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class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Conteúdo" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Conteúdo</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mover para a barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ocultar</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Início</div> </a> </li> <li id="toc-Postulado_de_Bohr" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Postulado_de_Bohr"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Postulado de Bohr</span> </div> </a> <ul id="toc-Postulado_de_Bohr-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Expressão_para_o_raio_de_Bohr" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Expressão_para_o_raio_de_Bohr"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Expressão para o raio de Bohr</span> </div> </a> <ul id="toc-Expressão_para_o_raio_de_Bohr-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equação_de_Rydberg" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equação_de_Rydberg"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Equação de Rydberg</span> </div> </a> <ul id="toc-Equação_de_Rydberg-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Níveis_energéticos_dos_elétrons_em_um_átomo_de_hidrogênio" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Níveis_energéticos_dos_elétrons_em_um_átomo_de_hidrogênio"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Níveis energéticos dos elétrons em um átomo de hidrogênio</span> </div> </a> <ul id="toc-Níveis_energéticos_dos_elétrons_em_um_átomo_de_hidrogênio-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Frequência" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Frequência"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Frequência</span> </div> </a> <ul id="toc-Frequência-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Problemas_e_contradições" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Problemas_e_contradições"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Problemas e contradições</span> </div> </a> <ul id="toc-Problemas_e_contradições-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referências" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referências"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Referências</span> </div> </a> <ul id="toc-Referências-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ver_também" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ver_também"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Ver também</span> </div> </a> <ul id="toc-Ver_também-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ligações_externas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ligações_externas"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Ligações externas</span> </div> </a> <ul id="toc-Ligações_externas-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Conteúdo" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Índice" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Alternar o índice" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Alternar o índice</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Átomo de Bohr</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Ir para um artigo noutra língua. Disponível em 70 línguas" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-70" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">70 línguas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Bohrsches_Atommodell" title="Bohrsches Atommodell — alemão suíço" lang="gsw" hreflang="gsw" data-title="Bohrsches Atommodell" data-language-autonym="Alemannisch" data-language-local-name="alemão suíço" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Modelo_atomico_de_Bohr" title="Modelo atomico de Bohr — aragonês" lang="an" hreflang="an" data-title="Modelo atomico de Bohr" data-language-autonym="Aragonés" data-language-local-name="aragonês" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D9%85%D9%88%D8%B0%D8%AC_%D8%A8%D9%88%D8%B1" title="نموذج بور — árabe" lang="ar" hreflang="ar" data-title="نموذج بور" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Modelu_at%C3%B3micu_de_Bohr" title="Modelu atómicu de Bohr — asturiano" lang="ast" hreflang="ast" data-title="Modelu atómicu de Bohr" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Bor_modeli" title="Bor modeli — azerbaijano" lang="az" hreflang="az" data-title="Bor modeli" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijano" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%91%D0%BE%D1%80%D0%B0%D1%9E%D1%81%D0%BA%D0%B0%D1%8F_%D0%BC%D0%B0%D0%B4%D1%8D%D0%BB%D1%8C_%D0%B0%D1%82%D0%B0%D0%BC%D0%B0" title="Бораўская мадэль атама — bielorrusso" lang="be" hreflang="be" data-title="Бораўская мадэль атама" data-language-autonym="Беларуская" data-language-local-name="bielorrusso" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%BE%D0%B4%D0%B5%D0%BB_%D0%BD%D0%B0_%D0%91%D0%BE%D1%80" title="Модел на Бор — búlgaro" lang="bg" hreflang="bg" data-title="Модел на Бор" data-language-autonym="Български" data-language-local-name="búlgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A7%8B%E0%A6%B0_%E0%A6%AE%E0%A6%A1%E0%A7%87%E0%A6%B2" title="বোর মডেল — bengalês" lang="bn" hreflang="bn" data-title="বোর মডেল" data-language-autonym="বাংলা" data-language-local-name="bengalês" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Bohrov_model_atoma" title="Bohrov model atoma — bósnio" lang="bs" hreflang="bs" data-title="Bohrov model atoma" data-language-autonym="Bosanski" data-language-local-name="bósnio" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Model_at%C3%B2mic_de_Bohr" title="Model atòmic de Bohr — catalão" lang="ca" hreflang="ca" data-title="Model atòmic de Bohr" data-language-autonym="Català" data-language-local-name="catalão" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bohr%C5%AFv_model_atomu" title="Bohrův model atomu — checo" lang="cs" hreflang="cs" data-title="Bohrův model atomu" data-language-autonym="Čeština" data-language-local-name="checo" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Bohrs_atommodel" title="Bohrs atommodel — dinamarquês" lang="da" hreflang="da" data-title="Bohrs atommodel" data-language-autonym="Dansk" data-language-local-name="dinamarquês" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Bohrsches_Atommodell" title="Bohrsches Atommodell — alemão" lang="de" hreflang="de" data-title="Bohrsches Atommodell" data-language-autonym="Deutsch" data-language-local-name="alemão" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CF%84%CE%BF%CE%BC%CE%B9%CE%BA%CF%8C_%CF%80%CF%81%CF%8C%CF%84%CF%85%CF%80%CE%BF_%CF%84%CE%BF%CF%85_%CE%9C%CF%80%CE%BF%CF%81" title="Ατομικό πρότυπο του Μπορ — grego" lang="el" hreflang="el" data-title="Ατομικό πρότυπο του Μπορ" data-language-autonym="Ελληνικά" data-language-local-name="grego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Bohr_model" title="Bohr model — inglês" lang="en" hreflang="en" data-title="Bohr model" data-language-autonym="English" data-language-local-name="inglês" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Modelo_at%C3%B3mico_de_Bohr" title="Modelo atómico de Bohr — espanhol" lang="es" hreflang="es" data-title="Modelo atómico de Bohr" data-language-autonym="Español" data-language-local-name="espanhol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Bohri_aatomiteooria" title="Bohri aatomiteooria — estónio" lang="et" hreflang="et" data-title="Bohri aatomiteooria" data-language-autonym="Eesti" data-language-local-name="estónio" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bohrren_eredu_atomikoa" title="Bohrren eredu atomikoa — basco" lang="eu" hreflang="eu" data-title="Bohrren eredu atomikoa" data-language-autonym="Euskara" data-language-local-name="basco" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%AF%D9%84_%D8%A8%D9%88%D8%B1" title="مدل بور — persa" lang="fa" hreflang="fa" data-title="مدل بور" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Bohrin_atomimalli" title="Bohrin atomimalli — finlandês" lang="fi" hreflang="fi" data-title="Bohrin atomimalli" data-language-autonym="Suomi" data-language-local-name="finlandês" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Mod%C3%A8le_de_Bohr" title="Modèle de Bohr — francês" lang="fr" hreflang="fr" data-title="Modèle de Bohr" data-language-autonym="Français" data-language-local-name="francês" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Samhail_Bohr" title="Samhail Bohr — irlandês" lang="ga" hreflang="ga" data-title="Samhail Bohr" data-language-autonym="Gaeilge" data-language-local-name="irlandês" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Modelo_at%C3%B3mico_de_Bohr" title="Modelo atómico de Bohr — galego" lang="gl" hreflang="gl" data-title="Modelo atómico de Bohr" data-language-autonym="Galego" data-language-local-name="galego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%AC%E0%AB%89%E0%AA%B9%E0%AA%B0%E0%AA%A8%E0%AB%8B_%E0%AA%B8%E0%AA%BF%E0%AA%A6%E0%AB%8D%E0%AA%A7%E0%AA%BE%E0%AA%82%E0%AA%A4" title="બૉહરનો સિદ્ધાંત — guzerate" lang="gu" hreflang="gu" data-title="બૉહરનો સિદ્ધાંત" data-language-autonym="ગુજરાતી" data-language-local-name="guzerate" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%95%D7%93%D7%9C_%D7%94%D7%90%D7%98%D7%95%D7%9D_%D7%A9%D7%9C_%D7%91%D7%95%D7%94%D7%A8" title="מודל האטום של בוהר — hebraico" lang="he" hreflang="he" data-title="מודל האטום של בוהר" data-language-autonym="עברית" data-language-local-name="hebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A5%8B%E0%A4%B0_%E0%A4%95%E0%A4%BE_%E0%A4%AA%E0%A4%B0%E0%A4%AE%E0%A4%BE%E0%A4%A3%E0%A5%81_%E0%A4%AE%E0%A5%89%E0%A4%A1%E0%A4%B2" title="बोर का परमाणु मॉडल — hindi" lang="hi" hreflang="hi" data-title="बोर का परमाणु मॉडल" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Bohrov_model_atoma" title="Bohrov model atoma — croata" lang="hr" hreflang="hr" data-title="Bohrov model atoma" data-language-autonym="Hrvatski" data-language-local-name="croata" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Bohr-f%C3%A9le_atommodell" title="Bohr-féle atommodell — húngaro" lang="hu" hreflang="hu" data-title="Bohr-féle atommodell" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%B8%D6%80%D5%AB_%D5%B4%D5%B8%D5%A4%D5%A5%D5%AC" title="Բորի մոդել — arménio" lang="hy" hreflang="hy" data-title="Բորի մոդել" data-language-autonym="Հայերեն" data-language-local-name="arménio" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Model_Bohr" title="Model Bohr — indonésio" lang="id" hreflang="id" data-title="Model Bohr" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Modello_atomico_di_Bohr" title="Modello atomico di Bohr — italiano" lang="it" hreflang="it" data-title="Modello atomico di Bohr" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9C%E3%83%BC%E3%82%A2%E3%81%AE%E5%8E%9F%E5%AD%90%E6%A8%A1%E5%9E%8B" title="ボーアの原子模型 — japonês" lang="ja" hreflang="ja" data-title="ボーアの原子模型" data-language-autonym="日本語" data-language-local-name="japonês" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%91%E1%83%9D%E1%83%A0%E1%83%98%E1%83%A1_%E1%83%9B%E1%83%9D%E1%83%93%E1%83%94%E1%83%9A%E1%83%98" title="ბორის მოდელი — georgiano" lang="ka" hreflang="ka" data-title="ბორის მოდელი" data-language-autonym="ქართული" data-language-local-name="georgiano" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%91%D0%BE%D1%80_%D2%9B%D0%B0%D2%93%D0%B8%D0%B4%D0%B0%D0%BB%D0%B0%D1%80%D1%8B" title="Бор қағидалары — cazaque" lang="kk" hreflang="kk" data-title="Бор қағидалары" data-language-autonym="Қазақша" data-language-local-name="cazaque" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B3%B4%EC%96%B4_%EB%AA%A8%ED%98%95" title="보어 모형 — coreano" lang="ko" hreflang="ko" data-title="보어 모형" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Boro_teorija" title="Boro teorija — lituano" lang="lt" hreflang="lt" data-title="Boro teorija" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Bora_atoma_uzb%C5%ABves_modelis" title="Bora atoma uzbūves modelis — letão" lang="lv" hreflang="lv" data-title="Bora atoma uzbūves modelis" data-language-autonym="Latviešu" data-language-local-name="letão" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%91%D0%BE%D1%80%D0%BE%D0%B2_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB" title="Боров модел — macedónio" lang="mk" hreflang="mk" data-title="Боров модел" data-language-autonym="Македонски" data-language-local-name="macedónio" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AC%E0%B5%8B%E0%B5%BC_%E0%B4%AE%E0%B4%BE%E0%B4%A4%E0%B5%83%E0%B4%95" title="ബോർ മാതൃക — malaiala" lang="ml" hreflang="ml" data-title="ബോർ മാതൃക" data-language-autonym="മലയാളം" data-language-local-name="malaiala" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AC%E0%A5%8B%E0%A4%B0%E0%A4%9A%E0%A5%80_%E0%A4%85%E0%A4%A3%E0%A5%82%E0%A4%9A%E0%A5%80_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A5%83%E0%A4%A4%E0%A5%80" title="बोरची अणूची प्रतिकृती — marata" lang="mr" hreflang="mr" data-title="बोरची अणूची प्रतिकृती" data-language-autonym="मराठी" data-language-local-name="marata" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Model_Bohr" title="Model Bohr — malaio" lang="ms" hreflang="ms" data-title="Model Bohr" data-language-autonym="Bahasa Melayu" data-language-local-name="malaio" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%98%E1%80%AD%E1%80%AF%E1%80%B8%E1%80%A1%E1%80%AC%E1%81%8F_%E1%80%A1%E1%80%80%E1%80%BA%E1%80%90%E1%80%99%E1%80%BA%E1%80%95%E1%80%AF%E1%80%B6%E1%80%85%E1%80%B6" title="ဘိုးအာ၏ အက်တမ်ပုံစံ — birmanês" lang="my" hreflang="my" data-title="ဘိုးအာ၏ အက်တမ်ပုံစံ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmanês" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Atoommodel_van_Bohr" title="Atoommodel van Bohr — neerlandês" lang="nl" hreflang="nl" data-title="Atoommodel van Bohr" data-language-autonym="Nederlands" data-language-local-name="neerlandês" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Atommodell" title="Atommodell — norueguês nynorsk" lang="nn" hreflang="nn" data-title="Atommodell" data-language-autonym="Norsk nynorsk" data-language-local-name="norueguês nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Bohrs_atommodell" title="Bohrs atommodell — norueguês bokmål" lang="nb" hreflang="nb" data-title="Bohrs atommodell" data-language-autonym="Norsk bokmål" data-language-local-name="norueguês bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AC%E0%A9%8B%E0%A8%B9%E0%A8%B0_%E0%A8%AE%E0%A8%BE%E0%A8%A1%E0%A8%B2" title="ਬੋਹਰ ਮਾਡਲ — panjabi" lang="pa" hreflang="pa" data-title="ਬੋਹਰ ਮਾਡਲ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Model_atomu_Bohra" title="Model atomu Bohra — polaco" lang="pl" hreflang="pl" data-title="Model atomu Bohra" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%A8%D9%88%DB%81%D8%B1_%D9%85%D8%A7%DA%88%D9%84" title="بوہر ماڈل — Western Punjabi" lang="pnb" hreflang="pnb" data-title="بوہر ماڈل" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Modelul_atomic_Bohr" title="Modelul atomic Bohr — romeno" lang="ro" hreflang="ro" data-title="Modelul atomic Bohr" data-language-autonym="Română" data-language-local-name="romeno" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%91%D0%BE%D1%80%D0%BE%D0%B2%D1%81%D0%BA%D0%B0%D1%8F_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB%D1%8C_%D0%B0%D1%82%D0%BE%D0%BC%D0%B0" title="Боровская модель атома — russo" lang="ru" hreflang="ru" data-title="Боровская модель атома" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Mud%C3%A8llu_at%C3%B2micu_di_Bohr" title="Mudèllu atòmicu di Bohr — siciliano" lang="scn" hreflang="scn" data-title="Mudèllu atòmicu di Bohr" data-language-autonym="Sicilianu" data-language-local-name="siciliano" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Bohr_model" title="Bohr model — scots" lang="sco" hreflang="sco" data-title="Bohr model" data-language-autonym="Scots" data-language-local-name="scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Bohrov_model_atoma" title="Bohrov model atoma — servo-croata" lang="sh" hreflang="sh" data-title="Bohrov model atoma" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="servo-croata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Bohr_model" title="Bohr model — Simple English" lang="en-simple" hreflang="en-simple" data-title="Bohr model" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Bohrov_model_at%C3%B3mu" title="Bohrov model atómu — eslovaco" lang="sk" hreflang="sk" data-title="Bohrov model atómu" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Bohrov_model_atoma" title="Bohrov model atoma — esloveno" lang="sl" hreflang="sl" data-title="Bohrov model atoma" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D0%BE%D1%80%D0%BE%D0%B2_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB_%D0%B0%D1%82%D0%BE%D0%BC%D0%B0" title="Боров модел атома — sérvio" lang="sr" hreflang="sr" data-title="Боров модел атома" data-language-autonym="Српски / srpski" data-language-local-name="sérvio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Bohrs_atommodell" title="Bohrs