{"id":30075,"date":"2021-03-10T13:00:06","date_gmt":"2021-03-10T13:00:06","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=30075"},"modified":"2025-10-20T07:34:38","modified_gmt":"2025-10-20T07:34:38","slug":"lequation-des-gaz-parfaits","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/lequation-des-gaz-parfaits\/","title":{"rendered":"L&#8217;\u00c9quation des Gaz Parfaits"},"content":{"rendered":"<style>\np, ul, ol{\n  text-align: justify;\n}\nh1{\n  text-align:center;\n  text-transform: uppercase;\n}\nh2{\n  text-align:center;\n  text-transform: uppercase;\n  font-size:24pt;\n}\nh3 { \n  text-align: center;\n  text-transform: uppercase;\n  font-size: 24px !important;\n}\n.example{\n  background:#f6f8fa; \n  border-left:4px solid #d00000; \n  padding:12px 14px; \n  margin:14px 0;\n}\n.small{\n  font-size: 0.95em;\n  color:#333;\n}\n<\/style>\n<h1>Formulation empirique du gaz id\u00e9al<\/h1>\n<p style=\"text-align:center;\" dir=\"ltr\">T\u2019es-tu d\u00e9j\u00e0 demand\u00e9 pourquoi un ballon se dilate lorsqu\u2019on le chauffe ou pourquoi la pression d\u2019un pneu varie lorsqu\u2019on change d\u2019altitude ? Dans cette le\u00e7on, nous examinerons les lois qui r\u00e9gissent ces comportements et comment elles conduisent \u00e0 l\u2019\u00e9quation des gaz parfaits, ainsi qu\u2019\u00e0 ses consid\u00e9rations et points importants.<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><b>Objectifs d\u2019apprentissage<\/b><br \/>\n\u00c0 la fin de cette le\u00e7on, l\u2019\u00e9tudiant sera capable de :<\/p>\n<ol>\n<li><b>Expliquer<\/b> les lois empiriques des gaz parfaits (Boyle\u2013Mariotte, Charles, Gay-Lussac) et leur synth\u00e8se dans l\u2019\u00e9quation d\u2019\u00e9tat (<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV = nRT<\/span><\/span>, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV = N k_B T<\/span><\/span>).<\/li>\n<li><b>Appliquer<\/b> l\u2019\u00e9quation des gaz parfaits et la relation <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV\/T = cte<\/span><\/span> pour r\u00e9soudre des changements d\u2019\u00e9tat avec des unit\u00e9s coh\u00e9rentes.<\/li>\n<li><b>Analyser<\/b> les processus isothermes, isobares et isochoriques et leurs trajectoires dans les diagrammes <i>P\u2013V<\/i>, <i>V\u2013T<\/i> et <i>P\u2013T<\/i>.<\/li>\n<li><b>Reconna\u00eetre<\/b> le domaine de validit\u00e9 du gaz parfait et choisir des mod\u00e8les alternatifs (van der Waals, quantique, relativiste) lorsque cela est appropri\u00e9.<\/li>\n<\/ol>\n<p style=\"text-align:center;\" dir=\"ltr\">\n<b>TABLE DES MATI\u00c8RES<\/b><br \/>\n<a href=\"#1\">Lois empiriques fondamentales<\/a><br \/>\n<a href=\"#2\">Combinaison des lois dans l\u2019\u00e9quation des gaz parfaits<\/a><br \/>\n<a href=\"#3\">D\u00e9monstrations par processus<\/a><br \/>\n<a href=\"#4\">Commentaires et contexte microscopique<\/a><br \/>\n<a href=\"#5\">Domaine de validit\u00e9 et limitations<\/a><br \/>\n<a href=\"#6\">Notes pratiques<\/a>\n<\/p>\n<p><center><br \/>\n <iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/7WkrH_FS290?si=xWJQ-VAtbWgzm9bQ\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><br \/>\n<\/center><br \/>\n<a name=\"1\"><\/a><\/p>\n<h2>Lois empiriques fondamentales<\/h2>\n<p>Les exp\u00e9riences sur les gaz montrent une d\u00e9pendance entre la pression <span class=\"katex-eq\" data-katex-display=\"false\">P<\/span>, le volume <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span> et la temp\u00e9rature <span class=\"katex-eq\" data-katex-display=\"false\">T<\/span>. Dans des conditions contr\u00f4l\u00e9es, trois lois empiriques fondamentales sont observ\u00e9es :<\/p>\n<ol>\n<li><strong>Loi de Boyle\u2013Mariotte (isotherme) :<\/strong> Dans un processus \u00e0 temp\u00e9rature constante, le volume et la pression d\u2019un gaz sont inversement proportionnels ; c\u2019est-\u00e0-dire :\n<div style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P \\propto \\dfrac{1}{V}\\quad\\Leftrightarrow\\quad PV=\\text{cte.