{"id":30775,"date":"2021-05-07T13:00:02","date_gmt":"2021-05-07T13:00:02","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=30775"},"modified":"2025-01-01T19:22:30","modified_gmt":"2025-01-01T19:22:30","slug":"equilibre-thermique-temperature-et-sa-definition-statistique","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/fr\/equilibre-thermique-temperature-et-sa-definition-statistique\/","title":{"rendered":"\u00c9quilibre thermique, temp\u00e9rature et sa d\u00e9finition statistique"},"content":{"rendered":"<style>\n\tp, ul, ol {\n\t\ttext-align: justify;\n\t}\n\th1, h2 {\n\t\ttext-align: center;\n\t}\n<\/style>\n<h1>\u00c9quilibre thermique, temp\u00e9rature et sa d\u00e9finition statistique<\/h1>\n<p style=\"text-align: center\"><em>La temp\u00e9rature thermodynamique, comprise comme la connexion entre \u00e9nergie et probabilit\u00e9, d\u00e9coule de l\u2019\u00e9quilibre entre macro\u00e9tats et micro\u00e9tats dans les syst\u00e8mes thermiques. Ce contenu explore comment la maximisation du nombre de micro\u00e9tats d\u00e9finit l\u2019\u00e9quilibre thermique, en utilisant des d\u00e9ductions math\u00e9matiques et le cadre de la m\u00e9canique statistique pour expliquer son lien avec la constante de Boltzmann et l\u2019\u00e9chelle Kelvin.<\/em><\/p>\n<p style=\"text-align: center;\"><strong>Objectifs d\u2019apprentissage :<\/strong><br \/>\n\u00c0 la fin de cette le\u00e7on, l\u2019\u00e9tudiant sera capable de :\n<\/p>\n<ol>\n<li><strong>Comprendre<\/strong> la relation entre macro\u00e9tats et micro\u00e9tats dans les syst\u00e8mes thermodynamiques et le concept de temp\u00e9rature.<\/li>\n<\/ol>\n<p style=\"text-align: center;\"><strong><u>TABLE DES MATI\u00c8RES<\/u> :<\/strong><br \/>\n<a href=\"#1\">Macro\u00e9tats et micro\u00e9tats des syst\u00e8mes en contact thermique<\/a><br \/>\n<a href=\"#2\">Temp\u00e9rature thermodynamique : La qualit\u00e9 commune des syst\u00e8mes en \u00e9quilibre<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/7LPUN_vBoXE\" title=\"Lecteur vid\u00e9o YouTube\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Macro\u00e9tats et micro\u00e9tats des syst\u00e8mes en contact thermique<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=7LPUN_vBoXE&amp;t=218s\" target=\"_blank\" rel=\"noopener\"><strong>Pour aborder le concept de temp\u00e9rature thermodynamique<\/strong><\/a>, nous devons d\u2019abord comprendre comment les syst\u00e8mes thermodynamiques interagissent et leur relation avec les macro\u00e9tats et les micro\u00e9tats. Consid\u00e9rons le cas de deux syst\u00e8mes en contact thermique entre eux, mais isol\u00e9s du reste de l\u2019univers.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-kL0Rr2L4wys\/YJPlhBUTueI\/AAAAAAAAFEw\/OWpuyASOdGANXsSG5O-d09ed034DtMTuwCLcBGAsYHQ\/s0\/contacto%2Btermico.PNG\" alt=\"Contact thermique\" class=\"aligncenter lazyload\" width=\"372\" height=\"158\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-kL0Rr2L4wys\/YJPlhBUTueI\/AAAAAAAAFEw\/OWpuyASOdGANXsSG5O-d09ed034DtMTuwCLcBGAsYHQ\/s0\/contacto%2Btermico.PNG\" alt=\"Contact thermique\" class=\"aligncenter lazyload\" width=\"372\" height=\"158\" \/><\/noscript><\/p>\n<p>D\u2019une part, \u00e9tant donn\u00e9 que les syst\u00e8mes restent isol\u00e9s, l\u2019\u00e9nergie totale <span class=\"katex-eq\" data-katex-display=\"false\">E=E_1+E_2<\/span> est constante, et par cons\u00e9quent, le macro\u00e9tat du syst\u00e8me est d\u00e9fini par l\u2019une des trois \u00e9nergies <span class=\"katex-eq\" data-katex-display=\"false\">E, E_1<\/span> ou <span class=\"katex-eq\" data-katex-display=\"false\">E_2<\/span>. D\u2019autre part, si nous parlons des micro\u00e9tats, le syst\u00e8me