{"id":30874,"date":"2021-07-03T13:00:16","date_gmt":"2021-07-03T13:00:16","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=30874"},"modified":"2025-01-02T05:04:02","modified_gmt":"2025-01-02T05:04:02","slug":"la-premiere-loi-de-la-thermodynamique","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/fr\/la-premiere-loi-de-la-thermodynamique\/","title":{"rendered":"La Premi\u00e8re Loi de la Thermodynamique"},"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>La Premi\u00e8re Loi de la Thermodynamique<\/h1>\n<p style=\"text-align:center;\"><em>La Premi\u00e8re Loi de la Thermodynamique est le fondement qui relie des concepts fondamentaux tels que la chaleur, le travail et l&#8217;\u00e9nergie interne, \u00e9tablissant que l&#8217;\u00e9nergie ne peut \u00eatre ni cr\u00e9\u00e9e ni d\u00e9truite, mais seulement transform\u00e9e. Ce document explore comment cette loi s&#8217;applique aux syst\u00e8mes ferm\u00e9s, approfondissant l&#8217;analyse du travail thermodynamique, des capacit\u00e9s calorifiques et des propri\u00e9t\u00e9s statistiques des gaz. \u00c0 travers une combinaison de formulations math\u00e9matiques et de raisonnements physiques, vous d\u00e9couvrirez des outils essentiels pour comprendre les processus \u00e9nerg\u00e9tiques dans des syst\u00e8mes complexes.<\/em><\/p>\n<p style=\"text-align:center;\"><strong>Objectifs d&#8217;apprentissage :<\/strong><br \/>\n\u00c0 la fin de ce cours, l&#8217;\u00e9tudiant sera capable de :\n<\/p>\n<ol>\n<li><strong>Justifier<\/strong> la Premi\u00e8re Loi de la Thermodynamique pour des syst\u00e8mes ferm\u00e9s, en expliquant les relations entre chaleur, travail et \u00e9nergie interne.<\/li>\n<li><strong>Analyser<\/strong> le concept de travail thermodynamique dans des processus de compression et d&#8217;expansion, en utilisant des formules diff\u00e9rentielles.<\/li>\n<li><strong>Calculer<\/strong> la capacit\u00e9 calorifique dans des conditions de volume et de pression constants, en appliquant des contraintes thermodynamiques.<\/li>\n<li><strong>Expliquer<\/strong> la distribution de Maxwell-Boltzmann et le principe de r\u00e9partition de l&#8217;\u00e9nergie dans les syst\u00e8mes mol\u00e9culaires.<\/li>\n<li><strong>D\u00e9montrer<\/strong> des relations sp\u00e9cifiques entre les capacit\u00e9s calorifiques, l&#8217;indice adiabatique et d&#8217;autres propri\u00e9t\u00e9s thermodynamiques pour des gaz parfaits.<\/li>\n<\/ol>\n<p style=\"text-align:center\"><strong><u>TABLE DES MATI\u00c8RES<\/u> :<\/strong><br \/>\n<a href=\"#1\">Formulation de la Premi\u00e8re Loi de la Thermodynamique<\/a><br \/>\n<a href=\"#2\">Travail thermodynamique<\/a><br \/>\n<a href=\"#3\">Capacit\u00e9 calorifique<\/a><br \/>\n<a href=\"#4\">Distribution de Maxwell-Boltzmann et r\u00e9partition de l&#8217;\u00e9nergie<\/a><br \/>\n<a href=\"#5\">Exercices<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/T6K1Nizc5NE\" title=\"Lecteur vid\u00e9o YouTube\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Formulation de la Premi\u00e8re Loi de la Thermodynamique<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=T6K1Nizc5NE&amp;t=140s\" target=\"_blank\" rel=\"noopener\"><strong>La Premi\u00e8re Loi de la Thermodynamique<\/strong><\/a> \u00e9tablit que :<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #c0ffc0;\"><span style=\"color: #000080;\">PREMI\u00c8RE LOI DE LA THERMODYNAMIQUE<br \/>\n<strong>L&#8217;\u00e9nergie ne peut \u00eatre ni cr\u00e9\u00e9e ni d\u00e9truite ; en outre, la chaleur et le travail sont des formes d&#8217;\u00e9nergie (\u00e9mises, absorb\u00e9es ou utilis\u00e9es dans un processus).