Energy Levels and Atomic Spectra: Hydrogen Spectrum Explained

Why Atoms Give Out Light of Particular Colours

Sodium street lamps glow orange-yellow, neon signs glow red, and fireworks owe their colours to different metals. When light from a heated gas is passed through a prism or a grating, it shows not a continuous rainbow but a set of sharp, bright lines at particular wavelengths, a unique "fingerprint" for each element. Explaining these lines led to the modern picture of the atom. This tutorial covers energy levels and atomic spectra as taught in PHY 102 and met in WAEC physics: the Rutherford and Bohr models, the energy levels of hydrogen, transitions and photon energies, emission and absorption spectra, the spectral series, ionisation, and applications. It builds on the photoelectric effect.

From Rutherford to Bohr

Rutherford's gold-foil experiment (1911) showed that an atom has a tiny, dense, positive nucleus, with electrons around it. But classical physics predicted that orbiting electrons should radiate energy continuously and spiral into the nucleus within a fraction of a second, giving a continuous spectrum. Neither happens.

In 1913, Niels Bohr proposed a new model for hydrogen:

Electrons move only in certain allowed orbits, or energy levels, without radiating energy.
Each level has a definite energy. The lowest is the ground state, and higher levels are excited states.
An electron can jump between levels. It emits a photon when it drops to a lower level and absorbs one when it rises to a higher level. The photon's energy equals the energy difference exactly.

Energy Levels of Hydrogen

The energy of the electron in level n (n = 1, 2, 3, …) of a hydrogen atom is:

Eₙ = −13.6 ÷ n² eV

E₁ = −13.6 eV (the ground state)
E₂ = −3.40 eV
E₃ = −1.51 eV
E₄ = −0.85 eV
E∞ = 0 eV (the electron is just free)

The energies are negative because the electron is bound: energy must be supplied to free it. The levels crowd closer together as n increases.

Transitions and Photon Energies

When an electron drops from a higher level to a lower one, it emits a photon with:

hf = E(higher) − E(lower), and λ = hc ÷ ΔE

A handy shortcut: with ΔE in eV, λ (in nm) ≈ 1240 ÷ ΔE, because hc ≈ 1240 eV nm.

Worked example 1

Find the wavelength of the photon emitted when a hydrogen electron drops from n = 3 to n = 2.

ΔE = −1.51 − (−3.40) = 1.89 eV, so λ ≈ 1240 ÷ 1.89 ≈ 656 nm.

This is the bright red line in the hydrogen spectrum (called H-alpha), which astronomers see in glowing gas clouds.

Worked example 2

Find the photon energy and wavelength for the transition n = 2 to n = 1.

ΔE = −3.40 − (−13.6) = 10.2 eV, so λ ≈ 1240 ÷ 10.2 ≈ 122 nm, which is ultraviolet, invisible to the eye.

Spectral Series of Hydrogen

Lines are grouped by the level the electron falls to:

Lyman series: transitions down to n = 1. These are high-energy photons in the ultraviolet.
Balmer series: transitions down to n = 2. Several lines are in the visible range: red (656 nm), blue-green (486 nm), violet (434 nm) and so on.
Paschen series: transitions down to n = 3. These are in the infrared.

Electrons in level n (in a sample of many atoms) can drop to the ground state by many routes. The number of different lines they can produce is n(n − 1)/2. From n = 4, that is 6 possible lines.

Emission and Absorption Spectra

Continuous spectrum: all colours with no gaps, emitted by hot solids and dense gases, such as a filament lamp.
Emission (line) spectrum: bright coloured lines on a dark background, emitted by a hot, low-pressure gas as its excited electrons fall to lower levels.
Absorption spectrum: dark lines on a continuous background, seen when white light passes through a cooler gas. The gas absorbs exactly the photon energies that lift its electrons to higher levels, so its dark lines appear at the same wavelengths as its emission lines.

The dark lines in sunlight (Fraunhofer lines) revealed the elements in the Sun's atmosphere. Helium was discovered this way, in the Sun, before it was found on Earth.

Absorbing only exact energies

An atom absorbs a photon only if the photon's energy exactly matches the gap between two levels. In hydrogen's ground state, the smallest jump needs 10.2 eV, so a 5 eV photon passes straight through without being absorbed. A 12.09 eV photon raises the electron exactly to n = 3, because −13.6 + 12.09 = −1.51 eV.

Electrons hitting atoms can also excite them. Unlike photons, they can give up just part of their energy.

Ionisation

Ionisation energy is the energy needed to remove an electron completely, raising it to E∞ = 0. From the ground state of hydrogen, it is 13.6 eV. From n = 2, it is only 3.40 eV. Any photon with more than the ionisation energy can free the electron, and the extra energy becomes the electron's kinetic energy.

Applications

Spectroscopy: identifying elements in chemical samples, in polluted air or water, and in stars and galaxies, from their line spectra.
Street and advertising lamps: sodium (yellow), neon (red) and mercury (blue-white) lamps each glow at their own characteristic lines.
Fluorescent lamps and LEDs: excited atoms emit light. In fluorescent tubes, UV from mercury is converted into visible light by a coating.
Lasers: based on stimulated emission, where an incoming photon triggers an excited atom to emit an identical photon, producing intense, coherent light.
Flame tests in chemistry: metal ions give characteristic colours, such as sodium yellow, potassium lilac and copper blue-green.

Common Mistakes with Energy Levels

Forgetting that the energy levels are negative, or subtracting in the wrong order. ΔE is always the higher level minus the lower level.
Thinking an atom absorbs any photon with enough energy. For transitions between bound levels, it must match exactly (ionisation is the exception).
Using the wrong units in λ = hc/ΔE. Either use joules with hc = 1.99 × 10⁻²⁵ J m, or use eV with 1240 eV nm.
Mixing up emission and absorption spectra. Emission gives bright lines on dark, and absorption gives dark lines on a bright continuous spectrum.
Assigning the Balmer series to the ultraviolet. It is partly visible. Lyman is in the ultraviolet.

Energy Levels: Practice Questions

Find the energy of the n = 4 level of hydrogen.
Find the wavelength of the photon emitted in the transition from n = 4 to n = 2.
What is the ionisation energy of hydrogen from its ground state?
How many different spectral lines can be produced by electrons falling from n = 3?
Can a 5 eV photon be absorbed by a hydrogen atom in its ground state? Explain.
State one difference between an emission spectrum and an absorption spectrum.

Answers: 1. −13.6 ÷ 16 = −0.85 eV. 2. ΔE = 3.40 − 0.85 = 2.55 eV, so λ ≈ 1240 ÷ 2.55 ≈ 486 nm (blue-green). 3. 13.6 eV. 4. 3 (3→2, 3→1 and 2→1). 5. No. The smallest possible jump from the ground state needs 10.2 eV, and 5 eV matches no energy gap. 6. An emission spectrum shows bright lines on a dark background. An absorption spectrum shows dark lines on a continuous spectrum.

Conclusion

Atoms have discrete energy levels. For hydrogen, Eₙ = −13.6/n² eV. Electrons emit or absorb photons only when they jump between levels, with hf = ΔE and λ ≈ 1240/ΔE nm (ΔE in eV). This produces each element's unique line spectrum: the Lyman (UV), Balmer (visible) and Paschen (infrared) series for hydrogen, bright emission lines from hot gases and matching dark absorption lines from cool gases. Energy levels explain spectroscopy, coloured lamps, flame tests and lasers, and they opened the door to quantum mechanics.