{"id":30766,"date":"2021-05-07T13:00:53","date_gmt":"2021-05-07T13:00:53","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=30766"},"modified":"2025-01-01T18:54:30","modified_gmt":"2025-01-01T18:54:30","slug":"equilibrio-termico-temperatura-e-sua-definicao-estatistica","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/pt\/equilibrio-termico-temperatura-e-sua-definicao-estatistica\/","title":{"rendered":"Equil\u00edbrio t\u00e9rmico, temperatura e sua defini\u00e7\u00e3o estat\u00edstica"},"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>Equil\u00edbrio t\u00e9rmico, temperatura e sua defini\u00e7\u00e3o estat\u00edstica<\/h1>\n<p style=\"text-align: center\"><em>A temperatura termodin\u00e2mica, entendida como a conex\u00e3o entre energia e probabilidade, surge do equil\u00edbrio entre macroestados e microestados em sistemas t\u00e9rmicos. Este conte\u00fado explora como a maximiza\u00e7\u00e3o do n\u00famero de microestados define o equil\u00edbrio t\u00e9rmico, utilizando dedu\u00e7\u00f5es matem\u00e1ticas e o arcabou\u00e7o da mec\u00e2nica estat\u00edstica para explicar sua rela\u00e7\u00e3o com a constante de Boltzmann e a escala Kelvin.<\/em><\/p>\n<p style=\"text-align: center;\"><strong>Objetivos de Aprendizagem:<\/strong><br \/>\nAo final desta aula, o estudante ser\u00e1 capaz de:\n<\/p>\n<ol>\n<li><strong>Compreender<\/strong> a rela\u00e7\u00e3o entre macroestados e microestados em sistemas termodin\u00e2micos e o conceito de temperatura.<\/li>\n<\/ol>\n<p style=\"text-align: center;\"><strong><u>\u00cdNDICE DE CONTE\u00daDOS<\/u>:<\/strong><br \/>\n<a href=\"#1\">Macroestados e microestados de sistemas em contato t\u00e9rmico<\/a><br \/>\n<a href=\"#2\">Temperatura termodin\u00e2mica: A qualidade comum dos sistemas em equil\u00edbrio<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/7LPUN_vBoXE\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Macroestados e microestados de sistemas em contato t\u00e9rmico<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=7LPUN_vBoXE&amp;t=218s\" target=\"_blank\" rel=\"noopener\"><strong>Para abordar o conceito de temperatura termodin\u00e2mica<\/strong><\/a>, precisamos primeiro analisar como os sistemas termodin\u00e2micos interagem e sua rela\u00e7\u00e3o com microestados e macroestados. Consideremos o caso de dois sistemas em contato t\u00e9rmico entre si, mas isolados do restante do universo.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-kL0Rr2L4wys\/YJPlhBUTueI\/AAAAAAAAFEw\/OWpuyASOdGANXsSG5O-d09ed034DtMTuwCLcBGAsYHQ\/s0\/contacto%2Btermico.PNG\" alt=\"Contato t\u00e9rmico\" class=\"aligncenter lazyload\" width=\"372\" height=\"158\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-kL0Rr2L4wys\/YJPlhBUTueI\/AAAAAAAAFEw\/OWpuyASOdGANXsSG5O-d09ed034DtMTuwCLcBGAsYHQ\/s0\/contacto%2Btermico.PNG\" alt=\"Contato t\u00e9rmico\" class=\"aligncenter lazyload\" width=\"372\" height=\"158\" \/><\/noscript><\/p>\n<p>Por um lado, j\u00e1 que os sistemas permanecem isolados, a energia total <span class=\"katex-eq\" data-katex-display=\"false\">E=E_1+E_2<\/span> ser\u00e1 constante, e por isso o macroestado do sistema \u00e9 definido por qualquer uma das tr\u00eas energias <span class=\"katex-eq\" data-katex-display=\"false\">E, E_1<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">E_2<\/span>. Por outro lado, ao falarmos de microestados, o sistema com energia <span class=\"katex-eq\" data-katex-display=\"false\">E_1<\/span> tem <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_1(E_1)<\/span> microestados, e de forma an\u00e1loga, o sistema com energia <span class=\"katex-eq\" data-katex-display=\"false\">E_2<\/span> possui <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_2(E_2)<\/span>. Assim, o sistema composto ser\u00e1 formado por um total de <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_1(E_1)\\Omega_2(E_2)<\/span> microestados.