<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>L&#8217;\u00e9nergie interne <span class=\"katex-eq\" data-katex-display=\"false\">U<\/span> est une fonction d&#8217;\u00e9tat car elle a une valeur bien d\u00e9finie pour chaque \u00e9tat d&#8217;\u00e9quilibre du syst\u00e8me. L&#8217;\u00e9nergie interne du syst\u00e8me peut \u00eatre modifi\u00e9e en appliquant de la chaleur <span class=\"katex-eq\" data-katex-display=\"false\">Q<\/span> ou en effectuant un travail <span class=\"katex-eq\" data-katex-display=\"false\">W<\/span> ; cependant, le travail et la chaleur ne sont pas des fonctions d&#8217;\u00e9tat. Cela s&#8217;explique par le fait qu&#8217;ils d\u00e9pendent du processus par lequel l&#8217;\u00e9nergie est ajout\u00e9e ou extraite, et qu&#8217;une fois le processus termin\u00e9, il est impossible de d\u00e9terminer la quantit\u00e9 de chaleur ou de travail n\u00e9cessaire pour atteindre cet \u00e9tat d&#8217;\u00e9quilibre.<\/p>\n<p>Le changement d&#8217;\u00e9nergie interne d&#8217;un syst\u00e8me peut \u00eatre exprim\u00e9 comme suit :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta U = \\Delta Q + \\Delta W<\/span><\/span>,<\/p>\n<p>o\u00f9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta Q<\/span><\/span> est la quantit\u00e9 de chaleur fournie et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta W<\/span><\/span> la quantit\u00e9 de travail effectu\u00e9e sur le syst\u00e8me. Par convention, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta Q<\/span><\/span> est positif lorsque de la chaleur est fournie au syst\u00e8me ; si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta Q<\/span><\/span> est n\u00e9gatif, cela signifie que la chaleur est extraite du syst\u00e8me ; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta W<\/span><\/span> est positif pour le travail effectu\u00e9 sur le syst\u00e8me ; si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta W<\/span><\/span> est n\u00e9gatif, le syst\u00e8me effectue un travail sur l&#8217;environnement.<\/p>\n<p>La relation entre le travail, la chaleur et l&#8217;\u00e9nergie interne peut \u00e9galement \u00eatre exprim\u00e9e de mani\u00e8re diff\u00e9rentielle comme suit :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dU = \\delta Q + \\delta W<\/span><\/span>.<\/p>\n<p>On utilise ici la lettre <span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span> pour repr\u00e9senter des diff\u00e9rentiels inexactes.<\/p>\n<p>Un syst\u00e8me thermiquement isol\u00e9 est d\u00e9fini comme un syst\u00e8me qui ne peut pas \u00e9changer de chaleur avec son environnement. Lorsque cela se produit, alors <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d U = \\delta W<\/span><\/span>. C&#8217;est la <strong>Premi\u00e8re Loi de la Thermodynamique<\/strong> restreinte \u00e0 un syst\u00e8me adiabatique.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Travail thermodynamique<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=T6K1Nizc5NE&amp;t=418s\" target=\"_blank\" rel=\"noopener\"><strong>Lorsque nous comprimons un ressort<\/strong><\/a> sur une distance <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dx<\/span><\/span>, il r\u00e9pond par une force \u00e9lastique de magnitude <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span>, et nous effectuons ainsi un travail :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta W = Fdx<\/span><\/span>.<\/p>\n<p>Lorsque nous comprimons un gaz, nous pouvons imaginer qu&#8217;il est constitu\u00e9 de nombreux ressorts plac\u00e9s les uns \u00e0 c\u00f4t\u00e9 des autres, remplissant un certain espace. De cette mani\u00e8re, si nous appliquons une force <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> sur une surface <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span>, nous exer\u00e7ons une pression <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P=F\/A<\/span><\/span>, et nous pouvons donc \u00e9crire que le travail effectu\u00e9 sera :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta W = PAdx = -PdV<\/span><\/span>.