Chapter Review

Atomic Structure

Subatomic Particles and Quantum Theory · Hydrogen Spectrum and Electronic Configuration

Subatomic Particles

Atoms consist of protons, neutrons (in the nucleus), and electrons (outside the nucleus). Their charge and mass properties define atomic number, mass number, and chemical behaviour.

Key Points

  • •
    Proton: charge +1.6022 × 10⁻¹⁹ C, mass ≈ 1.0073 amu, determines atomic number
  • •
    Neutron: charge 0, mass ≈ 1.0087 amu (slightly heavier than proton)
  • •
    Electron: charge −1.6022 × 10⁻¹⁹ C, mass ≈ 5.486 × 10⁻⁴ amu (1836× lighter than proton)
  • •
    Canal rays (positive ions) were discovered by Goldstein using a perforated cathode; only hydrogen gas yields protons
  • •
    e/m of canal rays is always smaller than e/m of cathode rays and depends on gas mass

Planck's Quantum Theory and Photon Energy

Energy is emitted or absorbed in discrete packets called quanta. For light, each quantum is a photon with energy proportional to its frequency.

Key Points

  • •
    Energy is quantized — only whole-number multiples of a quantum exist; fractional quanta are impossible
  • •
    Brighter light means more photons, not higher-energy individual photons
  • •
    Frequency and wavelength are inversely proportional: ν = c/λ
  • •
    Three equivalent forms: E = hν, E = hc/λ, E = hcν̄
  • •
    Greater wave number → shorter wavelength → higher photon energy
Formula

Quantum Numbers

Four quantum numbers completely describe an electron's state: shell, subshell, orbital orientation, and spin direction.

Key Points

  • •
    Principal (n): shell number, n = 1, 2, 3, …; max electrons = 2n²
  • •
    Azimuthal (l): subshell shape, l = 0 to n−1; s(0), p(1), d(2), f(3)
  • •
    Magnetic (ml): orbital orientation, ml = −l to +l; number of orbitals = 2l + 1
  • •
    Spin (ms): +½ (↑) or −½ (↓); two electrons in same orbital must have opposite spins
  • •
    K(n=1), L(n=2), M(n=3), N(n=4); s holds 2e⁻, p holds 6e⁻, d holds 10e⁻, f holds 14e⁻

Orbital Shapes and Nodes

Orbitals are probability distributions (not fixed paths) determined by the wave function ψ. Shape depends on the azimuthal quantum number l.

Key Points

  • •
    s-orbital: spherical, 0 angular nodes; p-orbital: dumbbell, 1 angular node; d-orbital: 5 shapes, 2 angular nodes
  • •
    Total nodes in any orbital = n − 1 (sum of radial and angular nodes)
  • •
    Radial nodes = n − l − 1; angular nodes = l
  • •
    Probability of finding an electron at a point is |ψ|²
  • •
    Within a subshell, all orbitals are degenerate (same energy, different orientation)

Bohr's Model of the Hydrogen Atom

Bohr proposed quantized circular orbits with fixed energies. The model successfully predicts hydrogen orbit radii, energies, and spectral lines but fails for multi-electron atoms.

Key Points

  • •
    Electron is stable only in permitted orbits; no energy radiation in a stationary state
  • •
    Angular momentum is quantized: mvr = nh/2π
  • •
    Radius scales as n²: rₙ = 0.529 n² Å for hydrogen (Bohr radius = 0.529 Å)
  • •
    Energy is negative and scales as 1/n²: Eₙ = −1313.31/n² kJ·mol⁻¹
  • •
    Ionization energy of hydrogen = E∞ − E₁ = 1313.31 kJ·mol⁻¹
  • •
    For He⁺ (Z=2), multiply energy by Z² = 4; radius by 1/Z
Formula

Hydrogen Spectral Series and Rydberg Formula

The hydrogen line spectrum arises from electron transitions between quantized energy levels. Each series corresponds to a different lower level (n₁).

Key Points

  • •
    Lyman (n₁=1, UV), Balmer (n₁=2, visible), Paschen (n₁=3, IR), Brackett (n₁=4, IR), Pfund (n₁=5, IR)
  • •
    Rydberg constant RH = 1.09678 × 10⁷ m⁻¹
  • •
    Limiting line of a series occurs when n₂ → ∞: ν̄limit = RH/n₁²
  • •
    The Balmer limiting line falls in UV; only inner Balmer lines are visible
  • •
    Bohr model cannot explain fine structure, Zeeman effect, or Stark effect
Formula

Electronic Configuration Rules

Three rules — Aufbau, Pauli exclusion, and Hund's — together with the (n+l) energy ordering determine the ground-state electron configuration of every element.

Key Points

  • •
    (n+l) rule: subshells fill by increasing n+l; tiebreak by lower n (e.g., 4s before 3d)
  • •
    Energy order: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s
  • •
    Aufbau: fill lowest-energy subshell completely before moving to the next
  • •
    Pauli: max 2 electrons per orbital, must have opposite spins (↑↓)
  • •
    Hund: fill degenerate orbitals singly with parallel spins before pairing
  • •
    Chromium (Z=24): Ar 4s¹ 3d⁵ (half-filled d exception); Copper (Z=29): Ar 4s¹ 3d¹⁰ (full d exception)
  • •
    Half-filled (d⁵, f⁷) and fully filled (d¹⁰, f¹⁴) subshells confer extra stability

Formulas

Photon Energy

Energy of a photon from its frequency. h = 6.626 × 10⁻³⁴ J·s.

Photon Energy from Wavelength

Use when wavelength is given. Convert λ to metres. c = 3.0 × 10⁸ m/s.

Bohr Orbit Radius (Hydrogen)

Radius grows as n². Bohr radius a₀ = 0.529 Å for n=1.

Bohr Orbit Energy (Joules)

Per atom. For He⁺ multiply by Z² = 4.

Bohr Orbit Energy (kJ/mol)

Per mole. Ionization energy = 1313.31 kJ·mol⁻¹ for hydrogen.

Rydberg Formula

RH = 1.09678 × 10⁷ m⁻¹. n₁ = lower level, n₂ = higher level. For ions, multiply RH by Z².

Angular Momentum Quantization

Angular momentum of electron is integer multiples of h/2π.

Max Electrons in Shell and Subshell

Shell capacity 2n², subshell capacity 2(2l+1), orbital capacity 2.