{"id":30067,"date":"2021-03-10T13:00:00","date_gmt":"2021-03-10T13:00:00","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=30067"},"modified":"2025-10-20T07:26:17","modified_gmt":"2025-10-20T07:26:17","slug":"a-equacao-dos-gases-ideais","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/pt\/a-equacao-dos-gases-ideais\/","title":{"rendered":"A Equa\u00e7\u00e3o dos Gases Ideais"},"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>Formula\u00e7\u00e3o emp\u00edrica do g\u00e1s ideal<\/h1>\n<p style=\"text-align:center;\" dir=\"ltr\">Voc\u00ea j\u00e1 se perguntou por que um bal\u00e3o se dilata quando \u00e9 aquecido ou por que a press\u00e3o de um pneu varia ao mudar de altitude? Nesta aula, revisaremos as leis que regem esses comportamentos e como isso conduz \u00e0 equa\u00e7\u00e3o dos gases ideais, suas considera\u00e7\u00f5es e pontos importantes.<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><b>Objetivos de Aprendizagem<\/b><br \/>\nAo final desta aula, o estudante ser\u00e1 capaz de:<\/p>\n<ol>\n<li><b>Explicar<\/b> as leis emp\u00edricas dos gases ideais (Boyle\u2013Mariotte, Charles, Gay-Lussac) e sua s\u00edntese na equa\u00e7\u00e3o de estado (<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>Aplicar<\/b> a equa\u00e7\u00e3o dos gases ideais e a rela\u00e7\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV\/T = cte<\/span><\/span> para resolver mudan\u00e7as de estado com unidades coerentes.<\/li>\n<li><b>Analisar<\/b> processos isot\u00e9rmicos, isob\u00e1ricos e isoc\u00f3ricos e suas trajet\u00f3rias em diagramas <i>P\u2013V<\/i>, <i>V\u2013T<\/i> e <i>P\u2013T<\/i>.<\/li>\n<li><b>Reconhecer<\/b> o \u00e2mbito de validade do g\u00e1s ideal e selecionar modelos alternativos (van der Waals, qu\u00e2ntico, relativ\u00edstico) quando apropriado.<\/li>\n<\/ol>\n<p style=\"text-align:center;\" dir=\"ltr\">\n<b>\u00cdNDICE DE CONTE\u00daDOS<\/b><br \/>\n<a href=\"#1\">Leis emp\u00edricas fundamentais<\/a><br \/>\n<a href=\"#2\">Combina\u00e7\u00e3o das leis na equa\u00e7\u00e3o dos gases ideais<\/a><br \/>\n<a href=\"#3\">Dedu\u00e7\u00e3o por processos<\/a><br \/>\n<a href=\"#4\">Coment\u00e1rios e contexto microsc\u00f3pico<\/a><br \/>\n<a href=\"#5\">\u00c2mbito de validade e limita\u00e7\u00f5es<\/a><br \/>\n<a href=\"#6\">Notas pr\u00e1ticas<\/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>Leis emp\u00edricas fundamentais<\/h2>\n<p>Os experimentos com gases mostram uma depend\u00eancia entre a press\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">P<\/span>, o volume <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span> e a temperatura <span class=\"katex-eq\" data-katex-display=\"false\">T<\/span>. Em condi\u00e7\u00f5es controladas, observam-se tr\u00eas leis emp\u00edricas fundamentais:<\/p>\n<ol>\n<li><strong>Lei de Boyle\u2013Mariotte (isot\u00e9rmico):<\/strong> Em um processo a temperatura constante, o volume e a press\u00e3o de um g\u00e1s s\u00e3o inversamente proporcionais; ou seja:\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>Exemplo:<\/strong> Um g\u00e1s que, a uma press\u00e3o inicial de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P_1 = 15\\ \\mathrm{MPa}<\/span><\/span>, se expande isotermicamente de um volume inicial <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V_1 = 1{,}00\\ \\mathrm{L}<\/span><\/span> at\u00e9 um volume final <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V_2 = 2{,}00\\ \\mathrm{L}<\/span><\/span> ver\u00e1 sua press\u00e3o reduzida \u00e0 metade. Como o produto <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV=\\text{cte.}<\/span><\/span>, cumpre-se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P_1 V_1 = P_2 V_2<\/span><\/span>, o que conduz a:<\/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=\"Diagrama PV para processo isot\u00e9rmico\" 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=\"Diagrama PV para processo isot\u00e9rmico\" 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>Lei de Charles (isob\u00e1rico):<\/strong> Em um processo a press\u00e3o constante, o volume e a temperatura de um g\u00e1s s\u00e3o diretamente proporcionais; ou seja:\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>Exemplo:<\/strong> Um g\u00e1s que, a uma temperatura inicial de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_1 = 300\\ \\mathrm{K}<\/span><\/span>, \u00e9 aquecido isobaricamente at\u00e9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_2 = 450\\ \\mathrm{K}<\/span><\/span> partindo de um volume inicial <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V_1 = 2{,}00\\ \\mathrm{L}<\/span><\/span> ver\u00e1 seu volume aumentado em 50&nbsp;% (um fator <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tfrac{3}{2}<\/span><\/span>). Como <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tfrac{V}{T}=\\text{cte.