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vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">przypnij</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ukryj</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">Początek</div> </a> </li> <li id="toc-Postulaty_Bohra_z_mechaniki_klasycznej" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Postulaty_Bohra_z_mechaniki_klasycznej"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Postulaty Bohra z mechaniki klasycznej</span> </div> </a> <ul id="toc-Postulaty_Bohra_z_mechaniki_klasycznej-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Postulaty_kwantowe_Bohra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Postulaty_kwantowe_Bohra"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Postulaty kwantowe Bohra</span> </div> </a> <ul id="toc-Postulaty_kwantowe_Bohra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kwantowanie_promieni_orbit_oraz_energii" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kwantowanie_promieni_orbit_oraz_energii"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Kwantowanie promieni orbit oraz energii</span> </div> </a> <ul id="toc-Kwantowanie_promieni_orbit_oraz_energii-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Podsumowanie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Podsumowanie"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Podsumowanie</span> </div> </a> <ul id="toc-Podsumowanie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zastąpienie_modelu_Bohra_przez_modele_kwantowe" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Zastąpienie_modelu_Bohra_przez_modele_kwantowe"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Zastąpienie modelu Bohra przez modele kwantowe</span> </div> </a> <ul id="toc-Zastąpienie_modelu_Bohra_przez_modele_kwantowe-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zobacz_też" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Zobacz_też"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Zobacz też</span> </div> </a> <ul id="toc-Zobacz_też-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Przypisy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Przypisy"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Przypisy</span> </div> </a> <ul id="toc-Przypisy-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="Spis treści" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Spis treści" > <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="Przełącz stan spisu treści" > <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">Przełącz stan spisu treści</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">Model atomu Bohra</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="Przejdź do artykułu w innym języku. Treść dostępna w 70 językach" > <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 języków</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 – szwajcarski niemiecki" lang="gsw" hreflang="gsw" data-title="Bohrsches Atommodell" data-language-autonym="Alemannisch" data-language-local-name="szwajcarski niemiecki" class="interlanguage-link-target"><span>Alemannisch</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 – arabski" lang="ar" hreflang="ar" data-title="نموذج بور" data-language-autonym="العربية" data-language-local-name="arabski" class="interlanguage-link-target"><span>العربية</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 – aragoński" lang="an" hreflang="an" data-title="Modelo atomico de Bohr" data-language-autonym="Aragonés" data-language-local-name="aragoński" class="interlanguage-link-target"><span>Aragonés</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 – asturyjski" lang="ast" hreflang="ast" data-title="Modelu atómicu de Bohr" data-language-autonym="Asturianu" data-language-local-name="asturyjski" 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 – azerbejdżański" lang="az" hreflang="az" data-title="Bor modeli" data-language-autonym="Azərbaycanca" data-language-local-name="azerbejdżański" class="interlanguage-link-target"><span>Azərbaycanca</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-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 – białoruski" lang="be" hreflang="be" data-title="Бораўская мадэль атама" data-language-autonym="Беларуская" data-language-local-name="białoruski" 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 – bułgarski" lang="bg" hreflang="bg" data-title="Модел на Бор" data-language-autonym="Български" data-language-local-name="bułgarski" class="interlanguage-link-target"><span>Български</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 – bośniacki" lang="bs" hreflang="bs" data-title="Bohrov model atoma" data-language-autonym="Bosanski" data-language-local-name="bośniacki" 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 – kataloński" lang="ca" hreflang="ca" data-title="Model atòmic de Bohr" data-language-autonym="Català" data-language-local-name="kataloński" 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 – czeski" lang="cs" hreflang="cs" data-title="Bohrův model atomu" data-language-autonym="Čeština" data-language-local-name="czeski" 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 – duński" lang="da" hreflang="da" data-title="Bohrs atommodel" data-language-autonym="Dansk" data-language-local-name="duński" 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 – niemiecki" lang="de" hreflang="de" data-title="Bohrsches Atommodell" data-language-autonym="Deutsch" data-language-local-name="niemiecki" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Bohri_aatomiteooria" title="Bohri aatomiteooria – estoński" lang="et" hreflang="et" data-title="Bohri aatomiteooria" data-language-autonym="Eesti" data-language-local-name="estoński" class="interlanguage-link-target"><span>Eesti</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 – grecki" lang="el" hreflang="el" data-title="Ατομικό πρότυπο του Μπορ" data-language-autonym="Ελληνικά" data-language-local-name="grecki" 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 – angielski" lang="en" hreflang="en" data-title="Bohr model" data-language-autonym="English" data-language-local-name="angielski" 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 – hiszpański" lang="es" hreflang="es" data-title="Modelo atómico de Bohr" data-language-autonym="Español" data-language-local-name="hiszpański" class="interlanguage-link-target"><span>Español</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 – perski" lang="fa" hreflang="fa" data-title="مدل بور" data-language-autonym="فارسی" data-language-local-name="perski" class="interlanguage-link-target"><span>فارسی</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 – irlandzki" lang="ga" hreflang="ga" data-title="Samhail Bohr" data-language-autonym="Gaeilge" data-language-local-name="irlandzki" 