Atomic Structure
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Direct answer
Atomic structure rests on two pictures. The Bohr model treats the hydrogen atom as an electron in allowed circular orbits with energies E_n = -13.6 Z^2/n^2 eV and radii r_n = 52.9 n^2/Z pm, explaining the line spectrum through jumps between levels. Quantum mechanics replaces orbits with orbitals described by four quantum numbers, the de Broglie relation lambda = h/(mv) and the Heisenberg principle delta x × delta p >= h/(4 pi), while Aufbau, Pauli and Hund rules fix how orbitals fill.
What you must remember
- Bohr relations: r_n = 52.9 n^2/Z pm; E_n = -13.6 Z^2/n^2 eV (-2.18 × 10^-18 J); hydrogen levels -13.6, -3.4, -1.51, -0.85 eV; ionisation energy of hydrogen = 13.6 eV.
- Rydberg formula: wavenumber = R_H Z^2 (1/n1^2 - 1/n2^2), R_H = 109677 cm^-1; Lyman series ends at n1 = 1 (ultraviolet), Balmer at n1 = 2 (visible), higher series infrared; total lines from level n = n(n-1)/2.
- Quantum numbers: n = 1, 2, 3...; l = 0 to n-1 (s, p, d, f); m_l = -l to +l; m_s = ±1/2. Shell n holds n^2 orbitals and 2n^2 electrons; a subshell holds 2(2l+1).
- Nodes: radial = n - l - 1, angular = l, total = n - 1.
- de Broglie lambda = h/p; Heisenberg delta x × delta p >= h/(4 pi); photoelectric: h nu = h nu0 + KE_max.
- Filling rules: Aufbau by (n + l) then lower n; Pauli limits an orbital to two opposite spins; Hund's rule gives maximum unpaired electrons in degenerate orbitals.
- Exceptions from exchange energy: Cr = [Ar]3d5 4s1 and Cu = [Ar]3d10 4s1 — half-filled and filled d subshells win.
Common confusion
The recurring error is stretching the Bohr model beyond hydrogen-like species (H, He+, Li2+): it fails for multi-electron atoms, where penetration and shielding split 2s and 2p energies. Students also quote a level's energy as the photon energy — the photon carries the gap between two levels — and miscount spectral lines by forgetting that every pair of levels below n contributes one. Half-filled stability is not a universal law but an exchange-energy effect that matters mainly for d subshells.
Exam-focused takeaway
JEE Main tests rapid-fire facts: allowed quantum numbers, nodes, configuration exceptions, direct Rydberg or Bohr-energy numericals. JEE Advanced prefers synthesis: identifying a transition from a wavelength, scaling to hydrogen-like ions, linking de Broglie wavelength to Bohr quantisation mvr = nh/(2 pi), and photoelectric problems binding KE_max to stopping potential. Sketch the energy ladder with numbers before answering transition questions — most errors are bookkeeping, not concept.
Frequently asked questions
What do the four quantum numbers specify?
n fixes the shell, l the subshell and shape, m_l the orbital orientation, m_s the spin — together they identify an electron uniquely.
Why do Cr and Cu have exceptional configurations?
Half-filled (3d5) and filled (3d10) subshells gain exchange energy and symmetry, outweighing the small cost of promoting a 4s electron.
What is the Heisenberg uncertainty principle?
delta x × delta p >= h/(4 pi): position and momentum of a microscopic particle cannot both be known exactly at once — it kills the idea of a definite orbit.
Which spectral series lies in the visible region?
Only the Balmer series, ending at n = 2; Lyman is ultraviolet, the higher series infrared.
How many spectral lines can an electron falling from level n emit?
Up to n(n-1)/2, one for each pair of levels between n and the ground state.
Where does the Bohr model fail?
For any atom or ion with more than one electron, and for fine structure and the Zeeman effect — cases where electron-electron repulsion and orbital shapes matter.