atommodell — sueco" lang="sv" hreflang="sv" data-title="Bohrs atommodell" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AF%8B%E0%AE%B0%E0%AF%8D_%E0%AE%85%E0%AE%A3%E0%AF%81_%E0%AE%AE%E0%AE%BE%E0%AE%A4%E0%AE%BF%E0%AE%B0%E0%AE%BF" title="போர் அணு மாதிரி — tâmil" lang="ta" hreflang="ta" data-title="போர் அணு மாதிரி" data-language-autonym="தமிழ்" data-language-local-name="tâmil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%AC%E0%B1%8B%E0%B0%B0%E0%B1%8D_%E0%B0%A8%E0%B0%AE%E0%B1%82%E0%B0%A8%E0%B0%BE" title="బోర్ నమూనా — telugu" lang="te" hreflang="te" data-title="బోర్ నమూనా" data-language-autonym="తెలుగు" data-language-local-name="telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%9A%E0%B8%9A%E0%B8%88%E0%B8%B3%E0%B8%A5%E0%B8%AD%E0%B8%87%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B9%82%E0%B8%9B%E0%B8%A3%E0%B9%8C" title="แบบจำลองของโปร์ — tailandês" lang="th" hreflang="th" data-title="แบบจำลองของโปร์" data-language-autonym="ไทย" data-language-local-name="tailandês" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Modelong_Bohr" title="Modelong Bohr — tagalo" lang="tl" hreflang="tl" data-title="Modelong Bohr" data-language-autonym="Tagalog" data-language-local-name="tagalo" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Bohr_modeli" title="Bohr modeli — turco" lang="tr" hreflang="tr" data-title="Bohr modeli" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D1%82%D0%BE%D0%BC%D0%BD%D0%B0_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB%D1%8C_%D0%91%D0%BE%D1%80%D0%B0" title="Атомна модель Бора — ucraniano" lang="uk" hreflang="uk" data-title="Атомна модель Бора" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Bor_postulatlari" title="Bor postulatlari — usbeque" lang="uz" hreflang="uz" data-title="Bor postulatlari" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbeque" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/M%C3%B4_h%C3%ACnh_Bohr" title="Mô hình Bohr — vietnamita" lang="vi" hreflang="vi" data-title="Mô hình Bohr" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%8E%BB%E5%B0%94%E6%A8%A1%E5%9E%8B" title="玻尔模型 — wu" lang="wuu" hreflang="wuu" data-title="玻尔模型" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%8E%BB%E5%B0%94%E6%A8%A1%E5%9E%8B" title="玻尔模型 — chinês" lang="zh" hreflang="zh" data-title="玻尔模型" data-language-autonym="中文" data-language-local-name="chinês" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%8E%BB%E7%88%BE%E6%A8%A1%E5%9E%8B" title="玻爾模型 — Literary Chinese" lang="lzh" hreflang="lzh" data-title="玻爾模型" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%8E%BB%E7%88%BE%E6%A8%A1%E5%9E%8B" title="玻爾模型 — cantonês" lang="yue" hreflang="yue" data-title="玻爾模型" data-language-autonym="粵語" data-language-local-name="cantonês" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q172948#sitelinks-wikipedia" title="Editar hiperligações interlínguas" class="wbc-editpage">Editar hiperligações</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espaços nominais"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%C3%81tomo_de_Bohr" title="Ver a página de conteúdo [c]" accesskey="c"><span>Artigo</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discuss%C3%A3o:%C3%81tomo_de_Bohr" rel="discussion" title="Discussão sobre o conteúdo da página [t]" accesskey="t"><span>Discussão</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" 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.tmbox.mbox-small{clear:right;float:right;margin:4px 0 4px 1em;width:238px}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-night .mw-parser-output .tmbox-speedy{background-color:#310402}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-os .mw-parser-output .tmbox-speedy{background-color:#310402}}body.skin--responsive .mw-parser-output table.tmbox img{max-width:none!important}</style><table class="box-Mais_notas plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/Ficheiro:Question_book-new.svg" class="mw-file-description"><img alt="Esta página cita fontes, mas não cobrem todo o conteúdo" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">Esta página <a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo/Cite_as_fontes" title="Wikipédia:Livro de estilo/Cite as fontes">cita fontes</a>, mas que <b><a href="/wiki/Wikip%C3%A9dia:V" class="mw-redirect" title="Wikipédia:V">não cobrem</a> todo o conteúdo</b>.<span class="hide-when-compact"> Ajude a <a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo/Refer%C3%AAncias_e_notas_de_rodap%C3%A9" title="Wikipédia:Livro de estilo/Referências e notas de rodapé">inserir referências</a> (<small><i>Encontre fontes:</i> <span class="plainlinks"><a rel="nofollow" class="external text" href="https://wikipedialibrary.wmflabs.org/">ABW</a>  •  <a rel="nofollow" class="external text" href="https://www.periodicos.capes.gov.br">CAPES</a>  •  <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&as_epq=%C3%81tomo+de+Bohr">Google</a> (<a rel="nofollow" class="external text" href="https://www.google.com/search?hl=pt&tbm=nws&q=%C3%81tomo+de+Bohr&oq=%C3%81tomo+de+Bohr">notícias</a> • <a rel="nofollow" class="external text" href="http://books.google.com/books?&as_brr=0&as_epq=%C3%81tomo+de+Bohr">livros</a> • <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?hl=pt&q=%C3%81tomo+de+Bohr">acadêmico</a>)</span></small>).</span> <small class="date-container"><i>(<span class="date">Dezembro de 2020</span>)</i></small></div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Bohr_atom_model_(mul).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Bohr_atom_model_%28mul%29.svg/langpt-320px-Bohr_atom_model_%28mul%29.svg.png" decoding="async" width="320" height="320" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Bohr_atom_model_%28mul%29.svg/langpt-480px-Bohr_atom_model_%28mul%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/Bohr_atom_model_%28mul%29.svg/langpt-640px-Bohr_atom_model_%28mul%29.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>Modelo do átomo de Bohr.</figcaption></figure> <p>Na <a href="/wiki/F%C3%ADsica_at%C3%B4mica" title="Física atômica">física atômica</a>, o <b>átomo de Bohr</b> é um modelo que descreve o <a href="/wiki/%C3%81tomo" title="Átomo">átomo</a> como um núcleo pequeno e carregado positivamente cercado por <a href="/wiki/El%C3%A9trons" class="mw-redirect" title="Elétrons">elétrons</a> em <a href="/wiki/%C3%93rbita" title="Órbita">órbita</a> circular.<sup id="cite_ref-Brasil_Escola_1-0" class="reference"><a href="#cite_note-Brasil_Escola-1"><span>[</span>1<span>]</span></a></sup> </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Bohr-model_v_Oxygen.png" class="mw-file-description"><img alt="Um modelo do átomo de Bohr" src="//upload.wikimedia.org/wikipedia/commons/8/80/Bohr-model_v_Oxygen.png" decoding="async" width="200" height="212" class="mw-file-element" data-file-width="200" data-file-height="212" /></a><figcaption> Um exemplo do modelo atômico de Bohr </figcaption></figure> <p><a href="/wiki/Ernest_Rutherford" title="Ernest Rutherford">Ernest Rutherford</a>, no início do século XX, realiza o experimento conhecido como <a href="/wiki/Espalhamento_de_Rutherford" class="mw-redirect" title="Espalhamento de Rutherford">espalhamento de Rutherford</a> ,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span>[</span>2<span>]</span></a></sup> no qual ele incidiu um feixe de <a href="/wiki/Part%C3%ADculas_alfa" class="mw-redirect" title="Partículas alfa">partículas alfa</a> (α) sobre uma folha de ouro e observou que, ao contrário do que era esperado - que as partículas deveriam ser refletidas pelos <a href="/wiki/%C3%81tomos" class="mw-redirect" title="Átomos">átomos</a> de ouro considerados maciços até então -, muitas partículas atravessaram a folha de ouro e outras sofreram desvios. A partir da análise dessa experiência, afirmou que <a href="/wiki/%C3%81tomos" class="mw-redirect" title="Átomos">átomos</a> eram constituídos de uma nuvem difusa de elétrons carregados negativamente que circundavam um <a href="/wiki/N%C3%BAcleo_at%C3%B4mico" class="mw-redirect" title="Núcleo atômico">núcleo atômico</a> denso, pequeno e carregado positivamente.