<\/p>\n<p>Le signe n\u00e9gatif qui appara\u00eet dans la derni\u00e8re \u00e9galit\u00e9 s&#8217;explique par le fait que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Adx=-dV<\/span><\/span>, ce qui signifie que lorsque les \u00abressorts\u00bb sont comprim\u00e9s d&#8217;une distance <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dx<\/span><\/span>, le volume qu&#8217;ils forment diminue. Lorsque du travail est effectu\u00e9 sur un syst\u00e8me thermodynamique, celui-ci r\u00e9agit en r\u00e9duisant son volume.<\/p>\n<p><a name=\"3\"><\/a> <\/p>\n<h2>Capacit\u00e9 calorifique<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=T6K1Nizc5NE&amp;t=677s\" target=\"_blank\" rel=\"noopener\"><strong>Supposons maintenant<\/strong><\/a> que nous souhaitons comprendre plus en d\u00e9tail comment l&#8217;\u00e9nergie interne d&#8217;un syst\u00e8me change lorsqu&#8217;on y ajoute de la chaleur. En g\u00e9n\u00e9ral, l&#8217;\u00e9nergie interne est une fonction de la temp\u00e9rature et du volume, ce que nous pouvons \u00e9crire sous la forme <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">U=U(T,V)<\/span><\/span>. Ensuite, comme l&#8217;\u00e9nergie est un diff\u00e9rentiel exact, il est possible d&#8217;exprimer la variation de <span class=\"katex-eq\" data-katex-display=\"false\">U<\/span> par rapport \u00e0 <span class=\"katex-eq\" data-katex-display=\"false\">T<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span> \u00e0 l&#8217;aide de la relation suivante :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle dU = \\left(\\frac{\\partial U}{\\partial T}\\right)_V dT + \\left(\\frac{\\partial U}{\\partial V}\\right)_T dV<\/span><\/span>.<\/p>\n<p>Maintenant, en utilisant les relations <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dU=\\delta Q + \\delta W<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta W=-PdV<\/span><\/span>, nous pouvons reformuler la <strong>Premi\u00e8re Loi de la Thermodynamique<\/strong> de la mani\u00e8re suivante :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\delta Q &amp; = dU + PdV\\\\ \\\\\n\n&amp; \\displaystyle  =\\left(\\frac{\\partial U}{\\partial T}\\right)_V dT + \\left(\\frac{\\partial U}{\\partial V}\\right)_T dV + PdV\\\\ \\\\\n\n&amp; \\displaystyle =\\left(\\frac{\\partial U}{\\partial T}\\right)_V dT + \\left[\\left(\\frac{\\partial U}{\\partial V}\\right)_T + P\\right]dV \\\\ \\\\\n\n\\displaystyle  \\frac{\\delta Q}{dT} &amp; \\displaystyle  =\\left(\\frac{\\partial U}{\\partial T}\\right)_V + \\left[\\left(\\frac{\\partial U}{\\partial V}\\right)_T + P\\right]\\frac{dV}{dT}.\n\n\\end{array}\n\n<\/span>\n<p>Cette derni\u00e8re relation est valable pour tout changement de temp\u00e9rature et de volume.<\/p>\n<p>\u00c0 partir de ces r\u00e9sultats, nous pouvons d\u00e9terminer la quantit\u00e9 de chaleur \u00e0 ajouter pour produire une variation de temp\u00e9rature sous certaines contraintes.<\/p>\n<h3>Contraintes \u00e0 volume constant<\/h3>\n<p>Pour examiner ce qui se passe \u00e0 volume constant, rappelons que la d\u00e9finition de la capacit\u00e9 calorifique \u00e0 volume constant est <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C_V=(\\partial Q\/ \\partial T)_V<\/span><\/span>. Ensuite, si nous nous limitons \u00e0 maintenir un volume constant dans l&#8217;analyse pr\u00e9c\u00e9dente, le terme <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dV\/dT<\/span><\/span> dans l&#8217;expression de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta Q\/dT<\/span><\/span> dispara\u00eet. Cela nous permet d&#8217;\u00e9crire :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle  C_V = \\left(\\frac{\\partial U}{\\partial T} \\right)_V<\/span><\/span>.