}<\/span><\/div>\n<div class=\"example\">\n  <strong>Exemple :<\/strong> Un gaz qui, \u00e0 une pression initiale de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P_1 = 15\\ \\mathrm{MPa}<\/span><\/span>, se d\u00e9tend isotermiquement d\u2019un volume initial <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V_1 = 1{,}00\\ \\mathrm{L}<\/span><\/span> jusqu\u2019\u00e0 un volume final <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V_2 = 2{,}00\\ \\mathrm{L}<\/span><\/span> verra sa pression r\u00e9duite de moiti\u00e9. Comme le produit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV=\\text{cte.}<\/span><\/span>, on a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P_1 V_1 = P_2 V_2<\/span><\/span>, ce qui conduit \u00e0 :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n  P_2 = \\dfrac{P_1 V_1}{V_2}\n\n      = 15\\ \\mathrm{MPa}\\left(\\dfrac{1{,}00\\ \\mathrm{L}}{2{,}00\\ \\mathrm{L}}\\right)\n\n      = 7{,}50\\ \\mathrm{MPa}\n\n  <\/span>\n<p>  <center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/pv-isotermico.jpg\" alt=\"Diagramme PV pour un processus isotherme\" width=\"480\" height=\"293\" class=\"aligncenter size-full wp-image-34975 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/pv-isotermico.jpg\" alt=\"Diagramme PV pour un processus isotherme\" width=\"480\" height=\"293\" class=\"aligncenter size-full wp-image-34975 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/pv-isotermico.jpg 480w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/pv-isotermico-300x183.jpg 300w\" sizes=\"(max-width: 480px) 100vw, 480px\" \/><\/noscript><\/center>\n<\/div>\n<\/li>\n<li><strong>Loi de Charles (isobare) :<\/strong> Dans un processus \u00e0 pression constante, le volume et la temp\u00e9rature d\u2019un gaz sont directement proportionnels ; c\u2019est-\u00e0-dire :\n<div style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V \\propto T \\quad\\Leftrightarrow\\quad \\dfrac{V}{T}=\\text{cte.}<\/span><\/div>\n<div class=\"example\">\n  <strong>Exemple :<\/strong> Un gaz qui, \u00e0 une temp\u00e9rature initiale de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_1 = 300\\ \\mathrm{K}<\/span><\/span>, est chauff\u00e9 isobariquement jusqu\u2019\u00e0 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_2 = 450\\ \\mathrm{K}<\/span><\/span> \u00e0 partir d\u2019un volume initial <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V_1 = 2{,}00\\ \\mathrm{L}<\/span><\/span> verra son volume augmenter de 50&nbsp;% (un facteur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tfrac{3}{2}<\/span><\/span>). Comme <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tfrac{V}{T}=\\text{cte.