<sup id="cite_ref-Brasil_Escola_1-1" class="reference"><a href="#cite_note-Brasil_Escola-1"><span>[</span>1<span>]</span></a></sup> </p><p>A partir dessa descrição, é fácil deixar-se induzir por uma concepção de um modelo planetário para o átomo, com elétrons orbitando ao redor do "núcleo-sol". Porém, a aberração mais séria desse modelo é a perda de energia dos elétrons através da <a href="/wiki/Radia%C3%A7%C3%A3o_s%C3%ADncrotron" title="Radiação síncrotron">radiação síncrotron</a>: uma partícula carregada eletricamente ao ser acelerada emite radiações eletromagnéticas que têm energia; fosse assim, ao orbitar em torno do núcleo atômico, o elétron deveria gradativamente emitir radiações e cada vez mais aproximar-se do núcleo, em uma órbita espiralada, até finalmente chocar-se contra ele. Um cálculo rápido mostra que isso deveria ocorrer quase que instantaneamente. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Postulado_de_Bohr">Postulado de Bohr</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%81tomo_de_Bohr&veaction=edit&section=1" title="Editar secção: Postulado de Bohr" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%81tomo_de_Bohr&action=edit&section=1" title="Editar código-fonte da secção: Postulado de Bohr"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Niels_Bohr.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Niels_Bohr.jpg/200px-Niels_Bohr.jpg" decoding="async" width="200" height="281" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Niels_Bohr.jpg/300px-Niels_Bohr.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Niels_Bohr.jpg/400px-Niels_Bohr.jpg 2x" data-file-width="1456" data-file-height="2048" /></a><figcaption>Niels Bohr</figcaption></figure> <p>Através das descrições quânticas da <a href="/wiki/Radia%C3%A7%C3%A3o_eletromagn%C3%A9tica" title="Radiação eletromagnética">radiação eletromagnética</a> propostas por <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> e <a href="/wiki/Max_Planck" title="Max Planck">Max Planck</a>, o físico dinamarquês <a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a> desenvolve seu modelo atômico a partir de quatro postulados:<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span>[</span>3<span>]</span></a></sup> </p> <ol><li>Os elétrons que circundam o núcleo atômico existem em órbitas que têm níveis de energia <a href="/wiki/Teoria_dos_quanta" class="mw-redirect" title="Teoria dos quanta">quantizados</a>.</li> <li>A energia total do elétron (cinética e potencial) não pode apresentar um valor qualquer e sim, valores múltiplos de um quantum.<sup id="cite_ref-Brasil_Escola_1-2" class="reference"><a href="#cite_note-Brasil_Escola-1"><span>[</span>1<span>]</span></a></sup></li> <li>Quando ocorre o salto de um elétron entre órbitas, a diferença de energia é emitida (ou suprida) por um simples <a href="/wiki/Teoria_dos_quanta" class="mw-redirect" title="Teoria dos quanta">quantum</a> de luz (também chamado de <a href="/wiki/F%C3%B3ton" class="mw-redirect" title="Fóton">fóton</a>), que tem energia exatamente igual à diferença de energia entre as órbitas em questão.</li> <li>As órbitas permitidas dependem de valores quantizados (bem definidos) de <a href="/wiki/Momento_angular" title="Momento angular">momento angular</a> orbital, <b>L</b>, de acordo com a equação</li></ol> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =n\cdot \hbar =n\cdot {h \over 2\pi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <mi>n</mi> <mo>⋅</mo> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mi>n</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =n\cdot \hbar =n\cdot {h \over 2\pi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34c35dfb5a494ac6f71484544eeda82afc631d2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.59ex; height:5.343ex;" alt="{\displaystyle \mathbf {L} =n\cdot \hbar =n\cdot {h \over 2\pi }}"></span> </p><p>onde n = 1, 2, 3, ... é chamado de <b>número quântico principal</b> e <i>h</i> é a <a href="/wiki/Constante_de_Planck" title="Constante de Planck">constante de Planck</a>.<sup id="cite_ref-ufsm_4-0" class="reference"><a href="#cite_note-ufsm-4"><span>[</span>4<span>]</span></a></sup> </p><p>A regra 4 afirma que o menor valor possível de <i>n</i> é 1. Isto corresponde ao menor raio atômico possível, de 0,0529 nm, valor também conhecido como <a href="/wiki/Raio_de_Bohr" title="Raio de Bohr">raio de Bohr</a>. Nenhum elétron pode aproximar-se mais do núcleo do que essa distância. </p><p>O modelo de átomo de Bohr é às vezes chamado de <b>modelo semi-clássico do átomo</b>, porque agrega algumas condições de quantização primitiva a um tratamento de <a href="/wiki/Mec%C3%A2nica_cl%C3%A1ssica" title="Mecânica clássica">mecânica clássica</a>. Este modelo certamente não é uma descrição mecânica quântica completa do átomo. A regra 2 diz que as leis da mecânica clássica não valem durante um salto quântico, mas não explica que leis devem substituir a mecânica clássica nesta circunstância. A regra 4 diz que o momento angular é quantizado, mas não diz por quê. </p> <div class="mw-heading mw-heading2"><h2 id="Expressão_para_o_raio_de_Bohr"><span id="Express.C3.A3o_para_o_raio_de_Bohr"></span>Expressão para o raio de Bohr</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%81tomo_de_Bohr&veaction=edit&section=2" title="Editar secção: Expressão para o raio de Bohr" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%81tomo_de_Bohr&action=edit&section=2" title="Editar código-fonte da secção: Expressão para o raio de Bohr"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Considere o caso de um íon com a carga do núcleo sendo <i>Ze</i> e um eléctron movendo-se com velocidade constante <i>v</i> ao longo de um círculo de raio r com centro no núcleo.<sup id="cite_ref-pt.scribd.com_5-0" class="reference"><a href="#cite_note-pt.scribd.com-5"><span>[</span>5<span>]</span></a></sup> </p><p>A <a href="/wiki/For%C3%A7a_de_Coulomb" class="mw-redirect" title="Força de Coulomb">força de Coulomb</a> sobre o electrão é </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F={\frac {Z.e^{2}}{4\pi \varepsilon _{o}r^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mo>.</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F={\frac {Z.e^{2}}{4\pi \varepsilon _{o}r^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/122184c2f92e7cc3c1e46e3d4e09ee93da050f86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.386ex; height:6.343ex;" alt="{\displaystyle F={\frac {Z.e^{2}}{4\pi \varepsilon _{o}r^{2}}}}"></span> </p><p>A força de Coulomb é a <a href="/wiki/For%C3%A7a_centr%C3%ADpeta" title="Força centrípeta">força centrípeta</a>. Logo: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F={\frac {Z.e^{2}}{4\pi \varepsilon _{o}r^{2}}}={\frac {m.v^{2}}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mo>.</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mo>.</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F={\frac {Z.e^{2}}{4\pi \varepsilon _{o}r^{2}}}={\frac {m.v^{2}}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbaf47ef6ab284304f7d2af5362253bd7eb00380" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.577ex; height:6.343ex;" alt="{\displaystyle F={\frac {Z.e^{2}}{4\pi \varepsilon _{o}r^{2}}}={\frac {m.v^{2}}{r}}}"></span> </p><p>Usando a regra de quantização do momento angular de Bohr: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=m.v.r={\frac {h}{2\pi }}=\hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>m</mi> <mo>.</mo> <mi>v</mi> <mo>.</mo> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=m.v.r={\frac {h}{2\pi }}=\hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b27d0c84f9238f57e4814ba18ce8eda2aaf89d48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.8ex; height:5.343ex;" alt="{\displaystyle L=m.v.r={\frac {h}{2\pi }}=\hbar }"></span> </p><p>Temos para o n-ésimo raio de Bohr: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{n}={\frac {\varepsilon _{o}n^{2}h^{2}}{\pi .mZe^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>π</mi> <mo>.</mo> <mi>m</mi> <mi>Z</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}={\frac {\varepsilon _{o}n^{2}h^{2}}{\pi .mZe^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/440c1e6ccc069b93164df671d96cc6611c675f00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.426ex; height:6.009ex;" alt="{\displaystyle r_{n}={\frac {\varepsilon _{o}n^{2}h^{2}}{\pi .mZe^{2}}}}"></span> </p><p>E a velocidade do electrão na n-ésima órbita: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{n}={\frac {Z.{e}^{2}}{2\varepsilon _{o}h.n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mo>.</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mi>h</mi> <mo>.</mo> <mi>n</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{n}={\frac {Z.