{"id":34620,"date":"2021-03-10T13:00:12","date_gmt":"2021-03-10T13:00:12","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34620"},"modified":"2025-10-20T07:36:22","modified_gmt":"2025-10-20T07:36:22","slug":"aequatio-gasorum-idealiumn","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/aequatio-gasorum-idealiumn\/","title":{"rendered":"Aequatio Gasorum Idealiumn"},"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>Formulatio empirica gasi idealis<\/h1>\n<p style=\"text-align:center;\" dir=\"ltr\">Numquamne quaesivisti cur globus calefactus dilatetur aut cur pressio rotae mutetur altitudine variata? In hac lectione leges recognoscemus quae hos mores regunt et quomodo haec ad aequationem gasorum idealium ducunt, eius considerationes et puncta momenti.<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><b>Proposita Discendi<\/b><br \/>\nIn fine huius lectionis discipulus poterit:<\/p>\n<ol>\n<li><b>Explicare<\/b> leges empiricas gasorum idealium (Boyle\u2013Mariotte, Charles, Gay-Lussac) et earum synthesen in aequatione status (<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>Applicare<\/b> aequationem gasorum idealium et relationem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV\/T = cte<\/span><\/span> ad solvendas mutationes status cum unitatibus coherentibus.<\/li>\n<li><b>Analysare<\/b> processus isothermos, isobaricos et isochoricos atque eorum trajectorias in diagrammatibus <i>P\u2013V<\/i>, <i>V\u2013T<\/i> et <i>P\u2013T<\/i>.<\/li>\n<li><b>Agnoscere<\/b> ambitum validitatis gasi idealis et eligere exempla alternativa (van der Waals, quanticum, relativisticum) cum opus sit.<\/li>\n<\/ol>\n<p style=\"text-align:center;\" dir=\"ltr\">\n<b>INDEX CONTENTORUM<\/b><br \/>\n<a href=\"#1\">Leges empiricae fundamentales<\/a><br \/>\n<a href=\"#2\">Combinatio legum in aequatione gasorum idealium<\/a><br \/>\n<a href=\"#3\">Deductiones per processus<\/a><br \/>\n<a href=\"#4\">Commentarii et fundamentum microscopicum<\/a><br \/>\n<a href=\"#5\">Ambitus validitatis et limitationes<\/a><br \/>\n<a href=\"#6\">Notae practicae<\/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>Leges empiricae fundamentales<\/h2>\n<p>Experimenta cum gasis ostendunt dependentiam inter pressionem <span class=\"katex-eq\" data-katex-display=\"false\">P<\/span>, volumen <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span> et temperaturam <span class=\"katex-eq\" data-katex-display=\"false\">T<\/span>. In condicionibus moderatis tres leges empiricae fundamentales observantur:<\/p>\n<ol>\n<li><strong>Lex Boyle\u2013Mariotte (isothermica):<\/strong> In processu ad temperaturam constantem, volumen et pressio gasi sunt proportionalia inverse; id est:\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>Exemplum:<\/strong> Gas quod, ad pressionem initialem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P_1 = 15\\ \\mathrm{MPa}<\/span><\/span>, isothermice expanditur ab volumine initiali <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V_1 = 1{,}00\\ \\mathrm{L}<\/span><\/span> usque ad volumen finale <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V_2 = 2{,}00\\ \\mathrm{L}<\/span><\/span> pressionem suam ad dimidium redigi videbit. Cum productum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV=\\text{cte.}<\/span><\/span>, valet <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P_1 V_1 = P_2 V_2<\/span><\/span>, quod ducit ad:<\/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=\"Diagramma PV pro processu isothermico\" 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=\"Diagramma PV pro processu isothermico\" 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>Lex Charles (isobarica):<\/strong> In processu ad pressionem constantem, volumen et temperatura gasi sunt proportionalia directe; id est:\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>Exemplum:<\/strong> Gas quod, ad temperaturam initialem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_1 = 300\\ \\mathrm{K}<\/span><\/span>, isobarice calefit usque ad <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_2 = 450\\ \\mathrm{K}<\/span><\/span> incipiens a volumine initiali <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V_1 = 2{,}00\\ \\mathrm{L}<\/span><\/span> volumen suum auget 50&nbsp;% (factor <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tfrac{3}{2}<\/span><\/span>). Cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tfrac{V}{T}=\\text{cte.}<\/span><\/span>, valet <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{V_1}{T_1}=\\dfrac{V_2}{T_2}<\/span><\/span>, quod ducit ad:<\/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>Lex Gay-Lussac (isochorica):<\/strong> In processu ad volumen constans, pressio et temperatura gasi sunt proportionalia directe; id est:\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>Exemplum:<\/strong> Gas quod, ad temperaturam initialem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_1 = 300\\ \\mathrm{K}<\/span><\/span>, isochorice calefit usque ad <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_2 = 450\\ \\mathrm{K}<\/span><\/span> incipiens a pressione initiali <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P_1 = 1{,}00\\ \\mathrm{MPa}<\/span><\/span> pressionem suam eadem proportione augeri videbit. Cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tfrac{P}{T}=\\text{cte.