{1}^{2}{\frac {1}{p}}\left({\frac {\partial p}{\partial T}}\right)_{V}\mathrm {d} T=-\ln {\frac {p_{2}}{p_{1}}}}" /></span></dd></dl> <p>On reconnait la <a href="/wiki/Loi_de_Gay-Lussac" title="Loi de Gay-Lussac">loi de Gay-Lussac</a> pour un <a href="/wiki/Gaz_parfait" title="Gaz parfait">gaz parfait</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 {\frac {T_{2}}{T_{1}}}={\frac {p_{2}}{p_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {T_{2}}{T_{1}}}={\frac {p_{2}}{p_{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e88dc552b5e35c4ee8e017e66c35bddc22b9b63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.406ex; height:5.676ex;" alt="{\displaystyle {\frac {T_{2}}{T_{1}}}={\frac {p_{2}}{p_{1}}}}" /></span></dd></dl> <p>L'<a href="/wiki/Loi_des_gaz_parfaits" title="Loi des gaz parfaits">équation d'état du gaz parfait</a> est assez naturellement le meilleur choix pour l'échelle de référence<sup id="cite_ref-H_11-0" class="reference"><a href="#cite_note-H-11"><span class="cite_crochet">[</span>7<span class="cite_crochet">]</span></a></sup>. </p> </div> <div class="NavEnd"> </div> </div> <p>On note que, <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> 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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" /></span> étant des <a href="/wiki/Extensivit%C3%A9_et_intensivit%C3%A9_(physique)" title="Extensivité et intensivité (physique)">quantités extensives</a>, la relation non linéaire les liant fait de <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> une quantité intensive. </p><p>Dans les années 1870, <a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Ludwig Boltzmann</a> énonce le <a href="/wiki/Th%C3%A9or%C3%A8me_H" title="Théorème H">théorème H</a> et l'<a href="/wiki/%C3%89quation_de_Boltzmann" title="Équation de Boltzmann">équation de Boltzmann</a> pour le développement temporel des fonctions de distribution microscopiques<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite_crochet">[</span>8<span class="cite_crochet">]</span></a></sup>. Ces relations fondamentales permettent le lien entre les descriptions <a href="/wiki/Microscopique_et_macroscopique" title="Microscopique et macroscopique">microscopique et macroscopique</a> de la physique. L'entropie représente le nombre de <a href="/wiki/Micro-%C3%A9tat_(physique_statistique)" title="Micro-état (physique statistique)">configurations microscopiques</a> possibles pour un système donné et la température le paramètre qui, avec le nombre de particules par unité de volume, caractérise entièrement ce système à l'équilibre thermodynamique au travers de la fonction de <a href="/wiki/Loi_de_distribution_des_vitesses_de_Maxwell" title="Loi de distribution des vitesses de Maxwell">distribution de Maxwell</a> pour la translation et <a href="/wiki/Statistique_de_Maxwell-Boltzmann" title="Statistique de Maxwell-Boltzmann">celle de Boltzmann</a> pour les autres <a href="/wiki/Degr%C3%A9_de_libert%C3%A9_(physique)" title="Degré de liberté (physique)">degrés de liberté</a>. </p><p>Si on considère la <a href="/wiki/Translation" title="Translation">translation</a>, la température est définie à partir de la <a href="/wiki/Vitesse_quadratique_moyenne" title="Vitesse quadratique moyenne">vitesse quadratique moyenne</a> des particules au niveau microscopique par l'expression suivante issue de la <a href="/wiki/Th%C3%A9orie_cin%C3%A9tique_des_gaz" title="Théorie cinétique des gaz">théorie cinétique des gaz</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 {\frac {3}{2}}kT={\frac {1}{2}}m{\overline {v^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mi>k</mi> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3}{2}}kT={\frac {1}{2}}m{\overline {v^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7cf46ac78d17ce833dbff5193a171be4831948d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.28ex; height:5.176ex;" alt="{\displaystyle {\frac {3}{2}}kT={\frac {1}{2}}m{\overline {v^{2}}}}" /></span></dd></dl> <p>Si on considère les populations <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57f87f905ba5a4d8c691ccaecd65fc47bd007ba4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.194ex; height:2.009ex;" alt="{\displaystyle n_{i}}" /></span> d'états microscopiques d'énergie <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 E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}" /></span>, la <a href="/wiki/Statistique_de_Maxwell-Boltzmann#Température_de_Boltzmann" title="Statistique de Maxwell-Boltzmann">température</a> est donnée 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 kT=-{\frac {E_{j}-E_{i}}{\ln \left({\frac {n_{j}}{n_{i}}}\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mi>T</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle kT=-{\frac {E_{j}-E_{i}}{\ln \left({\frac {n_{j}}{n_{i}}}\right)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77011765be12e9dd0d8006e9fd022e12fc9f0c08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:16.57ex; height:8.176ex;" alt="{\displaystyle kT=-{\frac {E_{j}-E_{i}}{\ln \left({\frac {n_{j}}{n_{i}}}\right)}}}" /></span>, <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 \forall i,j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall i,j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcf3dc95180d5758a13c4954b86affe61b54ba55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.087ex; height:2.509ex;" alt="{\displaystyle \forall i,j}" /></span></dd></dl> <p>Ces deux valeurs sont identiques en vertu du <a href="/wiki/%C3%89quipartition_de_l%27%C3%A9nergie#Théorème_d’équipartition_en_physique_statistique" title="Équipartition de l'énergie">théorème d'équipartition de l'énergie</a>. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r222902168" /><div class="NavFrame" style="" title="[afficher]/[masquer]"> <div class="NavHead" style="">La température au niveau microscopique.</div> <div class="NavContent" style=""> <p>En <a href="/wiki/Physique_statistique" title="Physique statistique">physique statistique</a>, la <a href="/wiki/Fonction_de_partition" title="Fonction de partition">fonction de partition</a> d'un système formé d'un ensemble de <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> états microscopiques d'énergie <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 E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}" /></span>, chacun étant occupé <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57f87f905ba5a4d8c691ccaecd65fc47bd007ba4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.194ex; height:2.009ex;" alt="{\displaystyle n_{i}}" /></span> fois, est donnée à une constante de normalisation près 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 \Omega ={\frac {N!}{\Pi _{i}n_{i}!}}\,,\;\;\;\;\;N=\sum _{i}n_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mo>!</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>!</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mo>,</mo> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mi>N</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega ={\frac {N!}{\Pi _{i}n_{i}!}}\,,\;\;\;\;\;N=\sum _{i}n_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bb04567a841330eb664ea381a84108d6d4e63b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.742ex; height:6.509ex;" alt="{\displaystyle \Omega ={\frac {N!}{\Pi _{i}n_{i}!}}\,,\;\;\;\;\;N=\sum _{i}n_{i}}" /></span></dd></dl> <p>L'<a href="/wiki/%C3%89nergie_interne" title="Énergie interne">énergie interne</a> de ce système est : </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 U={\frac {E}{N}}=\sum _{i}{\frac {n_{i}}{N}}E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mi>N</mi> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>N</mi> </mfrac> </mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U={\frac {E}{N}}=\sum _{i}{\frac {n_{i}}{N}}E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/051bd1fb7c72eba8c8f5d117975eac7d2d12023f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.167ex; height:6.343ex;" alt="{\displaystyle U={\frac {E}{N}}=\sum _{i}{\frac {n_{i}}{N}}E_{i}}" /></span></dd></dl> <p>On suppose que le nombre N est suffisamment grand pour que l'on puisse considérer Ω(n) comme une fonction et utiliser l'<a href="/wiki/Formule_de_Stirling" title="Formule de Stirling">approximation de Stirling</a>. On a donc : </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 {\begin{array}{lcl}\log \Omega &=&\log N!-\sum _{i}\log n_{i}!\\[0.6em]&\approx &N\log N-\sum _{i}n_{i}\log n_{i}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left center left" rowspacing="1em 0.4em" columnspacing="1em"> <mtr> <mtd> <mi>log</mi> <mo>⁡</mo> <mi mathvariant="normal">Ω</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mi>log</mi> <mo>⁡</mo> <mi>N</mi> <mo>!</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>log</mi> <mo>⁡</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>!</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mo>≈</mo> </mtd> <mtd> <mi>N</mi> <mi>log</mi> <mo>⁡</mo> <mi>N</mi> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>log</mi> <mo>⁡</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lcl}\log \Omega &=&\log N!-\sum _{i}\log n_{i}!\\[0.6em]&\approx &N\log N-\sum _{i}n_{i}\log n_{i}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5efcd4f615f148ee8d4d0bd7f812033f6dd8f1df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:34.731ex; height:7.843ex;" alt="{\displaystyle {\begin{array}{lcl}\log \Omega &=&\log N!-\sum _{i}\log n_{i}!