{e}^{2}}{2\varepsilon _{o}h.n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7edc46da6f3f66691415469d68ea1ce01138061" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.324ex; height:6.176ex;" alt="{\displaystyle v_{n}={\frac {Z.{e}^{2}}{2\varepsilon _{o}h.n}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Equação_de_Rydberg"><span id="Equa.C3.A7.C3.A3o_de_Rydberg"></span>Equação de Rydberg</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%81tomo_de_Bohr&veaction=edit&section=3" title="Editar secção: Equação de Rydberg" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%81tomo_de_Bohr&action=edit&section=3" title="Editar código-fonte da secção: Equação de Rydberg"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Equa%C3%A7%C3%A3o_de_Rydberg" class="mw-redirect" title="Equação de Rydberg">equação de Rydberg</a>, que era conhecida empiricamente antes da equação de Bohr, está agora na teoria de Bohr para descrever as energias de transições entre um <a href="/wiki/N%C3%ADvel_de_energia" title="Nível de energia">nível de energia</a> orbital e outro. A equação de Bohr dá o valor numérico da já conhecida e medida constante de Rydberg, e agora em termos de uma constante fundamental da natureza, inclui-se a carga do <a href="/wiki/El%C3%A9tron" title="Elétron">elétron</a> e a <a href="/wiki/Constante_de_Planck" title="Constante de Planck">constante de Planck</a>.<sup id="cite_ref-Brasil_Escola_1-3" class="reference"><a href="#cite_note-Brasil_Escola-1"><span>[</span>1<span>]</span></a></sup> Quando o elétron é movido do seu nível de energia original para um superior e, em seguida, recua um nível retornando à posição original, resulta num <a href="/wiki/F%C3%B3ton" class="mw-redirect" title="Fóton">fóton</a> a ser emitido. Usando a fórmula derivada para os diferentes níveis de energia de <a href="/wiki/Hidrog%C3%AAnio" class="mw-redirect" title="Hidrogênio">hidrogênio</a>, determinam-se os comprimentos de onda da luz que um átomo de hidrogênio pode emitir. A energia de um fóton emitido por um átomo de hidrogênio é determinado pela diferença de dois níveis de energia de hidrogênio:<sup id="cite_ref-Brasil_Escola_1-4" class="reference"><a href="#cite_note-Brasil_Escola-1"><span>[</span>1<span>]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=E_{f}-E_{i}=R_{E}\left({\frac {1}{n_{i}^{2}}}-{\frac {1}{n_{f}^{2}}}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=E_{f}-E_{i}=R_{E}\left({\frac {1}{n_{i}^{2}}}-{\frac {1}{n_{f}^{2}}}\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1721bdbef2ccdcb7a106c79e266a5e1f7de6c771" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:33.379ex; height:7.676ex;" alt="{\displaystyle E=E_{f}-E_{i}=R_{E}\left({\frac {1}{n_{i}^{2}}}-{\frac {1}{n_{f}^{2}}}\right)\,}"></span> </p><p>onde <i>n</i><sub><i>i</i></sub> é o nível inicial , e <i>n</i><sub><i>f</i></sub> é o nível final de energia. Uma vez que a energia de um fóton está </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\frac {hc}{\lambda }},\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mi>c</mi> </mrow> <mi>λ</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\frac {hc}{\lambda }},\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/152c0237b2ed4ba8ec478532d55e6240ed3d06fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.09ex; height:5.509ex;" alt="{\displaystyle E={\frac {hc}{\lambda }},\,}"></span> </p><p>o comprimento de onda do fóton emitido é dada pela </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\lambda }}=R\left({\frac {1}{n_{i}^{2}}}-{\frac {1}{n_{f}^{2}}}\right).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> </mrow> <mo>=</mo> <mi>R</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\lambda }}=R\left({\frac {1}{n_{i}^{2}}}-{\frac {1}{n_{f}^{2}}}\right).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a5a8a9ae7d468cdfcdf6f5c6e2d88d0f01c7253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.036ex; height:7.676ex;" alt="{\displaystyle {\frac {1}{\lambda }}=R\left({\frac {1}{n_{i}^{2}}}-{\frac {1}{n_{f}^{2}}}\right).\,}"></span> </p><p>Isto é conhecido como a equação de Rydberg, e o R da constante Rydberg é <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{E}/hc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>h</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{E}/hc}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df5534faeb57a7da4643fa7b4788d9f6af123938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.76ex; height:2.843ex;" alt="{\displaystyle R_{E}/hc}"></span> , ou <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{E}/2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{E}/2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca48a8e8a567abea035d28e6141ec8684035958a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.909ex; height:2.843ex;" alt="{\displaystyle R_{E}/2\pi }"></span> em <a href="/wiki/Unidades_naturais" title="Unidades naturais">unidades naturais</a> . Esta equação foi conhecida no século XIX pelos cientistas que estudavam a <a href="/wiki/Espectroscopia" title="Espectroscopia">espectroscopia</a>, mas não havia nenhuma explicação teórica para estas equações ou uma previsão teórica para o valor de R, até Bohr. A propósito, a derivação de Bohr da constante Rydberg, bem como o acordo concomitante da equação de Bohr com as experimentalmente observadas <a href="/wiki/Linhas_espectrais" class="mw-redirect" title="Linhas espectrais">linhas espectrais</a> de <a href="/wiki/S%C3%A9rie_de_Lyman" title="Série de Lyman">Lyman</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{f}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{f}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f83d37821026f7f0e89b101fbbfd1a3c129f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.792ex; height:2.843ex;" alt="{\displaystyle n_{f}=1}"></span>), <a href="/wiki/S%C3%A9rie_de_Balmer" title="Série de Balmer">Balmer</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{f}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{f}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/236ca4acf61a204f34691106085db74b73e0d527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.792ex; height:2.843ex;" alt="{\displaystyle n_{f}=2}"></span>), e <a href="/wiki/S%C3%A9rie_de_Paschen" title="Série de Paschen">Paschen</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{f}=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{f}=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24cbc96202009351a7b62640729a01119eb1b144" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.792ex; height:2.843ex;" alt="{\displaystyle n_{f}=3}"></span>), e a previsão teórica bem sucedida de outras linhas ainda não observadas, foi uma das razões para o seu modelo ser imediatamente aceito. Para aplicar em átomos com mais de um elétron, a equação de Rydberg pode ser modificada pela substituição de "Z" por "Z - b" ou "n" por "n - b", em que b é uma constante que representa o efeito de triagem devido a outros elétrons. Isto foi estabelecido empiricamente antes de Bohr apresentar seu modelo.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span>[</span>6<span>]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Níveis_energéticos_dos_elétrons_em_um_átomo_de_hidrogênio"><span id="N.C3.ADveis_energ.C3.A9ticos_dos_el.C3.A9trons_em_um_.C3.A1tomo_de_hidrog.C3.AAnio"></span>Níveis energéticos dos elétrons em um átomo de hidrogênio</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%81tomo_de_Bohr&veaction=edit&section=4" title="Editar secção: Níveis energéticos dos elétrons em um átomo de hidrogênio" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%81tomo_de_Bohr&action=edit&section=4" title="Editar código-fonte da secção: Níveis energéticos dos elétrons em um átomo de hidrogênio"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O modelo do <b>átomo de Bohr</b> explica bem o comportamento do átomo de <a href="/wiki/Hidrog%C3%AAnio" class="mw-redirect" title="Hidrogênio">hidrogênio</a> e do átomo de <a href="/wiki/H%C3%A9lio" title="Hélio">hélio</a> ionizado, mas é insuficiente para átomos com mais de um elétron. </p><p>Segue abaixo um desenvolvimento do modelo de Bohr que demonstra os níveis de energia no hidrogênio. </p><p>Sejam as seguintes convenções: </p><p>1. Todas as partículas são como ondas e, assim, o <a href="/wiki/Comprimento_de_onda" title="Comprimento de onda">comprimento de onda</a> do <a href="/wiki/El%C3%A9tron" title="Elétron">elétron</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>, está relacionado à sua velocidade por </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ={h \over m_{e}v}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mi>v</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ={h \over m_{e}v}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb2a93144d2080ff4baa5d149081c8ea711ca790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.456ex; height:5.676ex;" alt="{\displaystyle \lambda ={h \over m_{e}v}}"></span> </p><p>onde <i>h</i> é a <a href="/wiki/Constante_de_Planck" title="Constante de Planck">constante de Planck</a> e <i>m<sub>e</sub></i>, a <a href="/wiki/El%C3%A9tron" title="Elétron">massa do elétron</a>. Bohr não tinha levantado esta hipótese porque só depois é que foi proposto o conceito associado a esta afirmação (veja <a href="/wiki/Dualidade_onda-part%C3%ADcula" class="mw-redirect" title="Dualidade onda-partícula">dualidade onda-partícula</a>). Porém, permite chegar na próxima afirmação. </p><p>2. A <a href="/wiki/Circunfer%C3%AAncia" title="Circunferência">circunferência</a> da órbita do elétron deve ser um múltiplo inteiro de seu comprimento de onda: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi r=n\lambda \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mo>=</mo> <mi>n</mi> <mi>λ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi r=n\lambda \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f3fba71b630e07fc6212277a6c8bd5b16c01e7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.779ex; height:2.176ex;" alt="{\displaystyle 2\pi r=n\lambda \,}"></span> </p><p>onde <i>r</i> é o raio da órbita do elétron e <i>n</i>, um número inteiro positivo. </p><p>3. O elétron mantém-se em órbita por <a href="/wiki/Lei_de_Coulomb" title="Lei de Coulomb">forças eletrostáticas</a>. Isto é, a força eletrostática é igual à <a href="/wiki/For%C3%A7a_centr%C3%ADpeta" title="Força centrípeta">força centrípeta</a>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {kq_{e}^{2} \over r^{2}}={m_{e}v^{2} \over r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <msubsup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {kq_{e}^{2} \over r^{2}}={m_{e}v^{2} \over r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2571cf49a9821d2726639c0993ffa704f09e5f1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.336ex; height:6.009ex;" alt="{\displaystyle {kq_{e}^{2} \over r^{2}}={m_{e}v^{2} \over r}}"></span> </p><p>onde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={1 \over 4\pi \epsilon _{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={1 \over 4\pi \epsilon _{0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7499d5f27dd8684a5c3ad154a8bfccef7988b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.639ex; height:5.676ex;" alt="{\displaystyle k={1 \over 4\pi \epsilon _{0}}}"></span> e <i>q<sub>e</sub></i>, a <a href="/wiki/El%C3%A9tron" title="Elétron">carga elétrica do elétron</a>. </p><p>Temos três equações e três incógnitas: <i>v</i>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> e <i>r</i>. Depois de manipulações algébricas para obter <i>v</i> em função das outras variáveis, pode-se substituir as soluções na equação da energia total do elétron: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=E_{cinetica}+E_{potencial}={1 \over 2}m_{e}v^{2}-{kq_{e}^{2} \over r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>i</mi> <mi>n</mi> <mi>e</mi> <mi>t</mi> <mi>i</mi> <mi>c</mi> <mi>a</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>o</mi> <mi>t</mi> <mi>e</mi> <mi>n</mi> <mi>c</mi> <mi>i</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <msubsup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=E_{cinetica}+E_{potencial}={1 \over 2}m_{e}v^{2}-{kq_{e}^{2} \over r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04f1b8937c07e0b65cb90c1263f31b640076d5b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.332ex; height:5.676ex;" alt="{\displaystyle E=E_{cinetica}+E_{potencial}={1 \over 2}m_{e}v^{2}-{kq_{e}^{2} \over r}}"></span> </p><p>Pelo <a href="/wiki/Teorema_do_virial" title="Teorema do virial">teorema do virial</a>, a energia total simplifica-se para </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=-{1 \over 2}m_{e}v^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=-{1 \over 2}m_{e}v^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d72718185fdf81ac8d697194bce77434ff4d9884" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.901ex; height:5.176ex;" alt="{\displaystyle E=-{1 \over 2}m_{e}v^{2}}"></span> </p> <table> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c621cabe4418802f7f26e069a046cd2270bff41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.321ex; height:2.509ex;" alt="{\displaystyle E_{n}\,}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =-2\pi ^{2}k^{2}\left({\frac {m_{e}q_{e}^{4}}{h^{2}}}\right){\frac {1}{n^{2}}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mo>−</mo> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msubsup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =-2\pi ^{2}k^{2}\left({\frac {m_{e}q_{e}^{4}}{h^{2}}}\right){\frac {1}{n^{2}}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f19737c625c11c0be0fb48c35fc4e1bc7176d6de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.954ex; height:6.343ex;" alt="{\displaystyle =-2\pi ^{2}k^{2}\left({\frac {m_{e}q_{e}^{4}}{h^{2}}}\right){\frac {1}{n^{2}}}\,}"></span> </td></tr> <tr> <td> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {-m_{e}q_{e}^{4}}{8h^{2}\epsilon _{0}^{2}}}{\frac {1}{n^{2}}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msubsup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> </mrow> <mrow> <mn>8</mn> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {-m_{e}q_{e}^{4}}{8h^{2}\epsilon _{0}^{2}}}{\frac {1}{n^{2}}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ed016ae6e8b88071d4edc83d60962dda77b8d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.942ex; height:6.843ex;" alt="{\displaystyle ={\frac {-m_{e}q_{e}^{4}}{8h^{2}\epsilon _{0}^{2}}}{\frac {1}{n^{2}}}\,}"></span> </td></tr></tbody></table> <p>Ou, depois de substituídos os valores das constantes:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span>[</span>7<span>]</span></a></sup> </p> <table cellpadding="2" style="border:2px solid #ccccff"> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}={\frac {-13.6\ \mathbf {eV} }{n^{2}}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>13.6</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> <mi mathvariant="bold">V</mi> </mrow> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}={\frac {-13.6\ \mathbf {eV} }{n^{2}}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a41499d62103559a6524fd55b24c93b34d1dade7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.023ex; height:5.676ex;" alt="{\displaystyle E_{n}={\frac {-13.6\ \mathbf {eV} }{n^{2}}}\,}"></span> </td></tr></tbody></table> <p>Assim, o menor nível de energia do hidrogênio (n = 1) é cerca de -13.6 eV. O próximo nível de energia (n = 2) é -3.4 eV. O terceiro (n = 3), -1.51 eV, e assim por diante. Note que estas energias são menores que zero, o que significa que o elétron está em um estado de ligação com o próton presente no núcleo. Estados de energia positiva correspondem ao átomo ionizado, no qual o elétron não está mais ligado, mas em um estado desagregado. </p><p>O modelo atômico de Bohr pode ser facilmente usado para a composição do modelo atômico de <a href="/wiki/Linus_Pauling" title="Linus Pauling">Linus Pauling</a>. Apenas somando as camadas e as colocando na ordem de Pauling. </p> <div class="mw-heading mw-heading2"><h2 id="Frequência"><span id="Frequ.C3.AAncia"></span>Frequência</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%81tomo_de_Bohr&veaction=edit&section=5" title="Editar secção: Frequência" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%81tomo_de_Bohr&action=edit&section=5" title="Editar código-fonte da secção: Frequência"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A frequência orbital<sup id="cite_ref-pt.scribd.com_5-1" class="reference"><a href="#cite_note-pt.scribd.com-5"><span>[</span>5<span>]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f={\frac {\omega }{2\pi }}={\frac {v}{2\pi .r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mrow> <mn>2</mn> <mi>π</mi> <mo>.</mo> <mi>r</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f={\frac {\omega }{2\pi }}={\frac {v}{2\pi .r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64e1a19455cedef8288d12dbef4129c3c6184a51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.219ex; height:4.676ex;" alt="{\displaystyle f={\frac {\omega }{2\pi }}={\frac {v}{2\pi .r}}}"></span> (X) </p><p>Onde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> é a velocidade angular orbital do elétron. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {ke^{2}}{r^{2}}}={\frac {m.v^{2}}{r}}\Rightarrow \;m.v^{2}={\frac {ke^{2}}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mo>.</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>r</mi> </mfrac> </mrow> <mo stretchy="false">⇒</mo> <mspace width="thickmathspace" /> <mi>m</mi> <mo>.</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {ke^{2}}{r^{2}}}={\frac {m.v^{2}}{r}}\Rightarrow \;m.v^{2}={\frac {ke^{2}}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/578a527918135711f362b128933d1f4574c5bc61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:30.175ex; height:6.009ex;" alt="{\displaystyle {\frac {ke^{2}}{r^{2}}}={\frac {m.v^{2}}{r}}\Rightarrow \;m.v^{2}={\frac {ke^{2}}{r}}}"></span> </p><p>A partir da equação acima, que descreve o movimento orbital mantido pela força de Coulomb, podemos inferir: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {v}{r}}={\sqrt {\frac {k.e^{2}}{m.r^{3}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mi>r</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>k</mi> <mo>.</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>m</mi> <mo>.</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {v}{r}}={\sqrt {\frac {k.e^{2}}{m.r^{3}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5870b15e87b19183a6ebff98ccb9038b87677ca0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.399ex; height:7.509ex;" alt="{\displaystyle {\frac {v}{r}}={\sqrt {\frac {k.e^{2}}{m.r^{3}}}}}"></span> </p><p>Substituindo esta expressão na Equação (X) temos: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f={\frac {1}{2\pi }}{\sqrt {\frac {k.e^{2}}{m.r^{3}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>k</mi> <mo>.</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>m</mi> <mo>.</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f={\frac {1}{2\pi }}{\sqrt {\frac {k.e^{2}}{m.r^{3}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2ee4b22c501ee86829fcfb34679989c9d198e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.045ex; height:7.509ex;" alt="{\displaystyle f={\frac {1}{2\pi }}{\sqrt {\frac {k.e^{2}}{m.r^{3}}}}}"></span> (Z) </p><p>Para o átomo - <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Hf=7x10^{15}Hz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mi>f</mi> <mo>=</mo> <mn>7</mn> <mi>x</mi> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msup> <mi>H</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Hf=7x10^{15}Hz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0709252d5d12d4ea52ce8d6aa77c04434ae7af5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.286ex; height:3.009ex;" alt="{\displaystyle Hf=7x10^{15}Hz}"></span>, a qual está na região ultravioleta do <a href="/wiki/Espectro_electromagn%C3%A9tico" class="mw-redirect" title="Espectro electromagnético">espectro electromagnético</a>. </p><p>Se o elétron irradia, a energia E irá decrescer tornando-se cada vez negativa e a partir da Equação do raio da órbita <b>r</b> também diminui. O decréscimo em r na Equação (Z), provoca um aumento na frequência <i>f</i>. </p><p>De modo que temos um efeito de pista que quando a energia é irradiada, <i>E</i> diminui, o raio orbital <b>r</b> diminui, a qual por sua vez causa um aumento da frequência orbital <i>f</i> e aumentando continuamente a frequência irradiada. </p><p>Este modelo planetário prevê que o electrão se mova em espiral para dentro em direção ao núcleo, emitindo um espectro contínuo. Calcula-se que este processo não dure mais do que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\times 10-8s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>×</mo> <mn>10</mn> <mo>−</mo> <mn>8</mn> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times 10-8s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f803b257d574487beba28e7ce346c1d5bf3a00bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.421ex; height:2.343ex;" alt="{\displaystyle 1\times 10-8s}"></span>, um tempo muito curto na verdade. </p> <div class="mw-heading mw-heading2"><h2 id="Problemas_e_contradições"><span id="Problemas_e_contradi.C3.A7.C3.B5es"></span>Problemas e contradições</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%81tomo_de_Bohr&veaction=edit&section=6" title="Editar secção: Problemas e contradições" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%81tomo_de_Bohr&action=edit&section=6" title="Editar código-fonte da secção: Problemas e contradições"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r68971778"><table class="box-Sem_fontes plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/Ficheiro:Question_book.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Question_book.svg/40px-Question_book.svg.png" decoding="async" width="40" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Question_book.svg/60px-Question_book.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Question_book.svg/80px-Question_book.svg.png 2x" data-file-width="252" data-file-height="199" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">Esta seção <b>não cita <a href="/wiki/Wikip%C3%A9dia:Verificabilidade" title="Wikipédia:Verificabilidade">fontes confiáveis</a></b>.<span class="hide-when-compact"> Ajude a <a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo/Cite_as_fontes" title="Wikipédia:Livro de estilo/Cite as fontes">inserir referências</a>. Conteúdo não <a href="/wiki/Wikip%C3%A9dia:Verificabilidade" title="Wikipédia:Verificabilidade">verificável</a> pode ser removido.—<small><i>Encontre fontes:</i> <span class="plainlinks"><a rel="nofollow" class="external text" href="https://wikipedialibrary.wmflabs.org/">ABW</a>  •  <a rel="nofollow" class="external text" href="https://www.periodicos.capes.gov.br">CAPES</a>  •  <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&as_epq=%C3%81tomo+de+Bohr">Google</a> (<a rel="nofollow" class="external text" href="https://www.google.com/search?hl=pt&tbm=nws&q=%C3%81tomo+de+Bohr&oq=%C3%81tomo+de+Bohr">notícias</a> • <a rel="nofollow" class="external text" href="http://books.google.com/books?&as_brr=0&as_epq=%C3%81tomo+de+Bohr">livros</a> • <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?hl=pt&q=%C3%81tomo+de+Bohr">acadêmico</a>)</span></small></span> <small class="date-container"><i>(<span class="date">Maio de 2022</span>)</i></small></div></td></tr></tbody></table> <p>Algumas fragilidades e contradições do modelo ficaram claras na publicação de 1913. Outros foram mais tarde evidenciados com experimentos melhores (mais modernos) e teorias mais elaboradas da mecânica quântica. </p> <ul><li>Os postulados são justificados por qualquer princípio fundamental, mas apenas através de seu sucesso. Eles contradizem a eletrodinâmica clássica.</li> <li>O modelo de Bohr descreve o comportamento dos átomos de hidrogênio e íons com apenas um elétron. Sistemas de vários elétrons não estão incluídos.</li> <li>A teoria de relatividade não é considerada, embora seja atribuído ao elétron no estado fundamental do átomo hidrogênio, cerca de 1% da velocidade da luz.</li> <li>O átomo de hidrogênio no modelo de Bohr teria de ser um disco plano.</li> <li>Ligações químicas no modelo de Bohr não podem ser entendidas (ou seja, o modelo não explica ligações químicas).</li> <li>Em todos os estados estacionários o momento angular do elétron em torno de órbita de fora é muito grande. Em particular no estado fundamental, mesmo na realidade sendo 0 (nulo).</li> <li>Até mesmo o dividir de muitas linhas espectrais sob a influência de campos magnéticos (efeito anômalo de Zeeman) não pode ser explicado.</li> <li>Certas linhas espectrais do hidrogênio são capazes de resistir à medidas mais precisas do que as linhas duplas. Após isso, descobriram uma separação que não podia ser explicada pelo modelo de Bohr que foi chamada de Lamb-Shift .</li> <li>No radioastronomia a principal linha de 21 cm do hidrogênio pode ser obtida a partir do modelo de Bohr.</li> <li>A noção de uma órbita definida do elétron em torno do núcleo em 1927 conflitava com o princípio da incerteza descoberto por Werner Heisenberg.