<\/p>\n<h3>Pressupostos sobre o equil\u00edbrio termodin\u00e2mico<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=7LPUN_vBoXE&amp;t=392s\" target=\"_blank\" rel=\"noopener\"><strong>Se cada sistema puder trocar energia<\/strong><\/a> at\u00e9 alcan\u00e7ar o <strong>equil\u00edbrio termodin\u00e2mico,<\/strong> em algum momento ocorrer\u00e1 que <span class=\"katex-eq\" data-katex-display=\"false\">E_1<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">E_2<\/span> se aproximar\u00e3o de valores constantes. A ideia que devemos ter em mente \u00e9 que o macroestado com maior probabilidade de ocorrer ser\u00e1 aquele que maximiza o n\u00famero de microestados. A partir disso, seguem-se os seguintes pressupostos:<\/p>\n<ol>\n<li>Cada um dos microestados tem a mesma probabilidade de ocorrer.<\/li>\n<li>A din\u00e2mica interna do sistema \u00e9 tal que os microestados est\u00e3o em constante mudan\u00e7a.<\/li>\n<li>Ap\u00f3s algum tempo, o sistema explorar\u00e1 todos os microestados poss\u00edveis e passar\u00e1 a mesma quantidade de tempo em cada um deles.<\/li>\n<\/ol>\n<p>Esses pressupostos implicam que o sistema passar\u00e1 a maior parte do tempo nos macroestados que representam o maior n\u00famero de microestados, sendo, portanto, os mais prov\u00e1veis. Agora, nos sistemas normalmente estudados em termodin\u00e2mica, a express\u00e3o \u00abmais prov\u00e1vel\u00bb significa, na pr\u00e1tica, \u00abimposs\u00edvel de n\u00e3o ocorrer.\u00bb O que pode parecer uma simples afirma\u00e7\u00e3o estat\u00edstica \u00e9, na realidade, uma declara\u00e7\u00e3o quase absoluta de certeza.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Temperatura termodin\u00e2mica: A qualidade comum dos sistemas em equil\u00edbrio<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=7LPUN_vBoXE&amp;t=617s\" target=\"_blank\" rel=\"noopener\"><strong>Para o nosso problema de dois corpos colocados<\/strong><\/a> em contato t\u00e9rmico, a distribui\u00e7\u00e3o mais prov\u00e1vel de energia \u00e9 aquela que maximiza o n\u00famero <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_1(E_1)\\Omega_2(E_2)<\/span> de microestados. Como os sistemas s\u00e3o enormes, podemos usar ferramentas de c\u00e1lculo diferencial como uma boa aproxima\u00e7\u00e3o para fazer infer\u00eancias sobre o significado dessas afirma\u00e7\u00f5es e deduzir suas propriedades.