Atomic Theory IV | Chemistry

<!DOCTYPE html> <html lang="en" dir="ltr"> <head>  <meta http-equiv="X-UA-Compatible" content="ie=edge"> <meta charset="utf-8"> <base href="https://www.visionlearning.com"> <title>Atomic Theory IV | Chemistry | Visionlearning</title> <link rel="canonical" href="https://www.visionlearning.com/en/library/chemistry/1/atomic-theory-iv/231"> <meta name="description" content="Our Atomic Theory series continues, exploring the quantum model of the atom in greater detail. This module takes a closer look at the Schrödinger equation that defines the energies and probable positions of electrons within atoms. Using the hydrogen atom as an example, the module explains how orbitals can be described by type of wave function. Evidence for orbitals and the quantum model is provided by the absorption and emission spectra of hydrogen. Other concepts include multi-electron atoms, the Aufbau Principle, and Hund’s Rule."> <meta name="keywords" content="atomic theory, quantum model, Schrödinger equation, wave/particle duality"> <meta name="viewport" content="width=device-width, initial-scale=1.0, shrink-to-fit=no"> <meta name="msvalidate.01" content="D8E20F39AD48052260032E56DE409970"> <script type="application/ld+json"> { "@context": "http://schema.org/", "@type": "Article", "mainEntityOfPage": { "@type": "WebPage", "@id": "https://visionlearning.com/en/library/chemistry/1/atomic-theory-iv/231" }, "name": "Atomic Theory IV", "headline": "Atomic Theory IV: Quantum numbers and orbitals", "author": [ { "@type": "Person", "name": "Adrian Dingle, B.Sc." } , { "@type": "Person", "name": "Anthony Carpi, Ph.D." }], "datePublished": "2016-04-05 02:45:47", "dateModified": "2017-02-12T08:30:00+05:00", "image": { "@type": "ImageObject", "url": "/img/library/moduleImages/featured_image_231-23061209064512.jpeg", "width": 696, "height": 464 }, "publisher": { "@type": "Organization", "name": "Visionlearning, Inc.", "logo": { "@type": "ImageObject", "url": "http://visionlearning.com/images/logo.png", "width": 278, "height": 60 } }, "description": "Our Atomic Theory series continues, exploring the quantum model of the atom in greater detail. This module takes a closer look at the Schrödinger equation that defines the energies and probable positions of electrons within atoms. Using the hydrogen atom as an example, the module explains how orbitals can be described by type of wave function. Evidence for orbitals and the quantum model is provided by the absorption and emission spectra of hydrogen. Other concepts include multi-electron atoms, the Aufbau Principle, and Hund’s Rule.", "keywords": "atomic theory, quantum model, Schrödinger equation, wave/particle duality", "inLanguage": { "@type": "Language", "name": "English", "alternateName": "en" }, "copyrightHolder": { "@type": "Organization", "name": "Visionlearning, Inc." }, "copyrightYear": "2016"} </script> <meta property="og:url" content="https://visionlearning.com/en/library/chemistry/1/atomic-theory-iv/231"> <meta property="og:title" content="Atomic Theory IV | Chemistry | Visionlearning" /> <meta property="og:type" content="website"> <meta property="og:site_name" content="Visionlearning"> <meta property="og:description" content="Our Atomic Theory series continues, exploring the quantum model of the atom in greater detail. This module takes a closer look at the Schrödinger equation that defines the energies and probable positions of electrons within atoms. Using the hydrogen atom as an example, the module explains how orbitals can be described by type of wave function. Evidence for orbitals and the quantum model is provided by the absorption and emission spectra of hydrogen. Other concepts include multi-electron atoms, the Aufbau Principle, and Hund’s Rule."> <meta property="og:image" content="https://visionlearning.com/images/logo.png"> <meta property="fb:admins" content="100000299664514"> <link rel="stylesheet" type="text/css" href="/css/visionlearning.css">  <link rel="stylesheet" type="text/css" href="/css/visionlearning-icons.css">  <link rel="preload" href="https://fonts.gstatic.com"> <link rel="preload" href="https://fonts.googleapis.com/css2?family=Open+Sans:ital,wght@0,400;0,700;1,400;1,700&family=Schoolbell&display=swap"> <style> textarea.myEditor { width: 90%; height: 350px; } </style> <script type="text/x-mathjax-config" src="/js/mathjax-config.js"></script> <script id="MathJax-script" async src="/js/mathjax/tex-svg.js"></script> <script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js?client=ca-pub-9561344156007092" crossorigin="anonymous"></script> </head> <body>  <header class="box-shadow-1" id="global-header"> <div 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href="/en/library/earth-science/6/ocean-currents/282">Ocean Currents</a></li> <li><a href="/en/library/earth-science/6/water-in-the-atmosphere/289">Water in the Atmosphere</a></li> <li><a href="/en/library/earth-science/6/history-of-earths-atmosphere-i/202">History of Earth's Atmosphere I</a></li> <li><a href="/en/library/earth-science/6/history-of-earths-atmosphere-ii/203">History of Earth's Atmosphere II</a></li> <li><a href="/en/library/earth-science/6/earths-atmosphere/107">Earth's Atmosphere</a></li> <li><a href="/en/library/earth-science/6/factors-that-control-earths-temperature/234">Factors that Control Earth's Temperature</a></li> <li><a href="/en/library/earth-science/6/circulation-in-the-atmosphere/255">Circulation in the Atmosphere</a></li> </ul> </div> <button class="accordion__button" id="acc-button-hazards" data-accordion="button" aria-controls="acc-panel-hazards" aria-expanded="false"> <span class="accordion__button__label"> Hazards </span> </button> <div 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aria-controls="acc-panel-environmental-science" aria-expanded="false"> <span class="accordion__button__label"> Environmental Science </span> </button> <div class="accordion__panel" id="acc-panel-environmental-science" data-accordion="panel" aria-labelledby="acc-button-environmental-science" role="region"> <div class="accordion accordion--secondary"> <button class="accordion__button" id="acc-button-ecology" data-accordion="button" aria-controls="acc-panel-ecology" aria-expanded="false"> <span class="accordion__button__label"> Ecology </span> </button> <div class="accordion__panel" id="acc-panel-ecology" data-accordion="panel" aria-labelledby="acc-button-ecology" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/environmental-science/61/biodiversity-i/276">Biodiversity I</a></li> <li><a href="/en/library/environmental-science/61/biodiversity-ii/281">Biodiversity II</a></li> <li><a href="/en/library/environmental-science/61/ecosystem-services/279">Ecosystem Services</a></li> <li><a href="/en/library/environmental-science/61/population-biology/287">Population Biology</a></li> </ul> </div> <button class="accordion__button" id="acc-button-earth-cycles" data-accordion="button" aria-controls="acc-panel-earth-cycles" aria-expanded="false"> <span class="accordion__button__label"> Earth Cycles </span> </button> <div class="accordion__panel" id="acc-panel-earth-cycles" data-accordion="panel" aria-labelledby="acc-button-earth-cycles" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/environmental-science/61/the-nitrogen-cycle/98">The Nitrogen Cycle</a></li> <li><a href="/en/library/environmental-science/61/the-carbon-cycle/95">The Carbon Cycle</a></li> <li><a href="/en/library/environmental-science/61/the-phosphorus-cycle/197">The Phosphorus Cycle</a></li> </ul> </div> <button class="accordion__button" id="acc-button-scientific-research" data-accordion="button" aria-controls="acc-panel-scientific-research" aria-expanded="false"> <span class="accordion__button__label"> Scientific Research </span> </button> <div class="accordion__panel" id="acc-panel-scientific-research" data-accordion="panel" aria-labelledby="acc-button-scientific-research" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/environmental-science/61/collaborative-research-in-the-arctic-towards-understanding-climate-change/183">Collaborative Research in the Arctic Towards Understanding Climate Change</a></li> <li><a href="/en/library/environmental-science/61/atmospheric-chemistry-research-that-changed-global-policy/211">Atmospheric Chemistry Research that Changed Global Policy</a></li> </ul> </div> </div> </div> <button class="accordion__button" id="acc-button-general-science" data-accordion="button" aria-controls="acc-panel-general-science" aria-expanded="false"> <span class="accordion__button__label"> General Science </span> </button> <div class="accordion__panel" id="acc-panel-general-science" data-accordion="panel" aria-labelledby="acc-button-general-science" role="region"> <div class="accordion accordion--secondary"> <button class="accordion__button" id="acc-button-methods" data-accordion="button" aria-controls="acc-panel-methods" aria-expanded="false"> <span class="accordion__button__label"> Methods </span> </button> <div class="accordion__panel" id="acc-panel-methods" data-accordion="panel" aria-labelledby="acc-button-methods" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/general-science/3/the-scientific-method/45">The Scientific Method</a></li> </ul> </div> <button class="accordion__button" id="acc-button-measurement" data-accordion="button" aria-controls="acc-panel-measurement" aria-expanded="false"> <span class="accordion__button__label"> Measurement </span> </button> <div class="accordion__panel" id="acc-panel-measurement" data-accordion="panel" aria-labelledby="acc-button-measurement" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/general-science/3/the-metric-system/47">The Metric System</a></li> </ul> </div> <button class="accordion__button" id="acc-button-physical-properties" data-accordion="button" aria-controls="acc-panel-physical-properties" aria-expanded="false"> <span class="accordion__button__label"> Physical Properties </span> </button> <div class="accordion__panel" id="acc-panel-physical-properties" data-accordion="panel" aria-labelledby="acc-button-physical-properties" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/general-science/3/temperature/48">Temperature</a></li> <li><a href="/en/library/general-science/3/density-and-buoyancy/37">Density and Buoyancy</a></li> </ul> </div> </div> </div> <button class="accordion__button" id="acc-button-math-in-science" data-accordion="button" aria-controls="acc-panel-math-in-science" aria-expanded="false"> <span class="accordion__button__label"> Math in Science </span> </button> <div class="accordion__panel" id="acc-panel-math-in-science" data-accordion="panel" aria-labelledby="acc-button-math-in-science" role="region"> <div class="accordion accordion--secondary"> <button class="accordion__button" id="acc-button-equations" data-accordion="button" aria-controls="acc-panel-equations" aria-expanded="false"> <span class="accordion__button__label"> Equations </span> </button> <div class="accordion__panel" id="acc-panel-equations" data-accordion="panel" aria-labelledby="acc-button-equations" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/math-in-science/62/unit-conversion/144">Unit Conversion</a></li> <li><a href="/en/library/math-in-science/62/linear-equations/194">Linear Equations</a></li> <li><a href="/en/library/math-in-science/62/exponential-equations-i/206">Exponential Equations I</a></li> <li><a href="/en/library/math-in-science/62/exponential-equations-ii/210">Exponential Equations II</a></li> <li><a href="/en/library/math-in-science/62/scientific-notation/250">Scientific Notation</a></li> <li><a href="/en/library/math-in-science/62/measurement/257">Measurement</a></li> </ul> </div> <button class="accordion__button" id="acc-button-statistics" data-accordion="button" aria-controls="acc-panel-statistics" aria-expanded="false"> <span class="accordion__button__label"> Statistics </span> </button> <div class="accordion__panel" id="acc-panel-statistics" data-accordion="panel" aria-labelledby="acc-button-statistics" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/math-in-science/62/introduction-to-descriptive-statistics/218">Introduction to Descriptive Statistics</a></li> <li><a href="/en/library/math-in-science/62/introduction-to-inferential-statistics/224">Introduction to Inferential Statistics</a></li> <li><a href="/en/library/math-in-science/62/statistical-techniques/239">Statistical Techniques</a></li> </ul> </div> <button class="accordion__button" id="acc-button-trigonometric-functions" data-accordion="button" aria-controls="acc-panel-trigonometric-functions" aria-expanded="false"> <span class="accordion__button__label"> Trigonometric Functions </span> </button> <div class="accordion__panel" id="acc-panel-trigonometric-functions" data-accordion="panel" aria-labelledby="acc-button-trigonometric-functions" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/math-in-science/62/wave-mathematics/131">Wave Mathematics</a></li> </ul> </div> </div> </div> <button class="accordion__button" id="acc-button-physics" data-accordion="button" aria-controls="acc-panel-physics" aria-expanded="false"> <span class="accordion__button__label"> Physics </span> </button> <div class="accordion__panel" id="acc-panel-physics" data-accordion="panel" aria-labelledby="acc-button-physics" role="region"> <div class="accordion accordion--secondary"> <button class="accordion__button" id="acc-button-light-and-optics" data-accordion="button" aria-controls="acc-panel-light-and-optics" aria-expanded="false"> <span class="accordion__button__label"> Light and Optics </span> </button> <div class="accordion__panel" id="acc-panel-light-and-optics" data-accordion="panel" aria-labelledby="acc-button-light-and-optics" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/physics/24/the-nature-of-light/132">The Nature of Light</a></li> <li><a href="/en/library/physics/24/electromagnetism-and-light/138">Electromagnetism and Light</a></li> </ul> </div> <button class="accordion__button" id="acc-button-mechanics" data-accordion="button" aria-controls="acc-panel-mechanics" aria-expanded="false"> <span