The Hydrogen Emission Spectrum: Lyman, Balmer and Paschen Series

Pass an electric discharge through hydrogen gas and it glows a soft pinkish-purple. Look at that glow through a prism or a diffraction grating and something remarkable appears: not a continuous rainbow, but a handful of sharp, bright lines of specific colours on a dark background. This is hydrogen’s emission spectrum, and explaining it was one of the great puzzles that led to quantum theory. It’s still the clearest example of how atoms reveal their structure through light.

What an emission spectrum is

When atoms are given energy, by heating or by an electric discharge, their electrons are excited to higher energy levels. They quickly fall back to lower levels, releasing the energy difference as photons of light.

Because the energy levels in an atom are fixed (quantised), only certain energy differences are possible, so only certain photon energies, and therefore certain wavelengths, are emitted. Each element produces its own unique pattern of lines, a kind of atomic barcode. See photons and energy levels.

The relationship between photon energy and wavelength is:

E = hf = hc ÷ λ

Bigger energy drops produce higher-frequency, shorter-wavelength light.

Hydrogen’s energy levels

Hydrogen has just one electron, so its energy levels are especially simple:

Eₙ = −13.6 eV ÷ n² (n = 1, 2, 3…)

Level	Energy (eV)
n = 1	−13.6
n = 2	−3.40
n = 3	−1.51
n = 4	−0.85
n = 5	−0.54
n = ∞	0

Two key features:

• The gap from n = 1 to n = 2 is huge (10.2 eV).
• The levels get closer and closer together as n increases, crowding towards n = ∞.

See the Bohr model and Bohr model calculations.

The series of lines

Lines are grouped into series according to the level the electron falls to.

Series	Electron falls to	Region of spectrum	Discovered
Lyman	n = 1	ultraviolet	1906–1914, Theodore Lyman
Balmer	n = 2	visible (and near UV)	1885, Johann Balmer
Paschen	n = 3	infrared	1908, Friedrich Paschen
Brackett	n = 4	infrared	1922
Pfund	n = 5	far infrared	1924

The Lyman series (to n = 1)

Transitions ending on the ground state involve the biggest energy drops, so they emit ultraviolet light. The first Lyman line (n = 2 → 1) is at 122 nm. This series is invisible to the eye.

The Balmer series (to n = 2)

These are the lines you can actually see:

Transition	Wavelength (nm)	Colour	Name
3 → 2	656	red	H-alpha
4 → 2	486	blue-green	H-beta
5 → 2	434	violet	H-gamma
6 → 2	410	violet	H-delta

In 1885, the Swiss schoolteacher Johann Balmer found a simple formula that fitted these wavelengths exactly, although he had no idea why it worked. Nearly thirty years later, Bohr’s model explained it. The general formula, the Rydberg equation, covers all the series.

The Paschen series (to n = 3)

Smaller energy drops give infrared lines, beginning at 1,875 nm (n = 4 → 3).

Why the lines converge

Within each series, the lines get closer together as the wavelength decreases, eventually merging at a convergence limit. This happens because the upper energy levels themselves crowd together as n increases. Transitions from n = 10, 11, 12… down to n = 2 differ in energy by tiny amounts.

The convergence limit corresponds to an electron falling from n = ∞ (a free electron) to the lower level of that series.

Ionisation energy from the Lyman limit

The convergence limit of the Lyman series corresponds to the transition n = ∞ → n = 1. Its energy is exactly the energy needed to remove an electron from a ground-state hydrogen atom: the ionisation energy.

• Lyman convergence limit ≈ 91.2 nm
• E = hc ÷ λ = (6.626 × 10⁻³⁴ × 2.998 × 10⁸) ÷ (91.2 × 10⁻⁹) = 2.18 × 10⁻¹⁸ J per atom
• × 6.022 × 10²³ = 1,312 kJ/mol

This is a classic way to determine ionisation energies from spectra. See ionization energy trend.

Reading an energy level diagram

Energy level diagrams show horizontal lines for each level, with n = 1 at the bottom and n = ∞ at the top (energy 0). Downward arrows represent emission:

• Longer arrows = bigger energy drops = higher frequency = shorter wavelength.
• All arrows ending on n = 1 form the Lyman series; on n = 2, the Balmer series; on n = 3, the Paschen series.

A common exam question: “Which transition produces the line of longest wavelength in the Balmer series?” The smallest drop ending at n = 2: n = 3 → 2 (656 nm).

How many lines?

From a given upper level n, an electron can fall in several steps or one jump. The total number of different lines possible from level n down to level 1 is n(n − 1) ÷ 2. From n = 4: 4 × 3 ÷ 2 = 6 lines (4→3, 4→2, 4→1, 3→2, 3→1, 2→1).

Hydrogen’s spectrum in astronomy

Hydrogen is the most abundant element in the universe, and its lines are everywhere in astronomy:

• The red H-alpha line gives many nebulae their pink-red glow in photographs.
• Dark absorption lines at the Balmer wavelengths appear in the spectra of stars, where cooler hydrogen gas absorbs light from hotter layers beneath. See emission vs absorption spectra.
• Redshift: in distant galaxies, hydrogen’s lines are shifted to longer wavelengths because the galaxies are moving away from us. Measuring this shift revealed the expansion of the universe.
• The 21 cm radio line of hydrogen, from a tiny flip in the electron’s spin, lets radio astronomers map hydrogen gas throughout our galaxy.

Why the spectrum was so important

Hydrogen’s line spectrum provided three key pieces of evidence:

1. Energy levels are quantised. Continuous energies would give a continuous spectrum.
2. Energy is emitted in packets (photons) whose energy matches the gap between levels.
3. The pattern of levels follows 1/n², which Bohr explained and quantum mechanics later derived from first principles.

The same principles explain flame test colours, neon lights, sodium street lamps and fireworks, and they’re the basis of analytical techniques such as atomic absorption spectroscopy.

Seeing it yourself

Many schools have hydrogen discharge tubes and handheld spectroscopes or diffraction-grating glasses. Looking at a hydrogen tube through a grating, you can usually see the red H-alpha line and the blue-green H-beta line clearly, and the violet H-gamma line faintly. Comparing this with a filament bulb, which gives a continuous rainbow, is a memorable way to see the difference between line and continuous spectra. Handle discharge tubes only as instructed, because they run at high voltage.

Key takeaways

• Hydrogen’s emission spectrum consists of sharp lines because its energy levels are quantised.
• Lines are grouped by the final level: Lyman (to n = 1, UV), Balmer (to n = 2, visible), Paschen (to n = 3, IR).
• Visible Balmer lines are at 656, 486, 434 and 410 nm.
• Lines converge because the upper levels crowd together; the Lyman convergence limit gives the ionisation energy (1,312 kJ/mol).
• Hydrogen’s spectrum is fundamental to quantum theory and to astronomy.