Bohr Model of the Atom

• Historical context: Rutherford nucleus vs classical radiation problem for electrons.
• Bohr postulates — stationary orbits, angular momentum quantization, emission/absorption of photons.
• Formulas for radius, velocity, and energy of electron in hydrogen-like species.
• Hydrogen spectrum — Lyman, Balmer, Paschen series and Rydberg equation.

Level 1 — Postulates and allowed orbits

Verbal: Bohr proposed electrons revolve in certain stationary orbits without radiating energy. Energy is emitted/absorbed only when electron jumps between orbits.

Symbolic: E_n = −13.6 Z²/n² eV; r_n ∝ n²/Z; 1/λ = R_H(1/n₁² − 1/n₂²); mvr = nh/(2π).

Postulates:

1. Electrons move in circular orbits with fixed angular momentum mvr = nh/(2π), n = 1, 2, 3…
2. Orbits have fixed energy; no radiation in stationary state.
3. ΔE = hν when jumping between levels.

Hydrogen energy levels: E_n = −13.6/n² eV (ground n=1 at −13.6 eV).

Level 2 — Radii, spectra, and hydrogen-like ions

Rydberg formula (H spectrum): 1/λ = R_H (1/n₁² − 1/n₂²), n₂ > n₁

Series: Lyman (UV, to n₁=1), Balmer (visible, n₁=2), Paschen (IR, n₁=3).

Limitations: Bohr works for one-electron systems (H, He⁺, Li²⁺) but fails for multi-electron fine structure — leads to quantum mechanics.

NCERT spotlight — Hydrogen-like ions and spectra

He+ has Z = 2, so energy levels En = -13.6 times 4/n squared eV. Transitions to n = 1 give Lyman series in UV; to n = 2 give Balmer visible lines.

Rydberg constant: RH approximately 1.097 times 10^7 m^-1. Wavelength in nm equals 10^9 divided by (1/lambda in m).

Limitations: Bohr model fails for Zeeman effect, fine structure, and multi-electron atoms — motivates quantum mechanics chapter.

Calculate wavelength of light emitted when electron in H atom falls from n = 3 to n = 2 (Balmer series). R_H ≈ 1.097×10⁷ m⁻¹.

Step 1 — 1/λ = R_H (1/2² − 1/3²) = R_H (1/4 − 1/9) = R_H × 5/36.
Step 2 — 1/λ = 1.097×10⁷ × 5/36 = 1.524×10⁶ m⁻¹.
Step 3 — λ = 1/(1.524×10⁶) = 6.56×10⁻⁷ m = 656 nm (red, H-alpha line).
Step 4 — ΔE = hc/λ ≈ 3.03 eV (matches E₃ − E₂ = −13.6/9 + 13.6/4).
Step 5 — Visible Balmer line — explains hydrogen emission spectrum ✓

Applications — spectroscopy and lasers

Hydrogen emission lines fingerprint excited atoms — used in stellar spectroscopy to identify composition of distant stars. Balmer series visible lines distinguish hydrogen in discharge tubes. Laser action involves stimulated emission between discrete energy levels — Bohr picture qualitatively explains line spectra though quantum mechanics refines selection rules.

Deep dive — spectral series and hydrogen-like systems

Energy level spacing decreases as n increases — transitions to n=1 (Lyman series) release most energy per photon (UV). Balmer series (n→2) produces visible lines H-alpha 656 nm red, H-beta 486 nm blue-green, H-gamma 434 nm violet — astronomical spectra identify hydrogen in stars. Paschen and Brackett series lie in IR. For He+ (Z=2), all energies scale by Z² — ground state -54.4 eV, transitions four times more energetic than hydrogen at same n jump. Recoil and fine structure corrections Bohr ignored — explain why his model approximates but does not exact-match modern spectroscopy. de Broglie standing waves on orbit circumference n lambda = 2 pi r rationalises quantisation without ad hoc postulate in hindsight. Failure for helium atom (two electrons) — electron-electron repulsion and shielding require multi-electron quantum model. NCERT diagrams of energy level ladder should be memorised with transition arrows labelled for series names — common diagram-based NEET questions.

Review and practice drill

Review checklist: (1) E_n = -13.6 Z squared/n squared eV. (2) Rydberg 1/lambda = R(1/n1^2 - 1/n2^2). (3) Lyman UV, Balmer visible. (4) One-electron species only. Practice: Wavelength for n=4 to n=2 in H — Balmer line calculate with Rydberg.

For board exams, reproduce labelled diagrams where NCERT provides them and define every technical term in one precise sentence before using it in longer answers. Link this topic to adjacent units in your revision map so multi-chapter questions feel familiar rather than surprising on exam day.

• Write Bohr's quantization condition for angular momentum.
• Find energy of n = 4 level in hydrogen (eV).
• Which series includes the 656 nm line?

Open the Practice tab for graded questions on Bohr.