Comprehensive CH301/CH104M/N Quantum Numbers & Electron Configurations – Study Notes


Quantum Numbers & Orbitals (Page 2)

  • Orbitals and Schrödinger equation

    • Orbitals are solutions to the Schrödinger equation; each orbital has a characteristic shape and energy

  • Principal quantum number (n)

    • As n increases:

    • The orbital becomes larger

    • The electron spends more time farther from the nucleus

    • The electron energy increases; the electron is less tightly bound to the nucleus

  • Angular momentum quantum number (ℓ)

    • Values and corresponding orbital types:

    • ℓ = 0 → s orbitals

    • ℓ = 1 → p orbitals

    • ℓ = 2 → d orbitals

    • ℓ = 3 → f orbitals

    • This quantum number defines the shape of the orbital

  • Magnetic quantum number (mℓ)

    • Describes the orientation of the orbital in space

    • For a given ℓ, mℓ takes 2ℓ + 1 possible values: mℓ ∈ −ℓ,−ℓ+1,…,ℓ−1,ℓm_ℓ \,\in\,{-ℓ, -ℓ+1, \ldots, ℓ-1, ℓ}

  • Spin magnetic quantum number (ms)

    • Describes the spin orientation of the electron

    • Possible values: ms ∈ +12,−12m_s \,\in\, {+\tfrac{1}{2}, -\tfrac{1}{2}}

  • Practice 1: Possible quartet for an electron in a 4d orbital

    • Options: A) n = 3, ℓ = 1, mℓ = 0, m_s = -½

    • B) n = 4, ℓ = 2, mℓ = +3, m_s = +½

    • C) n = 4, ℓ = 1, mℓ = 0, m_s = -½

    • D) n = 4, ℓ = 2, mℓ = +1, m_s = +½

    • Correct answer: D. Explanation: For a 4d orbital, n = 4 and ℓ = 2; allowed mℓ values are −2, −1, 0, 1, 2; so mℓ = +3 is invalid; only mℓ = +1 with ms = +½ is valid among the choices.

  • Practice 2: Maximum number of electrons with n = 4 and ℓ = 2

    • In a given subshell (defined by n and ℓ), the maximum number of electrons is 2(2ℓ + 1)

    • For ℓ = 2 (d subshell): Nextmax=2(2⋅2+1)=2⋅5=10N_{ ext{max}} = 2(2\cdot 2 + 1) = 2\cdot 5 = 10

  • Summary formulas

    • Number of orbitals in a subshell: extorbitals=2ℓ+1ext{orbitals} = 2ℓ + 1

    • Maximum electrons in a subshell: Nextmax=2(2ℓ+1)N_{ ext{max}} = 2(2ℓ + 1)

Orbital Shapes (Page 3)

  • s-Orbitals

    • Angular momentum quantum number: ℓ = 0

    • Lowest-energy orbital

    • Electron density is spherically symmetric about the nucleus

  • p-Orbitals

    • ℓ = 1

    • More energetic than s orbitals in general

    • Electron density forms two lobes on either side of the nucleus with a node at the nucleus (no electron density at the center)

    • Shape: dumbbell-shaped

  • d-Orbitals

    • ℓ = 2

    • Complex shapes, often described as four-leaf clover or doughnut-and-two-lobed shapes depending on orientation

  • f-Orbitals

    • ℓ = 3

    • Even more complex shapes with multiple lobes

  • Visual note

    • The page includes illustrative icons; shapes are simplified representations of probability densities

Electron Configurations & Energy Levels (Page 4)

  • Electron configurations and ground state

    • Ground state: electrons occupy the lowest available energy states (most stable configuration)

  • Four guiding rules to determine ground-state configuration 1) Aufbau (Build-Up) Principle: fill in order of increasing energy 2) Pauli Exclusion Principle: each orbital can hold a maximum of 2 electrons, and the two electrons must have opposite spins 3) Hund’s Rule (two parts):

    • (i) Electrons remain unpaired as long as possible (one electron in each orbital before pairing)

    • (ii) For degenerate orbitals, singly occupied orbitals should have parallel spins (same spin)

  • Let’s Practice: ground-state electron configurations

    • Oxygen (O, Z = 8): 1s² 2s² 2p⁴

    • Valence electrons: outermost shell (n = 2) contains 2s² and 2p⁴ → total valence electrons = 6

    • Chlorine (Cl, Z = 17): 1s² 2s² 2p⁶ 3s² 3p⁵

    • Valence electrons: outermost shell (n = 3) contains 3s² and 3p⁵ → total valence electrons = 7