class="accordion__button__label"> Mechanics </span> </button> <div class="accordion__panel" id="acc-panel-mechanics" data-accordion="panel" aria-labelledby="acc-button-mechanics" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/physics/24/defining-energy/199">Defining Energy</a></li> <li><a href="/en/library/physics/24/waves-and-wave-motion/102">Waves and Wave Motion</a></li> <li><a href="/en/library/physics/24/gravity/118">Gravity</a></li> <li><a href="/en/library/physics/24/thermodynamics-i/200">Thermodynamics I</a></li> </ul> </div> </div> </div> <button class="accordion__button" id="acc-button-process-of-science" data-accordion="button" aria-controls="acc-panel-process-of-science" aria-expanded="false"> <span class="accordion__button__label"> Process of Science </span> </button> <div class="accordion__panel" id="acc-panel-process-of-science" data-accordion="panel" aria-labelledby="acc-button-process-of-science" role="region"> <div class="accordion accordion--secondary"> <button class="accordion__button" id="acc-button-introduction" data-accordion="button" aria-controls="acc-panel-introduction" aria-expanded="false"> <span class="accordion__button__label"> Introduction </span> </button> <div class="accordion__panel" id="acc-panel-introduction" data-accordion="panel" aria-labelledby="acc-button-introduction" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/process-of-science/49/the-process-of-science/176">The Process of Science</a></li> </ul> </div> <button class="accordion__button" id="acc-button-the-culture-of-science" data-accordion="button" aria-controls="acc-panel-the-culture-of-science" aria-expanded="false"> <span class="accordion__button__label"> The Culture of Science </span> </button> <div class="accordion__panel" id="acc-panel-the-culture-of-science" data-accordion="panel" aria-labelledby="acc-button-the-culture-of-science" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/process-of-science/49/the-nature-of-scientific-knowledge/185">The Nature of Scientific Knowledge</a></li> <li><a href="/en/library/process-of-science/49/scientists-and-the-scientific-community/172">Scientists and the Scientific Community</a></li> <li><a href="/en/library/process-of-science/49/scientific-ethics/161">Scientific Ethics</a></li> <li><a href="/en/library/process-of-science/49/scientific-institutions-and-societies/162">Scientific Institutions and Societies</a></li> </ul> </div> <button class="accordion__button" id="acc-button-ideas-in-science" data-accordion="button" aria-controls="acc-panel-ideas-in-science" aria-expanded="false"> <span class="accordion__button__label"> Ideas in Science </span> </button> <div class="accordion__panel" id="acc-panel-ideas-in-science" data-accordion="panel" aria-labelledby="acc-button-ideas-in-science" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/process-of-science/49/theories-hypotheses-and-laws/177">Theories, Hypotheses, and Laws</a></li> <li><a href="/en/library/process-of-science/49/scientific-controversy/181">Scientific Controversy</a></li> <li><a href="/en/library/process-of-science/49/creativity-in-science/182">Creativity in Science</a></li> </ul> </div> <button class="accordion__button" id="acc-button-research-methods" data-accordion="button" aria-controls="acc-panel-research-methods" aria-expanded="false"> <span class="accordion__button__label"> Research Methods </span> </button> <div class="accordion__panel" id="acc-panel-research-methods" data-accordion="panel" aria-labelledby="acc-button-research-methods" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/process-of-science/49/the-practice-of-science/148">The Practice of Science</a></li> <li><a href="/en/library/process-of-science/49/experimentation-in-scientific-research/150">Experimentation in Scientific Research</a></li> <li><a href="/en/library/process-of-science/49/description-in-scientific-research/151">Description in Scientific Research</a></li> <li><a href="/en/library/process-of-science/49/comparison-in-scientific-research/152">Comparison in Scientific Research</a></li> <li><a href="/en/library/process-of-science/49/modeling-in-scientific-research/153">Modeling in Scientific Research</a></li> </ul> </div> <button class="accordion__button" id="acc-button-data" data-accordion="button" aria-controls="acc-panel-data" aria-expanded="false"> <span class="accordion__button__label"> Data </span> </button> <div class="accordion__panel" id="acc-panel-data" data-accordion="panel" aria-labelledby="acc-button-data" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/process-of-science/49/data-analysis-and-interpretation/154">Data Analysis and Interpretation</a></li> <li><a href="/en/library/process-of-science/49/uncertainty-error-and-confidence/157">Uncertainty, Error, and Confidence</a></li> <li><a href="/en/library/process-of-science/49/statistics-in-science/155">Statistics in Science</a></li> <li><a href="/en/library/process-of-science/49/using-graphs-and-visual-data-in-science/156">Using Graphs and Visual Data in Science</a></li> </ul> </div> <button class="accordion__button" id="acc-button-scientific-communication" data-accordion="button" aria-controls="acc-panel-scientific-communication" aria-expanded="false"> <span class="accordion__button__label"> Scientific Communication </span> </button> <div class="accordion__panel" id="acc-panel-scientific-communication" data-accordion="panel" aria-labelledby="acc-button-scientific-communication" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/process-of-science/49/understanding-scientific-journals-and-articles/158">Understanding Scientific Journals and Articles</a></li> <li><a href="/en/library/process-of-science/49/utilizing-the-scientific-literature/173">Utilizing the Scientific Literature</a></li> <li><a href="/en/library/process-of-science/49/peer-review-in-scientific-publishing/159">Peer Review in Scientific Publishing</a></li> <li><a href="/en/library/process-of-science/49/the-how-and-why-of-scientific-meetings/186">The How and Why of Scientific Meetings</a></li> </ul> </div> </div> </div> <button class="accordion__button" id="acc-button-scientists-and-research" data-accordion="button" aria-controls="acc-panel-scientists-and-research" aria-expanded="false"> <span class="accordion__button__label"> Scientists and Research </span> </button> <div class="accordion__panel" id="acc-panel-scientists-and-research" data-accordion="panel" aria-labelledby="acc-button-scientists-and-research" role="region"> <div class="accordion accordion--secondary"> <button class="accordion__button" id="acc-button-scientific-research" data-accordion="button" aria-controls="acc-panel-scientific-research" aria-expanded="false"> <span class="accordion__button__label"> Scientific Research </span> </button> <div class="accordion__panel" id="acc-panel-scientific-research" data-accordion="panel" aria-labelledby="acc-button-scientific-research" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/scientists-and-research/58/collaborative-research-in-the-arctic-towards-understanding-climate-change/183">Collaborative Research in the Arctic Towards Understanding Climate Change</a></li> <li><a href="/en/library/scientists-and-research/58/from-stable-chromosomes-to-jumping-genes/184">From Stable Chromosomes to Jumping Genes</a></li> <li><a href="/en/library/scientists-and-research/58/an-elegant-experiment-to-test-the-process-of-dna-replication/187">An Elegant Experiment to Test the Process of DNA Replication</a></li> <li><a href="/en/library/scientists-and-research/58/the-founding-of-neuroscience/233">The Founding of Neuroscience</a></li> <li><a href="/en/library/scientists-and-research/58/tracking-endangered-jaguars-across-the-border/189">Tracking Endangered Jaguars across the Border</a></li> <li><a href="/en/library/scientists-and-research/58/atmospheric-chemistry-research-that-changed-global-policy/211">Atmospheric Chemistry Research that Changed Global Policy</a></li> <li><a href="/en/library/scientists-and-research/58/revolutionizing-medicine-with-monoclonal-antibodies/220">Revolutionizing Medicine with Monoclonal Antibodies</a></li> <li><a href="/en/library/scientists-and-research/58/uncovering-the-mysteries-of-chronic-mountain-sickness/238">Uncovering the Mysteries of Chronic Mountain Sickness</a></li> </ul> </div> <button class="accordion__button" id="acc-button-profiles-in-science" data-accordion="button" aria-controls="acc-panel-profiles-in-science" aria-expanded="false"> <span class="accordion__button__label"> Profiles in Science </span> </button> <div class="accordion__panel" id="acc-panel-profiles-in-science" data-accordion="panel" aria-labelledby="acc-button-profiles-in-science" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/scientists-and-research/58/luis-e.-miramontes/232">Luis E. Miramontes</a></li> <li><a href="/en/library/scientists-and-research/58/bernardo-houssay/237">Bernardo Houssay</a></li> <li><a href="/en/library/scientists-and-research/58/craig-lee/256">Craig Lee</a></li> <li><a href="/en/library/scientists-and-research/58/david-ho/241">David Ho</a></li> <li><a href="/en/library/scientists-and-research/58/louis-tompkins-wright/244">Louis Tompkins Wright</a></li> <li><a href="/en/library/scientists-and-research/58/carlos-j.-finlay/217">Carlos J. Finlay</a></li> <li><a href="/en/library/scientists-and-research/58/cecilia-payne/290">Cecilia Payne</a></li> <li><a href="/en/library/scientists-and-research/58/jazmin-scarlett/291">Jazmin Scarlett</a></li> <li><a href="/en/library/scientists-and-research/58/ramari-stewart/292">Ramari Stewart</a></li> <li><a href="/en/library/scientists-and-research/58/johnson-cerda/300">Johnson Cerda</a></li> <li><a href="/en/library/scientists-and-research/58/ellen-ochoa/201">Ellen Ochoa</a></li> <li><a href="/en/library/scientists-and-research/58/ruth-benerito/205">Ruth Benerito</a></li> <li><a href="/en/library/scientists-and-research/58/franklin-chang-díaz/219">Franklin Chang Díaz</a></li> <li><a href="/en/library/scientists-and-research/58/percy-lavon-julian/221">Percy Lavon Julian</a></li> <li><a href="/en/library/scientists-and-research/58/luis-walter-alvarez/229">Luis Walter Alvarez</a></li> <li><a href="/en/library/scientists-and-research/58/france-anne-dominic-córdova/230">France Anne-Dominic Córdova</a></li> </ul> </div> </div> </div> </div> </div> </li> <li>  <button class="button" data-toggle="dropdown">Chemistry </button> <div class="nav__dropdown box-shadow-1 padding-1"> <div class="accordion accordion--secondary font-size-sm"> <div class="accordion accordion--secondary"> <button class="accordion__button" id="acc-sub-button-atomic-theory-and-structure" data-accordion="button" aria-controls="acc-sub-panel-atomic-theory-and-structure" aria-expanded="false"> <span class="accordion__button__label"> Atomic Theory and Structure </span> </button> <div class="accordion__panel" id="acc-sub-panel-atomic-theory-and-structure" data-accordion="panel" aria-labelledby="acc-sub-button-atomic-theory-and-structure" role="region"> <ul class="nav text-color-link"> <li><a href="/en/library/chemistry/1/early-ideas-about-matter/49">Early Ideas about Matter</a></li> <li><a href="/en/library/chemistry/1/the-periodic-table-of-elements-i/52">The Periodic Table of Elements I</a></li> <li><a href="/en/library/chemistry/1/the-periodic-table-of-elements-ii/296">The Periodic Table of Elements II</a></li> <li><a href="/en/library/chemistry/1/the-periodic-table-of-elements-iii/297">The Periodic Table of Elements III</a></li> <li><a href="/en/library/chemistry/1/the-periodic-table-of-elements-iv/298">The Periodic Table of Elements IV</a></li> <li><a href="/en/library/chemistry/1/the-periodic-table-of-elements-v/299">The Periodic Table of Elements V</a></li> <li><a href="/en/library/chemistry/1/atomic-theory-i/50">Atomic Theory I</a></li> <li><a href="/en/library/chemistry/1/atomic-theory-ii/51">Atomic Theory II</a></li> <li><a href="/en/library/chemistry/1/atomic-theory-iii/223">Atomic Theory III</a></li> <li class="current">Atomic Theory IV</li> <li><a href="/en/library/chemistry/1/the-mole-and-atomic-mass/53">The Mole and Atomic Mass</a></li> </ul> </div> <button class="accordion__button" id="acc-sub-button-physical-states-and-properties" data-accordion="button" 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id="skip-header-content"> <div class="margin-bottom-5"> <article class="container wide module"> <header class="grid grid--sidebar-right module__header"> <div class="module__header__title"> <span class="subcategory"> <strong><em>Atomic Theory and Structure</em></strong> </span> <h1>Atomic Theory IV: <sub><em>Quantum numbers and orbitals</em></sub></h1> <p class="byline">by Adrian Dingle, B.Sc., Anthony Carpi, Ph.D.