\\[0.6em]&\approx &N\log N-\sum _{i}n_{i}\log n_{i}\end{array}}}" /></span></dd></dl> <p>L'état d'équilibre est donné par <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 \Omega ={\text{constante}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>constante</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega ={\text{constante}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdadd113fc6695afda58d49602e7a0fc98d790d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.477ex; height:2.176ex;" alt="{\displaystyle \Omega ={\text{constante}}}" /></span>, soit : </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 {\frac {\mathrm {d} \log \Omega }{\mathrm {d} n_{i}}}=-\sum _{i}\left(1+\log {\frac {n_{i}}{N}}\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>log</mi> <mo>⁡</mo> <mi mathvariant="normal">Ω</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>N</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} \log \Omega }{\mathrm {d} n_{i}}}=-\sum _{i}\left(1+\log {\frac {n_{i}}{N}}\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8330e56e1b27b60357b0be1673d83745b314116" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.017ex; height:6.509ex;" alt="{\displaystyle {\frac {\mathrm {d} \log \Omega }{\mathrm {d} n_{i}}}=-\sum _{i}\left(1+\log {\frac {n_{i}}{N}}\right)=0}" /></span></dd></dl> <p>Ce problème est traité comme un problème d'<a href="/wiki/Extremum" title="Extremum">extremum</a> de <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 \log \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b1fc655360467b09c14991a078fae31557f4917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.037ex; height:2.509ex;" alt="{\displaystyle \log \Omega }" /></span> avec les contraintes de conservation du nombre de particules <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 \mathrm {d} N=\sum _{i}\mathrm {d} n_{i}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>N</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} N=\sum _{i}\mathrm {d} n_{i}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ae31e3d29daf27e23f9900f111678ad3dddb121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.945ex; height:5.509ex;" alt="{\displaystyle \mathrm {d} N=\sum _{i}\mathrm {d} n_{i}=0}" /></span> et de l'énergie <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 \mathrm {d} E=\sum _{i}E_{i}\,\mathrm {d} n_{i}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>E</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} E=\sum _{i}E_{i}\,\mathrm {d} n_{i}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4cf5fa113b29b2789206347e7cd157b1f7f6a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.559ex; height:5.509ex;" alt="{\displaystyle \mathrm {d} E=\sum _{i}E_{i}\,\mathrm {d} n_{i}=0}" /></span>. En utilisant deux <a href="/wiki/Multiplicateur_de_Lagrange" title="Multiplicateur de Lagrange">multiplicateurs de Lagrange</a> <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> </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/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </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/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" /></span> pour les contraintes sur <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> 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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" /></span>, respectivement, il vient : </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 \sum _{i}\left(\log n_{i}+\alpha +\beta E_{i}\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mi>log</mi> <mo>⁡</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}\left(\log n_{i}+\alpha +\beta E_{i}\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5e19717986ed1fa33d0f079db176cc66dd074ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.381ex; height:5.509ex;" alt="{\displaystyle \sum _{i}\left(\log n_{i}+\alpha +\beta E_{i}\right)=0}" /></span></dd></dl> <p>Les états étant indépendants, cette équation n'est vérifiée que si elle est vraie pour chaque état. Sa solution est donc : </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 n_{i}=e^{-\alpha }e^{-\beta E_{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{i}=e^{-\alpha }e^{-\beta E_{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f19e0268ee95baddea3d46c6a1bc3e730c345376" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.313ex; height:3.009ex;" alt="{\displaystyle n_{i}=e^{-\alpha }e^{-\beta E_{i}}}" /></span></dd></dl> <p>L'énergie vaut : </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 U={\frac {\sum _{i}E_{i}e^{-\beta E_{i}}}{\sum _{i}e^{-\beta E_{i}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U={\frac {\sum _{i}E_{i}e^{-\beta E_{i}}}{\sum _{i}e^{-\beta E_{i}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c918ea2b5ec9192e23de0c21190fe3a0acb3be7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:17.247ex; height:7.009ex;" alt="{\displaystyle U={\frac {\sum _{i}E_{i}e^{-\beta E_{i}}}{\sum _{i}e^{-\beta E_{i}}}}}" /></span></dd></dl> <p>En différentiant, on fait apparaitre la redistribution d'une énergie donnée sur les divers niveaux d'occupation (chaleur) et la variation d'énergie liée à la variation de volume (<a href="/wiki/Travail_d%27une_force" title="Travail d'une force">travail</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 \mathrm {d} U=\underbrace {\sum _{i}E_{i}\mathrm {d} \left({\frac {n_{i}}{N}}\right)} _{\delta Q}+\underbrace {\sum _{i}{\frac {n_{i}}{N}}\mathrm {d} E_{i}} _{-\delta W}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>U</mi> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>N</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mi>Q</mi> </mrow> </munder> <mo>+</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>N</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>δ</mi> <mi>W</mi> </mrow> </munder> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} U=\underbrace {\sum _{i}E_{i}\mathrm {d} \left({\frac {n_{i}}{N}}\right)} _{\delta Q}+\underbrace {\sum _{i}{\frac {n_{i}}{N}}\mathrm {d} E_{i}} _{-\delta W}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e4a0553c57c611e26832f0e649afb238541e12a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; margin-right: -0.028ex; width:33.365ex; height:9.843ex;" alt="{\displaystyle \mathrm {d} U=\underbrace {\sum _{i}E_{i}\mathrm {d} \left({\frac {n_{i}}{N}}\right)} _{\delta Q}+\underbrace {\sum _{i}{\frac {n_{i}}{N}}\mathrm {d} E_{i}} _{-\delta W}}" /></span></dd></dl> <p>On définit l'entropie canonique 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 S(E_{i},\beta )=k(\ln \Omega +\beta U)+S_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi mathvariant="normal">Ω</mi> <mo>+</mo> <mi>β</mi> <mi>U</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(E_{i},\beta )=k(\ln \Omega +\beta U)+S_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d6fa5bf46ed3b88d587e9c780077c7b7b1c8bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.587ex; height:2.843ex;" alt="{\displaystyle S(E_{i},\beta )=k(\ln \Omega +\beta U)+S_{0}}" /></span></dd></dl> <p>On peut définir une <a href="/wiki/Diff%C3%A9rentielle_exacte" class="mw-redirect" title="Différentielle exacte">différentielle exacte</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 \mathrm {d} S=\sum _{i}\left({\frac {\partial S}{\partial E_{i}}}\right)_{\beta }\mathrm {d} E_{i}+\left({\frac {\partial S}{\partial \beta }}\right)_{E_{i}}\mathrm {d} \beta =k\beta \delta Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>β</mi> <mo>=</mo> <mi>k</mi> <mi>β</mi> <mi>δ</mi> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} S=\sum _{i}\left({\frac {\partial S}{\partial E_{i}}}\right)_{\beta }\mathrm {d} E_{i}+\left({\frac {\partial S}{\partial \beta }}\right)_{E_{i}}\mathrm {d} \beta =k\beta \delta Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1238c676f18bd31282261a424f24db4f0565890" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.842ex; height:6.676ex;" alt="{\displaystyle \mathrm {d} S=\sum _{i}\left({\frac {\partial S}{\partial E_{i}}}\right)_{\beta }\mathrm {d} E_{i}+\left({\frac {\partial S}{\partial \beta }}\right)_{E_{i}}\mathrm {d} \beta =k\beta \delta Q}" /></span></dd></dl> <p>D'où la valeur de <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> </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/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" /></span> par identification avec l'expression classique de la thermodynamique <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 \mathrm {d} S={\frac {\delta Q}{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} S={\frac {\delta Q}{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4434f0305e8d6aeab2abee298bfd3b068ceef5f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.613ex; height:5.343ex;" alt="{\displaystyle \mathrm {d} S={\frac {\delta Q}{T}}}" /></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 k\beta ={\frac {1}{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\beta ={\frac {1}{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6c80b4bfd3e86acf8da3407162e602a7d70aa35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.114ex; height:5.176ex;" alt="{\displaystyle k\beta ={\frac {1}{T}}}" /></span></dd></dl> <p>Le raisonnement s'applique de la même manière pour le système continu que constitue la translation des particules, les quantités discrètes étant remplacées par les distributions d'énergie et de populations de particules de masse <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" /></span> et de vitesse <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/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" /></span><sup id="cite_ref-H_11-1" class="reference"><a href="#cite_note-H-11"><span class="cite_crochet">[</span>7<span class="cite_crochet">]</span></a></sup> : </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 E_{i}={\frac {1}{2}}\,m\,{v_{i}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mi>m</mi> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}={\frac {1}{2}}\,m\,{v_{i}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72637d1ef481cd1536176b78e43b89be5dc48391" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.408ex; height:5.176ex;" alt="{\displaystyle E_{i}={\frac {1}{2}}\,m\,{v_{i}}^{2}}" /></span></dd></dl> <p>et, d'après la <a