<\/p>\n<p>Se fizermos varia\u00e7\u00f5es infinitesimais na energia de um dos sistemas e procurarmos o caso em que o n\u00famero de microestados \u00e9 maximizado, obtemos o seguinte racioc\u00ednio:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n(1) &amp; \\dfrac{d}{dE_1}\\left[  \\Omega_1(E_1) \\Omega_2(E_2) \\right] = 0 \\\\\n\n&amp; \\text{; Porque o n\u00famero de microestados foi maximizado} \\\\ \\\\\n\n\\equiv &amp; \\dfrac{d\\Omega_1(E_1)}{dE_1} \\Omega_2(E_2) + \\Omega_1(E_1) \\dfrac{d\\Omega_2(E_2)}{E_2}\\dfrac{dE_2}{dE_1} = 0 \\\\\n\n&amp; \\text{; Aplicando a regra do produto e a regra da cadeia} \\\\ \\\\\n\n(2) &amp; E = E_1 + E_2 = \\text{Constante} \\\\\n\n&amp; \\text{; Energia total dos dois sistemas em contato t\u00e9rmico} \\\\ \\\\\n\n\\equiv &amp; E_1 = E - E_2 \\\\ \\\\\n\n(3) &amp; \\dfrac{dE_2}{dE_1} = \\dfrac{d}{dE_1} (E - E_1) = -1\\;\\text{; De (2)} \\\\ \\\\\n\n(4) &amp; \\dfrac{d\\Omega_1}{dE_1}\\Omega_2 - \\Omega_2 \\dfrac{d\\Omega_2}{dE_2} = 0\\;\\text{; De (1) e (3)} \\\\ \\\\\n\n\\equiv &amp; \\dfrac{d\\Omega_1}{dE_1} \\Omega_2 = \\Omega_1 \\dfrac{d\\Omega_2}{dE_2} \\\\ \\\\\n\n\\equiv &amp; \\dfrac{1}{\\Omega_1} \\dfrac{d\\Omega_1}{dE_1} = \\dfrac{1}{\\Omega_2} \\dfrac{d\\Omega_2}{dE_2} \\\\ \\\\\n\n\\equiv &amp; \\dfrac{d\\ln(\\Omega_1)}{dE_1} = \\dfrac{d\\ln(\\Omega_2)}{dE_2}\n\n\\end{array}<\/span>\n<p>A partir disso, entendemos que, dado que o equil\u00edbrio t\u00e9rmico \u00e9 o macroestado mais prov\u00e1vel de dois corpos em contato t\u00e9rmico prolongado, este \u00e9 o que maximiza o n\u00famero de microestados. Se isso ocorrer, do racioc\u00ednio que acabamos de realizar, conclu\u00edmos que existe uma magnitude comum a ambos os sistemas, que chamamos de <strong>temperatura.<\/strong><\/p>\n<h3>Defini\u00e7\u00e3o da temperatura termodin\u00e2mica<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=7LPUN_vBoXE&amp;t=972s\" target=\"_blank\" rel=\"noopener\"><strong>Quando isso ocorre, dizemos que os corpos<\/strong><\/a> \u00abest\u00e3o \u00e0 mesma temperatura\u00bb e relacionamos <span class=\"katex-eq\" data-katex-display=\"false\">d\\ln\\Omega\/dE<\/span> com a temperatura <span class=\"katex-eq\" data-katex-display=\"false\">T<\/span> (de modo que <span class=\"katex-eq\" data-katex-display=\"false\">T_1 = T_2<\/span>). Assim, a temperatura termodin\u00e2mica \u00e9 definida como:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{1}{k_B T} = \\frac{d\\ln\\Omega}{dE}<\/span>\n<p>onde <span class=\"katex-eq\" data-katex-display=\"false\">k_B=1.3807\\cdot 10^{-23}[J\/K]<\/span> \u00e9 a <strong>constante de Boltzmann.<\/strong><\/p>\n<p>Com essa escolha de constantes, a temperatura <span class=\"katex-eq\" data-katex-display=\"false\">T<\/span> que definimos adquire sua interpreta\u00e7\u00e3o usual no que conhecemos como a <strong>escala Kelvin.<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Equil\u00edbrio t\u00e9rmico, temperatura e sua defini\u00e7\u00e3o estat\u00edstica A temperatura termodin\u00e2mica, entendida como a conex\u00e3o entre energia e probabilidade, surge do equil\u00edbrio entre macroestados e microestados em sistemas t\u00e9rmicos. Este conte\u00fado explora como a maximiza\u00e7\u00e3o do n\u00famero de microestados define o equil\u00edbrio t\u00e9rmico, utilizando dedu\u00e7\u00f5es matem\u00e1ticas e o arcabou\u00e7o da mec\u00e2nica estat\u00edstica para explicar sua rela\u00e7\u00e3o [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":30760,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":4,"footnotes":""},"categories":[637,921],"tags":[],"class_list":["post-30766","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>Equil\u00edbrio t\u00e9rmico, temperatura e sua defini\u00e7\u00e3o estat\u00edstica - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Compreende o equil\u00edbrio 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