Hydrogen Spectrum: Lyman, Balmer and Paschen Series

When hydrogen gas is excited and then emits light, it does not give a continuous rainbow. It gives a set of sharp coloured lines. These lines fall into series named after their discoverers: Lyman, Balmer, Paschen, Brackett and Pfund.

The hydrogen spectrum matters because it was the first great test of atomic theory. Bohr's model and later quantum mechanics explained exactly where these lines occur, linking them to electron jumps between energy levels.

Key Concepts

Energy levels: Electrons in hydrogen occupy levels n = 1, 2, 3, … with energies Eₙ = −13.6 / n² eV. Level n = 1 is the ground state.

Emission: When an electron falls from a higher level to a lower one, a photon is emitted with energy equal to the level difference.

Series: All transitions ending on the same final level form a series. Balmer (ending on n = 2) lies mostly in visible light and was observed first.

Formulas

Photon energy: h ν = E_initial − E_final = 13.6 (1/n_f² − 1/n_i²) eV, with n_i > n_f. Wavelength follows from λ = h c / (h ν). Assumes hydrogen (single electron); other atoms have more complex levels.

Series limits: Lyman ends at n_f = 1 (ultraviolet, 91–122 nm), Balmer at n_f = 2 (visible, 365–656 nm), Paschen at n_f = 3 (infrared, 820–1875 nm), Brackett at n_f = 4 and Pfund at n_f = 5 (deeper infrared).

Worked Example

Find the wavelength of the red Balmer line (H-alpha) for transition n = 3 → n = 2.

Step 1: ΔE = 13.6 × (1/2² − 1/3²) = 13.6 × (1/4 − 1/9) = 13.6 × 5/36 = 1.89 eV.

Step 2: λ = h c / ΔE ≈ 1240 eV·nm / 1.89 eV = 656 nm.

This 656 nm red line is the famous H-alpha line seen in nebulae and solar prominences. The next Balmer line (4 → 2) gives 486 nm (blue-green).

Common Misconceptions

• Misconception: Hydrogen emits white continuous light like a bulb. Correct: Low-pressure excited hydrogen emits only discrete line colours.
• Misconception: All series are visible. Correct: Only Balmer is mainly visible; Lyman is ultraviolet and the rest are infrared.
• Misconception: Lines get evenly spaced. Correct: Lines crowd together towards the series limit as levels get closer.

Real-World Applications

Astronomers identify hydrogen everywhere in the universe by these lines and measure cosmic redshifts from their shifts. H-alpha filters reveal solar activity and star-forming nebulae. In labs, hydrogen lamps calibrate spectrometers because their wavelengths are precisely known.

Recap: hydrogen lines come from electron falls to a fixed final level. Lyman is UV, Balmer is visible, and Paschen, Brackett and Pfund reach further into the infrared.