href="/wiki/Loi_de_distribution_des_vitesses_de_Maxwell" title="Loi de distribution des vitesses de Maxwell">loi de distribution des vitesses de Maxwell</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 \mathrm {d} \left({\frac {n_{i}}{N}}\right)={\frac {2}{\sqrt {\pi }}}\,\beta ^{\frac {3}{2}}\,{E_{i}}^{\frac {1}{2}}\,e^{-\beta E_{i}}\,\mathrm {d} E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>N</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msqrt> <mi>π</mi> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mspace width="thinmathspace"></mspace> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \left({\frac {n_{i}}{N}}\right)={\frac {2}{\sqrt {\pi }}}\,\beta ^{\frac {3}{2}}\,{E_{i}}^{\frac {1}{2}}\,e^{-\beta E_{i}}\,\mathrm {d} E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8110c3f6447a257f9cbb48013d70e69057375b6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:32.741ex; height:6.176ex;" alt="{\displaystyle \mathrm {d} \left({\frac {n_{i}}{N}}\right)={\frac {2}{\sqrt {\pi }}}\,\beta ^{\frac {3}{2}}\,{E_{i}}^{\frac {1}{2}}\,e^{-\beta E_{i}}\,\mathrm {d} E_{i}}" /></span></dd></dl> </div> <div class="NavEnd"> </div> </div> <div class="mw-heading mw-heading3"><h3 id="Thermodynamique_hors_équilibre"><span id="Thermodynamique_hors_.C3.A9quilibre"></span>Thermodynamique hors équilibre</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&veaction=edit&section=3" title="Modifier la section : Thermodynamique hors équilibre" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&action=edit&section=3" title="Modifier le code source de la section : Thermodynamique hors équilibre"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On parle ici de <a href="/wiki/Thermodynamique_hors_%C3%A9quilibre" title="Thermodynamique hors équilibre">thermodynamique hors équilibre</a>, c'est-à-dire des chemins permettant de passer d'un état d'équilibre à un autre par des processus de transport ou de relaxation dans un milieu proche de l'équilibre thermodynamique, cette dernière condition étant nécessaire pour exprimer les flux sous forme de lois linéaires. </p><p>Au <abbr class="abbr" title="18ᵉ siècle"><span class="romain">XVIII</span><sup style="font-size:72%">e</sup></abbr> siècle, <a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a> établit la <a href="/wiki/Conduction_thermique" title="Conduction thermique">loi portant son nom</a> qui semble faire de la température le <a href="/wiki/Potentiel_d%27un_champ_vectoriel" title="Potentiel d'un champ vectoriel">potentiel scalaire</a> permettant de connaitre le <a href="/wiki/Flux_thermique" title="Flux thermique">flux de chaleur</a><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite_crochet">[</span>9<span class="cite_crochet">]</span></a></sup> : </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 \mathbf {q} =-\lambda \,\nabla T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>=</mo> <mo>−</mo> <mi>λ</mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">∇</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {q} =-\lambda \,\nabla T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9adce6f735c84dac22a8a9e6d1aa07a49a4b245" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.637ex; height:2.509ex;" alt="{\displaystyle \mathbf {q} =-\lambda \,\nabla T}" /></span></dd></dl> <p>En fait, la <a href="/wiki/Thermodynamique_hors_%C3%A9quilibre" title="Thermodynamique hors équilibre">thermodynamique hors équilibre</a> montre que cette expression devrait être écrite : </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 \mathbf {q} =\lambda '\,\nabla \left({\frac {1}{T}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>=</mo> <msup> <mi>λ</mi> <mo>′</mo> </msup> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">∇</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {q} =\lambda '\,\nabla \left({\frac {1}{T}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c08e495b7edb8ec9b8340f83350bf069bafcb5a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.158ex; height:6.176ex;" alt="{\displaystyle \mathbf {q} =\lambda '\,\nabla \left({\frac {1}{T}}\right)}" /></span></dd></dl> <p><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/T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee06bfe8f48b840ea1c11f78977a90f661f2375e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.961ex; height:2.843ex;" alt="{\displaystyle 1/T}" /></span> est l'<a href="/wiki/Potentiel_thermodynamique" title="Potentiel thermodynamique">affinité</a>, qui constitue le potentiel lié à l'<a href="/wiki/%C3%89nergie_interne" title="Énergie interne">énergie interne</a> <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" /></span>. <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/T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee06bfe8f48b840ea1c11f78977a90f661f2375e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.961ex; height:2.843ex;" alt="{\displaystyle 1/T}" /></span> apparait également dans les phénomènes de <a href="/wiki/Diffusion_de_la_mati%C3%A8re" title="Diffusion de la matière">diffusion de la matière</a> par l'intermédiaire du gradient <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 \nabla \left({\frac {\mu }{T}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>μ</mi> <mi>T</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \left({\frac {\mu }{T}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d63aa50ad985de7d9ff9e9b2a66e769cf03c8676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.571ex; height:4.843ex;" alt="{\displaystyle \nabla \left({\frac {\mu }{T}}\right)}" /></span> où <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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" /></span> est le <a href="/wiki/Potentiel_chimique" title="Potentiel chimique">potentiel chimique</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Remarques_sur_la_notion_de_température"><span id="Remarques_sur_la_notion_de_temp.C3.A9rature"></span>Remarques sur la notion de température</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&veaction=edit&section=4" title="Modifier la section : Remarques sur la notion de température" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&action=edit&section=4" title="Modifier le code source de la section : Remarques sur la notion de température"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Zéro_absolu"><span id="Z.C3.A9ro_absolu"></span>Zéro absolu</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&veaction=edit&section=5" title="Modifier la section : Zéro absolu" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&action=edit&section=5" title="Modifier le code source de la section : Zéro absolu"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Z%C3%A9ro_absolu" title="Zéro absolu">Zéro absolu</a>.</div></div> <p>Le zéro absolu, qui correspondrait à une matière totalement figée, est rendu inaccessible par le <a href="/wiki/Principe_d%27incertitude" title="Principe d'incertitude">principe d'incertitude</a> de la <a href="/wiki/M%C3%A9canique_quantique" title="Mécanique quantique">mécanique quantique</a>, qui prohibe tout état où l'on connaitrait simultanément position et vitesse. La description continue du <a href="/wiki/Degr%C3%A9_de_libert%C3%A9_(physique)" title="Degré de liberté (physique)">degré de liberté</a> en <a href="/wiki/Translation" title="Translation">translation</a> n'est donc pas pertinente pour les très basses températures. On ne peut pas, pour compter les états, prendre de boîte de taille inférieure à la <a href="/wiki/Longueur_de_Planck" title="Longueur de Planck">longueur de Planck</a>. </p><p>La température étant indissociablement liée à l'entropie, cette valeur va guider une définition du zéro absolu comme température associée à l'entropie minimale : </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 \lim \limits _{T\to 0}S=k\ln g_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mi>S</mi> <mo>=</mo> <mi>k</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim \limits _{T\to 0}S=k\ln g_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f36509cf4b1a7478ee0d219aa4069121d9b8067" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.695ex; height:3.843ex;" alt="{\displaystyle \lim \limits _{T\to 0}S=k\ln g_{0}}" /></span></dd></dl> <p>où <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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32d13273b9af4564fa2c421c96d039c414db8628" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{0}}" /></span> est la <a href="/wiki/D%C3%A9g%C3%A9n%C3%A9rescence_(physique_quantique)" title="Dégénérescence (physique quantique)">dégénérescence</a> du niveau d'énergie le plus bas. Si l'on peut considérer que, en théorie, <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_{0}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{0}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8adcdbc326a0270e3ceb5e0bce18652111dff107" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.424ex; height:2.509ex;" alt="{\displaystyle g_{0}=1}" /></span>, il n'en va pas de même en pratique où de faibles perturbations du système mesuré vont lever la dégénérescence et empêcher l'obtention du zéro absolu. La valeur la plus basse de <span title="−273,15 °C ou −459,67 °F" style="cursor:help">450</span> <abbr class="abbr" title="picokelvin"><a href="/wiki/Pico_(pr%C3%A9fixe)" title="Pico (préfixe)">p</a>K</abbr> a été atteinte dans un gaz de <a href="/wiki/Sodium" title="Sodium">sodium</a> au <span class="lang-en" lang="en"><a href="/wiki/Massachusetts_Institute_of_Technology" title="Massachusetts Institute of Technology">Massachusetts Institute of Technology</a></span> par l'équipe de <a href="/wiki/Wolfgang_Ketterle" title="Wolfgang Ketterle">Wolfgang Ketterle</a><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite_crochet">[</span>10<span class="cite_crochet">]</span></a></sup>. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r222902168" /><div class="NavFrame" style="" title="[afficher]/[masquer]"> <div class="NavHead" style="">Troisième principe de la thermodynamique</div> <div class="NavContent" style=""> <p>L'entropie microscopique d'un système est donnée 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 S=k\ln \Omega +{\frac {U}{T}}+S_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>k</mi> <mi>ln</mi> <mo>⁡</mo> <mi mathvariant="normal">Ω</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>U</mi> <mi>T</mi> </mfrac> </mrow> <mo>+</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=k\ln \Omega +{\frac {U}{T}}+S_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/779c46337921a3bee69d72260778f199ab301449" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.979ex; height:5.176ex;" alt="{\displaystyle S=k\ln \Omega +{\frac {U}{T}}+S_{0}}" /></span></dd></dl> <p>où <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 \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" /></span> est la <a href="/wiki/Fonction_de_partition" title="Fonction de partition">fonction de partition</a> 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 S_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebe0ac45a38c4437bd2689a14ec434cd499e7e49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{0}}" /></span> une constante d'intégration arbitraire que l'on prendra égale à zéro<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite_crochet">[</span>11<span class="cite_crochet">]</span></a></sup>. </p><p>On a (voir encadré précédent) : </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 U=-{\frac {\partial \ln \Omega }{\partial \beta }}=kT\,{\frac {\partial \ln \Omega }{\partial \ln T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <mi mathvariant="normal">Ω</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>k</mi> <mi>T</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <mi mathvariant="normal">Ω</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <mi>T</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=-{\frac {\partial \ln \Omega }{\partial \beta }}=kT\,{\frac {\partial \ln \Omega }{\partial \ln T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50942361eb22575492c2f7b2a570abc63bc36e05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.114ex; height:5.843ex;" alt="{\displaystyle U=-{\frac {\partial \ln \Omega }{\partial \beta }}=kT\,{\frac {\partial \ln \Omega }{\partial \ln T}}}" /></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 \Omega =\sum _{i}{\frac {n_{i}}{N}}e^{-{\frac {E_{i}}{kT}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>N</mi> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =\sum _{i}{\frac {n_{i}}{N}}e^{-{\frac {E_{i}}{kT}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a85cd4523540836054f54ea45da96c422a15352b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.614ex; height:6.676ex;" alt="{\displaystyle \Omega =\sum _{i}{\frac {n_{i}}{N}}e^{-{\frac {E_{i}}{kT}}}}" /></span></dd></dl> <p>D'où la limite correspondant à l'énergie minimale : </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 \lim \limits _{T\to 0}S=k\ln g_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mi>S</mi> <mo>=</mo> <mi>k</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim \limits _{T\to 0}S=k\ln g_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f36509cf4b1a7478ee0d219aa4069121d9b8067" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.695ex; height:3.843ex;" alt="{\displaystyle \lim \limits _{T\to 0}S=k\ln g_{0}}" /></span></dd></dl> <p>où <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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32d13273b9af4564fa2c421c96d039c414db8628" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{0}}" /></span> est la <a href="/wiki/D%C3%A9g%C3%A9n%C3%A9rescence_(physique_quantique)" title="Dégénérescence (physique quantique)">dégénérescence</a> du niveau fondamental. </p> </div> <div class="NavEnd"> </div> </div> <div class="mw-heading mw-heading3"><h3 id="Échelle_absolue_de_température"><span id=".C3.89chelle_absolue_de_temp.C3.A9rature"></span>Échelle absolue de température</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&veaction=edit&section=6" title="Modifier la section : Échelle absolue de température" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&action=edit&section=6" title="Modifier le code source de la section : Échelle absolue de température"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Le <a href="/wiki/Deuxi%C3%A8me_principe_de_la_thermodynamique" title="Deuxième principe de la thermodynamique">deuxième principe de la thermodynamique</a> a permis de définir la température et une échelle de référence pour celle-ci, basée sur l'<a href="/wiki/Loi_des_gaz_parfaits" title="Loi des gaz parfaits">équation d'état du gaz parfait</a> (voir encadré « Température et entropie »). L'instrument primaire de mesure est donc assez naturellement un thermomètre à hydrogène, gaz dont le comportement est proche de celui du <a href="/wiki/Gaz_parfait" title="Gaz parfait">gaz parfait</a> et qui est peu sensible au phénomène d'<a href="/wiki/Adsorption" title="Adsorption">adsorption</a> pariétale comme le sont les <a href="/wiki/Gaz_nobles" class="mw-redirect" title="Gaz nobles">gaz nobles</a>. Comme on sait définir le zéro absolu, il reste à définir un point de référence. La norme actuelle choisit d'utiliser le <a href="/wiki/Point_triple" title="Point triple">point triple</a> d'une <a href="/wiki/Eau_oc%C3%A9anique_moyenne_normalis%C3%A9e_de_Vienne" title="Eau océanique moyenne normalisée de Vienne">eau de composition isotopique donnée</a> comme constituant par définition la valeur <span title="0,01 °C ou 32,018 °F" style="cursor:help">273,16</span> <abbr class="abbr" title="kelvin">K</abbr><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite_crochet">[</span>12<span class="cite_crochet">]</span></a></sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Température_de_rayonnement"><span id="Temp.C3.A9rature_de_rayonnement"></span>Température de rayonnement</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&veaction=edit&section=7" title="Modifier la section : Température de rayonnement" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&action=edit&section=7" title="Modifier le code source de la section : Température de rayonnement"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Gaz_de_photons" title="Gaz de photons">Gaz de photons</a>.</div></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Spectre_source_air.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Spectre_source_air.png/180px-Spectre_source_air.png" decoding="async" width="180" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Spectre_source_air.png/270px-Spectre_source_air.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Spectre_source_air.png/360px-Spectre_source_air.png 2x" data-file-width="603" data-file-height="560" /></a><figcaption>Source de rayonnement gazeux.</figcaption></figure> <p>Le <a href="/wiki/Rayonnement_%C3%A9lectromagn%C3%A9tique" title="Rayonnement électromagnétique">rayonnement électromagnétique</a> est constitué de <a href="/wiki/Photon" title="Photon">photons</a> qui se comportent au plan thermodynamique comme un <a href="/wiki/Gaz" title="Gaz">gaz</a>. Les <a href="/wiki/Interaction_rayonnement-mati%C3%A8re" title="Interaction rayonnement-matière">échanges matière-rayonnement</a> peuvent permettent à ce dernier d'acquérir un <a href="/wiki/%C3%89quilibre_thermodynamique" title="Équilibre thermodynamique">équilibre thermodynamique</a> caractérisé par la <a href="/wiki/Loi_de_Planck" title="Loi de Planck">distribution de Planck</a> à 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}"> <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> du milieu matériel, qui constitue la température du rayonnement. En l'absence d'<a href="/w/index.php?title=Interaction_photon-photon&action=edit&redlink=1" class="new" title="Interaction photon-photon (page inexistante)">interaction photon-photon</a> <a href="https://en.wikipedia.org/wiki/Two-photon_physics" class="extiw" title="en:Two-photon physics"><span class="indicateur-langue" title="Article en anglais : « Two-photon physics »">(en)</span></a> pour assurer des échanges d'énergie, il faut multiplier les interactions photon-matière pour aboutir à l'équilibre thermodynamique. Un tel rayonnement est appelé <a href="/wiki/Rayonnement_du_corps_noir" title="Rayonnement du corps noir">rayonnement du corps noir</a>. Il est obtenu par l'utilisation d'une <a href="/wiki/Corps_noir_(appareil)" title="Corps noir (appareil)">enceinte fermée</a>. Le rayonnement que l'on rencontre dans de nombreux domaines s'éloigne de celui-ci en comportant des <a href="/wiki/Raie_spectrale" title="Raie spectrale">raies</a> d'émission ou d'absorption. </p><p>L'équilibre thermodynamique du rayonnement correspond à une <a href="/wiki/Distribution_angulaire" title="Distribution angulaire">distribution angulaire</a> de celui-ci <a href="/wiki/Isotropie" title="Isotropie">isotrope</a>. À l'opposé, un <a href="/wiki/Rayon_lumineux" title="Rayon lumineux">rayon lumineux</a> de fréquence <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 \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" /></span> correspond à une <a href="/wiki/Gaz_de_photons#Propriétés_thermodynamiques" title="Gaz de photons">entropie minimale</a><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite_crochet">[</span>13<span class="cite_crochet">]</span></a></sup> : </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 S={\frac {4\pi \ln 2\,k\nu ^{2}}{c^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mi>k</mi> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={\frac {4\pi \ln 2\,k\nu ^{2}}{c^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b93fbea3c8daac3ddbe1cdf7be7b1d4a6fd76e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.714ex; height:6.009ex;" alt="{\displaystyle