Bohrov model atoma

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vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Sadržaj</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">premjesti na bočnu traku</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sakrij</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">Početak</div> </a> </li> <li id="toc-Historija" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Historija"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Historija</span> </div> </a> <ul id="toc-Historija-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Energetski_nivoi_elektrona" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Energetski_nivoi_elektrona"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Energetski nivoi elektrona</span> </div> </a> <button aria-controls="toc-Energetski_nivoi_elektrona-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Uključi/isključi podsekciju Energetski nivoi elektrona</span> </button> <ul id="toc-Energetski_nivoi_elektrona-sublist" class="vector-toc-list"> <li id="toc-Izračunavanje_poluprečnika_putanje" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Izračunavanje_poluprečnika_putanje"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Izračunavanje poluprečnika putanje</span> </div> </a> <ul id="toc-Izračunavanje_poluprečnika_putanje-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ridbergova_formula" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ridbergova_formula"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Ridbergova formula</span> </div> </a> <ul id="toc-Ridbergova_formula-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="Sadržaj" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Sadržaj" > <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="Uključi/isključi sadržaj" > <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">Uključi/isključi sadržaj</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">Bohrov model atoma</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="Idi na članak na drugom jeziku. Dostupno na 70 jezika" > <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 jezika</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 – njemački (Švicarska)" lang="gsw" hreflang="gsw" data-title="Bohrsches Atommodell" data-language-autonym="Alemannisch" data-language-local-name="njemački (Švicarska)" 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 – aragonski" lang="an" hreflang="an" data-title="Modelo atomico de Bohr" data-language-autonym="Aragonés" data-language-local-name="aragonski" 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 – arapski" lang="ar" hreflang="ar" data-title="نموذج بور" data-language-autonym="العربية" data-language-local-name="arapski" 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 – asturijski" lang="ast" hreflang="ast" data-title="Modelu atómicu de Bohr" data-language-autonym="Asturianu" data-language-local-name="asturijski" 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 – azerbejdžanski" lang="az" hreflang="az" data-title="Bor modeli" data-language-autonym="Azərbaycanca" data-language-local-name="azerbejdžanski" 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 – bjeloruski" lang="be" hreflang="be" data-title="Бораўская мадэль атама" data-language-autonym="Беларуская" data-language-local-name="bjeloruski" 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 – bugarski" lang="bg" hreflang="bg" data-title="Модел на Бор" data-language-autonym="Български" data-language-local-name="bugarski" 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 – bengalski" lang="bn" hreflang="bn" data-title="বোর মডেল" data-language-autonym="বাংলা" data-language-local-name="bengalski" class="interlanguage-link-target"><span>বাংলা</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 – katalonski" lang="ca" hreflang="ca" data-title="Model atòmic de Bohr" data-language-autonym="Català" data-language-local-name="katalonski" 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 – češki" lang="cs" hreflang="cs" data-title="Bohrův model atomu" data-language-autonym="Čeština" data-language-local-name="češki" 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 – danski" lang="da" hreflang="da" data-title="Bohrs atommodel" data-language-autonym="Dansk" data-language-local-name="danski" 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 – njemački" lang="de" hreflang="de" data-title="Bohrsches Atommodell" data-language-autonym="Deutsch" data-language-local-name="njemački" 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 – grčki" lang="el" hreflang="el" data-title="Ατομικό πρότυπο του Μπορ" data-language-autonym="Ελληνικά" data-language-local-name="grčki" 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 – engleski" lang="en" hreflang="en" data-title="Bohr model" data-language-autonym="English" data-language-local-name="engleski" 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 – španski" lang="es" hreflang="es" data-title="Modelo atómico de Bohr" data-language-autonym="Español" data-language-local-name="španski" 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 – estonski" lang="et" hreflang="et" data-title="Bohri aatomiteooria" data-language-autonym="Eesti" data-language-local-name="estonski" 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 – baskijski" lang="eu" hreflang="eu" data-title="Bohrren eredu atomikoa" data-language-autonym="Euskara" data-language-local-name="baskijski" 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 – perzijski" lang="fa" hreflang="fa" data-title="مدل بور" data-language-autonym="فارسی" data-language-local-name="perzijski" 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 – finski" lang="fi" hreflang="fi" data-title="Bohrin atomimalli" data-language-autonym="Suomi" data-language-local-name="finski" 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 – francuski" lang="fr" hreflang="fr" data-title="Modèle de Bohr" data-language-autonym="Français" data-language-local-name="francuski" 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 – irski" lang="ga" hreflang="ga" data-title="Samhail Bohr" data-language-autonym="Gaeilge" data-language-local-name="irski" 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 – galicijski" lang="gl" hreflang="gl" data-title="Modelo atómico de Bohr" data-language-autonym="Galego" data-language-local-name="galicijski" 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 – gudžarati" lang="gu" hreflang="gu" data-title="બૉહરનો સિદ્ધાંત" data-language-autonym="ગુજરાતી" data-language-local-name="gudžarati" 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 – hebrejski" lang="he" hreflang="he" data-title="מודל האטום של בוהר" data-language-autonym="עברית" data-language-local-name="hebrejski" 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 – hrvatski" lang="hr" hreflang="hr" data-title="Bohrov model atoma" data-language-autonym="Hrvatski" data-language-local-name="hrvatski" 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 – mađarski" lang="hu" hreflang="hu" data-title="Bohr-féle atommodell" data-language-autonym="Magyar" data-language-local-name="mađarski" 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 – armenski" lang="hy" hreflang="hy" data-title="Բորի մոդել" data-language-autonym="Հայերեն" data-language-local-name="armenski" 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 – indonezijski" lang="id" hreflang="id" data-title="Model Bohr" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezijski" 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 – italijanski" lang="it" hreflang="it" data-title="Modello atomico di Bohr" data-language-autonym="Italiano" data-language-local-name="italijanski" 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 – japanski" lang="ja" hreflang="ja" data-title="ボーアの原子模型" data-language-autonym="日本語" data-language-local-name="japanski" 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 – gruzijski" lang="ka" hreflang="ka" data-title="ბორის მოდელი" data-language-autonym="ქართული" data-language-local-name="gruzijski" 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 – kazaški" lang="kk" hreflang="kk" data-title="Бор қағидалары" data-language-autonym="Қазақша" data-language-local-name="kazaški" 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 – korejski" lang="ko" hreflang="ko" data-title="보어 모형" data-language-autonym="한국어" data-language-local-name="korejski" 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 – litvanski" lang="lt" hreflang="lt" data-title="Boro teorija" data-language-autonym="Lietuvių" data-language-local-name="litvanski" 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 – latvijski" lang="lv" hreflang="lv" data-title="Bora atoma uzbūves modelis" data-language-autonym="Latviešu" data-language-local-name="latvijski" 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 – makedonski" lang="mk" hreflang="mk" data-title="Боров модел" data-language-autonym="Македонски" data-language-local-name="makedonski" 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 – malajalam" lang="ml" hreflang="ml" data-title="ബോർ മാതൃക" data-language-autonym="മലയാളം" data-language-local-name="malajalam" 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 – marati" lang="mr" hreflang="mr" data-title="बोरची अणूची प्रतिकृती" data-language-autonym="मराठी" data-language-local-name="marati" 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 – malajski" lang="ms" hreflang="ms" data-title="Model Bohr" data-language-autonym="Bahasa Melayu" data-language-local-name="malajski" 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 – burmanski" lang="my" hreflang="my" data-title="ဘိုးအာ၏ အက်တမ်ပုံစံ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="burmanski" 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 – nizozemski" lang="nl" hreflang="nl" data-title="Atoommodel van Bohr" data-language-autonym="Nederlands" data-language-local-name="nizozemski" 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 – norveški (Nynorsk)" lang="nn" hreflang="nn" data-title="Atommodell" data-language-autonym="Norsk nynorsk" data-language-local-name="norveški (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 – norveški (Bokmal)" lang="nb" hreflang="nb" data-title="Bohrs atommodell" data-language-autonym="Norsk bokmål" data-language-local-name="norveški (Bokmal)" 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 – pandžapski" lang="pa" hreflang="pa" data-title="ਬੋਹਰ ਮਾਡਲ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="pandžapski" 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 – poljski" lang="pl" hreflang="pl" data-title="Model atomu Bohra" data-language-autonym="Polski" data-language-local-name="poljski" 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-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%81tomo_de_Bohr" title="Átomo de Bohr – portugalski" lang="pt" hreflang="pt" data-title="Átomo de Bohr" data-language-autonym="Português" data-language-local-name="portugalski" class="interlanguage-link-target"><span>Português</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 – rumunski" lang="ro" hreflang="ro" data-title="Modelul atomic Bohr" data-language-autonym="Română" data-language-local-name="rumunski" 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 – ruski" lang="ru" hreflang="ru" data-title="Боровская модель атома" data-language-autonym="Русский" data-language-local-name="ruski" 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 – sicilijanski" lang="scn" hreflang="scn" data-title="Mudèllu atòmicu di Bohr" data-language-autonym="Sicilianu" data-language-local-name="sicilijanski" 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 – škotski" lang="sco" hreflang="sco" data-title="Bohr model" data-language-autonym="Scots" data-language-local-name="škotski" 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 – srpskohrvatski" lang="sh" hreflang="sh" data-title="Bohrov model atoma" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srpskohrvatski" 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 – slovački" lang="sk" hreflang="sk" data-title="Bohrov model atómu" data-language-autonym="Slovenčina" data-language-local-name="slovački" 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 – slovenski" lang="sl" hreflang="sl" data-title="Bohrov model atoma" data-language-autonym="Slovenščina" data-language-local-name="slovenski" 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 – srpski" lang="sr" hreflang="sr" data-title="Боров модел атома" data-language-autonym="Српски / srpski" data-language-local-name="srpski" 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 – švedski" lang="sv" hreflang="sv" data-title="Bohrs atommodell" data-language-autonym="Svenska" data-language-local-name="švedski" 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 – tamilski" lang="ta" hreflang="ta" data-title="போர் அணு மாதிரி" data-language-autonym="தமிழ்" data-language-local-name="tamilski" 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 – tajlandski" lang="th" hreflang="th" data-title="แบบจำลองของโปร์" data-language-autonym="ไทย" data-language-local-name="tajlandski" 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 – tagalog" lang="tl" hreflang="tl" data-title="Modelong Bohr" data-language-autonym="Tagalog" data-language-local-name="tagalog" 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 – turski" lang="tr" hreflang="tr" data-title="Bohr modeli" data-language-autonym="Türkçe" data-language-local-name="turski" 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 – ukrajinski" lang="uk" hreflang="uk" data-title="Атомна модель Бора" data-language-autonym="Українська" data-language-local-name="ukrajinski" 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 – uzbečki" lang="uz" hreflang="uz" data-title="Bor postulatlari" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbečki" 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 – vijetnamski" lang="vi" hreflang="vi" data-title="Mô hình Bohr" data-language-autonym="Tiếng Việt" data-language-local-name="vijetnamski" 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 kineski" lang="wuu" hreflang="wuu" data-title="玻尔模型" data-language-autonym="吴语" data-language-local-name="Wu kineski" 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 – kineski" lang="zh" hreflang="zh" data-title="玻尔模型" data-language-autonym="中文" data-language-local-name="kineski" 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 – kantonski" lang="yue" hreflang="yue" data-title="玻爾模型" data-language-autonym="粵語" data-language-local-name="kantonski" 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="Uredi međujezičke linkove" class="wbc-editpage">Uredi veze</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="Imenski prostori"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet 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id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="bs" dir="ltr"><table class="plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/Datoteka:Question_book-new.svg" class="mw-file-description"><img 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"><b>Ovaj članak ili neki od njegovih odlomaka nije dovoljno potkrijepljen <a href="/wiki/Wikipedia:Provjerljivost" title="Wikipedia:Provjerljivost">izvorima</a></b> (literatura, veb-sajtovi ili drugi izvori).<span class="hide-when-compact"> Ako se pravilno ne potkrijepe <a href="/wiki/Wikipedia:Pouzdani_izvori" title="Wikipedia:Pouzdani izvori">pouzdanim</a> izvorima, sporne rečenice i navodi mogli bi biti izbrisani. Pomozite Wikipediji tako što ćete navesti validne izvore putem <a href="/wiki/Wikipedia:Citiranje_izvora" class="mw-redirect" title="Wikipedia:Citiranje izvora">referenci</a> te nakon toga možete ukloniti ovaj šablon.</span></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Bohr_atom_model.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Bohr_atom_model.svg/220px-Bohr_atom_model.svg.png" decoding="async" width="220" height="192" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Bohr_atom_model.svg/330px-Bohr_atom_model.