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 – galicyjski" lang="gl" hreflang="gl" data-title="Modelo atómico de Bohr" data-language-autonym="Galego" data-language-local-name="galicyjski" 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 – gudżarati" lang="gu" hreflang="gu" data-title="બૉહરનો સિદ્ધાંત" data-language-autonym="ગુજરાતી" data-language-local-name="gudżarati" class="interlanguage-link-target"><span>ગુજરાતી</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 – koreański" lang="ko" hreflang="ko" data-title="보어 모형" data-language-autonym="한국어" data-language-local-name="koreański" class="interlanguage-link-target"><span>한국어</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 – ormiański" lang="hy" hreflang="hy" data-title="Բորի մոդել" data-language-autonym="Հայերեն" data-language-local-name="ormiański" class="interlanguage-link-target"><span>Հայերեն</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 – chorwacki" lang="hr" hreflang="hr" data-title="Bohrov model atoma" data-language-autonym="Hrvatski" data-language-local-name="chorwacki" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Model_Bohr" title="Model Bohr – indonezyjski" lang="id" hreflang="id" data-title="Model Bohr" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezyjski" 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 – włoski" lang="it" hreflang="it" data-title="Modello atomico di Bohr" data-language-autonym="Italiano" data-language-local-name="włoski" class="interlanguage-link-target"><span>Italiano</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 – hebrajski" lang="he" hreflang="he" data-title="מודל האטום של בוהר" data-language-autonym="עברית" data-language-local-name="hebrajski" class="interlanguage-link-target"><span>עברית</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 – gruziński" lang="ka" hreflang="ka" data-title="ბორის მოდელი" data-language-autonym="ქართული" data-language-local-name="gruziński" 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 – kazachski" lang="kk" hreflang="kk" data-title="Бор қағидалары" data-language-autonym="Қазақша" data-language-local-name="kazachski" class="interlanguage-link-target"><span>Қазақша</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 – łotewski" lang="lv" hreflang="lv" data-title="Bora atoma uzbūves modelis" data-language-autonym="Latviešu" data-language-local-name="łotewski" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Boro_teorija" title="Boro teorija – litewski" lang="lt" hreflang="lt" data-title="Boro teorija" data-language-autonym="Lietuvių" data-language-local-name="litewski" class="interlanguage-link-target"><span>Lietuvių</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 – węgierski" lang="hu" hreflang="hu" data-title="Bohr-féle atommodell" data-language-autonym="Magyar" data-language-local-name="węgierski" class="interlanguage-link-target"><span>Magyar</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 – macedoński" lang="mk" hreflang="mk" data-title="Боров модел" data-language-autonym="Македонски" data-language-local-name="macedoński" 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 – marathi" lang="mr" hreflang="mr" data-title="बोरची अणूची प्रतिकृती" data-language-autonym="मराठी" data-language-local-name="marathi" 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 – birmański" lang="my" hreflang="my" data-title="ဘိုးအာ၏ အက်တမ်ပုံစံ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmański" 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 – niderlandzki" lang="nl" hreflang="nl" data-title="Atoommodel van Bohr" data-language-autonym="Nederlands" data-language-local-name="niderlandzki" class="interlanguage-link-target"><span>Nederlands</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 – japoński" lang="ja" hreflang="ja" data-title="ボーアの原子模型" data-language-autonym="日本語" data-language-local-name="japoński" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Bohrs_atommodell" title="Bohrs atommodell – norweski (bokmål)" lang="nb" hreflang="nb" data-title="Bohrs atommodell" data-language-autonym="Norsk bokmål" data-language-local-name="norweski (bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Atommodell" title="Atommodell – norweski (nynorsk)" lang="nn" hreflang="nn" data-title="Atommodell" data-language-autonym="Norsk nynorsk" data-language-local-name="norweski (nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Bor_postulatlari" title="Bor postulatlari – uzbecki" lang="uz" hreflang="uz" data-title="Bor postulatlari" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbecki" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</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 – pendżabski" lang="pa" hreflang="pa" data-title="ਬੋਹਰ ਮਾਡਲ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="pendżabski" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</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 – rumuński" lang="ro" hreflang="ro" data-title="Modelul atomic Bohr" data-language-autonym="Română" data-language-local-name="rumuński" 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 – rosyjski" lang="ru" hreflang="ru" data-title="Боровская модель атома" data-language-autonym="Русский" data-language-local-name="rosyjski" class="interlanguage-link-target"><span>Русский</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-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Mud%C3%A8llu_at%C3%B2micu_di_Bohr" title="Mudèllu atòmicu di Bohr – sycylijski" lang="scn" hreflang="scn" data-title="Mudèllu atòmicu di Bohr" data-language-autonym="Sicilianu" data-language-local-name="sycylijski" class="interlanguage-link-target"><span>Sicilianu</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 – słowacki" lang="sk" hreflang="sk" data-title="Bohrov model atómu" data-language-autonym="Slovenčina" data-language-local-name="słowacki" 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 – słoweński" lang="sl" hreflang="sl" data-title="Bohrov model atoma" data-language-autonym="Slovenščina" data-language-local-name="słoweński" 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 – serbski" lang="sr" hreflang="sr" data-title="Боров модел атома" data-language-autonym="Српски / srpski" data-language-local-name="serbski" class="interlanguage-link-target"><span>Српски / srpski</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 – serbsko-chorwacki" lang="sh" hreflang="sh" data-title="Bohrov model atoma" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbsko-chorwacki" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Bohrin_atomimalli" title="Bohrin atomimalli – fiński" lang="fi" hreflang="fi" data-title="Bohrin atomimalli" data-language-autonym="Suomi" data-language-local-name="fiński" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Bohrs_atommodell" title="Bohrs atommodell – szwedzki" lang="sv" hreflang="sv" data-title="Bohrs atommodell" data-language-autonym="Svenska" data-language-local-name="szwedzki" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Modelong_Bohr" title="Modelong Bohr – tagalski" lang="tl" hreflang="tl" data-title="Modelong Bohr" data-language-autonym="Tagalog" data-language-local-name="tagalski" class="interlanguage-link-target"><span>Tagalog</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 – tajski" lang="th" hreflang="th" data-title="แบบจำลองของโปร์" data-language-autonym="ไทย" data-language-local-name="tajski" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Bohr_modeli" title="Bohr modeli – turecki" lang="tr" hreflang="tr" data-title="Bohr modeli" data-language-autonym="Türkçe" data-language-local-name="turecki" 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 – ukraiński" lang="uk" hreflang="uk" data-title="Атомна модель Бора" data-language-autonym="Українська" data-language-local-name="ukraiński" class="interlanguage-link-target"><span>Українська</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 – wietnamski" lang="vi" hreflang="vi" data-title="Mô hình Bohr" data-language-autonym="Tiếng Việt" data-language-local-name="wietnamski" class="interlanguage-link-target"><span>Tiếng Việt</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 – chiński klasyczny" lang="lzh" hreflang="lzh" data-title="玻爾模型" data-language-autonym="文言" data-language-local-name="chiński klasyczny" class="interlanguage-link-target"><span>文言</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-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 – kantoński" lang="yue" hreflang="yue" data-title="玻爾模型" data-language-autonym="粵語" data-language-local-name="kantoński" class="interlanguage-link-target"><span>粵語</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 – chiński" lang="zh" hreflang="zh" data-title="玻尔模型" data-language-autonym="中文" data-language-local-name="chiński" class="interlanguage-link-target"><span>中文</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" 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Bohr_atom_model_%28mul%29.svg/langpl-450px-Bohr_atom_model_%28mul%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/Bohr_atom_model_%28mul%29.svg/langpl-600px-Bohr_atom_model_%28mul%29.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>Model budowy atomu Bohra</figcaption></figure> <p><b>Model budowy atomu Bohra</b> – <a href="/wiki/Model_fizyczny" title="Model fizyczny">model</a> <a href="/wiki/Atom" title="Atom">atomu</a> <a href="/wiki/Wod%C3%B3r" title="Wodór">wodoru</a> autorstwa <a href="/wiki/Niels_Bohr" title="Niels Bohr">Nielsa Bohra</a> opracowany w 1913 roku<sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup>. Bohr przyjął wprowadzony przez <a href="/wiki/Ernest_Rutherford" title="Ernest Rutherford">Ernesta Rutherforda</a> model atomu, według którego <a href="/wiki/Elektron" title="Elektron">elektron</a> krąży wokół <a href="/wiki/Fizyka_j%C4%85drowa" title="Fizyka jądrowa">jądra</a> jako naładowany <a href="/wiki/Punkt_materialny" title="Punkt materialny">punkt materialny</a>, przyciągany przez jądro <a href="/wiki/Pole_elektryczne" title="Pole elektryczne">siłami elektrycznymi</a>. </p><p><a href="/wiki/Model_atomu_Rutherforda" title="Model atomu Rutherforda">Model atomu Rutherforda</a> nie przewidywał dyskretnego charakteru widma promieniowania wysyłanego przez atomy oraz nie wyjaśniał ich stabilności. Niels Bohr usunął tę trudność proponując model atomu oparty na dwóch nowych postulatach, sprzecznych z <a href="/wiki/Elektrodynamika_klasyczna" title="Elektrodynamika klasyczna">klasyczną elektrodynamiką</a>. Dzięki temu jednak zdołał wyjaśnić pojawianie się <a href="/wiki/Serie_widmowe_wodoru" title="Serie widmowe wodoru">linii widmowych gazowego wodoru</a> oraz wyprowadził wzory, pozwalające teoretycznie obliczać długości fal promieniowania elektromagnetycznego emitowanego przez wodór<sup id="cite_ref-2" class="reference"><a href="#cite_note-2">[2]</a></sup>. </p><p>Zastosowanie modelu Bohra nie ogranicza się jedynie do atomu wodoru. Model ten jest uniwersalny w tym sensie, że jest słuszny dla układu dwóch dowolnych cząstek naładowanych, które krążą wokół wspólnego środka masy z prędkościami znaczne mniejszymi od <a href="/wiki/Pr%C4%99dko%C5%9B%C4%87_%C5%9Bwiat%C5%82a" title="Prędkość światła">prędkości światła</a>. Model ten jest więc poprawny dla układu jednoelektronowego takiego jak atom wodoru, ale też jest słuszny dla pojedynczo zjonizowanego atomu <a href="/wiki/Hel_(pierwiastek)" title="Hel (pierwiastek)"> helu</a>, podwójnie zjonizowanego atomu <a href="/wiki/Lit" title="Lit">litu</a> itd. Modelem tym można także opisać <a href="/wiki/Pozytonium" title="Pozytonium">pozytonium</a> (czyli układ elektron-pozyton) oraz <a href="/wiki/Atomy_rydbergowskie" title="Atomy rydbergowskie">stany Rydberga</a> jakiegokolwiek atomu, tj. takie stany atomowe, w których jeden elektron jest w znacznej odległości od reszty atomu. Model ten może być także użyty do obliczania linii K <a href="/wiki/Promieniowanie_rentgenowskie" title="Promieniowanie rentgenowskie">promieniowania X</a>, pod warunkiem, że doda się dodatkowe założenia (por. <a href="/wiki/Prawo_Moseleya" title="Prawo Moseleya">prawo Moseleya</a>). W <a href="/wiki/Fizyka_cz%C4%85stek_elementarnych" title="Fizyka cząstek elementarnych">fizyce wysokich energii</a> może być użyty do obliczania mas <a href="/wiki/Mezony" title="Mezony">mezonów</a> zbudowanych z <a href="/wiki/Kwark" title="Kwark">ciężkich kwarków</a>. </p><p>W modelu Bohra elektron krąży wokół jądra atomu po orbicie kołowej. Przez analogię do ruchu <a href="/wiki/Planeta" title="Planeta">planet</a> wokół <a href="/wiki/S%C5%82o%C5%84ce" title="Słońce">Słońca</a> model ten nazwano „modelem planetarnym atomu”. </p><p>Model ten nie wyjaśnia <a href="/wiki/Spin_(fizyka)" title="Spin (fizyka)">spinu</a> elektronu, nie nadaje się do opisu atomów wieloelektronowych. Został zastąpiony przez dokładniejsze modele kwantowe (patrz niżej). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Postulaty_Bohra_z_mechaniki_klasycznej">Postulaty Bohra z mechaniki klasycznej</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Model_atomu_Bohra&veaction=edit&section=1" title="Edytuj sekcję: Postulaty Bohra z mechaniki klasycznej" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Model_atomu_Bohra&action=edit&section=1" title="Edytuj kod źródłowy sekcji: Postulaty Bohra z mechaniki klasycznej"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Model Bohra jest modelem po części opartym na <a href="/wiki/Elektrodynamika_klasyczna" title="Elektrodynamika klasyczna">klasycznej elektrodynamice</a>, zakładającym że: </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="https://en.wikipedia.org/wiki/File:Blausen_0342_ElectronEnergyLevels.png"><img resource="/wiki/Plik:Blausen_0342_ElectronEnergyLevels.png" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Blausen_0342_ElectronEnergyLevels.png/220px-Blausen_0342_ElectronEnergyLevels.