Quantum Mechanical Model

• Limitations of Bohr model and need for wave–particle duality (de Broglie, photoelectric effect recap).
• Heisenberg uncertainty principle — Δx·Δp ≥ ℏ/2 and its meaning for electron orbits.
• Quantum numbers n, l, m_l, m_s and Pauli exclusion principle.
• Aufbau principle, Hund's rule, and electronic configuration up to Z = 30.

Level 1 — Dual nature and uncertainty

Verbal: Electrons show wave behaviour — de Broglie wavelength λ = h/p. You cannot simultaneously know position and momentum with arbitrary precision; "orbit" becomes orbital (probability region).

Symbolic: λ = h/p; Δx·Δp ≥ h/(4π); quantum numbers (n, l, m_l, m_s); orbitals per subshell = 2l + 1.

de Broglie: λ = h/(mv). Explains allowed Bohr orbits as standing waves.

Photoelectric effect (Einstein): E = hν − φ; light quantised as photons — supports particle nature of radiation.

Uncertainty: Δx·Δp ≥ h/(4π). Fixed orbit violating this → replaced by orbital probability |ψ|².

Level 2 — Quantum numbers and filling rules

Pauli: No two electrons in an atom have same four quantum numbers → max 2 electrons per orbital with opposite spins.

Aufbau: Fill lowest energy orbitals first (order: 1s 2s 2p 3s 3p 4s 3d 4p … use n+l rule).

Hund's: Degenerate orbitals (same subshell) get one electron each with parallel spin before pairing.

Configuration example: Fe (Z=26): 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d⁶.

NCERT spotlight — Electronic configuration exceptions

Chromium (Z=24): [Ar] 4s1 3d5 — half-filled d subshell stability. Copper (Z=29): [Ar] 4s1 3d10. Know these NEET favourites.

Aufbau order: Use n+l rule; when n+l equal, lower n fills first. Orbitals written as 1s 2s 2p 3s 3p 4s 3d 4p sequence.

Hund rule and paramagnetism: Unpaired electrons in degenerate orbitals increase paramagnetic character — linked to O2 molecule in later bonding discussion.

Write quantum numbers for the outermost electron of sodium (Z = 11) and identify the subshell.

Step 1 — Configuration: 1s² 2s² 2p⁶ 3s¹ → valence electron in 3s.
Step 2 — n = 3, l = 0 (s subshell), m_l = 0, m_s = +½ (convention for unpaired).
Step 3 — Orbital: 3s¹ — one electron in spherical 3s orbital.
Step 4 — Na loses 3s¹ easily → Na⁺ stable noble gas core [Ne].
Step 5 — Compare Mg (3s²): both n=3, l=0 but paired vs unpaired affects properties.

Applications — periodic trends and spectroscopy

Electron configuration explains why Na loses one electron easily (3s1) while Mg loses two (3s2) with higher second ionisation energy jump. UV-Vis spectroscopy excites electrons between quantised levels — colour of transition metal compounds from d-d transitions preview in coordination compounds Class 12.

Deep dive — filling order and spectroscopic notation

Aufbau (n+l) rule order: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p — write configurations up to Z=30 then learn Cr Cu exceptions. Hund's rule maximises unpaired spins in degenerate orbitals — half-filled and fully-filled subshells extra stable (Cr 4s1 3d5, Cu 4s1 3d10). Pauli exclusion — four quantum numbers unique per electron; orbital capacity 2 electrons opposite m_s. Magnetic quantum number m_l ranges −l to +l — orientation in magnetic field Zeeman effect splitting spectral lines preview. Heisenberg uncertainty prevents simultaneous exact position and momentum — electron cloud not orbit. Photoelectric equation K_max = h nu − phi links photon energy to ejected electron kinetic energy — particle nature of light experimental foundation for quantum model replacing Bohr ad hoc orbits with probability orbitals.

Review and practice drill

Review checklist: (1) Four quantum numbers n l m_l m_s. (2) Pauli exclusion. (3) Aufbau order 4s before 3d. (4) Hund's rule maximum multiplicity. Practice: Configuration of Fe Z=26 — [Ar] 4s2 3d6.

• State Pauli exclusion principle.
• How many electrons in a p subshell maximum?
• Write configuration of nitrogen (Z = 7) and show Hund's rule for 2p.

Open the Practice tab for graded questions on Quantum.