</p> <nav class="module__header__tabs"> <ul class="tabs-nav tabs-nav--horizontal library"> <li> <a href="/en/library/chemistry/1/atomic-theory-iv/231/reading" aria-current="page" >Reading</a> </li> <li> <a href="/en/library/chemistry/1/atomic-theory-iv/231/quiz">Quiz</a> </li> <li> <a href="/en/library/chemistry/1/atomic-theory-iv/231/resources">Teach with this</a> </li> </ul> </nav> </div> <script type="application/ld+json"> { "@context": "http://schema.org", "@type": "AudioObject", "contentUrl": "https://www.visionlearning.com/img/library/moduleAudio/module_231.mp3", "description": "Recording of Atomic Theory IV : Our Atomic Theory series continues, exploring the quantum model of the atom in greater detail. This module takes a closer look at the Schrödinger equation that defines the energies and probable positions of electrons within atoms. Using the hydrogen atom as an example, the module explains how orbitals can be described by type of wave function. Evidence for orbitals and the quantum model is provided by the absorption and emission spectra of hydrogen. Other concepts include multi-electron atoms, the Aufbau Principle, and Hund’s Rule.", "encodingFormat": "mp3", "name": "module_231.mp3" } </script> <div class="module_header_audio"> <div class="audio-player border border-radius"> <audio id="audio"> <source src="https://www.visionlearning.com/img/library/moduleAudio/module_231.mp3" type="audio/mpeg"> Your browser does not support the audio element. </audio> <div class="audio-player__title"> <p>Listen to this reading</p> <span class="audio-player__timestamp" id="timestamp"> 00:00 </span> </div> <div class="audio-player__controls" id="controls"> <button class="button button--icon-only" id="play-pause-button"> <span class="icon icon-play" aria-hidden="true"></span> </button> <div class="audio-player__progress" id="progress-bar" tabindex="0" aria-valuemin="0" aria-valuemax="100" aria-valuenow="0" aria-label="Use arrow keys to forward or rewind the audio" role="slider"> <div class="audio-player__progress__fill"> <span class="audio-player__thumb"></span> </div> </div> <div class="audio-player__volume-container"> <button id="mute-button"> <span class="icon icon-volume"></span> </button> <div class="audio-player__volume" tabindex="0" aria-valuemin="0" aria-valuemax="100" aria-valuenow="100" aria-label="Use arrow keys to adjust volume" role="slider"> <div class="audio-player__volume__fill"> <span class="audio-player__thumb"></span> </div> </div> </div> </div> </div> </div> </header> <hr class="divider"/>   <div class="grid grid--sidebar-right grid--divider"> <div class="order-2 order-1--lg module__main"> <div class="narrow margin-x-auto margin-y-5"> <div class="accordion margin-bottom-5">  <button class="accordion__button" id="acc-button-key-concepts" data-accordion="button" aria-controls="acc-panel-key-concepts" aria-expanded="true" tabindex="0"> Did you know? </button> <div class="accordion__panel shown show" id="acc-panel-key-concepts" data-accordion="panel" aria-labelledby="acc-button-key-concepts" role="region"> <div class="accordion__panel__content"> <p>Did you know that electrons are so tiny that when you shine light on them, the light itself changes the electron’s path? Because of this, we can’t know exactly where an electron is within an atom. Rather, it necessary to describe the position of an electron in terms of probability. Thus, scientists use a mathematical equation to describe how electrons are most likely distributed around the atom's nucleus.</p> </div> </div>  <button class="accordion__button" id="acc-button-table-of-contents" data-accordion="button" aria-controls="acc-panel-table-of-contents" aria-expanded="false" tabindex="0"> Key concepts </button> <div class="accordion__panel" id="acc-panel-table-of-contents" data-accordion="panel" aria-labelledby="acc-button-table-of-contents" role="region" aria-hidden="true"> <div class="accordion__panel__content"> <ul class="bulleted"><li><p>The wave-particle nature of electrons means that their position and momentum cannot be described in simple physical terms but must be described by wave functions.</p></li> <li><p>The Schrödinger equation describes how the wave function of a wave-particle changes with time in a similar fashion to the way Newton’s second law describes the motion of a classical particle. The equation allows the calculation of each of the three quantum numbers related to individual atomic orbitals (principal, azimuthal, and magnetic).</p></li> <li><p>The Heisenberg uncertainty principle establishes that an electron’s position and momentum cannot be precisely known together; instead we can only calculate statistical likelihood of an electron’s location.</p></li> <li><p>The discovery of electron spin defines a fourth quantum number independent of the electron orbital but unique to an electron. The Pauli exclusion principle states that no two electrons with the same spin can occupy the same orbital.</p></li> <li><p>Quantum numbers, when taken as a set of four (principal, azimuthal, magnetic and spin) describe acceptable solutions to the Schrödinger equation, and as such, describe the most probable positions of electrons within atoms.</p></li> <li><p>Orbitals can be thought of as the three dimensional areas of space, defined by the quantum numbers, that describe the most probable position and energy of an electron within an atom.</p></li> </ul> </div> </div>  <button class="accordion__button" id="acc-button-terms-you-should-know" data-accordion="button" aria-controls="acc-panel-terms-you-should-know" aria-expanded="false" tabindex="0"> Terms you should know </button> <div class="accordion__panel" id="acc-panel-terms-you-should-know" data-accordion="panel" aria-labelledby="acc-button-terms-you-should-know" role="region" aria-hidden="true"> <div class="accordion__panel__content"> <dl> <dt>degenerate </dt> <dd> referring to orbitals at the same energy level in an atom. </dd> <dt>integer </dt> <dd> a whole number that is positive, negative, or zero.</dd> </dl> </div> </div> </div> <hr class="border-color-dark" /> <section> <div class="container narrow"> <p>In <a href="http://www.visionlearning.com/en/library/Chemistry/1/Atomic-Theory-III/223">Atomic Theory III: Wave-Particle Duality and the Electron</a>, we discussed the advances that were made by Schrödinger, Born, Pauli, and others in the application of the quantum <mark class="term" data-term="model" data-term-def="A representation, pattern, or mathematical description that can help scientists replicate a system." data-term-url="/en/glossary/view/model/8236">model</mark> to atomic <mark class="term" data-term="theory" data-term-def="A scientific theory is an explanation inferred from multiple lines of evidence for some broad aspect of the natural world and&hellip;" data-term-url="/en/glossary/view/theory/4854">theory</mark>. The <mark class="term" data-term="Schrödinger's equation" data-term-url="/en/glossary/view/Schr%C3%B6dinger%27s+equation" data-term-def="A partial differential equation developed by Erwin Schrodinger that describes the change of the quantum state of a physical system over&hellip;">Schrödinger equation</mark> was seen as a key mathematical link between the theory and the application of the quantum model. Born took the wave <mark class="term" data-term="function" data-term-def="Adaptations that influence how the animal interacts with other species. For example, animal function typically serves genetic and reproductive success." data-term-url="/en/glossary/view/function/13151">functions</mark> that Schrödinger produced and said that the <mark class="term" data-term="solution" data-term-def="A mixture of more than one substance with properties that do not vary within the sample. Commonly used to describe a&hellip;" data-term-url="/en/glossary/view/solution/1571">solutions</mark> to the equation could define the energies and the most probable positions of <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electrons</mark> within <mark class="term" data-term="atom" data-term-def="The smallest unit of an element that retains the chemical properties of the element. Atoms can exist alone or in&hellip;" data-term-url="/en/glossary/view/atom/1509">atoms</mark>, thus allowing us to build a much more detailed description of where electrons might be found within an atom. This module further explores these solutions, the position of electrons, the shape of <mark class="term" data-term="atomic orbitals" data-term-def="See <a href="http://www.visionlearning.com/en/glossary/index/E#term-9061">electron orbitals</a>." data-term-url="/en/glossary/view/atomic+orbitals/9065">atomic orbitals</mark>, and the implications of these ideas.</p> <p><section id="toc_1" class=""> <h2>Waves and measurement uncertainty</h2></p> <p>As we saw in earlier reading, the <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electron</mark> is not a true <mark class="term" data-term="particle" data-term-def="A tiny piece of matter." data-term-url="/en/glossary/view/particle/8259">particle</mark>, but a wave-particle similar to the photon. Since we measure the position of objects with <mark class="term" data-term="light" data-term-def="A form of electromagnetic radiation. Visible light is that associated with stimulating the organs of sight, which for normal human&hellip;" data-term-url="/en/glossary/view/light/1498">light</mark>, the small size of the electron introduces a challenge. If we shine a light <mark class="term" data-term="beam" data-term-def="A ray or shaft of light from a source." data-term-url="/en/glossary/view/beam/8277">beam</mark> on a moving tennis ball, the light has little effect on the tennis ball and we can measure both its position and momentum with a high degree of <mark class="term" data-term="accuracy" data-term-def="In science, the term accuracy describes how well a measurement approximates the theoretically correct value of that measurement, for example, how&hellip;" data-term-url="/en/glossary/view/accuracy/4222">accuracy</mark>. However, the electron is so tiny that even a single photon will influence its trajectory – thus if we shine a beam of light on it to measure its position, the <mark class="term" data-term="energy" data-term-def="An abstract property defined as the capacity to do work. The basic forms of energy include chemical, electrical, mechanical, nuclear, and&hellip;" data-term-url="/en/glossary/view/energy/1497">energy</mark> of the photon will affect its momentum, and vice versa. <mark id="ngss-421" class="ngss"> <mark class="term" data-term="Werner Heisenberg" data-term-def="German physicist instrumental in the development of quantum mechanics for which he won the Nobel Prize in 1932 in physics for,&hellip;" data-term-url="/en/glossary/view/Heisenberg%2C+Werner/9069">Werner Heisenberg</mark> developed a <mark class="term" data-term="principle" data-term-def="In the sciences, a principle is a fundamental, primary, or general law or truth. For instance, one of the most basic&hellip;" data-term-url="/en/glossary/view/principle/5289">principle</mark> to describe this <mark class="term" data-term="uncertainty" data-term-def="The quantitative estimation of error, which indicates the precision of a measurement or value. All measurements include some amount of uncertainty&hellip;" data-term-url="/en/glossary/view/uncertainty/5219">uncertainty</mark>, called appropriately the <mark class="term" data-term="Heisenberg Uncertainty Principle" data-term-def="A mathematical assertion in quantum mechanics, that states that both the position and momentum of a particle cannot be simultaneously, accurately&hellip;" data-term-url="/en/glossary/view/Heisenberg+Uncertainty+Principle/9070">Heisenberg Uncertainty Principle</mark> (Heisenberg, 1927). It tells us that mathematically, the product of the uncertainty in position (Δx) and the uncertainty in momentum (Δp) of an electron cannot be less than the reduced Planck <mark class="term" data-term="constant" data-term-def="In mathematics, a quantity that has a fixed value; something that does not vary." data-term-url="/en/glossary/view/constant/8557">constant</mark> ℏ/2 (Equation 1).</mark></p><div class="figure"><figure> $$\Delta x \Delta p \geq \frac{\hbar}{2}$$ <figcaption> <p><strong>Equation 1:</strong> The Heisenberg Uncertainty Principle equation, where Δx is the product of the uncertainty in position, Δp is the uncertainty in momentum of an electron, and $\frac{\hbar}{2}$ is the reduced Planck constant.</p> </figcaption> </figure></div><p>This is a very small number that can usually be ignored, but when dealing with a <mark class="term" data-term="particle" data-term-def="A tiny piece of matter." data-term-url="/en/glossary/view/particle/8259">particle</mark> as small as an <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electron</mark>, it is significant. Because of this, it becomes necessary to describe the position of an electron in terms of <mark class="term" data-term="probability" data-term-def="The likelihood that a given event will occur. In statistics, probability is often expressed as a ratio of the number of&hellip;" data-term-url="/en/glossary/view/probability/3913">probability</mark>, rather than that of absolute certainty. We have to say that an electron is likely to be found within the <mark class="term" data-term="atom" data-term-def="The smallest unit of an element that retains the chemical properties of the element. Atoms can exist alone or in&hellip;" data-term-url="/en/glossary/view/atom/1509">atom</mark> in certain areas of high probability, but we cannot be 100% sure of its precise position.</p> <div class="comprehension-checkpoint margin-y-4"> <h6 class="comprehension-checkpoint__header"> <span> <span class="icon icon-question"></span> </span> Comprehension Checkpoint </h6> <form class="" name="cc9492"> <div class="form-entry"> <div class="form-entry__field"> <span class="form-entry__field__label">The Heisenberg Uncertainty Principle tells us that</span> <div class="form-entry__option"> <div class="form-entry__option__radio" data-answer="incorrect"> <label> <input id="q1-9492-0-option-a" name="quiz-option-9492" type="radio" value="the sum of uncertainty in position plus uncertainty in momentum of an electron must be greater than the Planck constant." > <span class="option__label"> <span class="screen-reader-only">a.</span> the sum of uncertainty in position plus uncertainty in momentum of an electron must be greater than the Planck constant. </span> </label> <span class="quiz__response" id="response-9492-0"> <strong>Incorrect.</strong> </span> </div> <div class="form-entry__option__radio" data-answer="correct"> <label> <input id="q1-9492-1-option-b" name="quiz-option-9492" type="radio" value="the product of uncertainty in position and momentum of an electron cannot be less than the reduced Planck constant." > <span class="option__label"> <span class="screen-reader-only">b.</span> the product of uncertainty in position and momentum of an electron cannot be less than the reduced Planck constant. </span> </label> <span class="quiz__response" id="response-9492-1"> <strong>Correct!