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Bohr_atom_model.svg/440px-Bohr_atom_model.svg.png 2x" data-file-width="310" data-file-height="270" /></a><figcaption>Bohrov model atoma <a href="/wiki/Vodik" title="Vodik">vodika</a> (Z=1)</figcaption></figure> <p><b>Bohrov model atoma</b> (fon. Borov model) predstavlja <a href="/wiki/Atom" title="Atom">atom</a> sa malim pozitivno naelektrisanim jezgrom oko kojeg se <a href="/wiki/Elektron" title="Elektron">elektroni</a> kreću u kružnim orbitama slično kretanju planeta oko <a href="/wiki/Sunce" title="Sunce">Sunca</a>. Dakle, po Borovom modelu atom je sličan planetarnom sistemu s tom razlikom što privlačna sila potiče od elektrostatičke interakcije a ne od <a href="/wiki/Gravitacija" title="Gravitacija">gravitacije</a>. Glavni uspjeh modela, koji je predložio <a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a> <a href="/wiki/1913" class="mw-redirect" title="1913">1913</a>. godine, je objašnjenje <a href="/w/index.php?title=Ridbergova_formula&action=edit&redlink=1" class="new" title="Ridbergova formula (stranica ne postoji)">Ridbergove formule</a> za <a href="/wiki/Spektar" class="mw-redirect mw-disambig" title="Spektar">spektralne</a> emisione linije atomskog <a href="/wiki/Vodonik" class="mw-redirect" title="Vodonik">vodonika</a>. Ridbergova formula je eksperimentalno od ranije bila poznata ali je tek Borovim modelom bila kvantitativno teorijski objašnjena i povezana sa osnovnim osobinama atoma. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historija">Historija</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bohrov_model_atoma&veaction=edit&section=1" title="Uredi odlomak "Historija"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bohrov_model_atoma&action=edit&section=1" title="Edit section's source code: Historija"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Iz Rutherfordovih eksperimenata postalo je jasno da su pozitivno naelektrisanje i masa atoma skoncentrisani u centru atoma oko kojeg se nalazi difuzni oblak eletrona, nosioca negativnog naelektrisanja. Iz toga se prirodno nametnuo planetarni model atoma u kojem se elektroni kreću oko jezgra poput planeta oko Sunca. Međutim, planetarni model atoma nailazio je na brojne poteškoće u pogledu objašnjenja stabilnosti atoma i prirode atomskih spektara. Na primer, prema klasičnim zakonima eletrodinamike, naeletrisanje u kružnoj putanji mora da emituje <a href="/wiki/Elektromagnetno_zra%C4%8Denje" title="Elektromagnetno zračenje">elektromagnetno zračenje</a> gubeći pri tome <a href="/wiki/Energija" title="Energija">energiju</a>. Tako bi i elektron u kružnoj orbiti oko jezgra trebalo neprekidno da emituje zračenje. Pri tome bi zbog gubitka energije njegova putanja trebalo da bude spiralni pad u <a href="/wiki/Atomsko_jezgro" title="Atomsko jezgro">atomsko jezgro</a> a emitovano zračenje kontinualno jer se energija emitera neprekidno smanjuje. Međutim još krajem <a href="/wiki/19._vijek" title="19. vijek">19. vijeka</a> u brojnim eksperimentima sa električnim pražnjenjem u razrijeđenim gasovima pokazano je da atomi emituju zračenje na diskretnim, dobro definisanim <a href="/wiki/Frekvencija" title="Frekvencija">frekvencijama</a>. Problem primjene klasične elektrodinamike na atomske sisteme Bohr je riješio predloživši teoriju koja je uspješno objasnila spektre jednoelektronskih atoma. </p> <div align="center"> <table style="border-collapse:collapse;border-style:none;background-color:transparent;max-width:40em;width:auto;margin:0 10%;"> <tbody><tr> <td valign="bottom" style="color:silver;font-size:3.6em;font-family:serif;font-weight:bold;text-align:left;padding:10px 0;">“ </td> <td valign="center" style="padding:0 1em;font-family:serif;font-style:italic;">Nemogućnost teorija klasične fizike da objasne atomske fenomene je potom još više došla do izražaja napretkom našeg znanja o strukturi atoma. Raderfordovo otkriće atomskog jezgra (1911 g.) je odjednom otkrilo nepogodnost pojmova klasične mehanike i klasičnog magnetizma da objasne stabilnost atoma. I ovdje je opet kvantna teorija dala ključ za razjašnjenje situacije, a, posebno, postalo je moguće objašnjenje atomske stabilnosti, kao i empirijskih zakona koji upravljaju spektrima elemenata, polazeći od pretpostavke da se svaka reakcija atoma u kojoj dolazi do izmjene energije događa u potpunosti preko prijelaza između tzv. stacionarnih stanja, i da spektri nastaju u stepenastim procesima u kojima je svaki prijelaz praćen emisijom monohromatskog svetlosnog kvanta. </td> <td valign="top" style="color:silver;font-size:3.6em;font-family:serif;font-weight:bold;text-align:right;padding:10px 0;">” </td></tr> <tr> <td colspan="3" style="padding:1ex 0 0 0;"><p style="line-height:1em;text-align:right;"><small><cite style="font-style:normal;">Niels Bohr</cite></small></p> </td></tr></tbody></table></div> <p>Bohrova teorija se može izložiti u obliku Bohrovih postulata (pretpostavki): </p> <ol><li>Elektron ne može kružiti oko jezgre po bilo kojim već samo pod tačno određenim kvanitziranim stanjima. To su tzv. dopuštene ili stacionarne staze. Krećući se po njima elektron se nalazi u stacionarnom stanju i ne emituje energiju.</li> <li>Atom <a href="/wiki/Apsorpcija" class="mw-disambig" title="Apsorpcija">apsorbira</a> zračenje samo kada primi određeni <a href="/wiki/Kvant" title="Kvant">kvant</a> energije i emituje određeni kvant energije kada prelazi iz jednog stacionarnog stanja u drugo (tj. kada prelazi iz stanja više energije u stanje niže). Atom ne može sponatno preći iz stanja niže u stanje više energije, nego tek kada biva pogođen sa određenim kvantom energije (<a href="/wiki/Foton" title="Foton">fotonom</a>). Prelazak iz višeg stanja u niže je spontan događaj, uokviren statističkom prirodom, pri čemu se emituje kvant energije (foton). Frekvencija emitovanog fotona pri prelasku iz višeg stanja u niže je prema Bohrovoj formuli:</li></ol> <dl><dd><dl><dd><dl><dd><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_{2}-E_{1}=h\nu \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>h</mi> <mi>ν</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{2}-E_{1}=h\nu \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e15eca67b2bcd69dfdf5fc6d5a1238b3ac328988" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.436ex; height:2.509ex;" alt="{\displaystyle E_{2}-E_{1}=h\nu \,}"></span></dd></dl></dd></dl></dd></dl> <p>gdje je h - <a href="/wiki/Planckova_konstanta" title="Planckova konstanta">Planckova konstanta</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Energetski_nivoi_elektrona">Energetski nivoi elektrona</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bohrov_model_atoma&veaction=edit&section=2" title="Uredi odlomak "Energetski nivoi elektrona"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bohrov_model_atoma&action=edit&section=2" title="Edit section's source code: Energetski nivoi elektrona"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bohrov model je tačan samo za jednoelektronske sisteme poput vodonikovog atoma ili jednostruko ionizovanog <a href="/wiki/Helijum" class="mw-redirect" title="Helijum">helijuma</a> ili dvostruko ionizovanog <a href="/wiki/Litijum" class="mw-redirect" title="Litijum">litijuma</a>. Izvođenje formula za energetske nivoe počinje sa tri jednostavne pretpostavke: </p><p>1) Energija elektrona u orbiti je suma njegove <a href="/wiki/Potencijalna_energija" title="Potencijalna energija">potencijalne</a> i <a href="/wiki/Kineti%C4%8Dka_energija" title="Kinetička energija">kinetičke energije</a>: </p> <dl><dd><dl><dd><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\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9123abddc2ec35f72035ec59f443c79ee052c9ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.163ex; height:2.176ex;" alt="{\displaystyle E\,}"></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 =E_{kin}+E_{pot}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =E_{kin}+E_{pot}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f45c315bc950c9cab7407030ab42c03f522cda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.204ex; height:2.843ex;" alt="{\displaystyle =E_{kin}+E_{pot}\,}"></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 ={\begin{matrix}{\frac {1}{2}}\end{matrix}}m_{e}v^{2}-{\frac {kq_{e}^{2}}{r}}\quad \quad \quad \quad \quad (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </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> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\begin{matrix}{\frac {1}{2}}\end{matrix}}m_{e}v^{2}-{\frac {kq_{e}^{2}}{r}}\quad \quad \quad \quad \quad (1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98bef21d6d444bcd107a0419f417f4c71b712210" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.69ex; height:5.676ex;" alt="{\displaystyle ={\begin{matrix}{\frac {1}{2}}\end{matrix}}m_{e}v^{2}-{\frac {kq_{e}^{2}}{r}}\quad \quad \quad \quad \quad (1)}"></span> </td></tr></tbody></table></dd></dl></dd></dl> <p>2) Moment impulsa elektrona može imati samo određene diskretne vrijednosti: </p> <dl><dd><dl><dd><dl><dd><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_{e}vr=n{\frac {h}{2\pi }}=n\hbar \quad \quad \quad \quad \quad (2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mi>v</mi> <mi>r</mi> <mo>=</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>n</mi> <mi class="MJX-variant">ℏ</mi> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=m_{e}vr=n{\frac {h}{2\pi }}=n\hbar \quad \quad \quad \quad \quad (2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9db3a04022b5154b017ddf21a45a853ff0ab1525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.104ex; height:5.343ex;" alt="{\displaystyle L=m_{e}vr=n{\frac {h}{2\pi }}=n\hbar \quad \quad \quad \quad \quad (2)}"></span></dd></dl></dd></dl></dd></dl> <p>gdje je n glavni <a href="/wiki/Kvantni_brojevi" class="mw-redirect" title="Kvantni brojevi">kvantni broj</a>, n = 1,2,3,…, h Planckova konstanta, 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 \hbar =h/(2\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar =h/(2\pi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c43c4e0cfef2829e9705d8da36c3f9093ad9b53e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.21ex; height:2.843ex;" alt="{\displaystyle \hbar =h/(2\pi )}"></span>. </p><p>3) Elektron u orbiti održava Kulonova sila, dakle, Kulonova sila jednaka je centripetalnoj sili: </p> <dl><dd><dl><dd><dl><dd><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 {kq_{e}^{2}}{r^{2}}}={\frac {m_{e}v^{2}}{r}}\quad \quad \quad \quad \quad (3)\,}"> <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> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {kq_{e}^{2}}{r^{2}}}={\frac {m_{e}v^{2}}{r}}\quad \quad \quad \quad \quad (3)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/705f7b570cbf6c5887658d50aa7706499ac62dee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.308ex; height:6.009ex;" alt="{\displaystyle {\frac {kq_{e}^{2}}{r^{2}}}={\frac {m_{e}v^{2}}{r}}\quad \quad \quad \quad \quad (3)\,}"></span></dd></dl></dd></dl></dd></dl> <p>Pomnožimo li obje strane jednačine (3) sa r: </p> <dl><dd><dl><dd><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 {kq_{e}^{2}}{r}}=m_{e}v^{2}\quad \quad \quad \quad \quad (4)\,}"> <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> <mi>r</mi> </mfrac> </mrow> <mo>=</mo> <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> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {kq_{e}^{2}}{r}}=m_{e}v^{2}\quad \quad \quad \quad \quad (4)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7e8407883ff121785020b882070b4f87b682c38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.472ex; height:5.676ex;" alt="{\displaystyle {\frac {kq_{e}^{2}}{r}}=m_{e}v^{2}\quad \quad \quad \quad \quad (4)\,}"></span></dd></dl></dd></dl> <p>sa lijeve strane dobijamo izraz za potencijalnu energiju čijom zamjenom u jednačini (1) nalazimo da je ukupna energija elektrona u orbiti: </p> <dl><dd><dl><dd><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={\begin{matrix}{\frac {1}{2}}\end{matrix}}m_{e}v^{2}-{\frac {kq_{e}^{2}}{r}}=-{\begin{matrix}{\frac {1}{2}}\end{matrix}}m_{e}v^{2}\,\quad \quad \quad \quad \quad (5)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </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> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </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> <mspace width="thinmathspace" /> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\begin{matrix}{\frac {1}{2}}\end{matrix}}m_{e}v^{2}-{\frac {kq_{e}^{2}}{r}}=-{\begin{matrix}{\frac {1}{2}}\end{matrix}}m_{e}v^{2}\,\quad \quad \quad \quad \quad (5)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b2541b169aa48b4a52047a5a6e6150f457c2773" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:47.034ex; height:5.676ex;" alt="{\displaystyle E={\begin{matrix}{\frac {1}{2}}\end{matrix}}m_{e}v^{2}-{\frac {kq_{e}^{2}}{r}}=-{\begin{matrix}{\frac {1}{2}}\end{matrix}}m_{e}v^{2}\,\quad \quad \quad \quad \quad (5)}"></span></dd></dl></dd></dl> <p>Brzina elektrona je iz jednačine (2): </p> <dl><dd><dl><dd><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={\frac {n\hbar }{m_{e}v}}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi class="MJX-variant">ℏ</mi> </mrow> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mi>v</mi> </mrow> </mfrac> </mrow> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\frac {n\hbar }{m_{e}v}}.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/178b3c32b3fd0175fe566f01093533fe02bea018" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.184ex; height:5.676ex;" alt="{\displaystyle r={\frac {n\hbar }{m_{e}v}}.\,}"></span></dd></dl></dd></dl> <p>što zamjenom u jednačinu (4) daje: </p> <dl><dd><dl><dd><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}{\frac {m_{e}v}{n\hbar }}=m_{e}v^{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mi>v</mi> </mrow> <mrow> <mi>n</mi> <mi class="MJX-variant">ℏ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle kq_{e}^{2}{\frac {m_{e}v}{n\hbar }}=m_{e}v^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c8927741991efe21da42c5b1af60ea02bccab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.054ex; height:4.843ex;" alt="{\displaystyle kq_{e}^{2}{\frac {m_{e}v}{n\hbar }}=m_{e}v^{2}\,}"></span></dd></dl></dd></dl> <p>odnosno: </p> <dl><dd><dl><dd><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 {kq_{e}^{2}}{n\hbar }}=v\,}"> <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> <mrow> <mi>n</mi> <mi class="MJX-variant">ℏ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>v</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {kq_{e}^{2}}{n\hbar }}=v\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa844e6e51a04f779eee12e8465bfe36c3511dd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.794ex; height:5.843ex;" alt="{\displaystyle {\frac {kq_{e}^{2}}{n\hbar }}=v\,}"></span></dd></dl></dd></dl> <p>Zamjenom ove vrijednosti za brzinu v u izrazu za energiju, (5), a također i vrijednosti za k i \hbar, nalazimo da energija elektrona u orbitama atoma vodonika ima diskretne vrijednosti koje zavise od glavnog kvantnog broja, n: </p> <dl><dd><dl><dd><dl><dd><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 ={\frac {-1}{2}}m_{e}\left({\frac {kq_{e}^{2}}{n\hbar }}\right)^{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msup> <mrow> <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> <mrow> <mi>n</mi> <mi class="MJX-variant">ℏ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {-1}{2}}m_{e}\left({\frac {kq_{e}^{2}}{n\hbar }}\right)^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49dd986976dff504e68cbd2175ec0e359c9dd0d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.342ex; height:6.843ex;" alt="{\displaystyle ={\frac {-1}{2}}m_{e}\left({\frac {kq_{e}^{2}}{n\hbar }}\right)^{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 {-1}{2}}m_{e}\left({\frac {1}{4\pi \epsilon _{0}}}q_{e}^{2}{\frac {2\pi }{nh}}\right)^{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow> <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> <msubsup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mrow> <mi>n</mi> <mi>h</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {-1}{2}}m_{e}\left({\frac {1}{4\pi \epsilon _{0}}}q_{e}^{2}{\frac {2\pi }{nh}}\right)^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d871f5287f8ef93be8bebdfecd1c6a549c5aa86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.194ex; height:6.509ex;" alt="{\displaystyle ={\frac {-1}{2}}m_{e}\left({\frac {1}{4\pi \epsilon _{0}}}q_{e}^{2}{\frac {2\pi }{nh}}\right)^{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></dd></dl></dd></dl></dd></dl> <p>Uvrštavanjem konstanti dobija se: </p> <dl><dd><dl><dd><dl><dd><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}=(-13.6\ \mathrm {eV} ){\frac {1}{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> <mo stretchy="false">(</mo> <mo>−</mo> <mn>13.6</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">V</mi> </mrow> <mo stretchy="false">)</mo> <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 