avec une \u00e9nergie <span class=\"katex-eq\" data-katex-display=\"false\">E_1<\/span> poss\u00e8de <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_1(E_1)<\/span> micro\u00e9tats, et de mani\u00e8re similaire, celui avec une \u00e9nergie <span class=\"katex-eq\" data-katex-display=\"false\">E_2<\/span> en poss\u00e8de <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_2(E_2)<\/span>. Par cons\u00e9quent, le syst\u00e8me combin\u00e9 est compos\u00e9 d\u2019un total de <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_1(E_1)\\Omega_2(E_2)<\/span> micro\u00e9tats.<\/p>\n<h3>Hypoth\u00e8ses sur l\u2019\u00e9quilibre thermodynamique<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=7LPUN_vBoXE&amp;t=392s\" target=\"_blank\" rel=\"noopener\"><strong>Si chaque syst\u00e8me peut \u00e9changer de l\u2019\u00e9nergie<\/strong><\/a> jusqu\u2019\u00e0 atteindre un <strong>\u00e9quilibre thermodynamique<\/strong>, alors \u00e0 un certain moment, <span class=\"katex-eq\" data-katex-display=\"false\">E_1<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">E_2<\/span> convergeront vers des valeurs constantes. L\u2019id\u00e9e \u00e0 retenir est que le macro\u00e9tat ayant la plus grande probabilit\u00e9 de se produire sera celui qui maximise le nombre de micro\u00e9tats. \u00c0 partir de cela, les hypoth\u00e8ses suivantes sont faites :<\/p>\n<ol>\n<li>Chaque micro\u00e9tat a la m\u00eame probabilit\u00e9 de se produire.<\/li>\n<li>La dynamique interne du syst\u00e8me est telle que les micro\u00e9tats changent continuellement.<\/li>\n<li>Apr\u00e8s un certain temps, le syst\u00e8me explorera tous les micro\u00e9tats possibles et passera le m\u00eame temps dans chacun d\u2019eux.<\/li>\n<\/ol>\n<p>Ces hypoth\u00e8ses impliquent que le syst\u00e8me passera la majeure partie de son temps dans les macro\u00e9tats repr\u00e9sentant le plus grand nombre de micro\u00e9tats, et seront donc les plus probables. Maintenant, dans les syst\u00e8mes g\u00e9n\u00e9ralement \u00e9tudi\u00e9s en thermodynamique, l\u2019expression \u00ab le plus probable \u00bb signifie en r\u00e9alit\u00e9 \u00ab pratiquement impossible que cela ne se produise pas \u00bb. Ce qui peut sembler \u00eatre une simple affirmation statistique est en fait une d\u00e9claration presque absolue de certitude.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Temp\u00e9rature thermodynamique : La qualit\u00e9 commune des syst\u00e8mes en \u00e9quilibre<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=7LPUN_vBoXE&amp;t=617s\" target=\"_blank\" rel=\"noopener\"><strong>Dans notre probl\u00e8me de deux corps plac\u00e9s<\/strong><\/a> en contact thermique, la distribution d\u2019\u00e9nergie la plus probable est celle qui maximise le nombre <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_1(E_1)\\Omega_2(E_2)<\/span> de micro\u00e9tats. \u00c9tant donn\u00e9 que les syst\u00e8mes sont vastes, nous pouvons utiliser les outils du calcul diff\u00e9rentiel comme une bonne approximation pour d\u00e9duire la signification de ces affirmations et leurs propri\u00e9t\u00e9s.<\/p>\n<p>Si nous effectuons des variations infinit\u00e9simales sur l\u2019\u00e9nergie de l\u2019un des syst\u00e8mes et cherchons le cas o\u00f9 le nombre de micro\u00e9tats est maximis\u00e9, nous obtenons le raisonnement suivant :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n(1) &amp; \\dfrac{d}{dE_1}\\left[  \\Omega_1(E_1) \\Omega_2(E_2) \\right] = 0 \\\\\n\n&amp; \\text{; car le nombre de micro\u00e9tats a \u00e9t\u00e9 maximis\u00e9} \\\\ \\\\\n\n\\equiv &amp; \\dfrac{d\\Omega_1(E_1)}{dE_1} \\Omega_2(E_2) + \\Omega_1(E_1) \\dfrac{d\\Omega_2(E_2)}{E_2}\\dfrac{dE_2}{dE_1} = 0 \\\\\n\n&amp; \\text{; en appliquant la r\u00e8gle du produit et la r\u00e8gle de cha\u00eene} \\\\ \\\\\n\n(2) &amp; E = E_1 + E_2 = \\text{Constante} \\\\\n\n&amp; \\text{; \u00c9nergie totale des deux syst\u00e8mes en contact thermique} \\\\ \\\\\n\n\\equiv &amp; E_1 = E - E_2 \\\\ \\\\\n\n(3) &amp; \\dfrac{dE_2}{dE_1} = \\dfrac{d}{dE_1} (E - E_1) = -1\\;\\text{; d\u2019apr\u00e8s (2)} \\\\ \\\\\n\n(4) &amp; \\dfrac{d\\Omega_1}{dE_1}\\Omega_2 - \\Omega_1 \\dfrac{d\\Omega_2}{dE_2} = 0\\;\\text{; d\u2019apr\u00e8s (1) et (3)} \\\\ \\\\\n\n\\equiv &amp; \\dfrac{d\\Omega_1}{dE_1} \\Omega_2 = \\Omega_1 \\dfrac{d\\Omega_2}{dE_2} \\\\ \\\\\n\n\\equiv &amp; \\dfrac{1}{\\Omega_1} \\dfrac{d\\Omega_1}{dE_1} = \\dfrac{1}{\\Omega_2} \\dfrac{d\\Omega_2}{dE_2} \\\\ \\\\\n\n\\equiv &amp; \\dfrac{d\\ln(\\Omega_1)}{dE_1} = \\dfrac{d\\ln(\\Omega_2)}{dE_2}\n\n\\end{array}<\/span>\n<p>Nous comprenons ainsi que, puisque l\u2019\u00e9quilibre thermique est le macro\u00e9tat le plus probable pour deux corps en contact thermique prolong\u00e9, il maximise le nombre de micro\u00e9tats. Si cela se produit, le raisonnement ci-dessus montre qu\u2019il existe une grandeur commune aux deux syst\u00e8mes que nous appelons <strong>temp\u00e9rature<\/strong>.<\/p>\n<h3>D\u00e9finition de la temp\u00e9rature thermodynamique<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=7LPUN_vBoXE&amp;t=972s\" target=\"_blank\" rel=\"noopener\"><strong>Lorsque cela se produit, nous disons que les corps<\/strong><\/a> \u00ab sont \u00e0 la m\u00eame temp\u00e9rature \u00bb et nous relions <span class=\"katex-eq\" data-katex-display=\"false\">d\\ln\\Omega\/dE<\/span> \u00e0 la temp\u00e9rature <span class=\"katex-eq\" data-katex-display=\"false\">T<\/span> (de sorte que <span class=\"katex-eq\" data-katex-display=\"false\">T_1 = T_2<\/span>). Ainsi, la temp\u00e9rature thermodynamique est d\u00e9finie comme suit :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{1}{k_B T} = \\frac{d\\ln\\Omega}{dE}<\/span>\n<p>o\u00f9 <span class=\"katex-eq\" data-katex-display=\"false\">k_B=1.3807\\cdot 10^{-23}[J\/K]<\/span> est la <strong>constante de Boltzmann<\/strong>.<\/p>\n<p>Avec ce choix de constantes, la temp\u00e9rature <span class=\"katex-eq\" data-katex-display=\"false\">T<\/span> que nous avons d\u00e9finie acquiert son interpr\u00e9tation habituelle dans ce que nous appelons l\u2019<strong>\u00e9chelle Kelvin<\/strong>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u00c9quilibre thermique, temp\u00e9rature et sa d\u00e9finition statistique La temp\u00e9rature thermodynamique, comprise comme la connexion entre \u00e9nergie et probabilit\u00e9, d\u00e9coule de l\u2019\u00e9quilibre entre macro\u00e9tats et micro\u00e9tats dans les syst\u00e8mes thermiques. Ce contenu explore comment la maximisation du nombre de micro\u00e9tats d\u00e9finit l\u2019\u00e9quilibre thermique, en utilisant des d\u00e9ductions math\u00e9matiques et le cadre de la m\u00e9canique statistique pour [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":30760,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":1,"footnotes":""},"categories":[647,931],"tags":[],"class_list":["post-30775","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-physique","category-thermodynamique"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u00c9quilibre thermique, temp\u00e9rature et sa d\u00e9finition statistique - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Comprend l\u2019\u00e9quilibre thermique, sa relation avec la temp\u00e9rature et sa 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Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. 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