</strong> </span> </div> </div> </div> </div> </form> </div> </section> <section id="toc_2"> <h2>Schrödinger solves the probability question</h2><p>Because the <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electron</mark> is not a true <mark class="term" data-term="particle" data-term-def="A tiny piece of matter." data-term-url="/en/glossary/view/particle/8259">particle</mark>, we cannot describe its movement or location in traditional terms. In other words, those that would normally be defined by simple x, y, and z coordinates. <mark id="ngss-422" class="ngss">The challenges raised by the combination of wave particle duality and the Heisenberg Uncertainty Principle are simplified and expressed by Schrödinger’s equation that, when solved, produces wave <mark class="term" data-term="function" data-term-def="Adaptations that influence how the animal interacts with other species. For example, animal function typically serves genetic and reproductive success." data-term-url="/en/glossary/view/function/13151">functions</mark> denoted by Ψ. When Ψ is squared, the resulting solution gives the <mark class="term" data-term="probability" data-term-def="The likelihood that a given event will occur. In statistics, probability is often expressed as a ratio of the number of&hellip;" data-term-url="/en/glossary/view/probability/3913">probability</mark> of finding an electron at a particular place in the <mark class="term" data-term="atom" data-term-def="The smallest unit of an element that retains the chemical properties of the element. Atoms can exist alone or in&hellip;" data-term-url="/en/glossary/view/atom/1509">atom</mark>. The Ψ<sup>2</sup> term describes how electron <mark class="term" data-term="density" data-term-def="A measure of the compactness of a substance given by the mass per unit volume (d = m/v). Common units of&hellip;" data-term-url="/en/glossary/view/density/863">density</mark> and electron probability are distributed in space around the <mark class="term" data-term="nucleus" data-term-def="1. [Atomic] A tiny, dense positively charged mass at the heart of an atom. The nucleus is composed of protons and&hellip;" data-term-url="/en/glossary/view/nucleus/1526">nucleus</mark> of an atom.</mark></p></section> <section id="toc2_1"><h3>The simplest case - a hydrogen atom with a single electron</h3><p><mark id="ngss-423" class="ngss">The application of the <mark class="term" data-term="Schrödinger's equation" data-term-url="/en/glossary/view/Schr%C3%B6dinger%27s+equation" data-term-def="A partial differential equation developed by Erwin Schrodinger that describes the change of the quantum state of a physical system over&hellip;">Schrödinger equation</mark> is most easily understood in the case of the hydrogen <mark class="term" data-term="atom" data-term-def="The smallest unit of an element that retains the chemical properties of the element. Atoms can exist alone or in&hellip;" data-term-url="/en/glossary/view/atom/1509">atom</mark> because the one-electron atom allows us to avoid the complex interactions of multiple <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electrons</mark>. The time-independent form of the Schrödinger equation (Equation 2) can be solved to provide solutions that correspond to the <mark class="term" data-term="energy" data-term-def="An abstract property defined as the capacity to do work. The basic forms of energy include chemical, electrical, mechanical, nuclear, and&hellip;" data-term-url="/en/glossary/view/energy/1497">energy</mark> levels in a hydrogen atom. In solving this equation, we find that there are multiple solutions for Ψ that we call Ψ<sub>1</sub>, Ψ<sub>2</sub>, Ψ<sub>3</sub>, etc.</mark></p><div class="figure"><figure> $$\frac{-h^2}{8\pi^2m} \nabla^2\Psi - \frac{e^2}{4\pi\epsilon_or}\Psi = E\Psi^2$$ <figcaption> <p><strong>Equation 2:</strong> The time independent Schrödinger equation for hydrogen's energy levels.</p> </figcaption> </figure></div><p>Each of these solutions has a different <mark class="term" data-term="energy" data-term-def="An abstract property defined as the capacity to do work. The basic forms of energy include chemical, electrical, mechanical, nuclear, and&hellip;" data-term-url="/en/glossary/view/energy/1497">energy</mark> that corresponds to what we think of as different energy levels, or <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electron</mark> shells, within the <mark class="term" data-term="atom" data-term-def="The smallest unit of an element that retains the chemical properties of the element. Atoms can exist alone or in&hellip;" data-term-url="/en/glossary/view/atom/1509">atom</mark>. <mark id="ngss-424" class="ngss">The lowest energy electron distribution is that which is closest to the <mark class="term" data-term="nucleus" data-term-def="1. [Atomic] A tiny, dense positively charged mass at the heart of an atom. The nucleus is composed of protons and&hellip;" data-term-url="/en/glossary/view/nucleus/1526">nucleus</mark>, and it is called the <mark class="term" data-term="ground state" data-term-def="The lowest energy state for an atom or molecule. When an atom is in its ground state, its electrons fill the&hellip;" data-term-url="/en/glossary/view/ground+state/1537">ground state</mark>. The ground state is the most stable state of the single electron within the hydrogen atom. Other, higher energy states exist and are called <mark class="term" data-term="excited state" data-term-def="An energy state for an atom in which electrons exist above the minimum or ground state configuration. In general, excited&hellip;" data-term-url="/en/glossary/view/excited+state/1536">excited states</mark>.</mark> As the energy of each level increases above the ground state, we refer them to them as the second, third, fourth, etc. energy levels, respectively.</p> <div class="comprehension-checkpoint margin-y-4"> <h6 class="comprehension-checkpoint__header"> <span> <span class="icon icon-question"></span> </span> Comprehension Checkpoint </h6> <form class="" name="cc9494"> <div class="form-entry"> <div class="form-entry__field"> <span class="form-entry__field__label">The energy of the electron shells ______ as you move away from the nucleus of the atom.</span> <div class="form-entry__option"> <div class="form-entry__option__radio" data-answer="correct"> <label> <input id="q1-9494-0-option-a" name="quiz-option-9494" type="radio" value="increases" > <span class="option__label"> <span class="screen-reader-only">a.</span> increases </span> </label> <span class="quiz__response" id="response-9494-0"> <strong>Correct!</strong> </span> </div> <div class="form-entry__option__radio" data-answer="incorrect"> <label> <input id="q1-9494-1-option-b" name="quiz-option-9494" type="radio" value="decreases" > <span class="option__label"> <span class="screen-reader-only">b.</span> decreases </span> </label> <span class="quiz__response" id="response-9494-1"> <strong>Incorrect.</strong> </span> </div> </div> </div> </div> </form> </div> </section> <section id="toc_3"> <h2>Quantum numbers</h2><p><mark id="ngss-425" class="ngss">Acceptable solutions to the wave equation for an <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electron</mark> cannot be just <em>any</em> <mark class="term" data-term="value" data-term-def="A number that is assigned based on measurement or a calculation. In mathematics, an unknown value that is commonly represented by&hellip;" data-term-url="/en/glossary/view/value/8254">values</mark>, but rather they are restricted to those that obey certain <mark class="term" data-term="parameter" data-term-def="In statistics, a parameter is a numerical value that represents a characteristic of a statistical population. Contrast with statistic." data-term-url="/en/glossary/view/parameter/9481">parameters</mark>. Those <mark class="term" data-term="parameter" data-term-url="/en/glossary/view/parameter" data-term-def="In statistics, a parameter is a numerical value that represents a characteristic of a statistical population. Contrast with statistic.">parameters</mark> are described by a set of three <mark class="term" data-term="quantum numbers" data-term-def="A set of four numbers that are commonly used to describe the position of an electron within an atom. These include:&hellip;" data-term-url="/en/glossary/view/quantum+numbers/9066">quantum numbers</mark> that are named the principal <mark class="term" data-term="quantum numbers" data-term-url="/en/glossary/view/quantum+numbers" data-term-def="A set of four numbers that are commonly used to describe the position of an electron within an atom. These include:&hellip;">quantum number</mark>, the azimuthal quantum number, and the magnetic quantum number. These quantum numbers are given the symbols <em>n</em>, <em>l</em>, and <em>m</em>, respectively.</mark></p><p>The <strong>principal <mark class="term" data-term="quantum numbers" data-term-url="/en/glossary/view/quantum+numbers" data-term-def="A set of four numbers that are commonly used to describe the position of an electron within an atom. These include:&hellip;">quantum number</mark> <em>n</em></strong> is a positive integer starting with a <mark class="term" data-term="value" data-term-def="A number that is assigned based on measurement or a calculation. In mathematics, an unknown value that is commonly represented by&hellip;" data-term-url="/en/glossary/view/value/8254">value</mark> of 1, where 1 corresponds to the first (lowest energy) shell known as the <mark class="term" data-term="ground state" data-term-def="The lowest energy state for an atom or molecule. When an atom is in its ground state, its electrons fill the&hellip;" data-term-url="/en/glossary/view/ground+state/1537">ground state</mark> in hydrogen. The principal quantum number then increases with the increasing <mark class="term" data-term="energy" data-term-def="An abstract property defined as the capacity to do work. The basic forms of energy include chemical, electrical, mechanical, nuclear, and&hellip;" data-term-url="/en/glossary/view/energy/1497">energy</mark> of the shells (the <mark class="term" data-term="excited state" data-term-def="An energy state for an atom in which electrons exist above the minimum or ground state configuration. In general, excited&hellip;" data-term-url="/en/glossary/view/excited+state/1536">excited states</mark> in hydrogen) to values of <em>n</em> = 2, <em>n</em> = 3, <em>n</em> = 4, etc., as one moves farther away from the <mark class="term" data-term="nucleus" data-term-def="1. [Atomic] A tiny, dense positively charged mass at the heart of an atom. The nucleus is composed of protons and&hellip;" data-term-url="/en/glossary/view/nucleus/1526">nucleus</mark> to higher energy levels.</p><p>The <strong>azimuthal <mark class="term" data-term="quantum numbers" data-term-url="/en/glossary/view/quantum+numbers" data-term-def="A set of four numbers that are commonly used to describe the position of an electron within an atom. These include:&hellip;">quantum number</mark> <em>l</em></strong> (also called the orbital <mark class="term" data-term="angular momentum" data-term-def="The momentum possessed by an object in rotation around a point, it is the analog to linear momentum. In physics, one&hellip;" data-term-url="/en/glossary/view/angular+momentum/8713">angular momentum</mark> quantum number) determines the physical, three-dimensional shape of the orbitals in any <mark class="term" data-term="subshell" data-term-def="Subdivision of a shell in which the electron orbitals all have the same azimuthal quantum number, a valut that determines the&hellip;" data-term-url="/en/glossary/view/subshell/9495">subshell</mark>. The <mark class="term" data-term="value" data-term-def="A number that is assigned based on measurement or a calculation. In mathematics, an unknown value that is commonly represented by&hellip;" data-term-url="/en/glossary/view/value/8254">value</mark> of <em>l</em> is dependent on the principal quantum number <em>n</em>, and there are multiple values of <em>l</em> for each value of <em>n</em>. Values of <em>l</em> are positive integers or 0, and are determined by subtracting integers from the corresponding value of <em>n</em>. For example, when <em>n</em> = 3 we can subtract 3, 2, and 1 from <em>n</em> to yield values for <em>l</em> of 0, 1, and 2, respectively. Thus, the <mark class="term" data-term="ground state" data-term-def="The lowest energy state for an atom or molecule. When an atom is in its ground state, its electrons fill the&hellip;" data-term-url="/en/glossary/view/ground+state/1537">ground state</mark> shell in hydrogen (principal quantum number 1) has only one s subshell, shell 2 has s and p <mark class="term" data-term="subshell" data-term-url="/en/glossary/view/subshell" data-term-def="Subdivision of a shell in which the electron orbitals all have the same azimuthal quantum number, a valut that determines the&hellip;">subshells</mark>, and so on. The subshells indicated by <em>l</em> are also given the letter designations s, p, d, and f where <em>l</em> = 0 corresponds to an s subshell, <em>l</em> = 1 a p subshell, <em>l</em> = 2 a d subshell, and <em>l</em> = 3 an f subshell. Each subshell has a unique shape in 3D-space.</p><p>The <strong>magnetic <mark class="term" data-term="quantum numbers" data-term-url="/en/glossary/view/quantum+numbers" data-term-def="A set of four numbers that are commonly used to describe the position of an electron within an atom. These include:&hellip;">quantum number</mark> <em>m</em></strong>, defines the orientation (i.e., the position) and the number of orbitals within any given <mark class="term" data-term="subshell" data-term-def="Subdivision of a shell in which the electron orbitals all have the same azimuthal quantum number, a valut that determines the&hellip;" data-term-url="/en/glossary/view/subshell/9495">subshell</mark>. <mark class="term" data-term="value" data-term-def="A number that is assigned based on measurement or a calculation. In mathematics, an unknown value that is commonly represented by&hellip;" data-term-url="/en/glossary/view/value/8254">Values</mark> of <em>m</em> depend upon values of <em>l</em>, the azimuthal quantum number, and <em>m</em> can take on values of +1, -1, and all integers (including 0) in between. So for example, when <em>l</em> = 1, <em>m</em> can be +1, 0, or -1, yielding three separate orbitals, on axes x, y, and z in space. So in the case of <em>l</em> = 1 (the p subshell), there are three orbitals oriented in different directions in space.