S={\frac {4\pi \ln 2\,k\nu ^{2}}{c^{3}}}}" /></span></dd></dl> <p>Celle-ci tend vers zéro lorsque la <a href="/wiki/Longueur_d%27onde" title="Longueur d'onde">longueur d'onde</a> augmente. </p><p>Pour un spectre de rayonnement différent de celui du corps noir on définit une 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 \textstyle T_{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle T_{R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea9227424c207b661e8c5f2fabe37ab524ade297" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.837ex; height:2.509ex;" alt="{\displaystyle \textstyle T_{R}}" /></span> par analogie avec la température du corps noir par <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 \textstyle E_{R}=aT_{R}^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle E_{R}=aT_{R}^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79ae9b5f1e970a1a044046379bbd084af13d3e89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.36ex; height:3.176ex;" alt="{\displaystyle \textstyle E_{R}=aT_{R}^{4}}" /></span> où <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 \textstyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7abc4e8a2ef67ff9b7ad050afa4cf286c3ec0cd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle \textstyle a}" /></span> est la <a href="/wiki/Constante_radiative" title="Constante radiative">constante radiative</a>. Si l'énergie radiative <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 \textstyle E_{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle E_{R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33db5c602035c3472ec0dde44ce2f2b93f16f6af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.195ex; height:2.509ex;" alt="{\displaystyle \textstyle E_{R}}" /></span> est une quantité physique parfaitement définie, la température radiative ainsi définie n'a pas de grande signification mais elle permet de se ramener à une notion habituelle. </p> <div class="mw-heading mw-heading3"><h3 id="Milieu_à_plusieurs_températures"><span id="Milieu_.C3.A0_plusieurs_temp.C3.A9ratures"></span>Milieu à plusieurs températures</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&veaction=edit&section=8" title="Modifier la section : Milieu à plusieurs températures" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&action=edit&section=8" title="Modifier le code source de la section : Milieu à plusieurs températures"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:T_TR_TV_detente.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/T_TR_TV_detente.png/180px-T_TR_TV_detente.png" decoding="async" width="180" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/T_TR_TV_detente.png/270px-T_TR_TV_detente.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/T_TR_TV_detente.png/360px-T_TR_TV_detente.png 2x" data-file-width="771" data-file-height="707" /></a><figcaption>Températures de vibration T<sub>v</sub>, de translation T et de rotation T<sub>r</sub> dans une détente d'azote.</figcaption></figure> <p>L'équilibre thermodynamique entraîne l'existence d'une température unique pour tous les degrés de liberté du milieu : translation, décrite par la <a href="/wiki/Loi_de_distribution_des_vitesses_de_Maxwell" title="Loi de distribution des vitesses de Maxwell">statistique de Maxwell</a>, <a href="/wiki/Temp%C3%A9rature_rotationnelle" title="Température rotationnelle">rotation</a>, <a href="/wiki/Temp%C3%A9rature_vibrationnelle" title="Température vibrationnelle">vibration</a> et énergie interne, décrites par la <a href="/wiki/Statistique_de_Maxwell-Boltzmann" title="Statistique de Maxwell-Boltzmann">statistique de Boltzmann</a>. Dans certaines situations créés par l'introduction d'énergie dans le système, il se crée des états proches de l'équilibre qui peuvent être raisonnablement bien décrits par plusieurs températures. C'est par exemple le cas d'un <a href="/wiki/Physique_des_plasmas#Température_des_espèces_(«_plasma_chaud_»,_«_plasma_froid_»)" title="Physique des plasmas">plasma froid</a> qui présente des températures de translation différentes pour les électrons et les particules lourdes : ions, atomes, molécules. On peut séparer localement les divers degrés de liberté en effectuant une forte <a href="/wiki/D%C3%A9tente_(thermodynamique)" title="Détente (thermodynamique)">détente</a> (voir figure) à partir des temps caractéristiques différents pour les divers couplages entre ceux-ci<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite_crochet">[</span>14<span class="cite_crochet">]</span></a></sup>. Un tel système relaxe rapidement vers l'équilibre thermodynamique. </p> <div class="mw-heading mw-heading3"><h3 id="Milieu_dans_lequel_la_température_n'est_pas_un_paramètre_pertinent"><span id="Milieu_dans_lequel_la_temp.C3.A9rature_n.27est_pas_un_param.C3.A8tre_pertinent"></span>Milieu dans lequel la température n'est pas un paramètre pertinent</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&veaction=edit&section=9" title="Modifier la section : Milieu dans lequel la température n'est pas un paramètre pertinent" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&action=edit&section=9" title="Modifier le code source de la section : Milieu dans lequel la température n'est pas un paramètre pertinent"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Comptonisation.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Comptonisation.png/180px-Comptonisation.png" decoding="async" width="180" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Comptonisation.png/270px-Comptonisation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/Comptonisation.png/360px-Comptonisation.png 2x" data-file-width="771" data-file-height="724" /></a><figcaption><a href="/wiki/Effet_Sunyaev-Zel%27dovich" class="mw-redirect" title="Effet Sunyaev-Zel'dovich">Effet Sunyaev-Zel'dovich</a> : le spectre de corps noir <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 L_{CN}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{CN}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc24f2a58649f09723b41dd6b05abce2515b2a93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.523ex; height:2.509ex;" alt="{\displaystyle L_{CN}}" /></span> est modifié par <a href="/wiki/Effet_Compton" class="mw-redirect" title="Effet Compton">comptonisation</a>.</figcaption></figure> <p>Dans le cas où l'énergie injectée dans le milieu est très importante, la distribution microscopique s'éloigne notablement des distributions d'équilibre. Il peut cependant se créer des états stationnaires comme dans l'<a href="/wiki/Effet_Sunyaev-Zel%27dovich" class="mw-redirect" title="Effet Sunyaev-Zel'dovich">effet Sunyaev-Zel'dovich</a> issu d'<a href="/wiki/Effet_Compton" class="mw-redirect" title="Effet Compton">interactions Compton</a> multiples (comptonisation). Ce milieu possède une entropie et une énergie interne pour chaque constituant, électrons et <a href="/wiki/Gaz_de_photons" title="Gaz de photons">photons</a>. On peut donc lui associer deux températures, mais celles-ci n'ont que peu d'intérêt car elle ne permettent pas de caractériser les distributions d'énergie correspondantes. </p> <div class="mw-heading mw-heading3"><h3 id="Température_négative"><span id="Temp.C3.A9rature_n.C3.A9gative"></span>Température négative</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&veaction=edit&section=10" title="Modifier la section : Température négative" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&action=edit&section=10" title="Modifier le code source de la section : Température négative"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Temp%C3%A9rature_n%C3%A9gative" title="Température négative">Température négative</a>.</div></div> <p>Certains systèmes quantiques liés à la <a href="/wiki/R%C3%A9sonance_magn%C3%A9tique_nucl%C3%A9aire" title="Résonance magnétique nucléaire">résonance magnétique nucléaire</a> dans les cristaux ou les <a href="/wiki/Refroidissement_d%27atomes_par_laser" title="Refroidissement d'atomes par laser">gaz ultrafroids</a> possèdent des distributions d'énergie particulière pouvant être entièrement peuplées dans l'état de plus basse énergie (zéro absolu) mais également dans l'état de plus haute énergie<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite_crochet">[</span>15<span class="cite_crochet">]</span></a></sup>. Dans les deux cas, l'entropie est nulle par définition. Entre les deux extrêmes, l'entropie est finie, ce qui fait que <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> est une fonction de <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" /></span> d'abord croissante, puis décroissante. Si on adopte la définition standard de la température, cela conduit à une valeur infinie au maximum d'entropie, suivie de valeurs négatives pour les hautes énergies<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite_crochet">[</span>16<span class="cite_crochet">]</span></a></sup>. Ce problème est lié à la définition de l'<a href="/wiki/Entropie_(thermodynamique)" title="Entropie (thermodynamique)">entropie de Boltzmann</a>, mais il disparait si l'on utilise une définition alternative donnée par <a href="/wiki/Willard_Gibbs" title="Willard Gibbs">Willard Gibbs</a><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite_crochet">[</span>17<span class="cite_crochet">]</span></a></sup>. </p><p>Ce problème illustre la difficulté de la définition de la température. </p> <div class="mw-heading mw-heading2"><h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&veaction=edit&section=11" title="Modifier la section : Notes et références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&action=edit&section=11" title="Modifier le code source de la section : Notes et références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&veaction=edit&section=12" title="Modifier la section : Notes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&action=edit&section=12" title="Modifier le code source de la section : Notes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small lower-alpha" style=""><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a> </span><span class="reference-text">Expression valide dans le cas d'une transformation réversible.