{"id":30838,"date":"2021-06-18T13:00:09","date_gmt":"2021-06-18T13:00:09","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=30838"},"modified":"2025-01-02T03:24:25","modified_gmt":"2025-01-02T03:24:25","slug":"equation-et-fonction-detat","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/fr\/equation-et-fonction-detat\/","title":{"rendered":"\u00c9quation et Fonction d&#8217;\u00c9tat"},"content":{"rendered":"<style>\n    p, ul, ol {\n        text-align: justify;\n    }\n    h1, h2 {\n        text-align: center;\n    }\n<\/style>\n<h1>\u00c9quation et Fonction d&#8217;\u00c9tat<\/h1>\n<p style=\"text-align:center;\"><em>La thermodynamique nous r\u00e9v\u00e8le comment d\u00e9crire et connecter l&#8217;\u00e9tat d&#8217;un syst\u00e8me \u00e0 travers des concepts tels que les fonctions et les \u00e9quations d&#8217;\u00e9tat. Comment des propri\u00e9t\u00e9s telles que la pression, la temp\u00e9rature et l&#8217;\u00e9quation des gaz parfaits sont-elles li\u00e9es ? Ce contenu vous guidera pour comprendre les r\u00e8gles math\u00e9matiques et physiques qui r\u00e9gissent les \u00e9quilibres de l&#8217;univers, mettant votre intuition au d\u00e9fi et \u00e9largissant votre perspective.<\/em><\/p>\n<p style=\"text-align:center;\"><strong>Objectifs d&#8217;Apprentissage :<\/strong><br \/>\n\u00c0 la fin de cette le\u00e7on, l&#8217;\u00e9tudiant sera capable de :<\/p>\n<ol>\n<li><strong>D\u00e9crire<\/strong> le concept de fonction d&#8217;\u00e9tat et sa relation avec l&#8217;\u00e9quilibre thermique dans les syst\u00e8mes thermodynamiques.<\/li>\n<li><strong>Distinguer<\/strong> entre les fonctions d&#8217;\u00e9tat et les grandeurs qui ne sont pas des fonctions d&#8217;\u00e9tat, en identifiant des exemples concrets pour chaque cat\u00e9gorie.<\/li>\n<li><strong>Analyser<\/strong> comment les \u00e9quations d&#8217;\u00e9tat, comme celle des gaz parfaits, relient les fonctions d&#8217;\u00e9tat dans les syst\u00e8mes en \u00e9quilibre.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>TABLE DES MATI\u00c8RES<\/u> :<\/strong><br \/>\n<a href=\"#1\">La fonction d&#8217;\u00e9tat, les syst\u00e8mes et l&#8217;\u00e9quilibre thermique<\/a><br \/>\n<a href=\"#2\">L&#8217;\u00c9quation et les variables d&#8217;\u00e9tat<\/a><br \/>\n<a href=\"#3\">Notion math\u00e9matique de fonction d&#8217;\u00e9tat<\/a><br \/>\n<a href=\"#4\">L&#8217;\u00e9quation des gaz parfaits est une \u00e9quation d&#8217;\u00e9tat<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/ZXX3KAS5WKs\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p>Un des concepts cl\u00e9s de la thermodynamique classique est celui des \u00e9quations d&#8217;\u00e9tat et des fonctions d&#8217;\u00e9tat. Gr\u00e2ce \u00e0 eux, nous serons capables de d\u00e9crire l&#8217;\u00e9tat des syst\u00e8mes en \u00e9quilibre thermodynamique.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>La fonction d&#8217;\u00e9tat, les syst\u00e8mes et l&#8217;\u00e9quilibre thermique<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=ZXX3KAS5WKs&amp;t=115s\" target=\"_blank\" rel=\"noopener\"><strong>En thermodynamique, un syst\u00e8me est d\u00e9fini<\/strong><\/a> comme la portion de l&#8217;univers choisie pour \u00eatre \u00e9tudi\u00e9e, et l&#8217;environnement est proche du syst\u00e8me. Nous savons qu&#8217;un syst\u00e8me est en \u00e9quilibre thermique avec son environnement si ses grandeurs macroscopiques observables (ex. : pression, temp\u00e9rature) restent invariantes au cours du temps. Par exemple, si nous prenons un gaz \u00e0 l&#8217;int\u00e9rieur d&#8217;un conteneur et que sa temp\u00e9rature reste stable dans le temps, nous disons que le gaz est en \u00e9quilibre thermodynamique, et cet ensemble de grandeurs macroscopiques <strong>d\u00e9termine son \u00e9tat.<\/strong> Au contraire, si une grande quantit\u00e9 de chaleur est soudainement appliqu\u00e9e \u00e0 une partie du conteneur, alors, pendant un certain temps, le gaz sera dans un \u00e9tat diff\u00e9rent de l&#8217;\u00e9quilibre thermique, et son \u00e9tat deviendra variable dans le temps.