</p><p>Thus each <mark class="term" data-term="energy" data-term-def="An abstract property defined as the capacity to do work. The basic forms of energy include chemical, electrical, mechanical, nuclear, and&hellip;" data-term-url="/en/glossary/view/energy/1497">energy</mark> level is given a unique set of <mark class="term" data-term="quantum numbers" data-term-def="A set of four numbers that are commonly used to describe the position of an electron within an atom. These include:&hellip;" data-term-url="/en/glossary/view/quantum+numbers/9066">quantum numbers</mark> to fully describe it. Only certain <mark class="term" data-term="value" data-term-def="A number that is assigned based on measurement or a calculation. In mathematics, an unknown value that is commonly represented by&hellip;" data-term-url="/en/glossary/view/value/8254">values</mark> for quantum numbers in any one energy level are allowed, and those combinations are summarized in Table 1. Analysis of all of the allowed solutions to the wave equation shows that orbitals can be grouped together in sets according to their <em>l</em> values: <em>s</em>, <em>p</em>, <em>d</em>, and <em>f</em> sets.</p><div class='"table-container"'><table class="table" aria-describedby="configDescription2"> <caption id="configDescription2"> <p><strong>Table 1</strong>: The allowed solutions to the wave equation shows that orbitals can be grouped together in sets according to their <em>l</em> values (i.e., the s, p, d, and f sets).</p> </caption> <thead> <tr> <th scope="col">n</th> <th scope="col">1</th> <th scope="col">m</th> </tr> </thead> <tbody> <tr> <td scope="row">(principal quantum number)</td> <td>azimuthal quantum number, with letter designations)</td> <td>(magnetic quantum number, with letter designations)</td> </tr> <tr> <td scope="row">1</td> <td>0 (s)</td> <td>0 (s)</td> </tr> <tr> <td scope="row">2</td> <td>0 (s), <br>1 (p)</td> <td>0 (s), <br>-1/0/+1 (p)</td> </tr> <tr> <td scope="row">3</td> <td>0 (s), <br>1 (p), <br>2 (d)</td> <td>0 (s), <br>-1/0/+1 (p), <br>-2/-1/0/+1/+2 (d)</td> </tr> <tr> <td scope="row">4</td> <td>0 (s), <br>1 (p), <br>2 (d), <br>3 (f)</td> <td>0 (s), <br>-1/0/+1 (p), <br>-2/-1/0/+1/+2 (d), <br>-3/-2/-1/0/+1/+2/+3 (f)</td> </tr> </tbody> </table></div></section> <section id="toc_4"> <h2>Describing orbitals by type of wave function</h2><p><mark id="ngss-426" class="ngss">The first set of sub-orbitals are described by a type of wave <mark class="term" data-term="function" data-term-def="Adaptations that influence how the animal interacts with other species. For example, animal function typically serves genetic and reproductive success." data-term-url="/en/glossary/view/function/13151">function</mark> whose <mark class="term" data-term="probability" data-term-def="The likelihood that a given event will occur. In statistics, probability is often expressed as a ratio of the number of&hellip;" data-term-url="/en/glossary/view/probability/3913">probability</mark> <mark class="term" data-term="density" data-term-def="A measure of the compactness of a substance given by the mass per unit volume (d = m/v). Common units of&hellip;" data-term-url="/en/glossary/view/density/863">density</mark> depends only on the distance from the <mark class="term" data-term="nucleus" data-term-def="1. [Atomic] A tiny, dense positively charged mass at the heart of an atom. The nucleus is composed of protons and&hellip;" data-term-url="/en/glossary/view/nucleus/1526">nucleus</mark>, and whose probability is the same in all directions. Thus they have a <mark class="term" data-term="spherical" data-term-def="Shaped like a sphere." data-term-url="/en/glossary/view/spherical/11235">spherical</mark> shape and are called <em>s</em> orbitals. These <mark class="term" data-term="wave function" data-term-def="A function that satisfies a wave equation and describes the properties of a wave. In quantum physics, a mathematical function (&#936;),&hellip;" data-term-url="/en/glossary/view/wave+function/8711">wave functions</mark> are all found to have <mark class="term" data-term="value" data-term-def="A number that is assigned based on measurement or a calculation. In mathematics, an unknown value that is commonly represented by&hellip;" data-term-url="/en/glossary/view/value/8254">values</mark> of <em>l</em> = 0 and therefore values of <em>m</em> = 0, and every <mark class="term" data-term="energy" data-term-def="An abstract property defined as the capacity to do work. The basic forms of energy include chemical, electrical, mechanical, nuclear, and&hellip;" data-term-url="/en/glossary/view/energy/1497">energy</mark> level has such a wave function starting with 1s, and moving to 2s, 3s, etc. (Figure 1).</mark></p> <div class="figure"> <figure> <button class="lightbox-button lightbox-button--icon" data-lightbox="" data-lightbox-src="/img/library/large_images/image_9510.jpg"> <img src="/img/library/modules/mid231/Image/VLObject-9510-160410110421.jpg" alt="Figure 1: The spherical shaped s orbital." /> </button> <figcaption> <p><strong>Figure 1</strong>: The spherical shaped <em>s</em> orbital.</p> <span class="credit">image ©UC Davis ChemWiki</span> </figcaption> </figure> </div> <p><mark id="ngss-427" class="ngss">The second type of wave <mark class="term" data-term="function" data-term-def="Adaptations that influence how the animal interacts with other species. For example, animal function typically serves genetic and reproductive success." data-term-url="/en/glossary/view/function/13151">function</mark> has a <mark class="term" data-term="probability" data-term-def="The likelihood that a given event will occur. In statistics, probability is often expressed as a ratio of the number of&hellip;" data-term-url="/en/glossary/view/probability/3913">probability</mark> <mark class="term" data-term="density" data-term-def="A measure of the compactness of a substance given by the mass per unit volume (d = m/v). Common units of&hellip;" data-term-url="/en/glossary/view/density/863">density</mark> that depends on both the distance from the <mark class="term" data-term="nucleus" data-term-def="1. [Atomic] A tiny, dense positively charged mass at the heart of an atom. The nucleus is composed of protons and&hellip;" data-term-url="/en/glossary/view/nucleus/1526">nucleus</mark> <em>and</em> the orientation along either the x-, y-, or z-axis in space. This leads to three, separate, equivalent (or degenerate) <mark class="term" data-term="wave function" data-term-def="A function that satisfies a wave equation and describes the properties of a wave. In quantum physics, a mathematical function (&#936;),&hellip;" data-term-url="/en/glossary/view/wave+function/8711">wave functions</mark> that have a "figure eight" shape in 3D-space. These wave functions are all found to have <mark class="term" data-term="value" data-term-def="A number that is assigned based on measurement or a calculation. In mathematics, an unknown value that is commonly represented by&hellip;" data-term-url="/en/glossary/view/value/8254">values</mark> of <em>l</em> = 1 and can take on three, different <em>m</em> values. These are called p orbitals and exist for every <mark class="term" data-term="energy" data-term-def="An abstract property defined as the capacity to do work. The basic forms of energy include chemical, electrical, mechanical, nuclear, and&hellip;" data-term-url="/en/glossary/view/energy/1497">energy</mark> level except the first, thus appearing as 2p, 3p, 4p, etc. (Figure 2).</mark></p> <div class="figure"> <figure> <button class="lightbox-button lightbox-button--icon" data-lightbox="" data-lightbox-src="/img/library/large_images/image_9511.jpg"> <img src="/img/library/modules/mid231/Image/VLObject-9511-160410110428.jpg" alt="Figure 2: The "figure eight" p orbitals." /> </button> <figcaption> <p><strong>Figure 2</strong>: The "figure eight" <em>p</em> orbitals.</p> <span class="credit">image ©UC Davis ChemWiki</span> </figcaption> </figure> </div> <p><mark id="ngss-428" class="ngss">The third type of wave <mark class="term" data-term="function" data-term-def="Adaptations that influence how the animal interacts with other species. For example, animal function typically serves genetic and reproductive success." data-term-url="/en/glossary/view/function/13151">function</mark> has a <mark class="term" data-term="probability" data-term-def="The likelihood that a given event will occur. In statistics, probability is often expressed as a ratio of the number of&hellip;" data-term-url="/en/glossary/view/probability/3913">probability</mark> <mark class="term" data-term="density" data-term-def="A measure of the compactness of a substance given by the mass per unit volume (d = m/v). Common units of&hellip;" data-term-url="/en/glossary/view/density/863">density</mark> that depends on both the distance from the <mark class="term" data-term="nucleus" data-term-def="1. [Atomic] A tiny, dense positively charged mass at the heart of an atom. The nucleus is composed of protons and&hellip;" data-term-url="/en/glossary/view/nucleus/1526">nucleus</mark> and the orientation along two of the x-, y-, and z-axes in space. This leads to five separate, equivalent <mark class="term" data-term="wave function" data-term-def="A function that satisfies a wave equation and describes the properties of a wave. In quantum physics, a mathematical function (&#936;),&hellip;" data-term-url="/en/glossary/view/wave+function/8711">wave functions</mark>. When the wave functions have equivalent <mark class="term" data-term="energy" data-term-def="An abstract property defined as the capacity to do work. The basic forms of energy include chemical, electrical, mechanical, nuclear, and&hellip;" data-term-url="/en/glossary/view/energy/1497">energy</mark>, they are given the label "degenerate." They have complex shapes in 3D-space. These wave functions are all found to have <mark class="term" data-term="value" data-term-def="A number that is assigned based on measurement or a calculation. In mathematics, an unknown value that is commonly represented by&hellip;" data-term-url="/en/glossary/view/value/8254">values</mark> of <em>l</em> = 2 and can take on five different <em>m</em> values. These are called d orbitals, and every energy level except the first and second has such a wave function (Figure 3).</mark></p> <div class="figure"> <figure> <button class="lightbox-button lightbox-button--icon" data-lightbox="" data-lightbox-src="/img/library/large_images/image_9512.jpg"> <img src="/img/library/modules/mid231/Image/VLObject-9512-160410110432.jpg" alt="Figure 3: The d orbitals, the beginning of more complex orbital shapes." /> </button> <figcaption> <p><strong>Figure 3</strong>: The <em>d</em> orbitals, the beginning of more complex orbital shapes.</p> <span class="credit">image ©UC Davis ChemWiki</span> </figcaption> </figure> </div> <p><mark id="ngss-429" class="ngss">The fourth type is a set of orbitals with even more complexity in terms of their wave <mark class="term" data-term="function" data-term-def="Adaptations that influence how the animal interacts with other species. For example, animal function typically serves genetic and reproductive success." data-term-url="/en/glossary/view/function/13151">function</mark>, positions, and shape in 3D-space. These <mark class="term" data-term="wave function" data-term-def="A function that satisfies a wave equation and describes the properties of a wave. In quantum physics, a mathematical function (&#936;),&hellip;" data-term-url="/en/glossary/view/wave+function/8711">wave functions</mark> are all found to have <mark class="term" data-term="value" data-term-def="A number that is assigned based on measurement or a calculation. In mathematics, an unknown value that is commonly represented by&hellip;" data-term-url="/en/glossary/view/value/8254">values</mark> of <em>l</em> = 3 and can take on seven different <em>m</em> values. They, too, are all degenerate. These are called f orbitals and every <mark class="term" data-term="energy" data-term-def="An abstract property defined as the capacity to do work. The basic forms of energy include chemical, electrical, mechanical, nuclear, and&hellip;" data-term-url="/en/glossary/view/energy/1497">energy</mark> level except the first, second, and third has such a wave function, starting with 4f and moving to 5f (Figure 4).</mark></p> <div class="figure"> <figure> <button class="lightbox-button lightbox-button--icon" data-lightbox="" data-lightbox-src="/img/library/large_images/image_9513.jpg"> <img src="/img/library/modules/mid231/Image/VLObject-9513-160410110441.jpg" alt="Figure 4: The f orbitals, which continue the complexity of shape as seen in the d orbitals." /> </button> <figcaption> <p><strong>Figure 4</strong>: The <em>f</em> orbitals, which continue the complexity of shape as seen in the <em>d</em> orbitals.</p> <span class="credit">image ©UC Davis ChemWiki</span> </figcaption> </figure> </div> <p><mark id="ngss-430" class="ngss">To summarize, the orbitals available in the first four <mark class="term" data-term="energy" data-term-def="An abstract property defined as the capacity to do work. The basic forms of energy include chemical, electrical, mechanical, nuclear, and&hellip;" data-term-url="/en/glossary/view/energy/1497">energy</mark> levels are as follows in Table 2:</mark></p><div class='"table-container"'><table class="table" aria-describedby="configDescription"> <caption id="configDescription"> <strong>Table 2</strong>: Orbitals associated with the first four energy levels. </caption> <thead> <tr> <th scope="col">n</th> <th scope="col">orbitals</th> </tr> </thead> <tbody> <tr> <td scope="row">1</td> <td>1s</td> </tr> <tr> <td scope="row">2</td> <td>2s, 2p</td> </tr> <tr> <td scope="row">3</td> <td>3s, 3p, 3d</td> </tr> <tr> <td scope="row">4</td> <td>4s, 4p, 4d, 4f</td> </tr> </tbody> </table></div> <div class="comprehension-checkpoint margin-y-4"> <h6 class="comprehension-checkpoint__header"> <span> <span class="icon icon-question"></span> </span> Comprehension Checkpoint </h6> <form class="" name="cc9509"> <div class="form-entry"> <div class="form-entry__field"> <span class="form-entry__field__label">The principal quantum number n</span> <div class="form-entry__option"> <div class="form-entry__option__radio" data-answer="correct"> <label> <input id="q1-9509-0-option-a" name="quiz-option-9509" type="radio" value="must be a positive integer." > <span class="option__label"> <span class="screen-reader-only">a.