<\/p>\n<h3>Contraintes \u00e0 pression constante<\/h3>\n<p>Si nous maintenons la pression constante, nous obtenons :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle C_p =\\left(\\frac{\\partial Q}{\\partial T}\\right)_P=\\left(\\frac{\\partial U}{\\partial T}\\right)_V + \\left[\\left(\\frac{\\partial U}{\\partial V}\\right)_T + P\\right]\\left(\\frac{\\partial V}{\\partial T}\\right)_p<\/span><\/span>.<\/p>\n<h3>Capacit\u00e9 calorifique d&#8217;un gaz monoatomique<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=T6K1Nizc5NE&amp;t=974s\" target=\"_blank\" rel=\"noopener\"><strong>Lorsqu&#8217;on consid\u00e8re<\/strong><\/a> un gaz monoatomique, l&#8217;\u00e9nergie interne due \u00e0 l&#8217;\u00e9nergie cin\u00e9tique de ses particules est donn\u00e9e par l&#8217;expression <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n U=\\frac{3}{2}Nk_BT<\/span>. Ce r\u00e9sultat est justifi\u00e9 par le principe de r\u00e9partition de l&#8217;\u00e9nergie, qui peut \u00eatre \u00e9tudi\u00e9 \u00e0 partir d&#8217;une approche statistique du mouvement des particules.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Distribution de Maxwell-Boltzmann et la r\u00e9partition de l&#8217;\u00e9nergie<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=T6K1Nizc5NE&amp;t=1027s\" target=\"_blank\" rel=\"noopener\"><strong>\u00c9tant donn\u00e9 que l&#8217;\u00e9nergie d&#8217;un syst\u00e8me<\/strong><\/a> est proportionnelle \u00e0 son <strong>Facteur de Boltzmann<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">e^{-E\/(k_BT)}<\/span><\/span>. En raisonnant \u00e0 partir de ce principe et en consid\u00e9rant que l&#8217;\u00e9nergie cin\u00e9tique des particules prend la forme <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle  E_{cin}=\\frac{1}{2}mv^2<\/span><\/span>, nous pouvons en d\u00e9duire que l&#8217;\u00e9nergie associ\u00e9e au mouvement des particules du syst\u00e8me projet\u00e9e sur l&#8217;un des trois axes de coordonn\u00e9es (focalisons-nous pour le moment sur l&#8217;axe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span><\/span>) suit une distribution des vitesses <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(v_x)<\/span><\/span> proportionnelle \u00e0 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">e^{-mv_x^2\/(2k_BT)}<\/span><\/span>. Autrement dit :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(v_x)= A e^{-mv_x^2\/(2k_BT)}<\/span><\/span>,<\/p>\n<p>o\u00f9 <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> est une constante \u00e0 d\u00e9terminer. Maintenant, \u00e9tant donn\u00e9 que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(v_x)<\/span><\/span> est une fonction de distribution, elle doit \u00eatre normalis\u00e9e de mani\u00e8re \u00e0 v\u00e9rifier :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\int_{-\\infty}^{+\\infty} g(v_x)dv_x= 1<\/span><\/span>.<\/p>\n<p>Un r\u00e9sultat utile pour cette analyse est l&#8217;int\u00e9grale gaussienne :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\int_{-\\infty}^{+\\infty} e^{-x^2}dx= \\sqrt{\\pi}<\/span><\/span>.<\/p>\n<p>En utilisant cela, on peut d\u00e9duire :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle 1= \\int_{-\\infty}^{+\\infty} Ae^{\\frac{-mv_x^2}{2k_BT}}dv_x= A\\sqrt{\\frac{\\pi}{m\/(2k_BT)}} = A\\sqrt{\\frac{2\\pi k_BT}{m}}<\/span><\/span>.<\/p>\n<p>Et donc :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle  g(v_x) = \\sqrt{\\frac{m}{2\\pi k_BT}}e^{-mv_x^2\/(2k_BT)}<\/span><\/span>.