}<\/span><\/span>, cumpre-se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{V_1}{T_1}=\\dfrac{V_2}{T_2}<\/span><\/span>, o que conduz a:<\/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>Lei de Gay-Lussac (isoc\u00f3rico):<\/strong> Em um processo a volume constante, a press\u00e3o e a temperatura de um g\u00e1s s\u00e3o diretamente proporcionais; ou seja:\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>Exemplo:<\/strong> Um g\u00e1s que, a uma temperatura inicial de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_1 = 300\\ \\mathrm{K}<\/span><\/span>, \u00e9 aquecido isocoricamente at\u00e9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_2 = 450\\ \\mathrm{K}<\/span><\/span> partindo de uma press\u00e3o inicial <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P_1 = 1{,}00\\ \\mathrm{MPa}<\/span><\/span> ver\u00e1 sua press\u00e3o aumentada na mesma propor\u00e7\u00e3o. Como <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tfrac{P}{T}=\\text{cte.}<\/span><\/span>, cumpre-se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{P_1}{T_1}=\\dfrac{P_2}{T_2}<\/span><\/span>, o que conduz a:<\/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>Combina\u00e7\u00e3o das leis na equa\u00e7\u00e3o dos gases ideais<\/h2>\n<p>Essas tr\u00eas leis podem ser sintetizadas em uma \u00fanica rela\u00e7\u00e3o de proporcionalidade:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV \\propto T<\/span>\n<p>Onde, a partir de considera\u00e7\u00f5es experimentais e microsc\u00f3picas, \u00e9 poss\u00edvel inferir a constante de proporcionalidade como o resultado do produto entre o n\u00famero de part\u00edculas <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> e a 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>, obtendo a rela\u00e7\u00e3o microsc\u00f3pica:<\/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 forma an\u00e1loga, em termos molares, a constante de proporcionalidade \u00e9 obtida como o produto entre o n\u00famero de mols <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> e a constante universal dos gases <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>, obtendo-se a rela\u00e7\u00e3o molar:<\/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>Independentemente do caso, \u00e9 certo que existe uma rela\u00e7\u00e3o de proporcionalidade direta entre o produto <span class=\"katex-eq\" data-katex-display=\"false\">PV<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">T<\/span>, o que equivale a dizer que, se um g\u00e1s ideal passa de um estado para outro, com valores iniciais <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\alpha, V_\\alpha, T_\\alpha)<\/span><\/span> e finais <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\omega, V_\\omega, T_\\omega)<\/span><\/span>, ent\u00e3o estes satisfazem a rela\u00e7\u00e3o<\/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>e, portanto, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">PV\/T = cte.<\/span><\/span><\/p>\n<p>Essa rela\u00e7\u00e3o, que pode ser usada como pilar experimental para formular tanto a rela\u00e7\u00e3o microsc\u00f3pica quanto a molar, pode ser inferida diretamente das leis experimentais de Boyle-Mariotte, de Charles e de Gay-Lussac. O racioc\u00ednio \u00e9 mostrado a seguir:<\/p>\n<p>Isso pode ser demonstrado por meio de tr\u00eas vias:<\/p>\n<ol>\n<li>Uma mudan\u00e7a de <b>volume<\/b> ap\u00f3s um processo isot\u00e9rmico e um processo isob\u00e1rico<\/li>\n<li>Uma mudan\u00e7a de <b>press\u00e3o<\/b> ap\u00f3s um processo isot\u00e9rmico e um processo isoc\u00f3rico<\/li>\n<li>Uma mudan\u00e7a de <b>temperatura<\/b> ap\u00f3s um processo isob\u00e1rico e um processo isoc\u00f3rico<\/li>\n<\/ol>\n<p>Para o desenvolvimento desses tr\u00eas casos, precisaremos de um estado intermedi\u00e1rio com valores <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>Dedu\u00e7\u00f5es por