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Blausen_0342_ElectronEnergyLevels.png/330px-Blausen_0342_ElectronEnergyLevels.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Blausen_0342_ElectronEnergyLevels.png/440px-Blausen_0342_ElectronEnergyLevels.png 2x" data-file-width="1600" data-file-height="1600" /></a><figcaption>Modele obrazujące liczby elektronów w atomach wodoru, helu, litu i neonu.</figcaption></figure> <p><b>(P1)</b> Elektron jest utrzymywany na orbicie kołowej za pomocą sił elektrycznego oddziaływania z jądrem. <a href="/wiki/Si%C5%82a_do%C5%9Brodkowa" title="Siła dośrodkowa">Siła dośrodkowa</a> jest więc <a href="/wiki/Prawo_Coulomba" title="Prawo Coulomba">siłą Coulomba</a> </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 {mv^{2}}{r}}=k{\frac {Ze^{2}}{r^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>=</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {mv^{2}}{r}}=k{\frac {Ze^{2}}{r^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2039d645b648b84e1a9a43c5b7069b20f79cd97e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.669ex; height:6.009ex;" alt="{\displaystyle {\frac {mv^{2}}{r}}=k{\frac {Ze^{2}}{r^{2}}},}"></span></dd></dl></dd></dl> <p>gdzie: </p> <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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> – masa elektronu,</dd> <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 v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> – prędkość elektronu,</dd> <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> – promień orbity elektronu,</dd> <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> – stała oddziaływań elektrycznych,</dd> <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 Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span> – ładunek jądra (równy <a href="/wiki/Liczba_atomowa" title="Liczba atomowa">liczbie atomowej</a>),</dd> <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </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/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> – ładunek elektronu.</dd></dl> <p><b>(P2)</b> Całkowita energia elektronu w atomie jest sumą <a href="/wiki/Energia_kinetyczna" title="Energia kinetyczna">energii kinetycznej</a> oraz <a href="/wiki/Energia_potencjalna" title="Energia potencjalna">energii potencjalnej</a> oddziaływań elektrycznych </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 {1}{2}}mv^{2}-k{\frac {Ze^{2}}{r}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\frac {1}{2}}mv^{2}-k{\frac {Ze^{2}}{r}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/982bf953794a19d98b078d790ac5b0280817d636" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.448ex; height:5.676ex;" alt="{\displaystyle E={\frac {1}{2}}mv^{2}-k{\frac {Ze^{2}}{r}}.}"></span></dd></dl></dd></dl> <p>Powyższe równania zakładają, że masa jądra jest o wiele większa niż masa elektronu (co jest słuszne, gdyż np. masa protonu jest około 1830 razy większa niż masa elektronu). </p><p>Z rozwiązania układu tych równań otrzymuje się, iż energia całkowita jest równa energii kinetycznej ze znakiem minus </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 {1}{2}}mv^{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> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=-{\frac {1}{2}}mv^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30ebe21d61883c288915a545d337e47de974d2be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.903ex; height:5.176ex;" alt="{\displaystyle E=-{\frac {1}{2}}mv^{2}}"></span></dd></dl></dd></dl> <p>oraz </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 {Zke^{2}}{2}}\cdot {\frac {1}{r}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>k</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=-{\frac {Zke^{2}}{2}}\cdot {\frac {1}{r}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4843c13eb186b199d11e28aa592cc8cf30fe6492" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.872ex; height:5.676ex;" alt="{\displaystyle E=-{\frac {Zke^{2}}{2}}\cdot {\frac {1}{r}},}"></span></dd></dl></dd></dl> <p>czyli: energia całkowita jest równa połowie energii potencjalnej; energia całkowita jest ujemna i jest odwrotnie proporcjonalna do <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.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10110093812676dd04a92ce4c8b75940c366330a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.695ex; height:1.676ex;" alt="{\displaystyle r.}"></span> Oznacza to, że trzeba dostarczyć elektronowi energii z zewnątrz, aby przenieść go dalej od protonu. Dla <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> dążące do nieskończoności energia rośnie do zera. (Własność, iż całkowita energia jest połową energii potencjalnej jest na mocy <a href="/wiki/Twierdzenie_o_wiriale" title="Twierdzenie o wiriale">twierdzenia o wiriale</a> słuszna dla dowolnych orbit, niekoniecznie kołowych). </p> <div class="mw-heading mw-heading2"><h2 id="Postulaty_kwantowe_Bohra">Postulaty kwantowe Bohra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Model_atomu_Bohra&veaction=edit&section=2" title="Edytuj sekcję: Postulaty kwantowe Bohra" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Model_atomu_Bohra&action=edit&section=2" title="Edytuj kod źródłowy sekcji: Postulaty kwantowe Bohra"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O rewolucyjności podejścia Bohra decydują sformułowane przez niego zasady kwantowe. Postulaty te były wprowadzone <i><a href="/wiki/Ad_hoc" title="Ad hoc">ad hoc</a>,</i> aby uzyskać zgodność przewidywań modelu z eksperymentami. </p><p><b>(P3)</b> Postulat o kwantyzacji orbitalnego momentu pędu elektronu. Z nieskończonej liczby orbit, które dopuszczałaby <a href="/wiki/Mechanika_klasyczna" title="Mechanika klasyczna">mechanika klasyczna</a>, elektron może poruszać się tylko po tych, dla których <a href="/wiki/Moment_p%C4%99du" title="Moment pędu">moment pędu</a> elektronu <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=mvr}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>m</mi> <mi>v</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=mvr}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a61ad95845fc979255d075922775f436ba910ece" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.898ex; height:2.176ex;" alt="{\displaystyle L=mvr}"></span> jest równy całkowitej wielokrotności <a href="/wiki/Sta%C5%82a_Plancka" title="Stała Plancka">stałej Plancka</a> podzielonej przez <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 ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/461d52fa6fb5a16ef8a24e871488584db5398489" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.141ex; height:2.509ex;" alt="{\displaystyle 