<\/p>\n<p>Avec cette formule, il est d\u00e9sormais possible de calculer la vitesse moyenne projet\u00e9e sur l&#8217;axe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span><\/span>, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left\\langle v_x^2\\right\\rangle<\/span><\/span>. Le r\u00e9sultat est :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left\\langle v_x^2\\right\\rangle = \\int_{-\\infty}^{+\\infty} v_x^2 g(v_x) dv_x = \\sqrt{\\frac{m}{2\\pi k_BT}} \\int_{-\\infty}^{+\\infty} v_x^2 e^{-mv_x^2\/(2k_BT)} = \\frac{k_BT}{m} <\/span><\/span>.<\/p>\n<p>Comme la vitesse quadratique moyenne peut \u00eatre d\u00e9compos\u00e9e sous la forme <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle  \\left\\langle v^2\\right\\rangle = \\left\\langle v_x^2\\right\\rangle + \\left\\langle v_y^2\\right\\rangle + \\left\\langle v_z^2\\right\\rangle<\/span><\/span>, et que chacun de ces composants suit un d\u00e9veloppement identique, on peut \u00e9crire l&#8217;\u00e9nergie cin\u00e9tique moyenne du syst\u00e8me de particules de la mani\u00e8re suivante :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle  \\left\\langle E_{cin}\\right\\rangle =\\frac{1}{2}m\\left\\langle v^2\\right\\rangle  = \\frac{1}{2}m \\cdot 3\\frac{k_BT}{m}= \\frac{3}{2}k_BT<\/span><\/span>.<\/p>\n<p>C&#8217;est ce qu&#8217;on appelle le \u00abprincipe de r\u00e9partition de l&#8217;\u00e9nergie\u00bb. En cons\u00e9quence, si le syst\u00e8me est compos\u00e9 de <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> particules avec une \u00e9nergie cin\u00e9tique moyenne <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left\\langle E_{cin}\\right\\rangle<\/span><\/span> et que l&#8217;\u00e9nergie totale du syst\u00e8me est uniquement d&#8217;origine cin\u00e9tique, alors non seulement l&#8217;\u00e9nergie interne du syst\u00e8me sera <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n U=3Nk_BT\/2<\/span> (comme cela a \u00e9t\u00e9 pr\u00e9dit), mais il est \u00e9galement clair que l&#8217;\u00e9nergie interne d\u00e9pend uniquement de la temp\u00e9rature du syst\u00e8me, ce qui signifie :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\frac{\\partial U}{\\partial V}\\right)_T = 0<\/span><\/span>.<\/p>\n<h3>D\u00e9veloppement pour le gaz id\u00e9al<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=T6K1Nizc5NE&amp;t=1027s\" target=\"_blank\" rel=\"noopener\"><strong>En rappelant l&#8217;\u00e9quation des gaz parfaits<\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV=Nk_BT =nRT<\/span><\/span>, on obtient pour le volume :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle  V= \\frac{nRT}{P}<\/span><\/span>.<\/p>\n<p>Par cons\u00e9quent, nous avons :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle  \\left(\\frac{\\partial V}{\\partial T} \\right)_P = \\frac{nR}{P}<\/span><\/span>.<\/p>\n<p>En appliquant cela aux expressions de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C_V<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C_P<\/span><\/span>, nous observons que :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\nC_P - C_V &amp; \\displaystyle = \\left[\\left(\\frac{\\partial U}{\\partial V} \\right)_T + P \\right]\\left(\\frac{\\partial V}{\\partial T} \\right)_P = P\\cdot \\frac{nR}{P} = nR\n\n\\end{array}<\/span>\n<p>.<\/p>\n<p>Maintenant, comme <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle  C_V=(\\partial U \/ \\partial T)_V<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">U=3Nk_BT\/2=3nRT\/2<\/span><\/span>, nous obtenons :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\displaystyle  C_V = \\frac{3}{2}nR\n\n<\/span>\n<p>.