[{"data":1,"prerenderedAt":221},["ShallowReactive",2],{"topic-electron-structure-and-light":3},{"topic":4,"prev":205,"next":214},{"id":5,"slug":6,"title":7,"shortTitle":8,"lesson":5,"lastAlignmentAudit":9,"category":10,"description":11,"metaDescription":12,"objectives":13,"conceptSections":41,"workedExample":103,"oerSources":112,"selfStudyQuestions":117,"relatedTopicSlugs":144,"relatedElements":148,"relatedReferences":155,"keyEquations":156,"howTo":173,"conceptProbe":182,"practiceTeaser":186,"gatedContent":194,"seoKeywords":197,"objectiveCount":204},12,"electron-structure-and-light","Electron Structure and Light","Electron Structure","2026-07-16","atomic-structure","Build the modern model of the atom: shells and orbitals, quantum numbers, electron configurations, orbital diagrams, common exceptions, and the connection to light through the Bohr model and de Broglie wavelength.","Learn electron structure: shells, subshells, orbitals, quantum numbers, electron configurations, orbital diagrams, Aufbau/Hund/Pauli, photon energy E=hv, Bohr model, de Broglie.",[14,17,20,23,26,29,32,35,38],{"id":15,"text":16},"12.1","Distinguish among shells, subshells, orbitals, and electrons and describe their hierarchical relationship",{"id":18,"text":19},"12.2","Describe the four quantum numbers (n, l, mₗ, mₛ) and their allowed values",{"id":21,"text":22},"12.3","Determine whether a given set of quantum numbers is valid and assign a complete set of four quantum numbers to an electron in an atom",{"id":24,"text":25},"12.4","Write full and noble-gas (abbreviated) electron configurations for atoms and monatomic ions",{"id":27,"text":28},"12.5","Draw orbital diagrams showing electron distribution among orbitals and apply the Aufbau principle, Hund's rule, and the Pauli exclusion principle",{"id":30,"text":31},"12.6","Identify common exceptions to expected electron configurations (e.g., Cr, Cu) and explain them qualitatively",{"id":33,"text":34},"12.7","Relate photon energy, frequency, and wavelength using E = hν and c = λν",{"id":36,"text":37},"12.8","Calculate the energy or wavelength of light emitted or absorbed during electron transitions in a hydrogen atom using the Bohr model",{"id":39,"text":40},"12.9","Explain wave-particle duality and calculate a de Broglie wavelength for matter",[42,52,61,69,80,91,99],{"heading":43,"content":44,"relatedObjectives":45,"deepDive":48},"Light, Energy, and the Bohr Model","\u003Cp>\u003Cem>Note on topic scope:\u003C/em> \u003Cstrong>This topic (T12) covers electron structure and light\u003C/strong> — shells, orbitals, quantum numbers, electron configurations, and the connection to photons via the Bohr model and de Broglie wavelength. The companion topic \u003Cstrong>T13: \u003Ca href=\"/learn/periodic-properties-and-trends\">Periodic Properties and Trends\u003C/a>\u003C/strong> covers atomic radius, ionization energy, electronegativity, electron affinity, ionic radius, isoelectronic species, and predicting bond type from periodic position. If you arrived here looking for periodic-trends content, follow the link to T13.\u003C/p>\u003Cp>Understanding electron structure begins with light. The \u003Cstrong>electromagnetic spectrum\u003C/strong> spans from low-energy radio waves to high-energy gamma rays. Two relationships connect a photon’s wavelength, frequency, and energy:\u003C/p>\u003Cul>\u003Cli>\u003Cstrong>c = λν\u003C/strong> — the speed of light c (2.998 × 10⁸ m/s in vacuum) equals wavelength λ times frequency ν. Wavelength and frequency are inversely proportional.\u003C/li>\u003Cli>\u003Cstrong>E = hν\u003C/strong> — the energy of one photon equals Planck’s constant h (6.626 × 10⁻³⁴ J·s) times frequency. Higher frequency means higher energy.