</span> must be a positive integer. </span> </label> <span class="quiz__response" id="response-9509-0"> <strong>Correct!</strong> </span> </div> <div class="form-entry__option__radio" data-answer="incorrect"> <label> <input id="q1-9509-1-option-b" name="quiz-option-9509" type="radio" value="can be a positive or negative integer." > <span class="option__label"> <span class="screen-reader-only">b.</span> can be a positive or negative integer. </span> </label> <span class="quiz__response" id="response-9509-1"> <strong>Incorrect.</strong> </span> </div> </div> </div> </div> </form> </div> </section> <section id="toc2_2"><h3>Evidence for orbitals and the quantum model</h3><p><mark id="ngss-431" class="ngss">Since orbitals are defined by very specific solutions to the <mark class="term" data-term="Schrödinger's equation" data-term-url="/en/glossary/view/Schr%C3%B6dinger%27s+equation" data-term-def="A partial differential equation developed by Erwin Schrodinger that describes the change of the quantum state of a physical system over&hellip;">Schrödinger equation</mark>, <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electrons</mark> must <mark class="term" data-term="absorb" data-term-def="Take in or soak up (energy, liquids, or other substances), usually gradually, through a chemical or physical action." data-term-url="/en/glossary/view/absorb/11219">absorb</mark> very specific quantities of <mark class="term" data-term="energy" data-term-def="An abstract property defined as the capacity to do work. The basic forms of energy include chemical, electrical, mechanical, nuclear, and&hellip;" data-term-url="/en/glossary/view/energy/1497">energy</mark> to be promoted from one energy level to another. By exposing electrons to an external source of energy, such as <mark class="term" data-term="light" data-term-def="A form of electromagnetic radiation. Visible light is that associated with stimulating the organs of sight, which for normal human&hellip;" data-term-url="/en/glossary/view/light/1498">light</mark>, it is possible to promote the electrons from their <mark class="term" data-term="ground state" data-term-def="The lowest energy state for an atom or molecule. When an atom is in its ground state, its electrons fill the&hellip;" data-term-url="/en/glossary/view/ground+state/1537">ground state</mark> to other positions that have higher potential energies, called <mark class="term" data-term="excited state" data-term-def="An energy state for an atom in which electrons exist above the minimum or ground state configuration. In general, excited&hellip;" data-term-url="/en/glossary/view/excited+state/1536">excited states</mark>. These ‘excited’ electrons quickly return to lower energy positions (in order to regain the stability associated with lower energies), and in doing so they release energy in specific frequencies that correspond to the energy differences between <mark class="term" data-term="electron orbitals" data-term-def="Three dimensional areas of space, defined by acceptable solutions to the <a href="http://www.visionlearning.com/en/glossary/index/S#term-9057">Schrödinger equation</a>, which determine the likely location of any&hellip;" data-term-url="/en/glossary/view/electron+orbitals/9061">electron orbitals</mark>, or shells. Light energy is related to frequency (f) and Planck’s <mark class="term" data-term="constant" data-term-def="In mathematics, a quantity that has a fixed value; something that does not vary." data-term-url="/en/glossary/view/constant/8557">constant</mark> (h) in the equation <strong><em>E</em> = <em>hf</em></strong>.</mark></p><p><mark id="ngss-432" class="ngss">Electrons within a hydrogen <mark class="term" data-term="atom" data-term-def="The smallest unit of an element that retains the chemical properties of the element. Atoms can exist alone or in&hellip;" data-term-url="/en/glossary/view/atom/1509">atom</mark> will <mark class="term" data-term="absorb" data-term-def="Take in or soak up (energy, liquids, or other substances), usually gradually, through a chemical or physical action." data-term-url="/en/glossary/view/absorb/11219">absorb</mark> specific frequencies of <mark class="term" data-term="energy" data-term-def="An abstract property defined as the capacity to do work. The basic forms of energy include chemical, electrical, mechanical, nuclear, and&hellip;" data-term-url="/en/glossary/view/energy/1497">energy</mark> that correspond to the energy gaps (ΔE) between two energy levels, thus allowing the promotion of an <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electron</mark> from a relatively low energy level to a relatively high energy level. Examining the absorption <mark class="term" data-term="spectrum" data-term-def="(plural: <b>spectra</b>) A continuing range such as of color or frequency; a series of colors arranged by wavelength as in a rainbow." data-term-url="/en/glossary/view/spectrum/8261">spectrum</mark> produced when hydrogen is irradiated, we observe a pattern of lines that supports this (Figure 5).</mark></p> <div class="figure"> <figure> <button class="lightbox-button lightbox-button--icon" data-lightbox="" data-lightbox-src="/img/library/large_images/image_9370.jpg"> <img src="/img/library/modules/mid231/Image/VLObject-9370-160229040205.jpg" alt="Figure 5: Hydrogen's emission and absorption spectra from the Balmer series." /> </button> <figcaption> <p><strong>Figure 5</strong>: Hydrogen's emission and absorption spectra from the Balmer series.</p> <span class="credit">image ©Chem1 Virtual Textbook, adapted from the Online Journey through Astronomy site</span> </figcaption> </figure> </div> <p><mark id="ngss-433" class="ngss">Bohr proposed that <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electrons</mark> can <mark class="term" data-term="transition" data-term-def="A change from one stage, form, or state to another." data-term-url="/en/glossary/view/transition/8520">transition</mark> to different positions, but only in discrete, defined steps. He thought that electrons were restricted to a specific area of space around the <mark class="term" data-term="nucleus" data-term-def="1. [Atomic] A tiny, dense positively charged mass at the heart of an atom. The nucleus is composed of protons and&hellip;" data-term-url="/en/glossary/view/nucleus/1526">nucleus</mark>, much like the planets in our solar <mark class="term" data-term="system" data-term-def="A group of interacting, interrelated or interdependent components that form a complex whole. The size of the system is defined for&hellip;" data-term-url="/en/glossary/view/system/3904">system</mark> are restricted to specific paths. The hydrogen absorption <mark class="term" data-term="spectrum" data-term-def="(plural: <b>spectra</b>) A continuing range such as of color or frequency; a series of colors arranged by wavelength as in a rainbow." data-term-url="/en/glossary/view/spectrum/8261">spectrum</mark> of discrete lines (rather than a continuous spectrum) shows that only very specific transitions can be made and is <mark class="term" data-term="evidence" data-term-def="Support for an idea, opinion, or hypothesis." data-term-url="/en/glossary/view/evidence/8243">evidence</mark> for the quantum <mark class="term" data-term="model" data-term-def="A representation, pattern, or mathematical description that can help scientists replicate a system." data-term-url="/en/glossary/view/model/8236">model</mark>.</mark></p><p>In addition to the absorption <mark class="term" data-term="spectrum" data-term-def="(plural: <b>spectra</b>) A continuing range such as of color or frequency; a series of colors arranged by wavelength as in a rainbow." data-term-url="/en/glossary/view/spectrum/8261">spectrum</mark> shown above, another identical spectrum can be observed, this time with the <mark class="term" data-term="energy" data-term-def="An abstract property defined as the capacity to do work. The basic forms of energy include chemical, electrical, mechanical, nuclear, and&hellip;" data-term-url="/en/glossary/view/energy/1497">energy</mark> being emitted rather than absorbed. In this case the <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electrons</mark> fall back from higher energies to lower energies rather than being promoted as in the absorption spectrum. This is called the emission spectrum of the <mark class="term" data-term="atom" data-term-def="The smallest unit of an element that retains the chemical properties of the element. Atoms can exist alone or in&hellip;" data-term-url="/en/glossary/view/atom/1509">atom</mark> and is once again made up of discrete, individual lines.</p> <div class="comprehension-checkpoint margin-y-4"> <h6 class="comprehension-checkpoint__header"> <span> <span class="icon icon-question"></span> </span> Comprehension Checkpoint </h6> <form class="" name="cc9500"> <div class="form-entry"> <div class="form-entry__field"> <span class="form-entry__field__label">When energy is emitted, electrons</span> <div class="form-entry__option"> <div class="form-entry__option__radio" data-answer="incorrect"> <label> <input id="q1-9500-0-option-a" name="quiz-option-9500" type="radio" value="jump from lower energies to higher energies." > <span class="option__label"> <span class="screen-reader-only">a.</span> jump from lower energies to higher energies. </span> </label> <span class="quiz__response" id="response-9500-0"> <strong>Incorrect.</strong> </span> </div> <div class="form-entry__option__radio" data-answer="correct"> <label> <input id="q1-9500-1-option-b" name="quiz-option-9500" type="radio" value="fall back from higher energies to lower energies." > <span class="option__label"> <span class="screen-reader-only">b.</span> fall back from higher energies to lower energies. </span> </label> <span class="quiz__response" id="response-9500-1"> <strong>Correct!</strong> </span> </div> </div> </div> </div> </form> </div> <p>Each series of lines that appear in such spectra are named after their discoverers. The Lyman series are those lines in which the <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electrons</mark> are either promoted from, or to, the 1s orbital. In the Balmer, Paschen, Brackett, and Pfund series, electrons travel either from or to the orbitals where n = 2, 3, 4, and 5, respectively. The differing <mark class="term" data-term="energy" data-term-def="An abstract property defined as the capacity to do work. The basic forms of energy include chemical, electrical, mechanical, nuclear, and&hellip;" data-term-url="/en/glossary/view/energy/1497">energy</mark> gaps (and therefore the differing frequencies) correspond to lines in different regions of the electromagnetic <mark class="term" data-term="spectrum" data-term-def="(plural: <b>spectra</b>) A continuing range such as of color or frequency; a series of colors arranged by wavelength as in a rainbow." data-term-url="/en/glossary/view/spectrum/8261">spectrum</mark> as a whole (Figure 6). Lyman lines are in the <mark class="term" data-term="ultraviolet" data-term-def="Wavelengths between 1 and 380 nanometers (nm) on the electromagnetic spectrum, falling between X-rays (10<sup>-2</sup> nm to 1 nm) and visible&hellip;" data-term-url="/en/glossary/view/ultraviolet/8233">ultraviolet</mark> region, Balmer in the visible, and Paschen, Brackett, and Pfund in various parts (near and far) of the infrared.</p> <div class="figure"> <figure> <button class="lightbox-button lightbox-button--icon" data-lightbox="" data-lightbox-src="/img/library/large_images/image_9373.jpg"> <img src="/img/library/modules/mid231/Image/VLObject-9373-160229040225.jpg" alt="Figure 6: An energy level transition diagram for hydrogen." /> </button> <figcaption> <p><strong>Figure 6</strong>: An energy level transition diagram for hydrogen.</p> </figcaption> </figure> </div> <p><mark id="ngss-434" class="ngss">In hydrogen <mark class="term" data-term="atom" data-term-def="The smallest unit of an element that retains the chemical properties of the element. Atoms can exist alone or in&hellip;" data-term-url="/en/glossary/view/atom/1509">atoms</mark>, and other single <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electron</mark> <mark class="term" data-term="species" data-term-def="1. In biological classifications, it is the lowest and most basic unit of the Linnaean taxonomic hierarchy (although it is also&hellip;" data-term-url="/en/glossary/view/species/893">species</mark> such as He<sup>+</sup> and Li<sup>2+</sup>, it is found that the <mark class="term" data-term="energy" data-term-def="An abstract property defined as the capacity to do work. The basic forms of energy include chemical, electrical, mechanical, nuclear, and&hellip;" data-term-url="/en/glossary/view/energy/1497">energy</mark> of the orbitals is only dependent on their distance from the <mark class="term" data-term="nucleus" data-term-def="1. [Atomic] A tiny, dense positively charged mass at the heart of an atom. The nucleus is composed of protons and&hellip;" data-term-url="/en/glossary/view/nucleus/1526">nucleus</mark>, with increasing energies as <em>n</em> increases. The lowest energy orbital is 1s, followed by 2s and 2p (that have the same energy), and 3s, 3p, and 3d (that have the same energy) etc. As such, the <mark class="term" data-term="ground state" data-term-def="The lowest energy state for an atom or molecule. When an atom is in its ground state, its electrons fill the&hellip;" data-term-url="/en/glossary/view/ground+state/1537">ground state</mark> (lowest energy state) for a hydrogen atom with only one electron is with the electron residing in the 1s orbital. The differences in energies (ΔE) required to promote electrons from one level to another can be determined by using one version of the Rydberg <mark class="term" data-term="formula" data-term-def="An expression of the composition of a chemical compound using symbols." data-term-url="/en/glossary/view/formula/8554">Formula</mark>, where R<sub>H</sub> is a <mark class="term" data-term="constant" data-term-def="In mathematics, a quantity that has a fixed value; something that does not vary." data-term-url="/en/glossary/view/constant/8557">constant</mark> for hydrogen and n<sub>i</sub> and n<sub>f</sub> are the initial and final energy levels respectively.