processos<\/h2>\n<h3>Dedu\u00e7\u00f5es por mudan\u00e7a de volume<\/h3>\n<p>Se o estado inicial <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\alpha,V_\\alpha,T_\\alpha)<\/span><\/span> se conecta com o estado intermedi\u00e1rio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_i,V_i,T_i)<\/span><\/span> por meio de um processo isot\u00e9rmico, e depois o estado intermedi\u00e1rio se conecta com o estado final <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\omega,V_\\omega,T_\\omega)<\/span><\/span> por um processo isob\u00e1rico, ent\u00e3o temos:<\/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{isot\u00e9rmico}&amp; &amp;\\text{isob\u00e1rico} &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>A partir disso, tem-se que:<\/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>Dedu\u00e7\u00f5es por mudan\u00e7a de press\u00e3o<\/h3>\n<p>Se o estado inicial <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\alpha,V_\\alpha,T_\\alpha)<\/span><\/span> se conecta com o estado intermedi\u00e1rio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_i,V_i,T_i)<\/span><\/span> por meio de um processo isot\u00e9rmico, e depois o estado intermedi\u00e1rio se conecta com o estado final <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\omega,V_\\omega,T_\\omega)<\/span><\/span> por um processo isoc\u00f3rico, ent\u00e3o temos:<\/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{isot\u00e9rmico}&amp; &amp;\\text{isoc\u00f3rico} &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>A partir disso, tem-se que:<\/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>Dedu\u00e7\u00f5es por mudan\u00e7a de temperatura<\/h3>\n<p>Se o estado inicial <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\alpha,V_\\alpha,T_\\alpha)<\/span><\/span> se conecta com o estado intermedi\u00e1rio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_i,V_i,T_i)<\/span><\/span> por meio de um processo isob\u00e1rico, e depois o estado intermedi\u00e1rio se conecta com o estado final <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(P_\\omega,V_\\omega,T_\\omega)<\/span><\/span> por um processo isoc\u00f3rico, ent\u00e3o temos:<\/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{isob\u00e1rico}&amp; &amp;\\text{isoc\u00f3rico} &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>A partir disso, tem-se que:<\/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>Coment\u00e1rios e fundamento microsc\u00f3pico<\/h2>\n<p>Embora a formula\u00e7\u00e3o anterior seja emp\u00edrica, ela pode ser derivada a partir de primeiros princ\u00edpios por meio da Teoria Cin\u00e9tica dos Gases. Nesse modelo, o g\u00e1s \u00e9 uma cole\u00e7\u00e3o de part\u00edculas que se movem e colidem entre si e com as paredes do recipiente. \u00c9 idealizado com os seguintes pressupostos:<\/p>\n<ol>\n<li>Aus\u00eancia de for\u00e7as de atra\u00e7\u00e3o ou repuls\u00e3o a dist\u00e2ncia entre as part\u00edculas.<\/li>\n<li>Part\u00edculas pontuais ou de tamanho desprez\u00edvel com forma esf\u00e9rica.<\/li>\n<li>Colis\u00f5es perfeitamente el\u00e1sticas entre as part\u00edculas e com as paredes.<\/li>\n<\/ol>\n<p>Essas idealiza\u00e7\u00f5es simplificam a an\u00e1lise e, ainda que nenhum g\u00e1s real as satisfa\u00e7a exatamente, descrevem bem muitos gases em uma ampla faixa de condi\u00e7\u00f5es e oferecem uma base para a <strong>Termodin\u00e2mica Cl\u00e1ssica<\/strong>, com aplica\u00e7\u00f5es que v\u00e3o desde motores t\u00e9rmicos at\u00e9 f\u00edsica atmosf\u00e9rica e astrof\u00edsica.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>\u00c2mbito de validade e limita\u00e7\u00f5es<\/h2>\n<p>A lei dos gases ideais n\u00e3o \u00e9 universal. Ela se desvia quando as hip\u00f3teses anteriores deixam de ser razo\u00e1veis ou quando surgem efeitos fora da f\u00edsica cl\u00e1ssica.<\/p>\n<ul>\n<li><strong>Altas press\u00f5es e baixas temperaturas:<\/strong> as intera\u00e7\u00f5es entre mol\u00e9culas deixam de ser desprez\u00edveis e o tamanho finito das part\u00edculas torna-se relevante. Uma corre\u00e7\u00e3o comum \u00e9 a equa\u00e7\u00e3o 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>    com par\u00e2metros <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> caracter\u00edsticos de cada g\u00e1s.