2.2 Quantum Numbers, Atomic Spectra & Electron Configurations

2.2 Quantum Numbers, Atomic Spectra & Electron Configurations

Classical physics could not explain atomic stability or discrete line spectra. Quantum mechanics resolved these phenomena by treating electrons through quantized energy states and probability wavefunctions.

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Wave-Particle Duality and Quantum Radiation

Electromagnetic Radiation and Photons

Electromagnetic radiation travels at c = 2.998 × 10⁸ m/s, governed by c = λν. In 1900, Max Planck proposed that energy is quantized in discrete packets:

E = hν = hc / λ [h = 6.626 × 10⁻³⁴ J·s]

Albert Einstein explained the photoelectric effect by modeling light as discrete photons. An electron is ejected only when photon energy exceeds the metal's work function Φ:

KE_max = hν - Φ = hν - hν₀

Higher light intensity increases electron yield, while higher frequency increases kinetic energy.

de Broglie Matter Waves

In 1924, Louis de Broglie showed matter possesses wave characteristics:

λ = h / p = h / (mv)

For electrons (m = 9.109 × 10⁻³¹ kg), λ is ~10⁻¹⁰ m, matching crystal lattice spacings and producing observable diffraction.

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The Bohr Model & Atomic Spectra

When high voltage energizes hydrogen gas, it emits discrete spectral lines rather than a continuous spectrum.

The Bohr Model (1913)

Niels Bohr proposed:

1. Electrons occupy stationary circular orbits with quantized angular momentum: L = mvr = n(h / 2π).
2. Energy levels are quantized: E_n = -R_H / n² = -2.179 × 10⁻¹⁸ J / n².
3. Transitions emit or absorb photons matching the energy difference:

ΔE = -R_H · (1/n_f² - 1/n_i²) = hν

Rydberg Equation & Spectral Series

Transition wavelengths follow the Rydberg formula:

1/λ = R_∞ · (1/n₁² - 1/n₂²) [R_∞ = 1.097 × 10⁷ m⁻¹; n₂ > n₁]

• Lyman Series (n₁ = 1): Ultraviolet transitions terminating at n = 1.
• Balmer Series (n₁ = 2): Visible transitions terminating at n = 2 (656 nm red, 486 nm cyan, 434 nm blue, 410 nm violet).
• Paschen Series (n₁ = 3): Infrared transitions terminating at n = 3.

The model succeeds for single-electron species (H, He⁺) but fails for polyelectronic atoms because it ignores electron repulsion and assumes classical planetary orbits.

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Quantum Mechanical Model & Four Quantum Numbers

Heisenberg's Uncertainty Principle establishes that position and momentum cannot be measured simultaneously with arbitrary precision:

Δx · Δp ≥ h / (4π)

Schrödinger wave mechanics uses wavefunctions ψ, where ψ² represents probability density. An atomic orbital bounds 90% of electron probability.

Four quantum numbers characterize each electron:

Subshell Capacities and Geometries

The five d orbitals include four four-lobed cloverleaf shapes (d_xy, d_yz, d_xz, d_x²-y²) and the d_z² orbital, which has two lobes along the z-axis plus a doughnut-shaped ring in the xy-plane.

Nodal Surfaces

Nodes are regions of zero probability density (ψ² = 0):

• Radial Nodes: n - l - 1 (spherical shells of zero probability)
• Angular Nodes: l (planar or conical surfaces passing through nucleus)
• Total Nodes: n - 1 A 3p orbital (n = 3, l = 1) has 1 radial node, 1 angular node, and 2 total nodes.

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Ground-State Configurations

1. Aufbau Principle: Electrons occupy lowest-energy orbitals first according to the (n + l) rule. If (n + l) is identical, the orbital with lower n fills first.
2. Pauli Exclusion Principle: No two electrons in an atom share identical four quantum numbers; an orbital holds at most two electrons with opposing spins.
3. Hund's Rule: In degenerate orbitals, electrons enter singly with parallel spins before pairing to minimize electron-electron repulsion. For carbon (1s² 2s² 2p²), the two 2p electrons occupy separate orbitals with parallel spins ([↑][↑][ ]) rather than pairing in one orbital ([↑↓][ ][ ]).

Magnetic Properties

• Paramagnetic: Contains unpaired electrons; attracted into magnetic fields (e.g., O: [He] 2s² 2p⁴ has two unpaired electrons).
• Diamagnetic: All electrons paired; weakly repelled by magnetic fields (e.g., Ne: [He] 2s² 2p⁶).

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Transition Metal Anomalies & Cation Formation

Anomalies

Promoting an s electron provides exchange stability in half-filled or filled d subshells:

• Chromium (Z = 24): [Ar] 4s¹ 3d⁵ (half-filled subshell stability with six parallel spins).
• Copper (Z = 29): [Ar] 4s¹ 3d¹⁰ (fully filled subshell stability).
• Molybdenum ([Kr] 5s¹ 4d⁵) and Silver ([Kr] 5s¹ 4d¹⁰) exhibit similar promotions.

Cation Configurations

Transition metals always lose outer 4s electrons before 3d electrons upon ionization because 3d orbitals drop below 4s in energy once occupied:

• Neutral Fe: [Ar] 4s² 3d⁶
• Fe²⁺: [Ar] 3d⁶ (loses 4s electrons first)
• Fe³⁺: [Ar] 3d⁵ (stable half-filled d subshell)