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a> </span><span class="reference-text">Expression valide dans le cas d'une transformation isochore.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a> </span><span class="reference-text">Mais pas sur l'équation d'état dont la justification relève de l'approche microscopique.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><a href="#cite_ref-10">↑</a> </span><span class="reference-text">On peut également utiliser l'<a href="/wiki/Enthalpie" title="Enthalpie">enthalpie</a> au lieu de l'énergie interne.</span> </li> </ol></div> </div> <div class="mw-heading mw-heading3"><h3 id="Références"><span id="R.C3.A9f.C3.A9rences"></span>Références</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&veaction=edit&section=13" title="Modifier la section : Références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Temp%C3%A9rature_thermodynamique&action=edit&section=13" title="Modifier le code source de la section : Références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small decimal" style=""><div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Thomson1848"><span class="ouvrage" id="William_Thomson1848"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="/wiki/William_Thomson_(Lord_Kelvin)" title="William Thomson (Lord Kelvin)">William Thomson</a>, « <cite style="font-style:normal" lang="en">On an absolute thermometric scale founded on Carnot's theory of the motive power of heat and calculated from Regnaut's observations</cite> », <i><span class="lang-en" lang="en"><a href="/wiki/Philosophical_Magazine" title="Philosophical Magazine">Philosophical Magazine</a></span></i>, <abbr class="abbr" title="volume">vol.</abbr> 33,‎ <time>1848</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=On+an+absolute+thermometric+scale+founded+on+Carnot%27s+theory+of+the+motive+power+of+heat+and+calculated+from+Regnaut%27s+observations&rft.jtitle=Philosophical+Magazine&rft.aulast=Thomson&rft.aufirst=William&rft.date=1848&rft.volume=33&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATemp%C3%A9rature+thermodynamique"></span></span></span>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Robison1803"><span class="ouvrage" id="J._Robison1803"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> J. Robison, <cite class="italique" lang="en">Lectures on the Elements of Chemistry by Joseph Black</cite>, Longman and Rees, <time>1803</time> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="https://gallica.bnf.fr/ark:/12148/bpt6k62375d;%20https://archive.org/download/2543060RX2.nlm.nih.gov/2543060RX2.pdf">lire en ligne</a> <abbr class="abbr indicateur-format format-pdf" title="Document au format Portable Document Format (PDF) d'Adobe">[PDF]</abbr>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+the+Elements+of+Chemistry+by+Joseph+Black&rft.pub=Longman+and+Rees&rft.aulast=Robison&rft.aufirst=J.&rft.date=1803&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATemp%C3%A9rature+thermodynamique"></span></span></span>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Clausius1850"><span class="ouvrage" id="Rudolf_Clausius1850"><abbr class="abbr indicateur-langue" title="Langue : allemand">(de)</abbr> <a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a>, « <cite style="font-style:normal" lang="de">Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen</cite> », <i><span class="lang-de" lang="de"><a href="/wiki/Annalen_der_Physik" title="Annalen der Physik">Annalen der Physik</a></span></i>, <abbr class="abbr" title="volume">vol.</abbr> 155, <abbr class="abbr" title="numéro">n<sup>o</sup></abbr> 3,‎ <time>1850</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">368–397</span> <small style="line-height:1em;">(<a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">DOI</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1002/andp.18501550306">10.1002/andp.18501550306</a></span>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Ueber+die+bewegende+Kraft+der+W%C3%A4rme+und+die+Gesetze%2C+welche+sich+daraus+f%C3%BCr+die+W%C3%A4rmelehre+selbst+ableiten+lassen&rft.jtitle=Annalen+der+Physik&rft.issue=3&rft.aulast=Clausius&rft.aufirst=Rudolf&rft.date=1850&rft.volume=155&rft.pages=368%E2%80%93397&rft_id=info%3Adoi%2F10.1002%2Fandp.18501550306&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATemp%C3%A9rature+thermodynamique"></span></span></span>.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a> </span><span class="reference-text"><span class="ouvrage" id="EhrenfestEhrenfest1990"><span class="ouvrage" id="Paul_EhrenfestTatiana_Ehrenfest1990"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="/wiki/Paul_Ehrenfest" title="Paul Ehrenfest">Paul Ehrenfest</a> et <a href="/wiki/Tatiana_Ehrenfest" class="mw-redirect" title="Tatiana Ehrenfest">Tatiana Ehrenfest</a> (<abbr class="abbr" title="traduction">trad.</abbr> de l'allemand), <cite class="italique" lang="en">The Conceptual Foundations of the Statistical Approach in Mechanics</cite>, New York, <a href="/wiki/Dover_Publications" title="Dover Publications">Dover</a>, <time>1990</time>, 114 <abbr class="abbr" title="pages">p.</abbr> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/0-486-66250-0" title="Spécial:Ouvrages de référence/0-486-66250-0"><span class="nowrap">0-486-66250-0</span></a>, <a rel="nofollow" class="external text" href="https://books.google.fr/books?id=pGK29T4t-D0C&pg=PA1">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Conceptual+Foundations+of+the+Statistical+Approach+in+Mechanics&rft.place=New+York&rft.pub=Dover&rft.aulast=Ehrenfest&rft.aufirst=Paul&rft.au=Tatiana+Ehrenfest&rft.date=1990&rft.tpages=114&rft.isbn=0-486-66250-0&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATemp%C3%A9rature+thermodynamique"></span></span></span>.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Henry"><span class="ouvrage" id="M._Henry">M. Henry, « <a rel="nofollow" class="external text" href="https://prmarchenry.blogspot.fr/2014/05/echelle-de-kelvin.html"><cite style="font-style:normal;">Échelle de Kelvin</cite></a> »</span></span>.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Chandrasekhar1939"><span class="ouvrage" id="S._Chandrasekhar1939"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="/wiki/Subrahmanyan_Chandrasekhar" title="Subrahmanyan Chandrasekhar">S. Chandrasekhar</a>, <cite class="italique" lang="en">An Introduction to the Study of Stellar Structure</cite>, Dover, <time>1939</time>, 512 <abbr class="abbr" title="pages">p.</abbr> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-0-486-60413-8" title="Spécial:Ouvrages de référence/978-0-486-60413-8"><span class="nowrap">978-0-486-60413-8</span></a>, <a rel="nofollow" class="external text" href="https://books.google.fr/books?id=VwPLAgAAQBAJ&pg=PP1">lire en ligne</a>)</small><span class="Z3988" 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.pub=Dover&rft.aulast=Chandrasekhar&rft.aufirst=S.&rft.date=1939&rft.tpages=512&rft.isbn=978-0-486-60413-8&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATemp%C3%A9rature+thermodynamique"></span></span></span>.</span> </li> <li id="cite_note-H-11"><span class="mw-cite-backlink noprint">↑ <sup><a href="#cite_ref-H_11-0">a</a> et <a href="#cite_ref-H_11-1">b</a></sup> </span><span class="reference-text"><span class="ouvrage" id="Joseph_Oakland_HirschfelderCharles_Francis_CurtissRobert_Byron_Bird1966"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <span class="nom_auteur"><a href="/wiki/Joseph_Oakland_Hirschfelder" title="Joseph Oakland Hirschfelder">Joseph Oakland Hirschfelder</a></span>, <span class="nom_auteur"><a href="/wiki/Charles_Francis_Curtiss" title="Charles Francis Curtiss">Charles Francis Curtiss</a></span> et <span class="nom_auteur"><a href="/wiki/Robert_Byron_Bird" title="Robert Byron Bird">Robert Byron Bird</a></span>, <cite class="italique" lang="en">Molecular Theory of Gases and Liquids</cite>, <a href="/wiki/John_Wiley_and_Sons" class="mw-redirect" title="John Wiley and Sons">John Wiley and Sons</a>, <time>1966</time> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-0-471-40065-3" title="Spécial:Ouvrages de référence/978-0-471-40065-3"><span class="nowrap">978-0-471-40065-3</span></a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Molecular+Theory+of+Gases+and+Liquids&rft.pub=John+Wiley+and+Sons&rft.aulast=Joseph+Oakland+Hirschfelder&rft.au=Charles+Francis+Curtiss&rft.au=Robert+Byron+Bird&rft.date=1966&rft.isbn=978-0-471-40065-3&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATemp%C3%A9rature+thermodynamique"></span></span>.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><a href="#cite_ref-12">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Ludwig_Boltzmann1964"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <span class="nom_auteur"><a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Ludwig Boltzmann</a></span> (<abbr class="abbr" title="traduction">trad.</abbr> de l'allemand), <cite class="italique" lang="en">Lectures on Gas Theory</cite>, New York, Dover, <time>1964</time>, 490 <abbr class="abbr" title="pages">p.</abbr> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/0-486-68455-5" title="Spécial:Ouvrages de référence/0-486-68455-5"><span class="nowrap">0-486-68455-5</span></a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-I7QzCXnstEC">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Gas+Theory&rft.place=New+York&rft.pub=Dover&rft.aulast=Ludwig+Boltzmann&rft.date=1964&rft.tpages=490&rft.isbn=0-486-68455-5&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATemp%C3%A9rature+thermodynamique"></span></span>.