</mark></p><div class="figure"><figure> $$\Delta E= R_H\frac{1}{n_{i}^2}-\frac{1}{n_{f}^2}$$ <figcaption> <p><strong>Equation 3:</strong> The Rydberg Formula</p> </figcaption> </figure></div><p><mark id="ngss-435" class="ngss">Since other single <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electron</mark> <mark class="term" data-term="species" data-term-def="1. In biological classifications, it is the lowest and most basic unit of the Linnaean taxonomic hierarchy (although it is also&hellip;" data-term-url="/en/glossary/view/species/893">species</mark> such as He<sup>+</sup> and Li<sup>2+</sup> contain more <mark class="term" data-term="proton" data-term-def="A subatomic (ß link to atom) particle with a positive charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 1.672&hellip;" data-term-url="/en/glossary/view/proton/854">protons</mark> than the hydrogen <mark class="term" data-term="atom" data-term-def="The smallest unit of an element that retains the chemical properties of the element. Atoms can exist alone or in&hellip;" data-term-url="/en/glossary/view/atom/1509">atom</mark>, the electron experiences a greater attraction from the <mark class="term" data-term="nucleus" data-term-def="1. [Atomic] A tiny, dense positively charged mass at the heart of an atom. The nucleus is composed of protons and&hellip;" data-term-url="/en/glossary/view/nucleus/1526">nucleus</mark>. As such, differing, larger amounts of <mark class="term" data-term="energy" data-term-def="An abstract property defined as the capacity to do work. The basic forms of energy include chemical, electrical, mechanical, nuclear, and&hellip;" data-term-url="/en/glossary/view/energy/1497">energy</mark> are required for promotion of electrons in these species and a modified Rydberg <mark class="term" data-term="constant" data-term-def="In mathematics, a quantity that has a fixed value; something that does not vary." data-term-url="/en/glossary/view/constant/8557">constant</mark> is required in the calculation of ΔE for single electron species other than the hydrogen atom.</mark></p> <div class="comprehension-checkpoint margin-y-4"> <h6 class="comprehension-checkpoint__header"> <span> <span class="icon icon-question"></span> </span> Comprehension Checkpoint </h6> <form class="" name="cc9503"> <div class="form-entry"> <div class="form-entry__field"> <span class="form-entry__field__label">In atoms that have only one electron, the energy of the orbitals is dependent on the</span> <div class="form-entry__option"> <div class="form-entry__option__radio" data-answer="correct"> <label> <input id="q1-9503-0-option-a" name="quiz-option-9503" type="radio" value="distance of the electron from the nucleus." > <span class="option__label"> <span class="screen-reader-only">a.</span> distance of the electron from the nucleus. </span> </label> <span class="quiz__response" id="response-9503-0"> <strong>Correct!</strong> </span> </div> <div class="form-entry__option__radio" data-answer="incorrect"> <label> <input id="q1-9503-1-option-b" name="quiz-option-9503" type="radio" value="orientation of the electron along an axis." > <span class="option__label"> <span class="screen-reader-only">b.</span> orientation of the electron along an axis. </span> </label> <span class="quiz__response" id="response-9503-1"> <strong>Incorrect.</strong> </span> </div> </div> </div> </div> </form> </div> </section> <section id="toc_5"> <h2>The more complex case of many-electron atoms</h2><p>The hydrogen <mark class="term" data-term="atom" data-term-def="The smallest unit of an element that retains the chemical properties of the element. Atoms can exist alone or in&hellip;" data-term-url="/en/glossary/view/atom/1509">atom</mark> is the simplest case to study since its single <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electron</mark> is both uninfluenced by other electrons and does not influence any other electrons. In this case, the wave <mark class="term" data-term="function" data-term-def="Adaptations that influence how the animal interacts with other species. For example, animal function typically serves genetic and reproductive success." data-term-url="/en/glossary/view/function/13151">function</mark> is relatively easy to compute. However, in more complex atoms the calculations are not so easy. For example, when two electrons are present, as in helium, things become considerably more complex. Rather than there being just one simple potential <mark class="term" data-term="energy" data-term-def="An abstract property defined as the capacity to do work. The basic forms of energy include chemical, electrical, mechanical, nuclear, and&hellip;" data-term-url="/en/glossary/view/energy/1497">energy</mark> attraction between the <mark class="term" data-term="nucleus" data-term-def="1. [Atomic] A tiny, dense positively charged mass at the heart of an atom. The nucleus is composed of protons and&hellip;" data-term-url="/en/glossary/view/nucleus/1526">nucleus</mark> and the single electron in a hydrogen atom, in helium there are now three <mark class="term" data-term="potential energy" data-term-def="The energy an object possesses by virtue of its position in relation to a field of force. For example, lifting&hellip;" data-term-url="/en/glossary/view/potential+energy/1504">potential energy</mark> terms to consider: the attractive <mark class="term" data-term="force" data-term-def="An influence (a "push or pull") that changes the motion of a moving object (e.g., slows it down, speeds it up,&hellip;" data-term-url="/en/glossary/view/force/883">forces</mark> between the nucleus and each electron, and the repulsion between the two electrons. The addition of even more electrons leads to the equation and the calculations becoming increasingly complex.</p><p><mark id="ngss-436" class="ngss">The complexity of the <mark class="term" data-term="Schrödinger's equation" data-term-url="/en/glossary/view/Schr%C3%B6dinger%27s+equation" data-term-def="A partial differential equation developed by Erwin Schrodinger that describes the change of the quantum state of a physical system over&hellip;">Schrödinger equation</mark> under these many <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electron</mark> circumstances means that it has to be solved by a series of approximations rather than directly. One <mark class="term" data-term="method" data-term-def="A procedure or process; a systematic way of performing a task or conducting research." data-term-url="/en/glossary/view/method/8238">method</mark> is to effectively apply the single electron <mark class="term" data-term="system" data-term-def="A group of interacting, interrelated or interdependent components that form a complex whole. The size of the system is defined for&hellip;" data-term-url="/en/glossary/view/system/3904">system</mark> over and over again to produce what amounts to an approximate answer. Although not entirely correct, such approximations prove to be very workable and produce reasonable answers to the multi-electron problem.</mark></p><p><mark id="ngss-437" class="ngss">The approximation works well enough to once again produce Ψ<sup>2</sup> wave <mark class="term" data-term="function" data-term-def="Adaptations that influence how the animal interacts with other species. For example, animal function typically serves genetic and reproductive success." data-term-url="/en/glossary/view/function/13151">functions</mark> that predict the <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electron</mark> <mark class="term" data-term="density" data-term-def="A measure of the compactness of a substance given by the mass per unit volume (d = m/v). Common units of&hellip;" data-term-url="/en/glossary/view/density/863">density</mark> in space of multi-electron <mark class="term" data-term="atom" data-term-def="The smallest unit of an element that retains the chemical properties of the element. Atoms can exist alone or in&hellip;" data-term-url="/en/glossary/view/atom/1509">atoms</mark>. The orbitals in the single electron situation and those in the multi-electron situation are still defined by the same set of restricted <mark class="term" data-term="quantum numbers" data-term-def="A set of four numbers that are commonly used to describe the position of an electron within an atom. These include:&hellip;" data-term-url="/en/glossary/view/quantum+numbers/9066">quantum numbers</mark>, but one difference arises regarding the specific energies of those orbitals. Previously, the <mark class="term" data-term="energy" data-term-def="An abstract property defined as the capacity to do work. The basic forms of energy include chemical, electrical, mechanical, nuclear, and&hellip;" data-term-url="/en/glossary/view/energy/1497">energy</mark> depended only on the distance from the <mark class="term" data-term="nucleus" data-term-def="1. [Atomic] A tiny, dense positively charged mass at the heart of an atom. The nucleus is composed of protons and&hellip;" data-term-url="/en/glossary/view/nucleus/1526">nucleus</mark> (that is, the energy <em>n</em> in multi-electron <mark class="term" data-term="system" data-term-def="A group of interacting, interrelated or interdependent components that form a complex whole. The size of the system is defined for&hellip;" data-term-url="/en/glossary/view/system/3904">systems</mark> is found to be dependent on the <mark class="term" data-term="value" data-term-def="A number that is assigned based on measurement or a calculation. In mathematics, an unknown value that is commonly represented by&hellip;" data-term-url="/en/glossary/view/value/8254">value</mark> of <em>l</em>). This complicates the idea that relates electron energy simply to distance from the nucleus (i.e., orbital energy follows the pattern 1s < (2s = 2p) < (3s = 3p = 3d), etc.). Instead, it produces the following sequence of relative energies for the orbitals.</mark></p><p><mark id="ngss-438" class="ngss">1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p, etc.</mark></p><p>Note the insertion of 4s and 5s earlier in the sequence than expected. This order is reflected on the periodic table. When reaching the end of <mark class="term" data-term="period" data-term-def="A row of elements in the periodic table." data-term-url="/en/glossary/view/period/8565">period</mark> three on the periodic table (element 18, argon), we will have filled the 1s, 2s, 2p, 3s, and 3p orbitals giving a total (as expected) of 18 <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electrons</mark>. One would perhaps expect the next orbitals to be filled to be the 3d, since after all, they are also in the third <mark class="term" data-term="energy" data-term-def="An abstract property defined as the capacity to do work. The basic forms of energy include chemical, electrical, mechanical, nuclear, and&hellip;" data-term-url="/en/glossary/view/energy/1497">energy</mark> level. However, the next <mark class="term" data-term="element" data-term-def="One of fewer than 118 pure chemical substances. An element is a substance composed of atoms with identical atomic number." data-term-url="/en/glossary/view/element/1510">elements</mark> on the periodic table, potassium and calcium (elements 19 and 20, respectively), fill their 4s orbitals before their 3d. Moving further along the fourth period to element 21, scandium, we find that the 3d <mark class="term" data-term="subshell" data-term-def="Subdivision of a shell in which the electron orbitals all have the same azimuthal quantum number, a valut that determines the&hellip;" data-term-url="/en/glossary/view/subshell/9495">subshell</mark> now begins to populate. When that subshell is full at element 30, zinc, we then fill the 4p subshell beginning with element 31, gallium. In short, the periodic table is ordered in such a way that it reflects the sequence of relative energies of orbitals given above.</p></section> <section id="toc2_3"><h3>Filling orbitals with many electrons</h3><p>As seen previously, the concept of <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electron</mark> spin introduced the need for the fourth and final <mark class="term" data-term="quantum numbers" data-term-url="/en/glossary/view/quantum+numbers" data-term-def="A set of four numbers that are commonly used to describe the position of an electron within an atom. These include:&hellip;">quantum number</mark>, the spin quantum number, <em>s</em>. Its introduction ensures that each electron has a unique set of <mark class="term" data-term="quantum numbers" data-term-def="A set of four numbers that are commonly used to describe the position of an electron within an atom. These include:&hellip;" data-term-url="/en/glossary/view/quantum+numbers/9066">quantum numbers</mark> according to the <mark class="term" data-term="Pauli exclusion principle" data-term-def="Principle developed by Wolfgang Pauli (1925) that states that no two electrons in an atom can have the same set of&hellip;" data-term-url="/en/glossary/view/Pauli+exclusion+principle/9075">Pauli Exclusion Principle</mark>. With there being only two <mark class="term" data-term="value" data-term-def="A number that is assigned based on measurement or a calculation. In mathematics, an unknown value that is commonly represented by&hellip;" data-term-url="/en/glossary/view/value/8254">values</mark> for <em>s</em> (+½ or -½), it follows that each orbital may only hold a maximum of two electrons.