Hydrogen Spectrum & Bohr Model: JEE Main Guide

The hydrogen spectrum and Bohr model sit at the intersection of Chemistry (Atomic Structure) and Physics (Modern Physics) and are tested from both angles in JEE Main. The same calculation — finding the wavelength of emitted light when an electron transitions between energy levels — can appear in either section. Mastering this unified topic therefore gives you double value: chemistry marks from Atomic Structure and physics marks from Atoms and Nuclei. This guide covers everything JEE Main asks about hydrogen's spectral behaviour.

Bohr's Postulates and the Energy Level Formula

Bohr's three postulates: (1) electrons orbit the nucleus in discrete, non-radiating circular orbits; (2) the angular momentum of each orbit is an integer multiple of ħ: mvr = nħ; (3) radiation is emitted or absorbed only when an electron transitions between orbits, with frequency given by hν = E_i − E_f. From these postulates, the radius of the nth orbit: r_n = 0.529n²/Z Å (n is the principal quantum number, Z is the atomic number). Energy of the nth level: E_n = −13.6Z²/n² eV. For hydrogen (Z=1): E₁ = −13.6 eV, E₂ = −3.4 eV, E₃ = −1.51 eV, E₄ = −0.85 eV.

Ionisation energy = |E₁| = 13.6Z²/n² eV from the current shell. For hydrogen from ground state: IE = 13.6 eV. The energy needed to excite hydrogen from n=1 to n=2: ΔE = −3.4 − (−13.6) = 10.2 eV. This 10.2 eV fact is tested multiple times per year in both Chemistry and Physics sections. Take a free atomic structure mock to test your energy level recall speed. For the complete atomic structure foundation, see our atomic structure guide.

Spectral Series and the Rydberg Formula

The Rydberg formula gives the wavenumber (1/λ) of spectral lines: 1/λ = R_H Z²(1/n₁² − 1/n₂²), where R_H = 1.097 × 10⁷ m⁻¹ is the Rydberg constant and n₂ > n₁. The spectral series, defined by their lower orbit n₁: Lyman (n₁=1, UV, transitions to ground state), Balmer (n₁=2, visible, historically first observed), Paschen (n₁=3, infrared), Brackett (n₁=4, far IR), Pfund (n₁=5, far IR). The Balmer series is the most important for JEE — its first line (n=3→2, λ = 656 nm, red) and second line (n=4→2, λ = 486 nm, blue-green) appear in problem numerical values.

Number of spectral lines when electron falls from n=N to n=1: N(N−1)/2. Lines in Lyman series from n=N: (N−1) lines. JEE uses these counting formulae regularly. Also: the highest-energy (shortest wavelength) line in any series is the series limit — when n₂ = ∞; for Lyman series limit, 1/λ = R_H and λ = 91.2 nm. For the full quantum mechanical atomic model, see our atomic orbitals guide.

Velocity, Time Period, and Frequency of Orbital Electrons

Speed of electron in nth orbit: v_n = 2.18 × 10⁶ × Z/n m/s. For n=1, Z=1: v₁ ≈ 2.18 × 10⁶ m/s ≈ c/137 (fine structure constant). Time period: T_n = 2πr_n/v_n ∝ n³/Z². Frequency: f_n = v_n/(2πr_n) ∝ Z²/n³. These are tested in comparative questions: "If the time period in orbit n=2 is T, find the time period in orbit n=4" — answer: T × 8 (since T ∝ n³).

De Broglie wavelength of orbital electron: λ = h/(mv_n) = 2πr_n/n (the circumference divided by n). This elegant result shows that exactly n wavelengths fit in the nth orbit — the standing wave interpretation of Bohr quantisation. JEE tests this as: "Show that the Bohr orbits are consistent with de Broglie's hypothesis" — a standard derivation question. For the quantum mechanical connection, see our dual nature of radiation guide and our matter waves guide.

Multi-Electron Atoms and Bohr's Limitations

Bohr's model works exactly for hydrogen and hydrogen-like ions (He⁺, Li²⁺, Be³⁺) because they have only one electron. For multi-electron atoms, electron-electron repulsions destroy the simple energy-level formula. Bohr's limitations (tested as JEE conceptual questions): cannot explain fine structure (spin-orbit coupling), cannot explain Zeeman effect (splitting in magnetic field), cannot explain the intensities of spectral lines, and contradicts Heisenberg's uncertainty principle (well-defined electron position and momentum simultaneously). These limitations led to the quantum mechanical model. JEE sometimes asks you to identify which limitation Bohr's model cannot explain — memorise all four.