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><a href="#cite_ref-13">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Fourier1822"><span class="ouvrage" id="Joseph_Fourier1822"><a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a>, <cite class="italique">Théorie analytique de la chaleur</cite>, <a href="/wiki/Firmin_Didot" title="Firmin Didot">Firmin Didot</a>, <time>1822</time> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="https://gallica.bnf.fr/ark:/12148/bpt6k1045508v">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9orie+analytique+de+la+chaleur&rft.pub=Firmin+Didot&rft.aulast=Fourier&rft.aufirst=Joseph&rft.date=1822&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATemp%C3%A9rature+thermodynamique"></span></span></span>.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><a href="#cite_ref-14">↑</a> </span><span class="reference-text"><span class="ouvrage" id="LeanhardtPasquiniSabaSchirotzek2003"><span class="ouvrage" id="A._E._LeanhardtT._A._PasquiniM._SabaA._Schirotzek2003"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> A. E. Leanhardt, T. A. Pasquini, M. Saba, A. Schirotzek, Y. Shin, D. Kielpinski, D. E. Pritchard et W. Ketterle, « <cite style="font-style:normal" lang="en">Cooling Bose-Einstein Condensates Below 500 Picokelvin</cite> », <i><span class="lang-en" lang="en"><a href="/wiki/Science" title="Science">Science</a></span></i>, <abbr class="abbr" title="volume">vol.</abbr> 301, <abbr class="abbr" title="numéro">n<sup>o</sup></abbr> 5639,‎ <time>2003</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">1513-1515</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Cooling+Bose-Einstein+Condensates+Below+500+Picokelvin&rft.jtitle=Science&rft.issue=5639&rft.aulast=Leanhardt&rft.aufirst=A.+E.&rft.au=T.+A.+Pasquini&rft.au=M.+Saba&rft.au=A.+Schirotzek&rft.au=Y.+Shin&rft.au=D.+Kielpinski&rft.au=D.+E.+Pritchard&rft.au=W.+Ketterle&rft.date=2003&rft.volume=301&rft.pages=1513-1515&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATemp%C3%A9rature+thermodynamique"></span></span></span>.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><a href="#cite_ref-15">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Ivanov"><span class="ouvrage" id="D._Ivanov">D. Ivanov, « <a rel="nofollow" class="external text" href="https://people.phys.ethz.ch/~ivanov/ps1/0708/Polycopie/lecture5.pdf"><cite style="font-style:normal;">Cours ETH Zurich : Physique statistique I</cite></a> »</span></span>.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><a href="#cite_ref-16">↑</a> </span><span class="reference-text"><span class="ouvrage" id="2019"><cite class="italique">Le Système international d’unités (SI)</cite>, <a href="/wiki/Bureau_international_des_poids_et_mesures" title="Bureau international des poids et mesures">Bureau international des poids et mesures</a>, <time>2019</time>, <abbr class="abbr" title="deux-mille-dix-neuvième">2019<sup>e</sup></abbr> <abbr class="abbr" title="édition">éd.</abbr>, 109 <abbr class="abbr" title="pages">p.</abbr> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="https://www.bipm.org/fr/publications/si-brochure">présentation en ligne</a>, <a rel="nofollow" class="external text" href="https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-FR.pdf">lire en ligne</a> <abbr class="abbr indicateur-format format-pdf" title="Document au format Portable Document Format (PDF) d'Adobe">[PDF]</abbr>)</small>, <abbr class="abbr" title="page">p.</abbr> 21<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Le+Syst%C3%A8me+international+d%E2%80%99unit%C3%A9s+%28SI%29&rft.pub=Bureau+international+des+poids+et+mesures&rft.edition=2019&rft.date=2019&rft.pages=21&rft.tpages=109&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATemp%C3%A9rature+thermodynamique"></span></span>.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><a href="#cite_ref-17">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Oxenius1986"><span class="ouvrage" id="Joachim_Oxenius1986"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Joachim Oxenius, <cite class="italique" lang="en">Kinetic Theory of Particles and Photons : Theoretical Foundations of Non-LTE Plasma Spectroscopy</cite>, <a href="/wiki/Springer_Verlag" class="mw-redirect" title="Springer Verlag">Springer Verlag</a>, <abbr class="abbr" title="collection">coll.</abbr> « Springer Series in Electrophysics », <time>1986</time>, 356 <abbr class="abbr" title="pages">p.</abbr> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-3-642-70728-5" title="Spécial:Ouvrages de référence/978-3-642-70728-5"><span class="nowrap">978-3-642-70728-5</span></a>, <a rel="nofollow" class="external text" href="https://books.google.fr/books?id=5-r1CAAAQBAJ&pg=PA1">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Kinetic+Theory+of+Particles+and+Photons&rft.pub=Springer+Verlag&rft.stitle=Theoretical+Foundations+of+Non-LTE+Plasma+Spectroscopy&rft.aulast=Oxenius&rft.aufirst=Joachim&rft.date=1986&rft.tpages=356&rft.isbn=978-3-642-70728-5&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATemp%C3%A9rature+thermodynamique"></span></span></span>.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><a href="#cite_ref-18">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Brun2009"><span class="ouvrage" id="R._Brun2009"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> R. Brun, <cite class="italique" lang="en">Introduction to Reactive Gas Dynamics</cite>, <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, <time>2009</time> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-0-19-955268-9" title="Spécial:Ouvrages de référence/978-0-19-955268-9"><span class="nowrap">978-0-19-955268-9</span></a>, <a rel="nofollow" class="external text" href="https://books.google.fr/books?id=maltEt8dzV4C&pg=PR5">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Reactive+Gas+Dynamics&rft.pub=Oxford+University+Press&rft.aulast=Brun&rft.aufirst=R.&rft.date=2009&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATemp%C3%A9rature+thermodynamique"></span></span></span>.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><a href="#cite_ref-19">↑</a> </span><span class="reference-text"><span class="ouvrage"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> « <a rel="nofollow" class="external text" href="http://ltl.tkk.fi/triennial/positive.html"><cite style="font-style:normal;" lang="en">O. V. Lounasmaa Laboratory of Aalto University : Positive and negative picokelvin temperatures</cite></a> »</span>.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><a href="#cite_ref-20">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Foster_Ramsey1965"><span class="ouvrage" id="Norman_Foster_Ramsey1965"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="/wiki/Norman_Foster_Ramsey" title="Norman Foster Ramsey">Norman Foster Ramsey</a>, « <cite style="font-style:normal" lang="en">Thermodynamics and Statistical Mechanics at Negative Absolute Temperatures</cite> », <i><span class="lang-en" lang="en"><a href="/wiki/Physical_Review" title="Physical Review">Physical Review</a></span></i>, <abbr class="abbr" title="volume">vol.</abbr> 103, <abbr class="abbr" title="numéro">n<sup>o</sup></abbr> 1,‎ <time>1965</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">20-27</span> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="https://fenix.tecnico.ulisboa.pt/downloadFichier/3779580641790/Ramsey1956.pdf">lire en ligne</a> <abbr class="abbr indicateur-format format-pdf" title="Document au format Portable Document Format (PDF) d'Adobe">[PDF]</abbr>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Thermodynamics+and+Statistical+Mechanics+at+Negative+Absolute+Temperatures&rft.jtitle=Physical+Review&rft.issue=1&rft.au=Norman+Foster+Ramsey&rft.date=1965&rft.volume=103&rft.pages=20-27&rft_id=https%3A%2F%2Ffenix.tecnico.ulisboa.pt%2FdownloadFichier%2F3779580641790%2FRamsey1956.pdf&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATemp%C3%A9rature+thermodynamique"></span></span></span>.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><a href="#cite_ref-21">↑</a> </span><span class="reference-text"><span class="ouvrage" id="DunkelHilbert2014"><span class="ouvrage" id="J._DunkelS._Hilbert2014"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> J. Dunkel et S. Hilbert, « <cite style="font-style:normal" lang="en">Consistent thermostatistics forbids negative absolute temperatures</cite> », <i><span class="lang-en" lang="en"><a href="/wiki/Nature_Physics" title="Nature Physics">Nature Physics</a></span></i>, <abbr class="abbr" title="volume">vol.</abbr> 10,‎ <time>2014</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">67-72</span> <small style="line-height:1em;">(<a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">DOI</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1038/nphys2815">10.1038/nphys2815</a></span>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Consistent+thermostatistics+forbids+negative+absolute+temperatures&rft.jtitle=Nature+Physics&rft.aulast=Dunkel&rft.aufirst=J.&rft.au=S.+Hilbert&rft.date=2014&rft.volume=10&rft.pages=67-72&rft_id=info%3Adoi%2F10.1038%2Fnphys2815&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATemp%C3%A9rature+thermodynamique"></span></span></span>.