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>L&#8217;\u00c9quation et les Variables d&#8217;\u00c9tat<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=ZXX3KAS5WKs&amp;t=115s\" target=\"_blank\" rel=\"noopener\"><strong>Lorsqu&#8217;un syst\u00e8me est en \u00e9quilibre thermique<\/strong><\/a>, nous pouvons reconna\u00eetre deux types de grandeurs : certaines d\u00e9pendent de la mani\u00e8re dont le syst\u00e8me a atteint cet \u00e9quilibre, tandis que d&#8217;autres en sont ind\u00e9pendantes. Ces derni\u00e8res sont ce que nous appelons des <strong>fonctions d&#8217;\u00e9tat<\/strong> (parfois appel\u00e9es <strong>variables d&#8217;\u00e9tat<\/strong>). Une fonction d&#8217;\u00e9tat est une grandeur physique qui poss\u00e8de une valeur bien d\u00e9finie pour chaque \u00e9tat d&#8217;\u00e9quilibre du syst\u00e8me. Ainsi, en \u00e9quilibre thermique, ces variables ne d\u00e9pendent pas du temps. Voici quelques exemples :<\/p>\n<ul>\n<li>la pression<\/li>\n<li>la temp\u00e9rature<\/li>\n<li>le volume<\/li>\n<li>l&#8217;\u00e9nergie interne<\/li>\n<\/ul>\n<p>Parmi les grandeurs qui ne sont pas des fonctions d&#8217;\u00e9tat, on trouve la position d&#8217;une particule sp\u00e9cifique du syst\u00e8me, le travail total et la chaleur totale appliqu\u00e9s au syst\u00e8me. Cela peut \u00eatre compris intuitivement en observant que vos mains peuvent atteindre la m\u00eame temp\u00e9rature (et donc le m\u00eame \u00e9tat) par deux processus diff\u00e9rents : l&#8217;un en appliquant une certaine quantit\u00e9 de travail en les frottant l&#8217;une contre l&#8217;autre, et l&#8217;autre en appliquant de la chaleur en les plongeant dans de l&#8217;eau ti\u00e8de.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Notion Math\u00e9matique de Fonction d&#8217;\u00c9tat<\/h2>\n<p>Avec ces id\u00e9es intuitives en t\u00eate, pour parvenir \u00e0 une compr\u00e9hension plus compl\u00e8te, il ne nous reste plus qu&#8217;\u00e0 d\u00e9velopper une explication math\u00e9matique plus pr\u00e9cise de ce qu&#8217;est une fonction d&#8217;\u00e9tat. Consid\u00e9rons un syst\u00e8me d\u00e9crit par un vecteur de param\u00e8tres <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1, x_2, x_3, \\cdots)<\/span>, et soit <span class=\"katex-eq\" data-katex-display=\"false\">f(\\vec{x})<\/span> une fonction d&#8217;\u00e9tat. Si les param\u00e8tres du syst\u00e8me passent d&#8217;une valeur initiale <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}_i<\/span> \u00e0 une valeur finale <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}_f<\/span>, alors la variation de la fonction <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> sera donn\u00e9e par :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta f = \\displaystyle \\int_{\\vec{x}_i}^{\\vec{x}_f}df = f(\\vec{x}_f) - f(\\vec{x}_i)<\/span>\n<p>Si les choses se passent ainsi, la variation de la fonction d&#8217;\u00e9tat d\u00e9pend uniquement des valeurs initiales et finales de <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}.<\/span> Cela se produit lorsque la quantit\u00e9 <span class=\"katex-eq\" data-katex-display=\"false\">df<\/span> est un <strong>diff\u00e9rentiel exact<\/strong>. Toutes les fonctions d&#8217;\u00e9tat poss\u00e8dent des diff\u00e9rentiels exacts ; \u00e0 l&#8217;inverse, une grandeur dont le diff\u00e9rentiel n&#8217;est pas exact ne peut pas \u00eatre une fonction d&#8217;\u00e9tat.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>L&#8217;\u00c9quation des Gaz Parfaits est une \u00c9quation d&#8217;\u00c9tat<\/h2>\n<p>En g\u00e9n\u00e9ral, il est toujours possible de trouver, au moins approximativement, une \u00e9quation d&#8217;\u00e9tat reliant les fonctions d&#8217;\u00e9tat. Un exemple est <a href=\"https:\/\/toposuranos.com\/la-ecuacion-de-los-gases-ideales\/\" rel=\"noopener\" target=\"_blank\">l&#8217;\u00e9quation d&#8217;\u00e9tat des gaz parfaits<\/a> <span class=\"katex-eq\" data-katex-display=\"false\">f(P,V,T)=0<\/span>, qui prend la forme suivante :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">f(P,V,T) = PV - nRT = 0<\/span>\n","protected":false},"excerpt":{"rendered":"<p>\u00c9quation et Fonction d&#8217;\u00c9tat La thermodynamique nous r\u00e9v\u00e8le comment d\u00e9crire et connecter l&#8217;\u00e9tat d&#8217;un syst\u00e8me \u00e0 travers des concepts tels que les fonctions et les \u00e9quations d&#8217;\u00e9tat. Comment des propri\u00e9t\u00e9s telles que la pression, la temp\u00e9rature et l&#8217;\u00e9quation des gaz parfaits sont-elles li\u00e9es ? Ce contenu vous guidera pour comprendre les r\u00e8gles math\u00e9matiques et physiques [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":30827,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":10,"footnotes":""},"categories":[647,931],"tags":[],"class_list":["post-30838","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>\u00c9quation et Fonction d&#039;\u00c9tat - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Qu&#039;est-ce que l&#039;\u00c9quation d&#039;\u00c9tat ? 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