}<\/span><\/span>, valet <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{P_1}{T_1}=\\dfrac{P_2}{T_2}<\/span><\/span>, quod ducit ad:<\/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>Combinatio legum in aequatione gasorum idealium<\/h2>\n<p>Hae tres leges in unam relationem proportionalitatis componi possunt:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV \\propto T<\/span>\n<p>Ubi, ex considerationibus experimentalibus et microscopicis, constans proportionalitatis inferri potest tamquam effectus producti inter numerum particularum <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> et constantem Boltzmannianam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k_B = 1{,}380\\,649\\times10^{-23}\\ \\mathrm{J\\,K^{-1}}<\/span><\/span>, obtinendo relationem microscopicam:<\/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>Similiter, in terminis molaribus, constans proportionalitatis obtinetur ut productum inter numerum molium <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> et constantem universalem gasorum <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>, obtinendo relationem molarem:<\/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>Non refert quis sit casus, constat esse relationem proportionalitatis directae inter productum <span class=\"katex-eq\" data-katex-display=\"false\">PV<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">T<\/span>, quod aequivalet dicere quod, si gas idealis inter duo status transit, unum cum valoribus initialibus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\alpha, V_\\alpha, T_\\alpha)<\/span><\/span> et alterum cum valoribus finalibus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\omega, V_\\omega, T_\\omega)<\/span><\/span>, tunc hi satisfaciunt relationi<\/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, propterea, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV\/T = cte.<\/span><\/span><\/p>\n<p>Haec relatio, quae uti potest tamquam fundamentum experimentale ad formandam tam microscopicam quam molarem relationem, directe inferri potest ex legibus experimentalibus Boyle-Mariotte, Charles et Gay-Lussac. Ratiocinatio talis est qualis infra monstratur:<\/p>\n<p>Hoc demonstrare possumus per tres vias<\/p>\n<ol>\n<li>Mutatio <b>voluminis<\/b> post processum isothermicum et processum isobaricum<\/li>\n<li>Mutatio <b>pressionis<\/b> post processum isothermicum et processum isochoricum<\/li>\n<li>Mutatio <b>temperaturae<\/b> post processum isobaricum et processum isochoricum<\/li>\n<\/ol>\n<p>Ad evolutionem horum trium casuum opus erit statu intermedio cum valoribus <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>Deductiones per processus<\/h2>\n<h3>Deductio per mutationem voluminis<\/h3>\n<p>Si status initialis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\alpha,V_\\alpha,T_\\alpha)<\/span><\/span> coniungitur cum statu intermedio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_i,V_i,T_i)<\/span><\/span> per processum isothermicum, et deinde status intermedius cum statu finali <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\omega,V_\\omega,T_\\omega)<\/span><\/span> per processum isobaricum, tunc habetur:<\/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{isothermicus}&amp; &amp;\\text{isobaricus} &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>Ex hoc habetur quod:<\/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>Deductio per mutationem pressionis<\/h3>\n<p>Si status initialis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\alpha,V_\\alpha,T_\\alpha)<\/span><\/span> coniungitur cum statu intermedio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_i,V_i,T_i)<\/span><\/span> per processum isothermicum, et deinde status intermedius cum statu finali <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\omega,V_\\omega,T_\\omega)<\/span><\/span> per processum isochoricum, tunc habetur:<\/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{isothermicus}&amp; &amp;\\text{isochoricus} &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>Ex hoc habetur quod:<\/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>Deductio per mutationem temperaturae<\/h3>\n<p>Si status initialis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\alpha,V_\\alpha,T_\\alpha)<\/span><\/span> coniungitur cum statu intermedio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_i,V_i,T_i)<\/span><\/span> per processum isobaricum, et deinde status intermedius cum statu finali <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\omega,V_\\omega,T_\\omega)<\/span><\/span> per processum isochoricum, tunc habetur:<\/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{isobaricus}&amp; &amp;\\text{isochoricus} &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>Ex hoc habetur quod:<\/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>Commentarii et fundamentum microscopicum<\/h2>\n<p>Quamvis formulatio praecedens sit empirica, ex principiis primis deduci potest per Theoriam Kineticam Gasorum. In hoc modelo, gas est collectio particularum quae moventur et inter se atque cum parietibus receptaculi colliduntur. Idealis fit per suppositiones huiusmodi:<\/p>\n<ol>\n<li>Absentia virium attractionis aut repulsionis ad distantiam inter particulas.