Atomic Structure (Physics)

Concept overview

The Bohr model of the hydrogen atom quantised the electron's angular momentum: mvr = nh/(2π) for integer n. Combined with the classical circular-motion balance mv²/r = ke²/r², this gives quantised radii r_n = a₀n² (a₀ = 0.529 Å is the Bohr radius) and energies E_n = −13.6/n² eV (for hydrogen).

Electrons can jump between levels by absorbing or emitting photons of energy equal to the level difference. Emission from n₂ to n₁ (n₂ > n₁) gives a photon of frequency ν = (E_{n₂} − E_{n₁})/h. The Rydberg formula summarises this as 1/λ = R(1/n₁² − 1/n₂²), with R = 1.097 × 10⁷ m⁻¹.

The hydrogen emission spectrum organises into series named for their discoverers: Lyman (n₁ = 1, UV), Balmer (n₁ = 2, visible), Paschen (n₁ = 3, IR), Brackett, Pfund. Each series has a limit as n₂ → ∞.

De Broglie extended wave-particle duality: every particle of momentum p has an associated wavelength λ = h/p. For an electron, this wavelength is small enough to be observed only for slow electrons — but it's real, as electron diffraction experiments (Davisson-Germer, 1927) confirmed. The Bohr quantisation condition can be reinterpreted as fitting an integer number of de Broglie wavelengths around the orbit.

Key formulas & concepts

• Bohr radius (H atom): r_n = a₀ n² = 0.529 × n² Å
• Bohr energy (H atom): E_n = −13.6/n² eV
• Ionisation energy of H: 13.6 eV (n = 1 → ∞)
• Rydberg formula: 1/λ = R (1/n₁² − 1/n₂²), R = 1.097 × 10⁷ m⁻¹
• de Broglie wavelength: λ = h/p = h/(mv)
• Bohr angular momentum: mvr = nh/(2π)
• Hydrogen-like atoms: E_n = −13.6 Z²/n² eV (Z = nuclear charge)

Common mistakes to avoid

• Applying the Bohr model to multi-electron atoms — it works only for H and hydrogen-like ions (He⁺, Li²⁺, etc.).
• Forgetting the Z² factor for hydrogen-like ions — He⁺ has ionisation energy 54.4 eV, not 13.6.
• Using the wrong sign — Bohr energies are negative (bound states); photon emission energies are positive.
• Confusing photon frequency with orbital frequency — they're different quantities.

JEE Main relevance

JEE Main asks 1–2 atomic-structure problems every paper — usually one Rydberg formula for hydrogen and one Bohr-energy or de Broglie wavelength calculation.

JEE Advanced relevance

Advanced problems apply Bohr theory to hydrogen-like ions with different Z, or combine de Broglie wavelength with photoelectric effect. Isotope-shift questions (mass-dependence of Rydberg constant) also appear.

Previous-year questions (PYQ)

Rydberg-formula spectral-line calculations are the highest-frequency PYQ archetype. Drill both hydrogen and He⁺ / Li²⁺ variants.