\u003C/li>\u003C/ul>\u003Cp>The \u003Cstrong>Bohr model\u003C/strong> explained the hydrogen emission spectrum by proposing that electrons occupy specific quantized energy levels (orbits). The electron does not radiate while in a stationary state, but it emits or absorbs a photon when it moves between levels: |ΔE| = hν. For a one-electron atom or hydrogen-like ion the orbit energies are E\u003Csub>n\u003C/sub> = −k Z² / n², where k = 2.179 × 10⁻¹⁸ J and Z is the nuclear charge (Z = 1 for H, Z = 2 for He\u003Csup>+\u003C/sup>, …). The energy of a transition from n\u003Csub>i\u003C/sub> to n\u003Csub>f\u003C/sub> is ΔE = −k Z² (1/n\u003Csub>f\u003C/sub>² − 1/n\u003Csub>i\u003C/sub>²). The model correctly reproduced the Rydberg formula and hydrogen’s spectral lines but failed for any atom with more than one electron, because Bohr did not yet account for electron-electron interactions.\u003C/p>\u003Cp>\u003Cstrong>Wave-particle duality\u003C/strong> resolves the apparent paradox: electrons (and other matter) behave as both particles and waves. The \u003Cstrong>de Broglie wavelength\u003C/strong> λ = h/p = h/(mv) gives the wavelength of any moving particle of mass m and velocity v. Because h is so small, the wavelength is only appreciable for very low-mass and/or very fast objects: an electron at typical atomic speeds has a de Broglie wavelength on the order of atomic dimensions (about 10⁻¹⁰ m), which is why crystals diffract electrons. A 100-g softball at 35 m/s has a de Broglie wavelength of about 10⁻³⁴ m, far below any measurable scale.\u003C/p>\u003Cp>Modern quantum mechanics (Schrödinger’s wave equation, ca. 1926) replaced Bohr’s circular orbits with \u003Cstrong>orbitals\u003C/strong> — three-dimensional probability regions where electrons are most likely found. Heisenberg’s \u003Cstrong>uncertainty principle\u003C/strong> says the position and momentum of an electron cannot both be known precisely at the same time, so we describe an electron’s location probabilistically rather than as a definite trajectory.\u003C/p>",[46,47],103,104,[49],{"label":50,"body":51},"Why this matters: every colored light is an electron falling","\u003Cp>Fireworks are electron transitions you can watch from a lawn chair: strontium salts glow red, barium green, copper blue, each color set by the exact energy gap its electrons fall across. The orange-yellow of older streetlights is sodium’s signature 589 nm emission; a neon sign’s red-orange is neon atoms relaxing after an electric jolt. Because each element’s energy levels are unique, its emission colors are a fingerprint, and astronomers read those fingerprints in starlight to inventory elements in stars they will never touch. The quantized levels in this section are not abstractions; they are visible across the night sky.\u003C/p>",{"heading":53,"content":54,"relatedObjectives":55,"deepDive":57},"Shells, Subshells, Orbitals, and Electrons","\u003Cp>The modern atomic model is hierarchical — four nested levels of organization, each with strict capacity rules:\u003C/p>\u003Col>\u003Cli>\u003Cstrong>Shells\u003C/strong> are labeled by the principal quantum number n = 1, 2, 3, … A shell sets the general region of space and the energy level for an electron occupying it. The shells of an atom can be thought of as concentric circles radiating out from the nucleus, with higher shell numbers corresponding to greater average distance from the nucleus and higher energy.\u003C/li>\u003Cli>\u003Cstrong>Subshells\u003C/strong> are labeled s, p, d, f within a shell, set by the angular-momentum quantum number l (with l = 0, 1, 2, 3 mapping to s, p, d, f). Shell n contains exactly n subshells: shell 1 has only 1s; shell 2 has 2s and 2p; shell 3 has 3s, 3p, 3d; and so on. The subshell determines the \u003Cem>shape\u003C/em> of the orbital.