<\/li>\n<li>Particulae punctiformes vel magnitudine neglegendae forma sphaerica.<\/li>\n<li>Collisiones perfecte elasticae inter particulas et cum parietibus.<\/li>\n<\/ol>\n<p>Haec idealizationes analysin simpliciorem reddunt et, quamquam nullus gas realis eas prorsus implet, multos gases bene describunt in lata condicione et fundamentum praebent <strong>Thermodynamicae Classicae<\/strong>, cum applicationibus quae a machinis thermalibus ad physicam atmosphericam atque astrophysicam extenduntur.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Ambitus validitatis et limitationes<\/h2>\n<p>Lex gasorum idealium universalis non est. Avertitur cum praedictae hypotheseos amplius rationabiles non sunt aut cum effectus extra physicam classicam emergunt.<\/p>\n<ul>\n<li><strong>Altae pressiones et humiles temperatura:<\/strong> interactiones inter moleculas iam non sunt neglegendae et magnitudo finita particularum refert. Correctio usitata est aequatio 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>    cum parametris <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> propriis cuique gasi.\n  <\/li>\n<li><strong>Regimen quanticum:<\/strong> ad temperaturas valde humiles aut densitates altas apparent statisticae Bose\u2013Einstein vel Fermi\u2013Dirac, postulantes exempla <em>gasorum quanticorum<\/em>.<\/li>\n<li><strong>Regimen relativisticum:<\/strong> si particulae celeritatibus prope lucis moventur, correctiones relativisticae necessariae sunt.<\/li>\n<\/ul>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Notae practicae<\/h2>\n<ul>\n<li>Utere semper temperatura in <strong>Kelvin<\/strong> in formulis: <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>Cura unitatum consistentiam: si operaris cum <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>, utere <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 uteris <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>, adhibe <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>Memento unamquamque legem empiricam obtentam esse una variabili fixa. Resultata coniungere postulat clare intellegere quis processus thermodynamicus in singulis gradibus fiat.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Formulatio empirica gasi idealis Numquamne quaesivisti cur globus calefactus dilatetur aut cur pressio rotae mutetur altitudine variata? In hac lectione leges recognoscemus quae hos mores regunt et quomodo haec ad aequationem gasorum idealium ducunt, eius considerationes et puncta momenti. Proposita Discendi In fine huius lectionis discipulus poterit: Explicare leges empiricas gasorum idealium (Boyle\u2013Mariotte, Charles, Gay-Lussac) [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":35113,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":2,"footnotes":""},"categories":[1250,1292],"tags":[],"class_list":["post-34620","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-physica","category-thermodynamica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Aequatio Gasorum Idealiumn - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"\ud83c\udf21\ufe0f Disce aequationem gasorum idealium mirabilem: fundamenta, applicationes in thermodynamica, et eius 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\/la\/aequatio-gasorum-idealiumn\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Aequatio Gasorum Idealiumn\" \/>\n<meta property=\"og:description\" content=\"\ud83c\udf21\ufe0f Disce aequationem gasorum idealium mirabilem: fundamenta, applicationes in thermodynamica, et eius limites \ud83d\ude80\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/la\/aequatio-gasorum-idealiumn\/\" \/>\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=\"2021-03-10T13:00:12+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-10-20T07:36:22+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/min3-8.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=\"Aequatio Gasorum Idealiumn\" \/>\n<meta name=\"twitter:description\" content=\"\ud83c\udf21\ufe0f Disce aequationem gasorum idealium mirabilem: fundamenta, applicationes in thermodynamica, et eius limites \ud83d\ude80\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/min3-8.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=\"3 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-gasorum-idealiumn\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-gasorum-idealiumn\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Aequatio Gasorum Idealiumn\",\"datePublished\":\"2021-03-10T13:00:12+00:00\",\"dateModified\":\"2025-10-20T07:36:22+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-gasorum-idealiumn\\\/\"},\"wordCount\":1653,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-gasorum-idealiumn\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2021\\\/03\\\/min3-8.jpg\",\"articleSection\":[\"Physica\",\"Thermodynamica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-gasorum-idealiumn\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-gasorum-idealiumn\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/aequatio-gasorum-idealiumn\\\/\",\"name\":\"Aequatio Gasorum Idealiumn - 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