\n  <\/li>\n<li><strong>Regime qu\u00e2ntico:<\/strong> em temperaturas muito baixas ou densidades elevadas surgem estat\u00edsticas de Bose\u2013Einstein ou Fermi\u2013Dirac, exigindo modelos de <em>gases qu\u00e2nticos<\/em>.<\/li>\n<li><strong>Regime relativ\u00edstico:<\/strong> se as part\u00edculas se movem a velocidades pr\u00f3ximas \u00e0 da luz, s\u00e3o necess\u00e1rias corre\u00e7\u00f5es relativ\u00edsticas.<\/li>\n<\/ul>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Notas pr\u00e1ticas<\/h2>\n<ul>\n<li>Sempre use a temperatura em <strong>Kelvin<\/strong> nas f\u00f3rmulas: <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>Mantenha a coer\u00eancia das unidades: se trabalhar com <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathrm{atm}<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathrm{L}<\/span><\/span>, utilize <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">R=0{,}082\\,057\\ \\mathrm{L\\,atm\\,mol^{-1}\\,K^{-1}}<\/span><\/span>; se usar <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathrm{Pa}<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathrm{m^3}<\/span><\/span>, empregue <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>Lembre-se de que cada lei emp\u00edrica foi obtida mantendo uma vari\u00e1vel fixa. Combinar resultados requer ter clareza sobre qual processo termodin\u00e2mico \u00e9 realizado em cada etapa.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Formula\u00e7\u00e3o emp\u00edrica do g\u00e1s ideal Voc\u00ea j\u00e1 se perguntou por que um bal\u00e3o se dilata quando \u00e9 aquecido ou por que a press\u00e3o de um pneu varia ao mudar de altitude? Nesta aula, revisaremos as leis que regem esses comportamentos e como isso conduz \u00e0 equa\u00e7\u00e3o dos gases ideais, suas considera\u00e7\u00f5es e pontos importantes. Objetivos [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":35095,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":9,"footnotes":""},"categories":[637,921],"tags":[],"class_list":["post-30067","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-fisica-pt","category-termodinamica-pt"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>A Equa\u00e7\u00e3o dos Gases Ideais - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"\ud83c\udf21\ufe0f Discover the fascinating Ideal Gas Equation: principles, thermodynamics applications, and its limitations \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\/pt\/a-equacao-dos-gases-ideais\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"A Equa\u00e7\u00e3o dos Gases Ideais\" \/>\n<meta property=\"og:description\" content=\"\ud83c\udf21\ufe0f Descubra a fascinante Equa\u00e7\u00e3o dos Gases Ideais: fundamentos, aplica\u00e7\u00f5es em termodin\u00e2mica e seus limites \ud83d\ude80\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/pt\/a-equacao-dos-gases-ideais\/\" \/>\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=\"2021-03-10T13:00:00+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-10-20T07:26:17+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/min3-2.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=\"A Equa\u00e7\u00e3o dos Gases Ideais\" \/>\n<meta name=\"twitter:description\" content=\"\ud83c\udf21\ufe0f Descubra a fascinante Equa\u00e7\u00e3o dos Gases Ideais: fundamentos, aplica\u00e7\u00f5es em termodin\u00e2mica e seus limites \ud83d\ude80\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/min3-2.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\\\/pt\\\/a-equacao-dos-gases-ideais\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/a-equacao-dos-gases-ideais\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"A Equa\u00e7\u00e3o dos Gases Ideais\",\"datePublished\":\"2021-03-10T13:00:00+00:00\",\"dateModified\":\"2025-10-20T07:26:17+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/a-equacao-dos-gases-ideais\\\/\"},\"wordCount\":2059,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/a-equacao-dos-gases-ideais\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2021\\\/03\\\/min3-2.jpg\",\"articleSection\":[\"F\u00edsica\",\"Termodin\u00e2mica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/a-equacao-dos-gases-ideais\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/a-equacao-dos-gases-ideais\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/a-equacao-dos-gases-ideais\\\/\",\"name\":\"A Equa\u00e7\u00e3o dos Gases Ideais - 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