\u003C/li>\u003Cli>\u003Cstrong>Orbitals\u003C/strong> are individual three-dimensional probability regions within a subshell, set by the magnetic quantum number m\u003Csub>l\u003C/sub>. The number of orbitals in a subshell is 2l + 1: the s subshell has 1 orbital, p has 3, d has 5, f has 7. Each orbital has a distinct spatial orientation.\u003C/li>\u003Cli>\u003Cstrong>Electrons\u003C/strong> are the particles that occupy orbitals, distinguished within an orbital by the spin quantum number m\u003Csub>s\u003C/sub>. Each orbital holds at most 2 electrons (with opposite spins).\u003C/li>\u003C/ol>\u003Cp>Capacity rules follow directly:\u003C/p>\u003Cul>\u003Cli>Subshell capacity: s = 2, p = 6, d = 10, f = 14 (twice the orbital count, 2(2l + 1)).\u003C/li>\u003Cli>Shell capacity: 2n² electrons. So shell 1 holds 2, shell 2 holds 8, shell 3 holds 18, shell 4 holds 32.\u003C/li>\u003C/ul>\u003Cp>Every electron in an atom “lives” in exactly one orbital, which sits inside one subshell, which sits inside one shell. The four quantum numbers in the next section make this hierarchy precise.\u003C/p>",[56],95,[58],{"label":59,"body":60},"An orbital is a probability map, not a path","\u003Cp>The word “orbital” invites a planetary picture: an electron circling the nucleus on a track. The modern model says something stranger: an electron has no defined path, and an orbital is a \u003Cem>probability map\u003C/em>, the region where measurement would find the electron some chosen fraction (conventionally 90%) of the time. The s orbital’s sphere and the p orbital’s dumbbell are boundary surfaces of likelihood, not racetracks.\u003C/p>\u003Cp>This is why chemists say an electron \u003Cem>occupies\u003C/em> an orbital rather than travels it. The shell picture from the Atomic Structure topic is a useful first model for organizing electrons by energy level; orbitals provide the more accurate quantum description.\u003C/p>",{"heading":62,"content":63,"relatedObjectives":64,"deepDive":65},"Quantum Numbers","\u003Cp>Each electron in an atom is described by four \u003Cstrong>quantum numbers\u003C/strong>:\u003C/p>\u003Cul>\u003Cli>\u003Cstrong>n\u003C/strong> (principal) — energy level / shell: 1, 2, 3, … Higher n means higher energy and larger orbital.\u003C/li>\u003Cli>\u003Cstrong>l\u003C/strong> (angular momentum / subshell) — sublevel shape, with allowed values 0 to (n−1). Values 0, 1, 2, 3 correspond to s, p, d, f.\u003C/li>\u003Cli>\u003Cstrong>m\u003Csub>l\u003C/sub>\u003C/strong> (magnetic / orbital) — orbital orientation, with allowed values −l to +l. For l = 1 (p), m\u003Csub>l\u003C/sub> = −1, 0, +1, giving three p orbitals. For l = 2 (d), m\u003Csub>l\u003C/sub> = −2, −1, 0, +1, +2 (five d orbitals).\u003C/li>\u003Cli>\u003Cstrong>m\u003Csub>s\u003C/sub>\u003C/strong> (spin) — electron spin: +½ or −½.\u003C/li>\u003C/ul>\u003Cp>The \u003Cstrong>Pauli exclusion principle\u003C/strong> states that no two electrons in the same atom can have the same set of all four quantum numbers. As a consequence, each orbital holds at most 2 electrons (with opposite spins).\u003C/p>",[56],[66],{"label":67,"body":68},"Four quantum numbers, one mailing address","\u003Cp>The four quantum numbers work like a postal address of increasing precision: n names the city (shell), l the neighborhood (subshell), m\u003Csub>l\u003C/sub> the street (specific orbital), and m\u003Csub>s\u003C/sub> which of the two occupants you mean (spin). The Pauli exclusion principle then reads naturally: no two electrons in an atom may share the complete address; at most the two spin-partners share everything down to the street.