<\/p>\n<p>Par cons\u00e9quent :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\nC_P = C_V + nR = \\displaystyle  \\frac{3}{2}nR + nR = \\frac{5}{2}nR\n\n<\/span>\n<p>.<\/p>\n<h3>L&#8217;indice adiabatique<\/h3>\n<p>Une grandeur fr\u00e9quemment utilis\u00e9e est le rapport entre <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C_P<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C_V<\/span><\/span>, qui porte un nom particulier. On d\u00e9finit l&#8217;<strong>indice adiabatique<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span> par :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\gamma = \\displaystyle  \\frac{C_P}{C_V}\n\n<\/span>\n<p>.<\/p>\n<p>Dans le cas des gaz parfaits, l&#8217;indice adiabatique prend une valeur exacte :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\gamma = \\displaystyle \\frac{5}{3}\n\n<\/span>\n<p>.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Exercices<\/h2>\n<ol>\n<li>Est-il toujours vrai que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dU=C_VdT<\/span><\/span>? Comparez le cas g\u00e9n\u00e9ral avec celui des gaz parfaits et justifiez votre r\u00e9ponse.<\/li>\n<li>En supposant que pour un gaz parfait <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">U=C_VT<\/span><\/span> est v\u00e9rifi\u00e9, calculez :\n<ul>\n<li>(i) L&#8217;\u00e9nergie interne par unit\u00e9 de masse<\/li>\n<li>(ii) L&#8217;\u00e9nergie interne par unit\u00e9 de volume.<\/li>\n<\/ul>\n<\/li>\n<li>Une mole de gaz parfait monoatomique est confin\u00e9e dans un cylindre par un piston et maintenue \u00e0 temp\u00e9rature constante <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_0<\/span><\/span> gr\u00e2ce \u00e0 un contact avec une r\u00e9serve thermique. Le gaz est lentement \u00e9tendu d&#8217;un volume <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V_1<\/span><\/span> \u00e0 un autre volume <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V_2<\/span><\/span>, la temp\u00e9rature restant constante \u00e0 tout moment.\n<ul>\n<li>(i) L&#8217;\u00e9nergie interne du gaz change-t-elle ?<\/li>\n<li>(ii) Calculez le travail effectu\u00e9 par le gaz et le flux de chaleur vers le gaz.<\/li>\n<\/ul>\n<\/li>\n<li>D\u00e9montrez que, pour un gaz parfait, les relations suivantes sont satisfaites :\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{R}{c_V} = \\gamma-1<\/span><\/span><\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{R}{c_P} = \\frac{\\gamma -1}{\\gamma}<\/span><\/span><\/p>\n<p>o\u00f9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c_V<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c_P<\/span><\/span> sont les capacit\u00e9s calorifiques molaires.<\/p>\n<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>La Premi\u00e8re Loi de la Thermodynamique La Premi\u00e8re Loi de la Thermodynamique est le fondement qui relie des concepts fondamentaux tels que la chaleur, le travail et l&#8217;\u00e9nergie interne, \u00e9tablissant que l&#8217;\u00e9nergie ne peut \u00eatre ni cr\u00e9\u00e9e ni d\u00e9truite, mais seulement transform\u00e9e. Ce document explore comment cette loi s&#8217;applique aux syst\u00e8mes ferm\u00e9s, approfondissant l&#8217;analyse du [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":30854,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":6,"footnotes":""},"categories":[647,931],"tags":[],"class_list":["post-30874","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>La Premi\u00e8re Loi de la Thermodynamique - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"\u0423\u0437\u043d\u0430\u0439\u0442\u0435, \u043a\u0430\u043a \u043f\u0435\u0440\u0432\u044b\u0439 \u0437\u0430\u043a\u043e\u043d \u0442\u0435\u0440\u043c\u043e\u0434\u0438\u043d\u0430\u043c\u0438\u043a\u0438 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