}<\/span><\/span>, on a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{V_1}{T_1}=\\dfrac{V_2}{T_2}<\/span><\/span>, ce qui conduit \u00e0 :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n  V_2 = V_1 \\cdot \\dfrac{T_2}{T_1}\n\n      = 2{,}00\\ \\mathrm{L}\\left(\\dfrac{450\\ \\mathrm{K}}{300\\ \\mathrm{K}}\\right)\n\n      = 3{,}00\\ \\mathrm{L}\n\n  <\/span>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/isobara-proc.jpg\" alt=\"\" width=\"480\" height=\"298\" class=\"aligncenter size-full wp-image-34984 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/isobara-proc.jpg\" alt=\"\" width=\"480\" height=\"298\" class=\"aligncenter size-full wp-image-34984 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/isobara-proc.jpg 480w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/isobara-proc-300x186.jpg 300w\" sizes=\"(max-width: 480px) 100vw, 480px\" \/><\/noscript><\/center><\/p>\n<\/div>\n<\/li>\n<li><strong>Loi de Gay-Lussac (isochore) :<\/strong> Dans un processus \u00e0 volume constant, la pression et la temp\u00e9rature d\u2019un gaz sont directement proportionnelles ; c\u2019est-\u00e0-dire :\n<div style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P \\propto T \\quad\\Leftrightarrow\\quad \\dfrac{P}{T}=\\text{cte.}<\/span><\/div>\n<div class=\"example\">\n  <strong>Exemple :<\/strong> Un gaz qui, \u00e0 une temp\u00e9rature initiale de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_1 = 300\\ \\mathrm{K}<\/span><\/span>, est chauff\u00e9 isochoriquement jusqu\u2019\u00e0 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_2 = 450\\ \\mathrm{K}<\/span><\/span> \u00e0 partir d\u2019une pression initiale <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P_1 = 1{,}00\\ \\mathrm{MPa}<\/span><\/span> verra sa pression augmenter dans la m\u00eame proportion. Comme <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tfrac{P}{T}=\\text{cte.}<\/span><\/span>, on a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{P_1}{T_1}=\\dfrac{P_2}{T_2}<\/span><\/span>, ce qui conduit \u00e0 :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n  P_2 = P_1 \\cdot \\dfrac{T_2}{T_1}\n\n      = 1{,}00\\ \\mathrm{MPa}\\left(\\dfrac{450\\ \\mathrm{K}}{300\\ \\mathrm{K}}\\right)\n\n      = 1{,}50\\ \\mathrm{MPa}\n\n  <\/span>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/icororico-proc.jpg\" alt=\"\" width=\"480\" height=\"291\" class=\"aligncenter size-full wp-image-34987 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/icororico-proc.jpg\" alt=\"\" width=\"480\" height=\"291\" class=\"aligncenter size-full wp-image-34987 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/icororico-proc.jpg 480w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/icororico-proc-300x182.jpg 300w\" sizes=\"(max-width: 480px) 100vw, 480px\" \/><\/noscript><\/center>\n<\/div>\n<\/li>\n<\/ol>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Combinaison des lois dans l\u2019\u00e9quation des gaz parfaits<\/h2>\n<p>Ces trois lois peuvent \u00eatre synth\u00e9tis\u00e9es en une seule relation de proportionnalit\u00e9 :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV \\propto T<\/span>\n<p>\u00c0 partir de consid\u00e9rations exp\u00e9rimentales et microscopiques, il est possible de d\u00e9duire la constante de proportionnalit\u00e9 comme le produit entre le nombre de particules <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> et la constante de Boltzmann <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k_B = 1{,}380\\,649\\times10^{-23}\\ \\mathrm{J\\,K^{-1}}<\/span><\/span>, obtenant ainsi la relation microscopique :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{PV = N\\,k_B\\,T}<\/span>\n<p>De mani\u00e8re analogue, en termes molaires, la constante de proportionnalit\u00e9 est obtenue comme le produit entre le nombre de moles <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> et la constante universelle des gaz <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">R=8{,}314\\,462\\,6\\ \\mathrm{J\\,mol^{-1}\\,K^{-1}}=0{,}082\\,057\\ \\mathrm{L\\,atm\\,mol^{-1}\\,K^{-1}}<\/span><\/span>, conduisant \u00e0 la relation molaire :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{PV = n\\,R\\,T}<\/span>\n<p>Quelle que soit la situation, il existe une relation de proportionnalit\u00e9 directe entre le produit <span class=\"katex-eq\" data-katex-display=\"false\">PV<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">T<\/span>, ce qui revient \u00e0 dire que si un gaz parfait passe entre deux \u00e9tats, l\u2019un ayant les valeurs initiales <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\alpha, V_\\alpha, T_\\alpha)<\/span><\/span> et l\u2019autre les valeurs finales <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\omega, V_\\omega, T_\\omega)<\/span><\/span>, alors ils satisfont la relation<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{P_\\alpha V_\\alpha}{T_\\alpha} =  \\dfrac{P_\\omega V_\\omega}{T_\\omega}<\/span>\n<p>et, par cons\u00e9quent, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV\/T = cte.<\/span><\/span><\/p>\n<p>Cette relation, qui peut \u00eatre utilis\u00e9e comme base exp\u00e9rimentale pour formuler \u00e0 la fois la relation microscopique et molaire, peut \u00eatre d\u00e9duite directement \u00e0 partir des lois exp\u00e9rimentales de Boyle\u2013Mariotte, de Charles et de Gay-Lussac. Le raisonnement est le suivant :<\/p>\n<p>Nous pouvons le d\u00e9montrer de trois mani\u00e8res :<\/p>\n<ol>\n<li>Une variation de <b>volume<\/b> \u00e0 travers un processus isotherme et un processus isobare<\/li>\n<li>Une variation de <b>pression<\/b> \u00e0 travers un processus isotherme et un processus isochore<\/li>\n<li>Une variation de <b>temp\u00e9rature<\/b> \u00e0 travers un processus isobare et un processus isochore<\/li>\n<\/ol>\n<p>Pour le d\u00e9veloppement de ces trois cas, nous aurons besoin d\u2019un \u00e9tat interm\u00e9diaire avec des valeurs <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_i,V_i,T_i)<\/span><\/span><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>D\u00e9monstrations par processus<\/h2>\n<h3>D\u00e9monstration par variation de volume<\/h3>\n<p>Si l\u2019\u00e9tat initial <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\alpha,V_\\alpha,T_\\alpha)<\/span><\/span> est reli\u00e9 \u00e0 l\u2019\u00e9tat interm\u00e9diaire <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_i,V_i,T_i)<\/span><\/span> par un processus isotherme, et que cet \u00e9tat interm\u00e9diaire est ensuite reli\u00e9 \u00e0 l\u2019\u00e9tat final <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\omega,V_\\omega,T_\\omega)<\/span><\/span> par un processus isobare, alors on a :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rclcl}\n\n &amp; P_\\alpha V_\\alpha= P_i V_i &amp; &amp; V_i\/T_i = V_\\omega\/T_\\omega   &amp; \\\\\n\n &amp;\\text{isotherme}&amp; &amp;\\text{isobare} &amp; \\\\\n\nP_\\alpha &amp; \\longrightarrow &amp; P_i = \\dfrac{P_\\alpha V_\\alpha}{V_i}  &amp; \\longrightarrow &amp; P_\\omega = P_i \\\\ \\\\\n\nV_\\alpha &amp; \\longrightarrow &amp; V_i = \\dfrac{P_\\alpha V_\\alpha}{P_i} &amp; \\longrightarrow &amp; V_\\omega = \\dfrac{V_i T_\\omega}{T_i} \\\\ \\\\\n\nT_\\alpha &amp; \\longrightarrow &amp; T_i = T_\\alpha &amp; \\longrightarrow &amp; T_\\omega = \\dfrac{V_\\omega T_i}{V_i}\n\n\\end{array}\n\n<\/span>\n<p>\u00c0 partir de cela, on obtient :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n&amp; V_\\omega = \\left(\\dfrac{T_\\omega}{T_i}\\right) V_i = \\left(\\dfrac{T_\\omega}{T_i}\\right) \\left(\\dfrac{P_\\alpha}{P_i} \\right) V_\\alpha = \\dfrac{T_\\omega P_\\alpha V_\\alpha}{T_\\alpha P_\\omega} \\\\ \\\\\n\n\\equiv &amp; \\dfrac{P_\\alpha V_\\alpha}{T_\\alpha} = \\dfrac{P_\\omega V_\\omega}{T_\\omega}\n\n\\end{array}<\/span>\n<h3>D\u00e9monstration par variation de pression<\/h3>\n<p>Si l\u2019\u00e9tat initial <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\alpha,V_\\alpha,T_\\alpha)<\/span><\/span> est reli\u00e9 \u00e0 l\u2019\u00e9tat interm\u00e9diaire <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_i,V_i,T_i)<\/span><\/span> par un processus isotherme, et que cet \u00e9tat interm\u00e9diaire est ensuite reli\u00e9 \u00e0 l\u2019\u00e9tat final <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\omega,V_\\omega,T_\\omega)<\/span><\/span> par un processus isochore, alors on a :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rclcl}\n\n &amp; P_\\alpha V_\\alpha= P_i V_i &amp; &amp; P_i\/T_i = P_\\omega\/T_\\omega   &amp; \\\\\n\n &amp;\\text{isotherme}&amp; &amp;\\text{isochore} &amp; \\\\\n\nP_\\alpha &amp; \\longrightarrow &amp; P_i = \\dfrac{P_\\alpha V_\\alpha}{V_i}  &amp; \\longrightarrow &amp; P_\\omega = \\dfrac{P_i T_\\omega}{T_i} \\\\ \\\\\n\nV_\\alpha &amp; \\longrightarrow &amp; V_i = \\dfrac{P_\\alpha V_\\alpha}{P_i} &amp; \\longrightarrow &amp; V_\\omega = V_i \\\\ \\\\\n\nT_\\alpha &amp; \\longrightarrow &amp; T_i = T_\\alpha &amp; \\longrightarrow &amp; T_\\omega = \\dfrac{V_\\omega T_i}{V_i}\n\n\\end{array}\n\n<\/span>\n<p>\u00c0 partir de cela, on obtient :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n &amp; P_\\omega = \\left(\\dfrac{T_\\omega}{T_i}\\right) P_i = \\left(\\dfrac{T_\\omega}{T_i}\\right) \\left(\\dfrac{V_\\alpha}{V_i}\\right)P_\\alpha = \\dfrac{T_\\omega V_\\alpha P_\\alpha}{T_\\alpha V_\\omega} \\\\ \\\\\n\n\\equiv &amp; \\dfrac{P_\\alpha V_\\alpha}{T_\\alpha} = \\dfrac{P_\\omega V_\\omega}{T_\\omega}\n\n\\end{array}<\/span>\n<h3>D\u00e9monstration par variation de temp\u00e9rature<\/h3>\n<p>Si l\u2019\u00e9tat initial <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\alpha,V_\\alpha,T_\\alpha)<\/span><\/span> est reli\u00e9 \u00e0 l\u2019\u00e9tat interm\u00e9diaire <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_i,V_i,T_i)<\/span><\/span> par un processus isobare, et que cet \u00e9tat interm\u00e9diaire est ensuite reli\u00e9 \u00e0 l\u2019\u00e9tat final <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\omega,V_\\omega,T_\\omega)<\/span><\/span> par un processus isochore, alors on a :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rclcl}\n\n &amp; V_\\alpha\/ T_\\alpha= V_i \/ T_i &amp; &amp; P_i\/T_i = P_\\omega\/T_\\omega   &amp; \\\\\n\n &amp;\\text{isobare}&amp; &amp;\\text{isochore} &amp; \\\\\n\nP_\\alpha &amp; \\longrightarrow &amp; P_i = P_\\alpha  &amp; \\longrightarrow &amp; P_\\omega = \\dfrac{P_i T_\\omega}{T_i} \\\\ \\\\\n\nV_\\alpha &amp; \\longrightarrow &amp; V_i = \\dfrac{V_\\alpha T_i}{T_\\alpha} &amp; \\longrightarrow &amp; V_\\omega = V_i \\\\ \\\\\n\nT_\\alpha &amp; \\longrightarrow &amp; T_i = \\dfrac{V_i T_\\alpha}{V_\\alpha} &amp; \\longrightarrow &amp; T_\\omega = \\dfrac{P_\\omega T_i}{P_i}\n\n\\end{array}\n\n<\/span>\n<p>\u00c0 partir de cela, on obtient :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n &amp; T_\\omega = \\left(\\dfrac{P_\\omega}{P_i}\\right) T_i = \\left(\\dfrac{P_\\omega}{P_i}\\right) \\left(\\dfrac{V_i}{V_\\alpha}\\right)T_\\alpha = \\dfrac{P_\\omega V_\\omega T_\\alpha}{P_\\alpha V_\\alpha}  \\\\ \\\\\n\n\\equiv &amp; \\dfrac{P_\\alpha V_\\alpha}{T_\\alpha} = \\dfrac{P_\\omega V_\\omega}{T_\\omega}\n\n\\end{array}<\/span>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Commentaires et contexte microscopique<\/h2>\n<p>Bien que la formulation pr\u00e9c\u00e9dente soit empirique, elle peut \u00eatre d\u00e9riv\u00e9e \u00e0 partir des premiers principes gr\u00e2ce \u00e0 la Th\u00e9orie Cin\u00e9tique des Gaz. Dans ce mod\u00e8le, le gaz est une collection de particules qui se d\u00e9placent et entrent en collision entre elles et avec les parois du r\u00e9cipient. Il est id\u00e9alis\u00e9 \u00e0 l\u2019aide d\u2019hypoth\u00e8ses telles que :<\/p>\n<ol>\n<li>Absence de forces d\u2019attraction ou de r\u00e9pulsion \u00e0 distance entre les particules.<\/li>\n<li>Particules ponctuelles ou de taille n\u00e9gligeable, de forme sph\u00e9rique.<\/li>\n<li>Collisions parfaitement \u00e9lastiques entre les particules et avec les parois.<\/li>\n<\/ol>\n<p>Ces id\u00e9alisations simplifient l\u2019analyse et, bien qu\u2019aucun gaz r\u00e9el ne les respecte exactement, elles d\u00e9crivent correctement de nombreux gaz sur une large gamme de conditions et constituent une base pour la <strong>Thermodynamique Classique<\/strong>, avec des applications allant des moteurs thermiques \u00e0 la physique atmosph\u00e9rique et \u00e0 l\u2019astrophysique.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Domaine de validit\u00e9 et limitations<\/h2>\n<p>La loi des gaz parfaits n\u2019est pas universelle. Elle diverge lorsque les hypoth\u00e8ses pr\u00e9c\u00e9dentes cessent d\u2019\u00eatre raisonnables ou lorsque des effets hors du cadre de la physique classique apparaissent.<\/p>\n<ul>\n<li><strong>Hautes pressions et basses temp\u00e9ratures :<\/strong> les interactions entre mol\u00e9cules ne sont plus n\u00e9gligeables et la taille finie des particules devient importante. Une correction courante est l\u2019\u00e9quation de van der Waals :\n<div style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(P + a\\left(\\dfrac{n}{V}\\right)^2\\right)\\,(V - nb)=nRT<\/span><\/div>\n<p>    avec des param\u00e8tres <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> caract\u00e9ristiques de chaque gaz.\n  <\/li>\n<li><strong>R\u00e9gime quantique :<\/strong> \u00e0 tr\u00e8s basse temp\u00e9rature ou \u00e0 densit\u00e9 \u00e9lev\u00e9e, apparaissent les statistiques de Bose\u2013Einstein ou de Fermi\u2013Dirac, n\u00e9cessitant des mod\u00e8les de <em>gaz quantiques<\/em>.<\/li>\n<li><strong>R\u00e9gime relativiste :<\/strong> si les particules se d\u00e9placent \u00e0 des vitesses proches de celle de la lumi\u00e8re, des corrections relativistes sont n\u00e9cessaires.<\/li>\n<\/ul>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Notes pratiques<\/h2>\n<ul>\n<li>Utilise toujours la temp\u00e9rature en <strong>Kelvin<\/strong> dans les formules : <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T(\\mathrm{K}) = T(^{\\circ}\\mathrm{C}) + 273{,}15<\/span><\/span>.<\/li>\n<li>Assure-toi de la coh\u00e9rence des unit\u00e9s : si tu travailles avec <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathrm{atm}<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathrm{L}<\/span><\/span>, utilise <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">R=0{,}082\\,057\\ \\mathrm{L\\,atm\\,mol^{-1}\\,K^{-1}}<\/span><\/span> ; si tu utilises <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathrm{Pa}<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathrm{m^3}<\/span><\/span>, emploie <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">R=8{,}314\\,462\\,6\\ \\mathrm{J\\,mol^{-1}\\,K^{-1}}<\/span><\/span>.