\u003C/p>\u003Cp>The addressing also explains the capacity arithmetic: shell n contains n² orbitals, each housing 2 electrons, giving the 2, 8, 18, 32 shell capacities that structure the entire periodic table.\u003C/p>",{"heading":70,"content":71,"relatedObjectives":72,"deepDive":76},"Validity and Assignment of Quantum-Number Sets","\u003Cp>A \u003Cstrong>valid\u003C/strong> quantum-number set respects all the allowed-value rules. Three checks decide validity:\u003C/p>\u003Col>\u003Cli>\u003Cstrong>n\u003C/strong> must be a positive integer (1, 2, 3, …).\u003C/li>\u003Cli>\u003Cstrong>l\u003C/strong> must be an integer in the range 0 ≤ l ≤ n−1.\u003C/li>\u003Cli>\u003Cstrong>m\u003Csub>l\u003C/sub>\u003C/strong> must be an integer in the range −l ≤ m\u003Csub>l\u003C/sub> ≤ +l. \u003Cstrong>m\u003Csub>s\u003C/sub>\u003C/strong> must be +½ or −½.\u003C/li>\u003C/ol>\u003Cp>Examples:\u003C/p>\u003Cul>\u003Cli>(n, l, m\u003Csub>l\u003C/sub>, m\u003Csub>s\u003C/sub>) = (3, 2, −2, +½) → \u003Cstrong>valid\u003C/strong> (3d orbital).\u003C/li>\u003Cli>(2, 2, 0, +½) → \u003Cstrong>invalid\u003C/strong>: l cannot equal n.\u003C/li>\u003Cli>(3, 1, +2, −½) → \u003Cstrong>invalid\u003C/strong>: m\u003Csub>l\u003C/sub> cannot exceed l.\u003C/li>\u003Cli>(4, 0, 0, +1) → \u003Cstrong>invalid\u003C/strong>: m\u003Csub>s\u003C/sub> can only be ±½.\u003C/li>\u003C/ul>\u003Cp>To \u003Cstrong>assign\u003C/strong> a complete quantum-number set to a specific electron, locate the electron in the orbital diagram, then read off (n, l, m\u003Csub>l\u003C/sub>, m\u003Csub>s\u003C/sub>). For the last electron added to nitrogen (1s² 2s² 2p³, last electron in 2p), a valid assignment is (n, l, m\u003Csub>l\u003C/sub>, m\u003Csub>s\u003C/sub>) = (2, 1, m\u003Csub>l\u003C/sub>, +½) where m\u003Csub>l\u003C/sub> can be −1, 0, or +1: n = 2 because the electron sits in the second shell, l = 1 because it is in a p subshell, and m\u003Csub>l\u003C/sub> can take any of the three allowed values for l = 1 because Hund’s rule (one electron in each degenerate orbital before any pairing) does not single out a specific m\u003Csub>l\u003C/sub> value. The m\u003Csub>s\u003C/sub> = +½ assignment for the third 2p electron follows from Hund’s rule itself, which says the three half-filled 2p electrons all carry parallel spins; conventionally the parallel-spin convention for unpaired electrons is m\u003Csub>s\u003C/sub> = +½. Many textbook orbital diagrams draw the three 2p boxes left-to-right as m\u003Csub>l\u003C/sub> = −1, 0, +1 and fill them in that order, but that diagrammatic ordering is a drawing convention — not a consequence of Hund’s rule.\u003C/p>",[73,74,75],100,101,102,[77],{"label":78,"body":79},"Test yourself: which addresses are legal?","\u003Cp>Valid or invalid? Decide for each set (n, l, m\u003Csub>l\u003C/sub>, m\u003Csub>s\u003C/sub>), then check below.\u003C/p>\u003Cul>\u003Cli>(2, 2, 0, +½)\u003C/li>\u003Cli>(3, 1, −1, −½)\u003C/li>\u003Cli>(4, 0, 0, 1)\u003C/li>\u003C/ul>\u003Cp>\u003Cstrong>Answers:\u003C/strong> The first is \u003Cstrong>invalid\u003C/strong>: l may only run from 0 to n−1, so n = 2 allows l = 0 or 1, never 2. The second is \u003Cstrong>valid\u003C/strong>: a 3p electron with m\u003Csub>l\u003C/sub> = −1 and spin down breaks no rule. The third is \u003Cstrong>invalid\u003C/strong>: m\u003Csub>s\u003C/sub> can only be +½ or −½; a spin of 1 does not exist for an electron. Every invalid set fails at exactly one rule, so check the rules in order and you cannot miss.