2\pi ,}"></span> tj. </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 mvr=n\,\hbar ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mi>v</mi> <mi>r</mi> <mo>=</mo> <mi>n</mi> <mspace width="thinmathspace" /> <mi class="MJX-variant">ℏ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mvr=n\,\hbar ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53dd3eed853b15d684fc98f526cac44d5e12b5c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.05ex; height:2.509ex;" alt="{\displaystyle mvr=n\,\hbar ,}"></span></dd></dl></dd></dl> <p>gdzie: </p> <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 n=1,2,3\dots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>…</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1,2,3\dots ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449efa828e1a9a7f9c4d297a5a026c1ca0f85cbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.193ex; height:2.509ex;" alt="{\displaystyle n=1,2,3\dots ,}"></span></dd> <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 \hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de68de3a92517953436c93b5a76461d49160cc41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }"></span> – stała Plancka podzielona przez <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 .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a597bc393cb175d7b44b9bdfdbadc1fb7dc23aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.141ex; height:2.176ex;" alt="{\displaystyle 2\pi .}"></span></dd></dl> <figure typeof="mw:File/Thumb"><a href="/wiki/Plik:Bohr_atom_animation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Bohr_atom_animation.gif/300px-Bohr_atom_animation.gif" decoding="async" width="300" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Bohr_atom_animation.gif/450px-Bohr_atom_animation.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/3/31/Bohr_atom_animation.gif 2x" data-file-width="550" data-file-height="400" /></a><figcaption>Przy zmianie orbity elektron <a href="/wiki/Emisja" class="mw-disambig" title="Emisja">emituje</a> lub pochłania foton o ściśle określonej energii</figcaption></figure> <p><b>(P4)</b> Postulat o promieniowaniu. Elektron poruszając się po danej orbicie nie emituje ani nie pochłania energii. Elektron emituje foton, jeżeli spada z orbity wyższej na niższą, a pochłania foton, jeżeli przechodzi z orbity niższej na wyższą; energia <a href="/wiki/Emisja_promieniowania" title="Emisja promieniowania">emitowanego</a> (pochłanianego) fotonu jest równa różnicy między energią wyższą i niższą elektronu </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_{f}=h\nu =E_{2}-E_{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mi>h</mi> <mi>ν</mi> <mo>=</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{f}=h\nu =E_{2}-E_{1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33994fca0dbc9eeae208f3abdc5f2bf25caefebf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.646ex; height:2.843ex;" alt="{\displaystyle E_{f}=h\nu =E_{2}-E_{1},}"></span></dd></dl></dd></dl> <p>gdzie: </p> <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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{2},E_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04f6a7fad1cec0bf11575649ec7279d39a34561" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.573ex; height:2.509ex;" alt="{\displaystyle E_{2},E_{1}}"></span> – energie elektronu, odpowiednio, wyższa i niższa,</dd> <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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> – stała Plancka,</dd> <dd><a href="/wiki/Ny_(litera)" title="Ny (litera)"><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 \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span></a> – częstotliwość fotonu.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Kwantowanie_promieni_orbit_oraz_energii">Kwantowanie promieni orbit oraz energii</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Model_atomu_Bohra&veaction=edit&section=3" title="Edytuj sekcję: Kwantowanie promieni orbit oraz energii" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Model_atomu_Bohra&action=edit&section=3" title="Edytuj kod źródłowy sekcji: Kwantowanie promieni orbit oraz energii"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Z postulatu kwantowania momentu pędu wynika, iż promienie orbit oraz energia są skwantowane, tj. mogą przyjmować tylko dyskretne wartości. </p><p>Aby to pokazać zauważmy, że pierwsze równanie implikuje zależność prędkości od promienia </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 v={\sqrt {\frac {Zke^{2}}{mr}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>Z</mi> <mi>k</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>m</mi> <mi>r</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\sqrt {\frac {Zke^{2}}{mr}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa36fba1235068a7cfaa966f49c2ccea931453a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.062ex; height:6.176ex;" alt="{\displaystyle v={\sqrt {\frac {Zke^{2}}{mr}}}.}"></span></dd></dl></dd></dl> <p>Wstawiając tę zależność do wzoru na kwantowanie momentu pędu otrzyma się równanie zależne tylko od <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;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b785304382188b4d91249466d20cba1e544b1160" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.695ex; height:2.009ex;" alt="{\displaystyle r;}"></span> stąd dostaje się dozwolone promienie orbit </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_{n}={\frac {\hbar ^{2}}{Zke^{2}m}}\cdot n^{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> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>Z</mi> <mi>k</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}={\frac {\hbar ^{2}}{Zke^{2}m}}\cdot n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/326f40824f6aa44acb93390ec9a0e456bbdd3336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.4ex; height:6.009ex;" alt="{\displaystyle r_{n}={\frac {\hbar ^{2}}{Zke^{2}m}}\cdot n^{2}}"></span></dd></dl></dd></dl> <p>Widać, że promienie orbit rosną proporcjonalnie do kwadratu numeru <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> orbity. </p><p>Najmniejsza możliwa wartość promienia orbity w atomie wodoru <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 (Z=1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>Z</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Z=1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c5f9384a91d760d3f6e62dac0b032447773c3b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.751ex; height:2.843ex;" alt="{\displaystyle (Z=1)}"></span> jest nazywana <a href="/wiki/Promie%C5%84_atomu_Bohra" title="Promień atomu Bohra">promieniem Bohra</a> i wynosi </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_{1}={\frac {\hbar ^{2}}{ke^{2}m}}\approx 5{,}29\cdot 10^{-11}\mathrm {m} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>k</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>≈</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>29</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>11</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}={\frac {\hbar ^{2}}{ke^{2}m}}\approx 5{,}29\cdot 10^{-11}\mathrm {m} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19de774a0ed2a5090f09b24548561eb51bb8f055" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.401ex; height:6.009ex;" alt="{\displaystyle r_{1}={\frac {\hbar ^{2}}{ke^{2}m}}\approx 5{,}29\cdot 10^{-11}\mathrm {m} .