E_{n}=(-13.6\ \mathrm {eV} ){\frac {1}{n^{2}}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/809298b88ef5c312bdf50c4051ca5d59f5ab8f4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.812ex; height:5.509ex;" alt="{\displaystyle E_{n}=(-13.6\ \mathrm {eV} ){\frac {1}{n^{2}}}\,}"></span> </td></tr></tbody></table></dd></dl></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Izračunavanje_poluprečnika_putanje"><span id="Izra.C4.8Dunavanje_polupre.C4.8Dnika_putanje"></span>Izračunavanje poluprečnika putanje</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bohrov_model_atoma&veaction=edit&section=3" title="Uredi odlomak "Izračunavanje poluprečnika putanje"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bohrov_model_atoma&action=edit&section=3" title="Edit section's source code: Izračunavanje poluprečnika putanje"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Iz jednačine (3) pošto je <a href="/wiki/Koli%C4%8Dina_kretanja" title="Količina kretanja">količina kretanja</a> jednaka <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 p^{2}=v^{2}m_{e}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{2}=v^{2}m_{e}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c70ddac06e3092a15bbc111622c99140cc11e871" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.688ex; height:3.009ex;" alt="{\displaystyle p^{2}=v^{2}m_{e}^{2}}"></span>, slijedi da je: </p> <dl><dd><dl><dd><dl><dd><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}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{r^{2}}}={\frac {p^{2}}{m_{e}r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mi>r</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{r^{2}}}={\frac {p^{2}}{m_{e}r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c62a745e33e036b2cab210f5af42bc5711e610a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.464ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{r^{2}}}={\frac {p^{2}}{m_{e}r}}}"></span></dd></dl></dd></dl></dd></dl> <p>Preuređivanjem se dobije: </p> <dl><dd><dl><dd><dl><dd><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}}{4\pi \varepsilon _{0}}}e^{2}r=p^{2}r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {m_{e}}{4\pi \varepsilon _{0}}}e^{2}r=p^{2}r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/806bdf77ad339db397e7f8c9b97f52f122b9f6be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.08ex; height:5.176ex;" alt="{\displaystyle {\frac {m_{e}}{4\pi \varepsilon _{0}}}e^{2}r=p^{2}r^{2}}"></span></dd></dl></dd></dl></dd></dl> <p>Prema kvantnom uslovu, na desnu stranu jednačine umjesto <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 p^{2}r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{2}r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/313ff418b485589e41593716c24ef1d217d16c9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.416ex; height:3.009ex;" alt="{\displaystyle p^{2}r^{2}}"></span> uvrstimo <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{\frac {h}{2\pi }})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n{\frac {h}{2\pi }})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9e00355469ff517faaa2c5e8f95266c4bf006a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.589ex; height:5.343ex;" alt="{\displaystyle (n{\frac {h}{2\pi }})^{2}}"></span> jer je <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 p=m_{e}v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=m_{e}v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65ad78bc714ea75ce191cbef897f5cc380c76ed5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.524ex; height:2.009ex;" alt="{\displaystyle p=m_{e}v}"></span>, a reducirana konstanta <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{h}{2\pi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>r</mi> <mi>a</mi> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle frac{h}{2\pi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80a827750ab036a3ca1e484a98f000bb4ce2a84a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.397ex; height:2.509ex;" alt="{\displaystyle frac{h}{2\pi }}"></span> </p> <dl><dd><dl><dd><dl><dd><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}}{4\pi \varepsilon _{0}e^{2}r}}=(n{\frac {h}{2\pi }}^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {m_{e}}{4\pi \varepsilon _{0}e^{2}r}}=(n{\frac {h}{2\pi }}^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41ecd01031c066b4038d0c60398c691364fad241" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.342ex; height:6.343ex;" alt="{\displaystyle {\frac {m_{e}}{4\pi \varepsilon _{0}e^{2}r}}=(n{\frac {h}{2\pi }}^{2})}"></span></dd></dl></dd></dl></dd></dl> <p>Iz te jednačine izvučimo poluprečnik (r): </p> <dl><dd><dl><dd><dl><dd><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^{2}{\frac {h^{2}}{4\pi ^{2}}}{\frac {4\pi \varepsilon _{0}}{m_{e}e^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <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^{2}{\frac {h^{2}}{4\pi ^{2}}}{\frac {4\pi \varepsilon _{0}}{m_{e}e^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/021b30ed983625357c3f2fcec378d60000b916ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.996ex; height:6.343ex;" alt="{\displaystyle r=n^{2}{\frac {h^{2}}{4\pi ^{2}}}{\frac {4\pi \varepsilon _{0}}{m_{e}e^{2}}}}"></span></dd></dl></dd></dl></dd></dl> <p>gdje je <i>n</i> kvantni broj koji može imati vrijednosti <i>n</i>=1,2,3, ... </p><p>Bohr je ovu formulu skratio uvevši oznaku <i>α</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 a={\frac {h^{2}\varepsilon _{0}}{\pi