\u003C/p>",{"heading":81,"content":82,"relatedObjectives":83,"deepDive":87},"Electron Configurations and Orbital Diagrams","\u003Cp>An \u003Cstrong>electron configuration\u003C/strong> lists the sublevels occupied by an atom’s electrons in order of energy. Three rules govern the filling order:\u003C/p>\u003Cul>\u003Cli>\u003Cstrong>Aufbau principle\u003C/strong> — fill lowest-energy orbitals first: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p.\u003C/li>\u003Cli>\u003Cstrong>Pauli exclusion principle\u003C/strong> — maximum 2 electrons per orbital (opposite spins).\u003C/li>\u003Cli>\u003Cstrong>Hund’s rule\u003C/strong> — within a sublevel, place one electron in each orbital before pairing any.\u003C/li>\u003C/ul>\u003Cp>Example: Nitrogen (Z = 7) is 1s² 2s² 2p³. The three 2p electrons occupy three separate orbitals (one each), all with parallel spins.\u003C/p>\u003Cp>An \u003Cstrong>orbital diagram\u003C/strong> uses boxes (or lines) for orbitals and arrows (↑↓) for electrons, visually showing Hund’s rule. A \u003Cstrong>noble gas abbreviation\u003C/strong> replaces the inner-shell configuration with the preceding noble gas symbol in brackets: Na = [Ne] 3s¹.\u003C/p>\u003Cp>\u003Cstrong>For ions\u003C/strong>: when a transition metal forms a cation, electrons are removed from the ns orbital before the (n−1)d orbital (even though ns filled first). For example, Fe²⁺ = [Ar] 3d⁶ — not [Ar] 4s² 3d⁴ — because both 4s electrons leave before any 3d electron does. Main-group cations and anions are simpler: remove the highest-energy electrons (cations) or add to the next available orbital (anions). Cl⁻ = [Ne] 3s² 3p⁶ = [Ar].\u003C/p>",[84,85,86,73],96,97,98,[88],{"label":89,"body":90},"Why this matters: the periodic table is the filling order, drawn","\u003Cp>The table’s odd shape is not decoration; it is the Aufbau order made visible. The two-column block on the left is s filling, the six-column block on the right is p, the ten-column middle is d, and the fourteen-column strip below is f. Period lengths 2, 8, 8, 18, 18, 32 are exactly the electron capacities of the subshells filling across each row.\u003C/p>\u003Cp>This gives you a shortcut better than memorizing the Aufbau list: read the configuration straight off the table by walking to your element and noting the blocks you crossed. The table was assembled from chemical behavior decades before quantum mechanics; the fact that electron configurations reproduce its shape is one of the great confirmations in science.\u003C/p>",{"heading":92,"content":93,"relatedObjectives":94,"deepDive":95},"Exceptions to the Aufbau Principle","\u003Cp>A few elements have electron configurations that deviate from the standard filling order because a half-filled or fully filled d sublevel provides extra stability:\u003C/p>\u003Cul>\u003Cli>\u003Cstrong>Chromium (Z=24)\u003C/strong>: Expected [Ar] 4s² 3d⁴, actual [Ar] 4s¹ 3d⁵ (half-filled d).\u003C/li>\u003Cli>\u003Cstrong>Copper (Z=29)\u003C/strong>: Expected [Ar] 4s² 3d⁹, actual [Ar] 4s¹ 3d¹⁰ (fully filled d).\u003C/li>\u003Cli>Similar exceptions occur with Mo, Ag, Au, and other transition metals.\u003C/li>\u003C/ul>\u003Cp>The qualitative explanation: a half-filled or fully filled d subshell maximizes exchange energy (the stabilization that comes from electrons of parallel spin in different orbitals), and 4s and 3d are close enough in energy that promoting one 4s electron to 3d pays off.\u003C/p>",[84],[96],{"label":97,"body":98},"Why half-filled shells cut in line","\u003Cp>Chromium and copper break the Aufbau order for a qualitative reason you can reason about: electrons repel each other, and a set of orbitals that is exactly half-filled (one electron in each, all spins parallel) or completely filled spreads the electrons as evenly and symmetrically as possible. That even, parallel-spin spread minimizes their mutual repulsion and maximizes the exchange-energy stabilization named in the section above, enough that promoting one 4s electron into 3d becomes a net energy win: [Ar] 4s¹ 3d⁵ beats 4s² 3d⁴, and 4s¹ 3d¹⁰ beats 4s² 3d⁹.