}"></span></dd></dl></dd></dl> <p>Dowolny promień orbity można więc wyrazić w zależności od liczby kwantowej <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> oraz <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}"></span> </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_{n}=r_{1}\cdot n^{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> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}=r_{1}\cdot n^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ccae0a34f3c7e7d50a8ea93e4cd7ef1feff079c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.243ex; height:3.009ex;" alt="{\displaystyle r_{n}=r_{1}\cdot n^{2}.}"></span></dd></dl></dd></dl> <p>Ponieważ energia zależy <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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57784cdf7f49f2c46baad4eb7b6a7d5c14eb5fa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.267ex; height:2.009ex;" alt="{\displaystyle r_{n}}"></span> </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_{n}=-{\frac {Zke^{2}}{2}}{\frac {1}{r_{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> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>k</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}=-{\frac {Zke^{2}}{2}}{\frac {1}{r_{n}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a056b39cba24c523051761156907e4c33d917732" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.456ex; height:6.009ex;" alt="{\displaystyle E_{n}=-{\frac {Zke^{2}}{2}}{\frac {1}{r_{n}}},}"></span></dd></dl></dd></dl> <p>więc ostatecznie otrzymamy, iż energia <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-tego poziomu jest odwrotnie proporcjonalna do kwadratu liczby kwantowej <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> oraz energii <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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac42446bcd2cbb76ec8fe2895635d328da22e26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{1}}"></span> </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_{n}=E_{1}\cdot {\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> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}=E_{1}\cdot {\frac {1}{n^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/026dbef91a8b3e685f818ac1139e40da6b1325fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.413ex; height:5.509ex;" alt="{\displaystyle E_{n}=E_{1}\cdot {\frac {1}{n^{2}}},}"></span></dd></dl></dd></dl> <p>gdzie <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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac42446bcd2cbb76ec8fe2895635d328da22e26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{1}}"></span> – energia stanu podstawowego atomu </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_{1}=-{\frac {Zke^{2}}{2r_{1}}}\approx -13{,}6\,Z^{2}\cdot \mathrm {eV} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>k</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>≈</mo> <mo>−</mo> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>6</mn> <mspace width="thinmathspace" /> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">V</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1}=-{\frac {Zke^{2}}{2r_{1}}}\approx -13{,}6\,Z^{2}\cdot \mathrm {eV} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8152f0aefe1dd13b1cc697a606d99a003eb09bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:30.834ex; height:6.009ex;" alt="{\displaystyle E_{1}=-{\frac {Zke^{2}}{2r_{1}}}\approx -13{,}6\,Z^{2}\cdot \mathrm {eV} .}"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Podsumowanie">Podsumowanie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Model_atomu_Bohra&veaction=edit&section=4" title="Edytuj sekcję: Podsumowanie" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Model_atomu_Bohra&action=edit&section=4" title="Edytuj kod źródłowy sekcji: Podsumowanie"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Z postulatu <b>(P3)</b> o kwantowaniu momentu pędu wynika, że promienie orbit elektronu oraz energie elektronu w atomie są również skwantowane. Promienie te, w szczególności promień atomu wodoru w stanie podstawowym (tzw. promień atomu Bohra) oraz energie można wyznaczyć zakładając, iż elektron podlega prawu klasycznego oddziaływania elektrycznego klasycznego (postulaty (<b>P1</b>) oraz (<b>P2</b>)). </p><p>Postulat kwantowy (<b>P4</b>) o promieniowaniu atomu zakłada, że elektron krążąc po określonej orbicie nie emituje promieniowania, wbrew prawu promieniowania klasycznej elektrodynamiki (głoszącemu, iż cząstka naładowana krążąc po orbicie powinna emitować promieniowanie elektromagnetyczne i tracić energię), ale promieniowanie jest emitowane tylko wówczas, gdy elektron zmienia orbitę. Zgodnie z elektrodynamiką klasyczną elektron poruszając się ruchem niejednostajnym wokół jadra powinien promieniować energię w postaci fali elektromagnetycznej, w wyniku czego musiałby „spaść” na jądro już po czasie rzędu 10<sup>−6</sup> sekundy i atom przestałby istnieć. </p><p>Model Bohra, jakkolwiek będący sztucznym połączeniem mechaniki klasycznej i nowych postulatów kwantowych, daje prawidłowe wyniki dotyczące wartości <a href="/wiki/Energia_(fizyka)" class="mw-redirect" title="Energia (fizyka)">energii</a> elektronu w atomie wodoru oraz długości fal emitowanych przez wodór (<a href="/wiki/Serie_widmowe_wodoru" title="Serie widmowe wodoru">linie widmowe wodoru</a>). Jest też poprawny w odniesieniu do tzw. atomów wodoropodobnych, tj. atomów zjonizowanych tak, że posiadają tylko jeden elektron. </p><p>Mimo ogromnego sukcesu modelu, za który Bohr otrzymał w 1922 r. <a href="/wiki/Laureaci_Nagrody_Nobla_w_dziedzinie_fizyki" title="Laureaci Nagrody Nobla w dziedzinie fizyki">nagrodę Nobla w dziedzinie fizyki</a>, model ten pozostawiał wiele nie rozwiązanych problemów. M.in. nie dało się go zaadaptować do opisu atomów posiadających więcej niż dwa elektrony. Nie można też było za jego pomocą stworzyć zgodnej ze faktami eksperymentalnymi teorii <a href="/wiki/Wi%C4%85zanie_chemiczne" title="Wiązanie chemiczne">wiązań chemicznych</a>. Także wielkim problemem był fakt, iż model łączył w sobie sprzeczne