<\/li>\n<li>Souviens-toi que chaque loi empirique a \u00e9t\u00e9 obtenue en maintenant une variable fixe. Combiner les r\u00e9sultats n\u00e9cessite de bien comprendre quel processus thermodynamique est r\u00e9alis\u00e9 \u00e0 chaque \u00e9tape.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Formulation empirique du gaz id\u00e9al T\u2019es-tu d\u00e9j\u00e0 demand\u00e9 pourquoi un ballon se dilate lorsqu\u2019on le chauffe ou pourquoi la pression d\u2019un pneu varie lorsqu\u2019on change d\u2019altitude ? Dans cette le\u00e7on, nous examinerons les lois qui r\u00e9gissent ces comportements et comment elles conduisent \u00e0 l\u2019\u00e9quation des gaz parfaits, ainsi qu\u2019\u00e0 ses consid\u00e9rations et points importants. Objectifs [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":35110,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":17,"footnotes":""},"categories":[647,931],"tags":[],"class_list":["post-30075","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 v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>L&#039;\u00c9quation des Gaz Parfaits - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"\ud83c\udf21\ufe0f D\u00e9couvrez l&#039;incroyable \u00e9quation des gaz parfaits : principes, applications en thermodynamique et ses limites \ud83d\ude80\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/fr\/lequation-des-gaz-parfaits\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"L&#039;\u00c9quation des Gaz Parfaits\" \/>\n<meta property=\"og:description\" content=\"\ud83c\udf21\ufe0f D\u00e9couvrez l&#039;incroyable \u00e9quation des gaz parfaits : principes, applications en thermodynamique et ses limites \ud83d\ude80\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/fr\/lequation-des-gaz-parfaits\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-03-10T13:00:06+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-10-20T07:34:38+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/min3-7.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1536\" \/>\n\t<meta property=\"og:image:height\" content=\"1024\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"L&#039;\u00c9quation des Gaz Parfaits\" \/>\n<meta name=\"twitter:description\" content=\"\ud83c\udf21\ufe0f D\u00e9couvrez l&#039;incroyable \u00e9quation des gaz parfaits : principes, applications en thermodynamique et ses limites \ud83d\ude80\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/min3-7.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/lequation-des-gaz-parfaits\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/lequation-des-gaz-parfaits\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"L&#8217;\u00c9quation des Gaz Parfaits\",\"datePublished\":\"2021-03-10T13:00:06+00:00\",\"dateModified\":\"2025-10-20T07:34:38+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/lequation-des-gaz-parfaits\\\/\"},\"wordCount\":2024,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/lequation-des-gaz-parfaits\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2021\\\/03\\\/min3-7.jpg\",\"articleSection\":[\"Physique\",\"Thermodynamique\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/lequation-des-gaz-parfaits\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/lequation-des-gaz-parfaits\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/lequation-des-gaz-parfaits\\\/\",\"name\":\"L'\u00c9quation des Gaz Parfaits - 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