</p><p>The notation used to describe the electronic <mark class="term" data-term="configuration" data-term-def="The way parts are arranged, such as how electrons are distributed in orbitals, or electron shells, around the nucleus of an atom." data-term-url="/en/glossary/view/configuration/8262">configuration</mark> of an <mark class="term" data-term="atom" data-term-def="The smallest unit of an element that retains the chemical properties of the element. Atoms can exist alone or in&hellip;" data-term-url="/en/glossary/view/atom/1509">atom</mark> involves the use of a superscript to denote the number of <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electrons</mark> in any given orbital. For example, in the case of beryllium that has a total of four electrons, 1s<sup>2</sup> 2s<sup>2</sup> shows that both the 1s and the 2s orbitals are each filled with two electrons.</p></section> <section id="toc2_4"><h3>Hund's Rule - filling degenerate orbitals</h3><p><mark id="ngss-439" class="ngss">When identifying which orbitals the <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electrons</mark> in an <mark class="term" data-term="atom" data-term-def="The smallest unit of an element that retains the chemical properties of the element. Atoms can exist alone or in&hellip;" data-term-url="/en/glossary/view/atom/1509">atom</mark> occupy, we generally follow the <mark class="term" data-term="principle" data-term-def="In the sciences, a principle is a fundamental, primary, or general law or truth. For instance, one of the most basic&hellip;" data-term-url="/en/glossary/view/principle/5289">principle</mark> referred to as the Aufbau principle, where lower <mark class="term" data-term="energy" data-term-def="An abstract property defined as the capacity to do work. The basic forms of energy include chemical, electrical, mechanical, nuclear, and&hellip;" data-term-url="/en/glossary/view/energy/1497">energy</mark> orbitals are filled first. So, for example, 1s is filled before 2s, and 2s before 2p, etc. But what about degenerate orbitals, those that have the same energy, such as the three orbitals that exist in the 2p sub-level?</mark></p><p><mark id="ngss-440" class="ngss">Hund’s rule states that when <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electrons</mark> are placed into any set of degenerate orbitals, one electron is placed into each orbital before any spin pairing takes place.</mark> This means, for example, if we consider three electrons being placed into the degenerate 2p orbitals, one would see an electronic <mark class="term" data-term="configuration" data-term-def="The way parts are arranged, such as how electrons are distributed in orbitals, or electron shells, around the nucleus of an atom." data-term-url="/en/glossary/view/configuration/8262">configuration</mark> of 2p<sub>x</sub><sup>1</sup> 2p<sub>y</sub><sup>1</sup> 2p<sub>z</sub><sup>1</sup> (recall the p orbitals are oriented along x, y, and z axes, respectively) before we see any pairing. The fourth electron to enter the 2p sub-shell would create the need for an unavoidable pairing, hence 2p<sub>x</sub><sup>2</sup> 2p<sub>y</sub><sup>1</sup> 2p<sub>z</sub><sup>1</sup> (Figure 7).</p> <div class="figure"> <figure> <button class="lightbox-button lightbox-button--icon" data-lightbox="" data-lightbox-src="/img/library/large_images/image_9389.jpg"> <img src="/img/library/modules/mid231/Image/VLObject-9389-160229050220.jpg" alt="Figure 7: An illustration of Hund's Rule showing the placement of electrons in various orbitals of nitrogen (N) and oxygen (O)." /> </button> <figcaption> <p><strong>Figure 7</strong>: An illustration of Hund's Rule showing the placement of electrons in various orbitals of nitrogen (N) and oxygen (O).</p> </figcaption> </figure> </div> <p>Now that we have a sense of <mark class="term" data-term="electron" data-term-def="A subatomic particle with a negative charge of 1.60 × 10<sup>-19</sup> coulombs and a mass of 9.11 × 10<sup>-31</sup> kg. Electrons&hellip;" data-term-url="/en/glossary/view/electron/852">electron</mark> <mark class="term" data-term="configuration" data-term-def="The way parts are arranged, such as how electrons are distributed in orbitals, or electron shells, around the nucleus of an atom." data-term-url="/en/glossary/view/configuration/8262">configuration</mark>, their energies, and their most probable positions within <mark class="term" data-term="atom" data-term-def="The smallest unit of an element that retains the chemical properties of the element. Atoms can exist alone or in&hellip;" data-term-url="/en/glossary/view/atom/1509">atoms</mark>, the next step is to describe the behavior of electrons when atoms interact with one another. Electron interaction forms the basis of a key chemical <mark class="term" data-term="process" data-term-def="Method, procedure; series of actions or steps." data-term-url="/en/glossary/view/process/8256">process</mark> called chemical <mark class="term" data-term="bonding" data-term-def="The act of fastening two atoms together." data-term-url="/en/glossary/view/bonding/8295">bonding</mark>. And understanding the behavior of electrons provides us with a better understanding of the chemical behavior of the <mark class="term" data-term="element" data-term-def="One of fewer than 118 pure chemical substances. An element is a substance composed of atoms with identical atomic number." data-term-url="/en/glossary/view/element/1510">elements</mark> and their <mark class="term" data-term="compound" data-term-def="A material formed by the chemical combination of elements in defined proportions. Compounds can be chemically decomposed into simpler substances." data-term-url="/en/glossary/view/compound/1517">compounds</mark>.</p> </div> </section> <hr class="border-color-dark" /> <footer class="module__footer"> <p class="citation"> <em> Adrian Dingle, B.Sc., Anthony Carpi, Ph.D. “Atomic Theory IV” Visionlearning Vol. CHE-3 (7), 2016. </em> </p>  <div class="title-list" id="refs" name="refs"> <p class="h6 title-list__title"> References </p> <ul class="title-list__list"> <li><p>Heisenberg, W. (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. <em>Zeitschrift für Physik, 43</em>(3–4), 172-198.</li> <li>Pauli, W. (1925). Ueber den Einfluss der Geschwindigkeitsabhaengigkeit der Elektronenmasse auf den Zeeman-Effekt. <em>Zeitschrift für Physik 31</em>(1), 373-385.</li> <li>Pauli, W. (1925). Ueber den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren. <em>Zeitschrift für Physik 31</em>(1), 765-783.</li> <li>Pauli, W. (1946). Remarks on the history of the Exclusion Principle. <em>Science, New Series, 103</em>(2669), 213-215.</li> <li>Schrödinger, E. (1926). Quantisierung als Eigenwertproblem. <em>Annalen der Physik, 384</em>(4), 361-376.</li> <li>Stoner, E.C. (1924). The distribution of electrons among atomic energy levels. <em>The London, Edinburgh and Dublin Philosophical Magazine </em>(6th series),<em> 48</em>, 719-736.</p></li> </ul> </div>  <div class="title-list" name="further"> <p class="h6 title-list__title"> Further Reading </p> <ul class="grid grid--column-2--md grid--column-3--md gap-1"> <li> <a class="no-hover-focus height-100" href="/en/library/Chemistry/1/Chemical-Bonding/55"> <article class="flex-row align-items-center flex-column--md align-items-start--md height-100 theme-light padding-2 gap-2"> <div class="width-30 width-auto--md"> <img class="border-radius box-shadow-1" src="/img/library/moduleImages/featured_image_55-23061209065024.jpeg" alt="Chemical Bonding"> </div> <div class="flex-grow-shrink"> <h2 class="h6 font-weight-normal"> Chemical Bonding: <em>Ionic and covalent bonds and polarity</em> </h2> </div> </article> </a> </li> </ul> </div> </footer> </div>   </div>  <div class="order-1 order-2--lg module__tools"> <div class="narrow margin-x-auto position-sticky-top font-size-md"> <div class="padding-2 border-radius box-shadow-1--lg"> <div class="tabs" role="tablist"> <nav> <button class="button button--icon-label" id="tab-button-in-this-module" aria-label="Table of Contents" aria-controls="tab-panel-module__tools" aria-selected="true" role="tab"> <span class="icon icon-list" aria-hidden="true"></span> <span class="button__text">Contents</span> </button> <button class="button button--icon-label" id="tab-button-toggle-terms" aria-controls="tab-panel-toggle-terms" aria-selected="false" role="tab"> <span class="icon icon-glossary-highlight"></span> <span class="button__text">Glossary Terms</span> </button> <button class="button button--icon-label" id="tab-button-toggle-ngss" aria-controls="tab-panel-toggle-ngss" aria-selected="false" role="tab"> <span class="icon icon-ngss"></span> <span class="button__text">NGSS</span> </button> </nav> <hr class="divider" /> <div class="tabs__panel shown" id="tab-panel-module__tools" aria-labelledby="tab-button-module__tools" role="tabpanel"> <p class="font-weight-bold margin-bottom-1"> Table of Contents </p> <div class="table-of-contents" id="module-toc"> <ul> <li><a href="/en/library/chemistry/1/atomic-theory-iv/231#toc_1">Waves and measurement uncertainty</a> </li> <li><a href="/en/library/chemistry/1/atomic-theory-iv/231#toc_2">Schrödinger solves the probability question</a> </li> <li> <ul> <li><a href="/en/library/chemistry/1/atomic-theory-iv/231#toc2_1">The simplest case - a hydrogen atom with a single electron</a> </li> </ul> </li> <li><a href="/en/library/chemistry/1/atomic-theory-iv/231#toc_3">Quantum numbers</a> </li> <li><a href="/en/library/chemistry/1/atomic-theory-iv/231#toc_4">Describing orbitals by type of wave function</a> </li> <li> <ul> <li><a href="/en/library/chemistry/1/atomic-theory-iv/231#toc2_2">Evidence for orbitals and the quantum model</a> </li> </ul> </li> <li><a href="/en/library/chemistry/1/atomic-theory-iv/231#toc_5">The more complex case of many-electron atoms</a> </li> <li> <ul> <li><a href="/en/library/chemistry/1/atomic-theory-iv/231#toc2_3">Filling orbitals with many electrons</a> </li> </ul> </li> <li> <ul> <li><a href="/en/library/chemistry/1/atomic-theory-iv/231#toc2_4">Hund's Rule - filling degenerate orbitals</a> </li> </ul> </li> </ul> </div> </div>   <div class="tabs__panel" id="tab-panel-toggle-terms" aria-labelledby="tab-button-toggle-terms" role="tabpanel"> <div class="reading-toggle"> <div class="reading-toggle__switch"> <div class="form-entry__option__switch"> <label> <input type="checkbox" name="termsToggleSwitch" id="terms-toggle-switch" /> <span class="switch__slider"></span> <span class="option__label text-decoration-none font-size-md"> Highlight Glossary Terms </span> </label> </div> </div> <div class="reading-toggle__help"> <p> <em> Activate glossary term highlighting to easily identify key terms within the module. Once highlighted, you can click on these terms to view their definitions. </em> </p> </div> </div> </div> <div class="tabs__panel" id="tab-panel-toggle-ngss" aria-labelledby="tab-button-toggle-ngss" role="tabpanel"> <div class="reading-toggle"> <div class="reading-toggle__switch"> <div class="form-entry__option__switch"> <label> <input type="checkbox" name="ngssToggleSwitch" id="ngss-toggle-switch" /> <span class="switch__slider"></span> <span class="option__label text-decoration-none font-size-md"> Show NGSS Annotations </span> </label> </div> </div> <div class="reading-toggle__help"> <p> <em> Activate NGSS annotations to easily identify NGSS standards within the module. Once highlighted, you can click on them to view these standards. </em> </p> </div> </div> </div> <div class="reading-annotation-container"></div>  </div> </div> <div class="margin-3"> <script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js?client=ca-pub-9561344156007092" crossorigin="anonymous"></script>  <ins class="adsbygoogle" style="display:block" data-ad-client="ca-pub-9561344156007092" data-ad-slot="7634263342" data-ad-format="auto" data-full-width-responsive="true"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script> </div>   </article> </div> </main> <script id="ngssCommentdata" type="application/json"> [{"ngss_tag_id":null,"type":"p","tag":"","name":null,"description":null,"comment":"<p>Heisenberg Uncertainty Principle is a mathematical model supporting the location and momentum of an electron.<\/p>\r\n\r\n<p><strong>HS-SEP.5 Using Mathematics and Computational Thinking:<\/strong>\r\n\r\nUse mathematical, computational, and\/or algorithmic representations of phenomena or design solutions to describe and\/or support claims and\/or explanations.<\/p>","is_public":"1","mod_ngss_comment_id":"421","display_order":"1","dimension":"p","dimension_full":"Science and Engineering Practices"},{"ngss_tag_id":null,"type":"p","tag":"","name":null,"description":null,"comment":"<p>Equations can be applied to model electrons.<\/p>\r\n\r\n<p><strong>HS-SEP.5 Using Mathematics and Computational Thinking:<\/strong>\r\nUse mathematical, computational, and\/or algorithmic representations of phenomena or design solutions to describe and\/or support claims and\/or explanations.<\/p>","is_public":"1","mod_ngss_comment_id":"422","display_order":"2","dimension":"p","dimension_full":"Science and Engineering Practices"},{"ngss_tag_id":null,"type":"p","tag":"","name":null,"description":null,"comment":"<p><strong>HS-SEP.5 Using Mathematics and Computational Thinking:<\/strong>\r\n\r\nCreate and\/or revise a computational model or simulation of a phenomenon, designed device, process, or system.<\/p>","is_public":"1","mod_ngss_comment_id":"423","display_order":"3","dimension":"p","dimension_full":"Science and Engineering Practices"},{"ngss_tag_id":null,"type":"dci","tag":"","name":null,"description":null,"comment":"<p>This knowledge supports the DCI identified below.<\/p>\r\n\r\n<p><strong>PS1 Matter and Its Interactions;\r\nHS-PS1.A Structure and Properties of Matter:<\/strong> Stable forms of matter are those in which the electric and magnetic field energy is minimized. 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