idee, tj. zakładał postulaty kwantowe sprzeczne z prawami klasycznej elektrodynamiki, mimo iż częściowo na nich opierał się. </p> <div class="mw-heading mw-heading2"><h2 id="Zastąpienie_modelu_Bohra_przez_modele_kwantowe"><span id="Zast.C4.85pienie_modelu_Bohra_przez_modele_kwantowe"></span>Zastąpienie modelu Bohra przez modele kwantowe</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Model_atomu_Bohra&veaction=edit&section=5" title="Edytuj sekcję: Zastąpienie modelu Bohra przez modele kwantowe" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Model_atomu_Bohra&action=edit&section=5" title="Edytuj kod źródłowy sekcji: Zastąpienie modelu Bohra przez modele kwantowe"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kolejne, lepsze modele atomu wodoru oraz atomów wieloelektronowych zostały dostarczone przez <a href="/wiki/Mechanika_kwantowa" title="Mechanika kwantowa">mechanikę kwantową</a> (<a href="/wiki/R%C3%B3wnanie_Schr%C3%B6dingera" title="Równanie Schrödingera">równanie Schrödingera</a> 1926 r., <a href="/wiki/R%C3%B3wnanie_Pauliego" title="Równanie Pauliego">równanie Pauliego</a> 1927 r., <a href="/wiki/R%C3%B3wnanie_Diraca" title="Równanie Diraca">równanie Diraca</a> 1928 r.) oraz <a href="/wiki/Elektrodynamika_kwantowa" title="Elektrodynamika kwantowa">elektrodynamikę kwantową</a>. Modele te pozwalały z coraz lepszą precyzją obliczać długości <a href="/wiki/Widmo_(spektroskopia)" title="Widmo (spektroskopia)">linii widmowych</a> atomów i cząsteczek odpowiadającą coraz bardziej precyzyjnym pomiarom (np. pomiar <a href="/wiki/Przesuni%C4%99cie_Lamba" title="Przesunięcie Lamba">przesunięcia Lamba</a>) czy obliczać <a href="/wiki/Magnetyczny_moment_dipolowy" title="Magnetyczny moment dipolowy">moment magnetyczny</a> elektronu. Dzięki temu wytłumaczono też <a href="/wiki/Spin_(fizyka)" title="Spin (fizyka)">spin</a> elektronu i wiązania chemiczne. </p><p>Zagadnienie opisu atomu wodoru w ramach mechaniki kwantowej omówiono w artykule Operator Hamiltona – rozdział <a href="/wiki/Operator_Hamiltona#Hamiltonian_elektronu_w_polu_centralnym_–_zagadnienie_atomu_wodoru" title="Operator Hamiltona">Hamiltonian elektronu w polu centralnym – zagadnienie atomu wodoru</a> </p> <div class="mw-heading mw-heading2"><h2 id="Zobacz_też"><span id="Zobacz_te.C5.BC"></span>Zobacz też</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Model_atomu_Bohra&veaction=edit&section=6" title="Edytuj sekcję: Zobacz też" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Model_atomu_Bohra&action=edit&section=6" title="Edytuj kod źródłowy sekcji: Zobacz też"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Elektrodynamika_kwantowa" title="Elektrodynamika kwantowa">elektrodynamika kwantowa</a></li> <li><a href="/wiki/Model_atomu_Gryzi%C5%84skiego" title="Model atomu Gryzińskiego">model atomu Gryzińskiego</a></li> <li><a href="/wiki/Model_atomu_Rutherforda" title="Model atomu Rutherforda">model atomu Rutherforda</a></li> <li><a href="/wiki/Model_atomu_Thomsona" title="Model atomu Thomsona">model atomu Thomsona</a></li> <li><a href="/wiki/R%C3%B3wnanie_Diraca" title="Równanie Diraca">równanie Diraca</a></li> <li><a href="/wiki/R%C3%B3wnanie_Pauliego" title="Równanie Pauliego">równanie Pauliego</a></li> <li><a href="/wiki/R%C3%B3wnanie_Schr%C3%B6dingera" title="Równanie Schrödingera">równanie Schrödingera</a></li> <li><a href="/wiki/Zasada_odpowiednio%C5%9Bci" title="Zasada odpowiedniości">zasada odpowiedniości</a></li> <li><a href="/wiki/Paczka_troja%C5%84ska" title="Paczka trojańska">paczka trojańska</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Przypisy">Przypisy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Model_atomu_Bohra&veaction=edit&section=7" title="Edytuj sekcję: Przypisy" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Model_atomu_Bohra&action=edit&section=7" title="Edytuj kod źródłowy sekcji: Przypisy"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="do-not-make-smaller refsection"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">N. Bohr, <a href="/wiki/Philosophical_Magazine" title="Philosophical Magazine">Philosophical Magazine</a> 26 (1913) 1, 476, 857.</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 web open-access"><a rel="nofollow" class="external text" href="https://encyklopedia.pwn.pl/haslo/;3879060"><i>Bohra teoria atomu</i></a>, [w:] <i><a href="/wiki/Encyklopedia_PWN_(internetowa)" title="Encyklopedia PWN (internetowa)">Encyklopedia PWN</a></i> [online], <a href="/wiki/Wydawnictwo_Naukowe_PWN" title="Wydawnictwo Naukowe PWN">Wydawnictwo Naukowe PWN</a><span class="accessdate"> [dostęp 2021-07-29]</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft.gengre=unknown&rft.atitle=Bohra+teoria+atomu&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.jtitle=%5B%5BWydawnictwo+Naukowe+PWN%5D%5D&rft_id=https%3A%2F%2Fencyklopedia.pwn.pl%2Fhaslo%2F%3B3879060" style="display:none"> </span>.</cite></span> 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.mw-parser-output .navbox .title,html.skin-theme-clientpref-night .mw-parser-output .navbox>.mw-collapsible-content>.above,html.skin-theme-clientpref-night .mw-parser-output .navbox>.mw-collapsible-content>.below,html.skin-theme-clientpref-night .mw-parser-output .navbox .opis{background-color:#303234}html.skin-theme-clientpref-night .mw-parser-output .navbox.medaliści .opis.a1,html.skin-theme-clientpref-night .mw-parser-output .navbox.medaliści .a1 .opis{background:#715f00}html.skin-theme-clientpref-night .mw-parser-output .navbox.medaliści .opis.a2,html.skin-theme-clientpref-night .mw-parser-output .navbox.medaliści .a2 .opis{background:#5f5f5f}html.skin-theme-clientpref-night .mw-parser-output .navbox.medaliści .opis.a3,html.skin-theme-clientpref-night .mw-parser-output .navbox.medaliści .a3 .opis{background:#764617}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbox>.caption,html.skin-theme-clientpref-os .mw-parser-output .navbox>tbody>tr>th{background-color:#3a3c3e}html.skin-theme-clientpref-os .mw-parser-output .navbox .title,html.skin-theme-clientpref-os .mw-parser-output .navbox>.mw-collapsible-content>.above,html.skin-theme-clientpref-os .mw-parser-output .navbox>.mw-collapsible-content>.below,html.skin-theme-clientpref-os .mw-parser-output .navbox .opis{background-color:#303234}html.skin-theme-clientpref-os .mw-parser-output .navbox.medaliści .opis.a1,html.skin-theme-clientpref-os .mw-parser-output .navbox.medaliści .a1 .opis{background:#715f00}html.skin-theme-clientpref-os .mw-parser-output .navbox.medaliści .opis.a2,html.skin-theme-clientpref-os .mw-parser-output .navbox.medaliści .a2 .opis{background:#5f5f5f}html.skin-theme-clientpref-os .mw-parser-output .navbox.medaliści .opis.a3,html.skin-theme-clientpref-os .mw-parser-output .navbox.medaliści .a3 .opis{background:#764617}}</style><ul class="tnavbar noprint plainlinks hlist"><li><a href="/wiki/Szablon:Modele_atom%C3%B3w" title="Szablon:Modele atomów"><span title="Pokaż ten szablon">p</span></a></li><li><a href="/wiki/Dyskusja_szablonu:Modele_atom%C3%B3w" title="Dyskusja szablonu:Modele atomów"><span title="Dyskusja na temat tego szablonu">d</span></a></li><li title="Możesz edytować ten szablon. Użyj przycisku podglądu przed zapisaniem zmian."