  • Valence shell concept

    • The valence shell is the outermost principal energy level containing electrons capable of chemical bonding and reactivity

Electron Configurations & Periodic Table: Valence Electrons (Page 5)

  • Valence electrons

    • Definition: electrons on the outermost shell (highest principal quantum number, n)

  • Practice: Electron configuration of Arsenic (As, Z = 33)

    • Full configuration: 1s2  2s2  2p6  3s2  3p6  4s2  3d10  4p31s^2\;2s^2\;2p^6\;3s^2\;3p^6\;4s^2\;3d^{10}\;4p^3

    • Condensed form: [Ar]  3d10  4s2  4p3[\mathrm{Ar}]\;3d^{10}\;4s^2\;4p^3

    • Valence electrons: outermost shell is n = 4, consisting of 4s² and 4p³ → total valence electrons = 5

  • Notes on order

    • The typical condensed form often shown is [Ar] 3d¹⁰ 4s² 4p³; the energy ordering historically places 4s before 3d, but the electron count is what matters for valence

Case Studies (Page 6)

  • Case Study #1: Bismuth (Bi, Z = 83)

    • Condensed electron configuration: [Xe]  4f14  5d10  6s2  6p3[\mathrm{Xe}]\;4f^{14}\;5d^{10}\;6s^2\;6p^3

    • Unpaired electrons

    • In Bi, the valence p orbitals (6p) contain 3 electrons; following Hund’s rule, the three 6p electrons are unpaired

    • Therefore total unpaired electrons = 3

  • Case Study #2: Selenium (Se, Z = 34)

    • Electron configuration (full): [Ar]  3d10  4s2  4p4[\mathrm{Ar}]\;3d^{10}\;4s^2\;4p^4

    • Condensed electron configuration: [Ar]  3d10  4s2  4p4[\mathrm{Ar}]\;3d^{10}\;4s^2\;4p^4

    • Unpaired electrons in Se

    • In the 4p^4 subshell, Hund’s rule yields two unpaired electrons (the first three electrons singly occupy each p orbital with parallel spins; the fourth electron pairs one of these orbitals, leaving two orbitals with a single electron each)

    • Total unpaired electrons = 2

Connections, Implications & Practical Notes

  • Connections to foundational principles

    • Electron configurations embody Aufbau, Pauli, and Hund’s rules in a concrete way

    • Orbital shapes (s, p, d, f) reflect angular momentum and electron density distribution, influencing chemical bonding and spectroscopy

  • Real-world relevance

    • Quantum numbers n, ℓ, mℓ, ms determine atomic energy levels, spectral lines, and magnetic properties

    • Valence electrons dictate chemical reactivity and periodic trends (e.g., valence shell occupancy explains group behavior)

  • Ethical/philosophical/practical implications (from Page 1 content)

    • Integrity and honesty in academics build trust, relationships, and long-term professional opportunities

    • Shortcuts like cheating or plagiarism undermine learning and can have lasting consequences

Quick Reference Formulas & Key Values

  • Orbital types by ℓ:

    • s: ℓ = 0

    • p: ℓ = 1

    • d: ℓ = 2

    • f: ℓ = 3

  • Magnetic quantum numbers mℓ:

    • For a given ℓ, mℓ ∈ {−ℓ, −ℓ+1, …, ℓ−1, ℓ}

  • Spin quantum numbers m_s:

    • m_s ∈ {+1/2, −1/2}

  • Number of orbitals in a subshell:

    • extorbitals=2ℓ+1ext{orbitals} = 2ℓ + 1

  • Maximum electrons in a subshell:

    • Nextmax=2(2ℓ+1)N_{ ext{max}} = 2(2ℓ + 1)

  • Example electron configurations mentioned:

    • Oxygen: 1s2  2s2  2p41s^2\;2s^2\;2p^4

    • Chlorine: 1s2  2s2  2p6  3s2  3p51s^2\;2s^2\;2p^6\;3s^2\;3p^5

    • Arsenic (condensed): [Ar]  3d10  4s2  4p3[\mathrm{Ar}]\;3d^{10}\;4s^2\;4p^3

    • Bismuth (condensed): [Xe]  4f14  5d10  6s2  6p3[\mathrm{Xe}]\;4f^{14}\;5d^{10}\;6s^2\;6p^3

    • Selenium (condensed): [Ar]  3d10  4s2  4p4[\mathrm{Ar}]\;3d^{10}\;4s^2\;4p^4

  • Notes on practical interpretation

    • Condensed configurations emphasize the outermost valence electrons for chemistry

    • Ground-state configurations follow the Aufbau sequence but may require exceptions in transition metals and lanthanides/actinides due to near-degenerate energies