</li></ul> <p>Na física quântica, com todos as teorias e resultados obtidos até os dias de hoje e com os registros dos dados experimentais, o modelo orbital possui uma imagem fundamentalmente diferente do átomo. Ao contrário do que aceita pelo modelo de Bohr, os elétrons no átomo possuem probabilidade finita de estarem até mesmo no núcleo. Atualmente, sabemos que eles não se movem em órbitas. Razoavelmente aceitamos a ideia de uma nuvem de elétrons. </p> <h2 id="Referências" style="cursor: help;" title="Esta seção foi configurada para não ser editável diretamente. Edite a página toda ou a seção anterior em vez disso."><span id="Refer.C3.AAncias"></span>Referências</h2> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Brasil_Escola-1"><span class="mw-cite-backlink">↑ <sup><i><b><a href="#cite_ref-Brasil_Escola_1-0">a</a></b></i></sup> <sup><i><b><a href="#cite_ref-Brasil_Escola_1-1">b</a></b></i></sup> <sup><i><b><a href="#cite_ref-Brasil_Escola_1-2">c</a></b></i></sup> <sup><i><b><a href="#cite_ref-Brasil_Escola_1-3">d</a></b></i></sup> <sup><i><b><a href="#cite_ref-Brasil_Escola_1-4">e</a></b></i></sup></span> <span class="reference-text"><cite class="citation web">Líria Alves. <a rel="nofollow" class="external text" href="http://www.brasilescola.com/quimica/o-atomo-bohr.htm">«O átomo de Bohr»</a>. <i>R7</i>. Brasil Escola<span class="reference-accessdate">. Consultado em 10 de novembro de 2012</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3A%C3%81tomo+de+Bohr&rft.atitle=O+%C3%A1tomo+de+Bohr&rft.au=L%C3ria+Alves&rft.genre=unknown&rft.jtitle=R7&rft_id=http%3A%2F%2Fwww.brasilescola.com%2Fquimica%2Fo-atomo-bohr.htm&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><cite class="citation journal">Rutherford, Ernest (1911). «<span style="padding-left:0.2em;">"</span>The Scattering of α and β Particles by Matter and the Structure of the Atom<span style="padding-right:0.2em;">"</span>». <i>Philos. Mag</i>. <b>6</b>. 21 páginas</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3A%C3%81tomo+de+Bohr&rft.atitle=%22The+Scattering+of+%CE%B1+and+%CE%B2+Particles+by+Matter+and+the+Structure+of+the+Atom%22&rft.aufirst=Ernest&rft.aulast=Rutherford&rft.date=1911&rft.genre=article&rft.jtitle=Philos.+Mag.&rft.volume=6&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">D. J. Maia; J. C. de A. Bianchi. Química Geral – Fundamentos. 5ª edição – São Paulo. Editora Pearson, 2007. Pp. 32 a 35</span> </li> <li id="cite_note-ufsm-4"><span class="mw-cite-backlink"><a href="#cite_ref-ufsm_4-0">↑</a></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130203054242/http://www.ufsm.br/gef/Moderna/moderna08.pdf">«Modelo Atômico de Bohr»</a> <span style="font-size:85%;">(PDF)</span>. Grupo de Ensino de Física da Universidade Federal de Santa Maria<span class="reference-accessdate">. Consultado em 20 de janeiro de 2013</span>. Arquivado do <a rel="nofollow" class="external text" href="http://www.ufsm.br/gef/Moderna/moderna08.pdf">original</a> <span style="font-size:85%;">(PDF)</span> em 3 de fevereiro de 2013</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3A%C3%81tomo+de+Bohr&rft.btitle=Modelo+At%C3%B4mico+de+Bohr&rft.genre=unknown&rft.pub=Grupo+de+Ensino+de+F%C3sica+da+Universidade+Federal+de+Santa+Maria&rft_id=http%3A%2F%2Fwww.ufsm.br%2Fgef%2FModerna%2Fmoderna08.pdf&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-pt.scribd.com-5"><span class="mw-cite-backlink">↑ <sup><i><b><a href="#cite_ref-pt.scribd.com_5-0">a</a></b></i></sup> <sup><i><b><a href="#cite_ref-pt.scribd.com_5-1">b</a></b></i></sup></span> <span class="reference-text"><cite class="citation book">KIWANGA, Christopher Amelye (2013). Christopher Amelye. KIWANGA, ed. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140110214308/http://pt.scribd.com/doc/48312717/Fisica-Atomica"><i>Física Nuclear</i></a>. Introdução à Física Nuclear. <b>1</b> 1 ed. Reino Unido: [s.n.] 133 páginas<span class="reference-accessdate">. Consultado em 21 de agosto de 2013</span>. Arquivado do <a rel="nofollow" class="external text" href="https://pt.scribd.com/doc/48312717/Fisica-Atomica">original</a> em 10 de janeiro de 2014</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3A%C3%81tomo+de+Bohr&rft.aufirst=Christopher+Amelye&rft.aulast=KIWANGA&rft.btitle=F%C3sica+Nuclear&rft.date=2013&rft.edition=1&rft.genre=book&rft.place=Reino+Unido&rft_id=http%3A%2F%2Fpt.scribd.com%2Fdoc%2F48312717%2FFisica-Atomica&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">D. J. Maia; J. C. de A. Bianchi. Química Geral – Fundamentos. 5ª edição – São Paulo. Editora Pearson, 2007. Pp. 36 a 42</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120910150904/http://www.seara.ufc.br/tintim/fisica/hidrogenio/hidrogenio4.htm">«O Átomo de Bohr (Hidrogênio 1.0).»</a>. Seara da ciência<span class="reference-accessdate">. Consultado em 20 de janeiro de 2013</span>. Arquivado do <a rel="nofollow" class="external text" href="http://www.seara.ufc.br/tintim/fisica/hidrogenio/hidrogenio4.htm">original</a> em 10 de setembro de 2012</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3A%C3%81tomo+de+Bohr&rft.btitle=O+%C3%81tomo+de+Bohr+%28Hidrog%C3%AAnio+1.0%29.&rft.genre=unknown&rft.pub=Seara+da+ci%C3%AAncia&rft_id=http%3A%2F%2Fwww.seara.ufc.br%2Ftintim%2Ffisica%2Fhidrogenio%2Fhidrogenio4.htm&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Ver_também"><span id="Ver_tamb.C3.A9m"></span>Ver também</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%81tomo_de_Bohr&veaction=edit&section=7" title="Editar secção: Ver também" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%81tomo_de_Bohr&action=edit&section=7" title="Editar código-fonte da secção: Ver também"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="-moz-column-count: 2; -webkit-column-count: 2; column-count: 2;"> <ul><li><a href="/wiki/Modelo_at%C3%B4mico_de_Rutherford" title="Modelo atômico de Rutherford">Modelo atômico de Rutherford</a></li> <li><a href="/wiki/Modelo_at%C3%B4mico_de_Thomson" title="Modelo atômico de Thomson">Modelo atômico de Thomson</a></li> <li><a href="/wiki/Modelo_at%C3%B4mico_de_Dalton" title="Modelo atômico de Dalton">Modelo atômico de Dalton</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Ligações_externas"><span id="Liga.C3.A7.C3.B5es_externas"></span>Ligações externas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C3%81tomo_de_Bohr&veaction=edit&section=8" title="Editar secção: Ligações externas" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C3%81tomo_de_Bohr&action=edit&section=8" title="Editar código-fonte da secção: Ligações externas"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation web"><a rel="nofollow" class="external text" href="http://www.if.ufrgs.br/tex/fis142/modelobohr/modelobohrindex.html">«Modelo Atômico de Bohr - <b>www.if.ufrgs.br</b>»</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3A%C3%81tomo+de+Bohr&rft.btitle=Modelo+At%C3%B4mico+de+Bohr+-+www.if.ufrgs.br&rft.genre=unknown&rft_id=http%3A%2F%2Fwww.if.ufrgs.br%2Ftex%2Ffis142%2Fmodelobohr%2Fmodelobohrindex.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="http://www.rossetti.eti.br/aula-menu.asp">«MODELO ATÔMICO DE BOHR - <b>www.rossetti.eti.br</b>»</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3A%C3%81tomo+de+Bohr&rft.btitle=MODELO+AT%C3%94MICO+DE+BOHR+-+www.rossetti.eti.br&rft.genre=unknown&rft_id=http%3A%2F%2Fwww.rossetti.eti.br%2Faula-menu.asp&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="http://physicsact.wordpress.com/2008/01/02/modelo-atomico-de-bohr/">«Modelo Atômico de Bohr - <b>physicsact.wordpress.com</b>»</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3A%C3%81tomo+de+Bohr&rft.btitle=Modelo+At%C3%B4mico+de+Bohr+-+physicsact.wordpress.com&rft.genre=unknown&rft_id=http%3A%2F%2Fphysicsact.wordpress.com%2F2008%2F01%2F02%2Fmodelo-atomico-de-bohr%2F&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="http://www.if.ufrgs.br/tex/fis142/fismod/mod04/m_s03.html">«Modelo de Bohr - <b>www.if.ufrgs.br</b>»</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3A%C3%81tomo+de+Bohr&rft.btitle=Modelo+de+Bohr+-+www.if.ufrgs.br&rft.genre=unknown&rft_id=http%3A%2F%2Fwww.if.ufrgs.br%2Ftex%2Ffis142%2Ffismod%2Fmod04%2Fm_s03.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://www.if.ufrgs.br/public/tapf/tapf_v32n5.pdf">«Seria o átomo um novo sistema solar? 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Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2006-03-20T00:26:02Z","dateModified":"2025-02-14T01:02:41Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/7\/76\/Bohr_atom_model_%28mul%29.svg","headline":"modelo que descreve \u00e1tomos como n\u00facleo cercado por el\u00e9trons em \u00f3rbita, proposto por Niels Bohr"}</script> </body> </html>