><a class="external text" href="https://pl.wikipedia.org/w/index.php?title=Szablon:Modele_atom%C3%B3w&action=edit">e</a></li></ul><div class="navbox-title caption">Modele atomów</div><div class="mw-collapsible-content"><div class="hlist navbox-above above"> <ul><li><a href="/wiki/Atom" title="Atom">Atom</a></li> <li><a href="/wiki/J%C4%85dro_atomowe" title="Jądro atomowe">Jądro atomowe</a></li></ul> </div><div class="flex"><table class="navbox-main-content inner-standard"><tbody><tr class="a1"><th class="navbox-group opis" scope="row"><a href="/wiki/Fizyka_klasyczna" title="Fizyka klasyczna">klasyczne</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Teoria_atomistyczna_Daltona" title="Teoria atomistyczna Daltona">model Daltona</a></li> <li><a href="/wiki/Model_atomu_Thomsona" title="Model atomu Thomsona">model Thomsona</a></li> <li><a href="/wiki/Model_atomu_Lewisa" title="Model atomu Lewisa">model Lewisa</a></li> <li><a href="/w/index.php?title=Model_atomu_Nagaoki&action=edit&redlink=1" class="new" title="Model atomu Nagaoki (strona nie istnieje)">model Nagaoki</a></li> <li><a href="/wiki/Model_atomu_Rutherforda" title="Model atomu Rutherforda">model Rutherforda</a></li> <li><a href="/wiki/Model_atomu_Gryzi%C5%84skiego" title="Model atomu Gryzińskiego">model Gryzińskiego</a></li></ul> </td></tr><tr class="a2"><th class="navbox-group opis" scope="row"><a href="/wiki/Mechanika_kwantowa" title="Mechanika kwantowa">kwantowe</a></th><td class="navbox-list spis hlist navbox-even"> <ul><li><a class="mw-selflink selflink">model Bohra</a></li> <li><a href="/w/index.php?title=Model_atomu_Bohra-Sommerfelda&action=edit&redlink=1" class="new" title="Model atomu Bohra-Sommerfelda (strona nie istnieje)">model Bohra-Sommerfelda</a><span class="noprint link-interwiki" style="position:relative;top:-.4em; white-space: nowrap; font-size:x-small;"><a href="https://www.wikidata.org/wiki/Q360686#sitelinks-wikipedia" class="extiw" title="d:Q360686">(inne języki)</a></span></li> <li><a href="/wiki/Orbital" title="Orbital">model Schrödingera</a></li></ul> </td></tr><tr class="a3"><th class="navbox-group opis" scope="row"><a href="/wiki/Naukowiec" title="Naukowiec">uczeni</a> według<br />daty narodzin</th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a3_1"><th class="navbox-group opis" scope="row"><a href="/wiki/XVIII_wiek" title="XVIII wiek">XVIII wiek</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/John_Dalton" title="John Dalton">John Dalton</a></li></ul> </td></tr><tr class="a3_2"><th class="navbox-group opis" scope="row"><a href="/wiki/XIX_wiek" title="XIX wiek">XIX wiek</a></th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Joseph_John_Thomson" title="Joseph John Thomson">Joseph John Thomson</a></li> <li><a href="/w/index.php?title=Hantaro_Nagaoka&action=edit&redlink=1" class="new" title="Hantaro Nagaoka (strona nie istnieje)">Hantaro Nagaoka</a><span class="noprint link-interwiki" style="position:relative;top:-.4em; white-space: nowrap; font-size:x-small;"><a href="https://www.wikidata.org/wiki/Q1936356#sitelinks-wikipedia" class="extiw" title="d:Q1936356">(inne języki)</a></span></li> <li><a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Arnold Sommerfeld</a></li> <li><a href="/wiki/Ernest_Rutherford" title="Ernest Rutherford">Ernest Rutherford</a></li> <li><a href="/wiki/Gilbert_Lewis" title="Gilbert Lewis">Gilbert Lewis</a></li> <li><a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a></li> <li><a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a></li></ul> </td></tr><tr class="a3_3"><th class="navbox-group opis" scope="row"><a href="/wiki/XX_wiek" title="XX wiek">XX wiek</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Micha%C5%82_Gryzi%C5%84ski" title="Michał Gryziński">Michał Gryziński</a></li></ul> </td></tr></tbody></table></td></tr></tbody></table><div class="navbox-after after"> <p><span typeof="mw:File"><a href="/wiki/Plik:Atom_symbol.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Atom_symbol.svg/70px-Atom_symbol.svg.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Atom_symbol.svg/105px-Atom_symbol.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/58/Atom_symbol.svg/140px-Atom_symbol.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span> </p> </div></div></div></div> <style data-mw-deduplicate="TemplateStyles:r75562624">.mw-parser-output #normdaten>div+div{margin-top:0.5em}.mw-parser-output #normdaten>div>div{background:var(--background-color-neutral,#eaecf0);padding:.2em .5em}.mw-parser-output #normdaten ul{margin:0;padding:0}.mw-parser-output #normdaten ul li:first-child{padding-left:.5em;border-left:1px solid var(--border-color-base,#a2a9b1)}</style> <div id="normdaten" class="catlinks"><div class="normdaten-typ-w"><div><a href="/wiki/Kontrola_autorytatywna" title="Kontrola autorytatywna">Kontrola autorytatywna</a> (<span class="description">model atomu</span>):</div><ul><li><a href="/wiki/Centralna_Biblioteka_Narodowa_we_Florencji" title="Centralna Biblioteka Narodowa we Florencji">BNCF</a>: <span class="uid"><a rel="nofollow" class="external text" href="http://thes.bncf.firenze.sbn.it/termine.php?id=42793">42793</a></span></li></ul></div><div class="normdaten-andere"><div><a href="/wiki/Encyklopedia_internetowa" title="Encyklopedia internetowa">Encyklopedie internetowe</a>:</div> <ul><li><a href="/wiki/Encyklopedia_PWN_(internetowa)" title="Encyklopedia PWN (internetowa)">PWN</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://encyklopedia.pwn.pl/haslo/;3879060.html">3879060</a></span></li> <li><a href="/wiki/Encyklopedia_Britannica" title="Encyklopedia Britannica">Britannica</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://www.britannica.com/science/Bohr-model">science/Bohr-model</a></span></li> <li><a href="https://www.wikidata.org/wiki/Property:P3219" class="extiw" title="d:Property:P3219">Universalis</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://www.universalis.fr/encyclopedie/atome-de-bohr/">atome-de-bohr</a></span></li> <li><a href="/wiki/Den_Store_Danske_Encyklop%C3%A6di" title="Den Store Danske Encyklopædi">DSDE</a>: <span class="uid"><a class="external text" href="https://wikidata-externalid-url.toolforge.org/?p=8313&url_prefix=https://lex.dk/&id=Bohrs_atommodel">Bohrs_atommodel</a></span></li></ul> </div></div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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