e^{2}m_{e}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mi>π</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\frac {h^{2}\varepsilon _{0}}{\pi e^{2}m_{e}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88f33d9bc57cd7780c9a3342c6940ee1f03ec304" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.673ex; height:6.343ex;" alt="{\displaystyle a={\frac {h^{2}\varepsilon _{0}}{\pi e^{2}m_{e}}}}"></span> , koja označava najmanji mogući poluprečnik kružne putanje, tako da se formula može izraziti kao <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}=n^{2}a}"> <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> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}=n^{2}a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3f7535b05cbb72f39f6a2e973d44de2763b0397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.044ex; height:3.009ex;" alt="{\displaystyle r_{n}=n^{2}a}"></span>. </p><p>Vidimo da su vrijednosti koje sačinjavaju a konstante, pa je samim tim i α konstanta i iznosi <i>α=0,53*10<sup>-10</sup></i> <a href="/wiki/Metar" title="Metar">metara</a> i naziva se <b><a href="/wiki/Bohrov_radijus" title="Bohrov radijus">Bohrov poluprečnik</a></b>. </p> <div class="mw-heading mw-heading2"><h2 id="Ridbergova_formula">Ridbergova formula</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bohrov_model_atoma&veaction=edit&section=4" title="Uredi odlomak "Ridbergova formula"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bohrov_model_atoma&action=edit&section=4" title="Edit section's source code: Ridbergova formula"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Rydbergova formula (fon. Ridbergova formula), koja je bila empirijski poznata prije Bohrovih jednačina, u svjetlu Bohrove teorije može se shvatiti da opisuje energije kvantnih prijelaza između dobro definisanih energijskih nivoa. Bohrova formula daje numeričke vrijednosti već poznate i mjerene Ridbergove konstante, ali sada izražene preko fundamentalnih konstanti prirode, uključujući i naelektrisanje eletrona i Planckovu konstantu. </p><p>Kada elektron prelazi sa višeg energetskog nivoa na niži dolazi do emisije fotona. Korištenjem izvedenih formula za energetske nivoe vodonikovog atoma moguće je izračunati <a href="/wiki/Talasna_du%C5%BEina" title="Talasna dužina">talasne dužine</a> fotona koje ovaj atom emituje. </p><p>Energija emitovanog fotona jednaka je razlici energija nivoa među kojima dolazi do elektronskog prelaza: </p> <dl><dd><dl><dd><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_{i}-E_{f}={\frac {m_{e}q_{e}^{4}}{8h^{2}\epsilon _{0}^{2}}}\left({\frac {1}{n_{f}^{2}}}-{\frac {1}{n_{i}^{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>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <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> <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> <mo>(</mo> <mrow> <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> <mo>−</mo> <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> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=E_{i}-E_{f}={\frac {m_{e}q_{e}^{4}}{8h^{2}\epsilon _{0}^{2}}}\left({\frac {1}{n_{f}^{2}}}-{\frac {1}{n_{i}^{2}}}\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/079788022ebadb9c32d933917765abea5999ef74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:36.518ex; height:7.676ex;" alt="{\displaystyle E=E_{i}-E_{f}={\frac {m_{e}q_{e}^{4}}{8h^{2}\epsilon _{0}^{2}}}\left({\frac {1}{n_{f}^{2}}}-{\frac {1}{n_{i}^{2}}}\right)\,}"></span></dd></dl></dd></dl> <p>gdje je q<sub>e</sub> naelektrisanje elektrona (1,60 × 10<sup>−19</sup> C), n<sub>f</sub> je konačni nivo (<b>f</b>inal) a n<sub>i</sub> glavni kvantni broj početnog (<b>i</b>nitial) energetskog nivoa. Normalno, početni je nivo sa većom energijom. </p><p>Pošto je energija fotona: </p> <dl><dd><dl><dd><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></dd></dl></dd></dl> <p>njegova talasna dužina je: </p> <dl><dd><dl><dd><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 }}={\frac {m_{e}q_{e}^{4}}{8ch^{3}\epsilon _{0}^{2}}}\left({\frac {1}{n_{f}^{2}}}-{\frac {1}{n_{i}^{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> <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> <mrow> <mn>8</mn> <mi>c</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</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> <mo>(</mo> <mrow> <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> <mo>−</mo> <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> </mrow> <mo>)</mo> </mrow> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\lambda }}={\frac {m_{e}q_{e}^{4}}{8ch^{3}\epsilon _{0}^{2}}}\left({\frac {1}{n_{f}^{2}}}-{\frac {1}{n_{i}^{2}}}\right).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd84b54b17b1bb3a94c087f9d706d08f048a27f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:27.669ex; height:7.676ex;" alt="{\displaystyle {\frac {1}{\lambda }}={\frac {m_{e}q_{e}^{4}}{8ch^{3}\epsilon _{0}^{2}}}\left({\frac {1}{n_{f}^{2}}}-{\frac {1}{n_{i}^{2}}}\right).\,}"></span></dd></dl></dd></dl> <p>što predstavlja Ridbergovu formulu. Ova formula, gde su sve numeričke konstante bile stopljene u jednu, Ridbergovu konstantu, R, je empirijski bila tačno izmjerena i poznata u <a href="/wiki/Spektroskopija" title="Spektroskopija">spektroskopiji</a> još krajem <a href="/wiki/19._vijek" title="19. vijek">19. vijeka</a>. Međutim, teorijskog objašnjenja i njene veze sa drugim fundamentalnim konstantama nije bilo dok Bohr nije izveo navedene jednačine. 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