</span> </li> </ol></div> </div> <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=Temp%C3%A9rature_thermodynamique&veaction=edit&section=14" 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=Temp%C3%A9rature_thermodynamique&action=edit&section=14" title="Modifier le code source de la section : Voir aussi"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r194021218">.mw-parser-output .autres-projets>.titre{text-align:center;margin:0.2em 0}.mw-parser-output .autres-projets>ul{margin:0;padding:0}.mw-parser-output .autres-projets>ul>li{list-style:none;margin:0.2em 0;text-indent:0;padding-left:24px;min-height:20px;text-align:left;display:block}.mw-parser-output .autres-projets>ul>li>a{font-style:italic}@media(max-width:720px){.mw-parser-output .autres-projets{float:none}}</style><div class="autres-projets boite-grise boite-a-droite noprint js-interprojets"> <p class="titre">Sur les autres projets Wikimedia :</p> <ul class="noarchive plainlinks"> <li class="wiktionary"><a href="https://fr.wiktionary.org/wiki/temp%C3%A9rature_thermodynamique" class="extiw" title="wikt:température thermodynamique">température thermodynamique</a>, <span class="nowrap">sur le <span class="project">Wiktionnaire</span></span></li> </ul> </div> <ul><li><a href="/wiki/Chaleur_(thermodynamique)" title="Chaleur (thermodynamique)">Chaleur (thermodynamique)</a></li> <li><a href="/wiki/Temp%C3%A9rature_absolue" title="Température absolue">Température absolue</a></li> <li><a href="/wiki/%C3%89quilibre_thermique" title="Équilibre thermique">Équilibre thermique</a></li></ul> <div class="navbox-container" style="clear:both;"> <table class="navbox collapsible noprint autocollapse" style=""> <tbody><tr><th class="navbox-title" colspan="2" style=""><div style="float:left; width:6em; text-align:left"><div class="noprint plainlinks nowrap tnavbar" style="padding:0; font-size:xx-small; color:var(--color-emphasized, #000000);"><a href="/wiki/Mod%C3%A8le:Palette_Grandeurs_et_unit%C3%A9s" title="Modèle:Palette Grandeurs et unités"><abbr class="abbr" title="Voir ce modèle.">v</abbr></a> · <a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Mod%C3%A8le:Palette_Grandeurs_et_unit%C3%A9s&action=edit"><abbr class="abbr" title="Modifier ce modèle. Merci de prévisualiser avant de sauvegarder.">m</abbr></a></div></div><div style="font-size:110%"><a href="/wiki/Grandeur_physique" title="Grandeur physique">Grandeurs physiques</a> <a href="/wiki/Unit%C3%A9_fondamentale" title="Unité fondamentale">fondamentales</a> et <a href="/wiki/Unit%C3%A9_de_mesure" title="Unité de mesure">unités de mesure</a> usuelles associées</div></th> </tr> <tr> <th class="navbox-group" style=""><a href="/wiki/Longueur" title="Longueur">Longueur</a> (L)</th> <td class="navbox-list" style=""><table class="navbox-subgroup" style=""> <tbody><tr> <th class="navbox-group" style="white-space:nowrap; text-align:left; width:10em;"><a href="/wiki/Unit%C3%A9_de_longueur" title="Unité de longueur">Unités de longueur</a></th> <td class="navbox-list" style="text-align:left;;"><div class="liste-horizontale"> <ul><li><b><a href="/wiki/M%C3%A8tre" title="Mètre">mètre (m)</a></b>, <a href="/wiki/Centim%C3%A8tre" title="Centimètre">centimètre (cm)</a> et <a href="/wiki/Millim%C3%A8tre" title="Millimètre">millimètre (mm)</a></li> <li><a href="/wiki/Kilom%C3%A8tre" title="Kilomètre">kilomètre (km)</a></li> <li><a href="/wiki/Verge_(unit%C3%A9)" title="Verge (unité)">verge ou yard</a>, <a href="/wiki/Pied_(unit%C3%A9)" title="Pied (unité)">pied</a> et <a href="/wiki/Pouce_(unit%C3%A9)" title="Pouce (unité)">pouce</a></li> <li><a href="/wiki/Mille_terrestre_international" title="Mille terrestre international">mille terrestre (mi)</a></li> <li><a href="/wiki/Mille_marin" title="Mille marin">mille marin</a></li> <li><a href="/wiki/Milli%C3%A8me_de_pouce" title="Millième de pouce">thou ou mil</a></li> <li><a href="/wiki/Microm%C3%A8tre" title="Micromètre">micromètre (μm)</a> (<abbr class="abbr" title="anciennement">anc.</abbr> micron (µ))</li> <li><a href="/wiki/Nanom%C3%A8tre" title="Nanomètre">nanomètre (nm)</a></li> <li><a href="/wiki/%C3%85ngstr%C3%B6m" title="Ångström">ångström (Å)</a></li> <li><a href="/wiki/Picom%C3%A8tre" title="Picomètre">picomètre (pm)</a></li> <li><a href="/wiki/Femtom%C3%A8tre" title="Femtomètre">femtomètre (fm)</a></li> <li><a href="/wiki/Ann%C3%A9e-lumi%C3%A8re" title="Année-lumière">année-lumière (al)</a></li> <li><a href="/wiki/Parsec" title="Parsec">parsec (pc)</a></li> <li><a href="/wiki/Unit%C3%A9_astronomique" title="Unité astronomique">unité astronomique (au)</a></li></ul> </div></td> </tr> </tbody></table></td> </tr> <tr> <th class="navbox-group" style=""><a href="/wiki/Masse" title="Masse">Masse</a> (M)</th> <td class="navbox-list navbox-even" style=""><table class="navbox-subgroup" style=""> <tbody><tr> <th class="navbox-group" style="white-space:nowrap; text-align:left; width:10em;"><a href="/wiki/Unit%C3%A9_de_masse" title="Unité de masse">Unités de masse</a></th> <td class="navbox-list" style="text-align:left;;"><div class="liste-horizontale"> <ul><li><b><a href="/wiki/Kilogramme" title="Kilogramme">kilogramme (kg)</a></b> et <a href="/wiki/Gramme" title="Gramme">gramme (g)</a></li> <li><a href="/wiki/Livre_(unit%C3%A9)" title="Livre (unité)">livre (lb)</a> et <a href="/wiki/Once_(unit%C3%A9)" title="Once (unité)">once (oz)</a></li> <li><a href="/wiki/Tonne" title="Tonne">tonne (t)</a></li></ul> </div></td> </tr> </tbody></table></td> </tr> <tr> <th class="navbox-group" style=""><a href="/wiki/Temps" title="Temps">Temps</a> (T)</th> <td class="navbox-list" style=""><table class="navbox-subgroup" style=""> <tbody><tr> <th class="navbox-group" style="white-space:nowrap; text-align:left; width:10em;"><a href="/wiki/Unit%C3%A9_de_temps" title="Unité de temps">Unité de temps</a></th> <td class="navbox-list" style="text-align:left;;"><div class="liste-horizontale"> <ul><li><b><a href="/wiki/Seconde_(temps)" title="Seconde (temps)">seconde (s)</a></b></li></ul> </div></td> </tr> </tbody></table></td> </tr> <tr> <th class="navbox-group" style=""><a href="/wiki/Courant_%C3%A9lectrique" title="Courant électrique">Intensité électrique</a> (I)</th> <td class="navbox-list navbox-even" style=""><table class="navbox-subgroup" style=""> <tbody><tr> <th class="navbox-group" style="white-space:nowrap; text-align:left; width:10em;"><a href="/wiki/Grandeur_%C3%A9lectrique" title="Grandeur électrique">Unité électrique</a></th> <td class="navbox-list" style="text-align:left;;"><div class="liste-horizontale"> <ul><li><b><a href="/wiki/Amp%C3%A8re" title="Ampère">ampère (A)</a></b></li></ul> </div></td> </tr> </tbody></table></td> </tr> <tr> <th class="navbox-group" style=""><a class="mw-selflink selflink">Température</a> (<a href="/wiki/Th%C3%AAta" title="Thêta">Θ</a>)</th> <td class="navbox-list" style=""><table class="navbox-subgroup" style=""> <tbody><tr> <th class="navbox-group" style="white-space:nowrap; text-align:left; width:16em;"><a href="/wiki/Unit%C3%A9_de_temp%C3%A9rature" title="Unité de température">Unités de température</a></th> <td class="navbox-list" style="text-align:left;;"><div class="liste-horizontale"> <ul><li><b><a href="/wiki/Kelvin" title="Kelvin">kelvin (K)</a></b></li> <li><a href="/wiki/Degr%C3%A9_Celsius" title="Degré Celsius">degré Celsius (°C)</a></li> <li><a href="/wiki/Degr%C3%A9_Fahrenheit" title="Degré Fahrenheit">degré Fahrenheit (°F)</a></li> <li><a href="/wiki/%C3%89chelle_Rankine" title="Échelle Rankine">degré Rankine (°Ra)</a></li></ul> </div></td> </tr> </tbody></table></td> </tr> <tr> <th class="navbox-group" style=""><a href="/wiki/Quantit%C3%A9_de_mati%C3%A8re" title="Quantité de matière">Qté de matière</a> (N)</th> <td class="navbox-list navbox-even" style=""><table class="navbox-subgroup" style=""> <tbody><tr> <th class="navbox-group" style="white-space:nowrap; text-align:left; width:16em;"><a href="/wiki/Unit%C3%A9_de_quantit%C3%A9_de_mati%C3%A8re" title="Unité de quantité de matière">Unité de quantité de matière</a></th> <td class="navbox-list" style="text-align:left;;"><div class="liste-horizontale"> <ul><li><b><a href="/wiki/Mole_(unit%C3%A9)" title="Mole (unité)">mole (mol)</a></b></li></ul> </div></td> </tr> </tbody></table></td> </tr> <tr> <th class="navbox-group" style=""><a href="/wiki/Intensit%C3%A9_lumineuse" title="Intensité lumineuse">Intensité lumineuse</a> (J)</th> <td class="navbox-list" style=""><table class="navbox-subgroup" style=""> <tbody><tr> <th class="navbox-group" style="white-space:nowrap; text-align:left; width:16em;"><a href="/wiki/Photom%C3%A9trie_(optique)" title="Photométrie (optique)">Unités photométriques</a></th> <td class="navbox-list" style="text-align:left;;"><div class="liste-horizontale"> <ul><li><b><a href="/wiki/Candela" title="Candela">candela (cd)</a></b></li> <li><a href="/wiki/Bougie_(unit%C3%A9)" title="Bougie (unité)">bougie</a></li> <li><a href="/wiki/Bertrand_Guillaume_Carcel#Une_ancienne_unité_de_mesure" title="Bertrand Guillaume Carcel">carcel</a></li></ul> </div></td> </tr> </tbody></table></td> </tr> <tr> <td class="navbox-banner" style="" colspan="2"><div class="liste-horizontale"> <ul><li><a href="/wiki/Syst%C3%A8me_international_d%27unit%C3%A9s" title="Système international d'unités">Système international d'unités</a> (SI)</li> <li><a href="/wiki/Unit%C3%A9s_de_base_du_Syst%C3%A8me_international" title="Unités de base du Système international">Unités de base du SI</a></li> <li><a href="/wiki/Analyse_dimensionnelle" title="Analyse dimensionnelle">Analyse dimensionnelle</a></li> <li><a href="/wiki/Ordre_de_grandeur" title="Ordre de grandeur">Ordre de grandeur</a></li></ul> </div> <hr /> <small>Remarques : En <b>gras</b> : les <span class="nowrap">7 unités</span> de base légales du SI ; les grandeurs et unités photométriques sont les seules références subjectives du SI --</small></td></tr></tbody></table> </div> <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> <li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:Chimie" title="Portail de la chimie"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Nuvola_apps_edu_science.svg/24px-Nuvola_apps_edu_science.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Nuvola_apps_edu_science.svg/36px-Nuvola_apps_edu_science.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/59/Nuvola_apps_edu_science.svg/48px-Nuvola_apps_edu_science.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Chimie" title="Portail:Chimie">Portail de la chimie</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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