\u003C/p>\u003Cp>The pattern repeats directly below: molybdenum takes 5s¹ 4d⁵ and silver 5s¹ 4d¹⁰. Learn Cr and Cu as the type specimens, and the same half-filled and filled-d preference reappears in the rows just beneath them.\u003C/p>",{"heading":100,"content":101,"relatedObjectives":102},"Electron Configuration: Quick-Check Strategy","\u003Cp>Use this sequence when students get stuck:\u003C/p>\u003Col>\u003Cli>\u003Cstrong>Locate the element\u003C/strong> (period/group/block) first; this anchors expected valence behavior.\u003C/li>\u003Cli>\u003Cstrong>Write the configuration systematically\u003C/strong> (Aufbau order), then verify Pauli and Hund.\u003C/li>\u003Cli>\u003Cstrong>For ions, remove electrons from the highest principal quantum number first\u003C/strong> (for transition metals, remove 4s before 3d; main-group elements lose ns/np electrons before any (n−1)d).\u003C/li>\u003Cli>\u003Cstrong>Validate quantum-number sets\u003C/strong> against n > 0, 0 ≤ l ≤ n−1, −l ≤ m\u003Csub>l\u003C/sub> ≤ +l, m\u003Csub>s\u003C/sub> = ±½.\u003C/li>\u003C/ol>\u003Cp>High-frequency errors: treating Cr/Cu exceptions as random, removing d electrons before 4s in cations, and forgetting the Hund-rule parallel-spin convention in orbital diagrams. A reliable final check is to confirm the total electron count matches the atomic number (or Z − charge for an ion).\u003C/p>",[84,86],{"title":104,"problem":105,"steps":106,"answer":111},"Writing Electron Configuration for an Ion","Write the full and abbreviated electron configurations for Fe²⁺ (Z = 26). How many unpaired electrons does it have?",[107,108,109,110],"Write the configuration for neutral Fe (Z=26): 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d⁶.","To form Fe²⁺, remove 2 electrons. For transition metals, remove from the 4s orbital first: Fe²⁺ = 1s² 2s² 2p⁶ 3s² 3p⁶ 3d⁶.","Abbreviated: Fe²⁺ = [Ar] 3d⁶.","Count unpaired electrons using Hund's rule: 3d⁶ fills as ↑↓ ↑ ↑ ↑ ↑ (one paired, four unpaired). So Fe²⁺ has 4 unpaired electrons.","Fe²⁺ = [Ar] 3d⁶ with 4 unpaired electrons.",[113,114,115,116],"OpenStax Chemistry 2e, Ch 6.1: Electromagnetic Energy (CC BY 4.0)","OpenStax Chemistry 2e, Ch 6.2: The Bohr Model (CC BY 4.0)","OpenStax Chemistry 2e, Ch 6.3: Development of Quantum Theory (CC BY 4.0)","OpenStax Chemistry 2e, Ch 6.4: Electronic Structure of Atoms (CC BY 4.0)",[118,121,123,125,127,129,131,134,136,138,140,142],{"question":119,"hint":120},"What is the difference between a shell, a subshell, an orbital, and an electron?","These are nested levels of the modern atomic model — each one fits inside the previous.",{"question":122},"What are the four quantum numbers and what does each describe?",{"question":124},"What allowed values can each quantum number take?",{"question":126},"What is the Aufbau principle?",{"question":128},"What is Hund's rule?",{"question":130},"What is the Pauli exclusion principle?",{"question":132,"hint":133},"How do you write the electron configuration of a neutral atom?","Follow the Aufbau filling order and apply Hund's rule at each sublevel.",{"question":135},"How does writing the configuration of a transition-metal cation differ from a neutral atom?",{"question":137},"What is a noble-gas (core) electron configuration?",{"question":139},"Why do chromium and copper deviate from the expected Aufbau filling?",{"question":141},"How are photon energy, frequency, and wavelength related?",{"question":143},"What does the de Broglie wavelength represent?",[145,146,147],"atomic-structure-and-isotopes","periodic-properties-and-trends","chemical-bonding-and-lewis-structures",[149,150,151,152,153,154],"H","He","Na","Fe","Cu","Cr",[],[157,161,165,169],{"label":158,"equation":159,"note":160},"Photon Energy","E = h × v","h = 6.626 × 10^-34 J·s; v is frequency in Hz",{"label":162,"equation":163,"note":164},"Wavelength–Frequency","c = λ × v","c = 3.00 × 10^8 m/s",{"label":166,"equation":167,"note":168},"Hydrogen Energy Level","E_n = -2.18 × 10^-18 J / n^2","Bohr model; n is the principal quantum number",{"label":170,"equation":171,"note":172},"de Broglie Wavelength","λ = h / (m × v)","Applies to any moving particle of mass m and velocity v",{"title":174,"steps":175},"How to Write an Electron Configuration",[176,177,178,179,180,181],"Determine the atomic number (Z) of the element, which equals the total number of electrons in a neutral atom.","Fill orbitals in order of increasing energy using the Aufbau sequence: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, and so on.","Place a maximum of 2 electrons per orbital (Pauli exclusion principle), with opposite spins.","Within a sublevel (e.g., 2p), place one electron in each orbital before pairing any (Hund's rule).","For a noble-gas abbreviation, replace the filled inner shells with the symbol of the preceding noble gas in brackets (e.g., [Ar] for elements after argon).","For transition-metal cations, remove electrons from the ns orbital before the (n-1)d orbital.",{"question":183,"answer":184,"type":185},"Two electrons in a nitrogen atom are described by the quantum-number sets (2, 1, 0, +1/2) and (2, 1, +1, +1/2). Are both sets valid? What does Hund's rule tell you about whether two p-block electrons in the same atom can share the m_s = +1/2 spin?","Both sets are valid: in shell n=2, l=1 (p subshell) is allowed, and m_l values of 0 and +1 are within the allowed range -1 to +1. The two electrons sit in different 2p orbitals (different m_l), so the Pauli exclusion principle is not violated even though both have m_s = +1/2. Hund's rule actually predicts this: the three 2p electrons in nitrogen each occupy a separate p orbital, all with parallel spins (+1/2), which maximizes exchange energy and gives the lowest-energy configuration.","conceptual",[187,191],{"id":188,"problem":189,"type":190},"pt-12-1","Write the full and abbreviated electron configurations for the Mn²⁺ ion (Z = 25). How many unpaired electrons does it have?","calculation",{"id":192,"problem":193,"type":185},"pt-12-2","Determine whether each of the following quantum-number sets is valid: (a) (3, 2, -1, +1/2), (b) (2, 2, 0, -1/2), (c) (4, 0, 0, +1/2), (d) (1, 0, +1, -1/2). For each invalid set, identify the rule that is violated.",{"workedExampleCount":195,"hasWorksheets":196},14,true,[198,199,200,201,202,203],"electron configuration","quantum numbers","orbital diagrams","Aufbau principle","Bohr model","de Broglie wavelength",9,{"id":206,"slug":207,"lesson":206,"title":208,"shortTitle":209,"description":210,"category":211,"objectiveCount":212,"problemCount":213},11,"solution-stoichiometry-and-titrations","Solution Stoichiometry and Titrations","Solution Stoichiometry","Apply stoichiometric principles to reactions in aqueous solution, including titrations (acid-base and redox), gravimetric analysis, and purity calculations.","quantitative",4,36,{"id":215,"slug":146,"lesson":215,"title":216,"shortTitle":217,"description":218,"category":10,"objectiveCount":219,"problemCount":220},13,"Periodic Properties and Trends","Periodic Trends","Read the periodic table as a predictive tool: valence electrons, atomic and ionic radius, ionization energy, electronegativity, electron affinity, and using trends to predict ionic vs covalent bonding.",6,30,1785108608100]