All subjects · Physics · WPH12 (Unit 2)

Energy levels and line spectra

Atoms emit and absorb light only at particular wavelengths. These line spectra show that electrons in atoms have discrete energy levels.

Discrete energy levels

Electrons in an atom can only have certain energies, called energy levels. They are usually given as negative values, with 0 eV meaning the electron has just escaped (ionisation). The lowest level is the ground state; higher levels are excited states.

Emission and absorption

An electron dropping from a higher level E1 to a lower level E2 emits a photon with energy hf = E1 - E2.
An electron can move up by absorbing a photon whose energy exactly matches the gap between two levels (or by gaining energy in a collision).

The frequency of the radiation is f = (E1 - E2) / h.

Line spectra

Emission spectrum: a hot, low-pressure gas gives bright coloured lines on a dark background.
Absorption spectrum: white light through a cool gas gives a continuous spectrum crossed by dark lines.

The lines are at the same wavelengths for both, and each element has its own set, so spectra identify elements (for example in stars). Because the levels are discrete, only certain photon energies are possible, so the spectrum is lines rather than continuous.

Worked example

An electron in hydrogen falls from -3.40 eV to -13.6 eV. Find the wavelength of the photon emitted.

ΔE = -3.40 - (-13.6) = 10.2 eV = 10.2 x 1.60 x 10^-19 = 1.63 x 10^-18 J
λ = hc / ΔE = 1.989 x 10^-25 / 1.63 x 10^-18

Answer: λ = 1.22 x 10^-7 m (122 nm, ultraviolet)

Key idea: Photon energy = difference between levels: hf = E1 - E2. Discrete levels give line spectra.

Check yourself

An electron falls from the -1.51 eV level to the -3.40 eV level. Find the energy of the photon emitted, in eV.
Show answer
1.89 eV. ΔE = -1.51 - (-3.40) = 1.89 eV.
Find the wavelength of a 1.89 eV photon in nm, to 3 s.f.
Show answer
658 nm. E = 1.89 x 1.60 x 10^-19 = 3.02 x 10^-19 J; λ = hc / E = 6.58 x 10^-7 m = 658 nm.
An atom has four energy levels. An electron is in the highest level. How many different photon energies could be emitted as it returns to the ground state by any route?
Show answer
6. Each pair of levels gives one transition: 4 x 3 / 2 = 6.
Why do atoms produce line spectra rather than continuous spectra?
- Atoms emit photons of every energy
- Light is a transverse wave
- Electron energy levels are discrete, so only certain photon energies are possible
- Electrons have a de Broglie wavelength
Show answer
Electron energy levels are discrete, so only certain photon energies are possible. Each line corresponds to a particular gap between two levels.
What does an absorption spectrum look like?
- Dark lines on a continuous spectrum
- Bright lines on a dark background
- A continuous spectrum with no lines
- A single bright line
Show answer
Dark lines on a continuous spectrum. The cool gas absorbs photons of particular energies from the white light.
The ground state of hydrogen is -13.6 eV. What does this mean?
- The electron has 13.6 eV of kinetic energy
- Hydrogen emits only 13.6 eV photons
- 13.6 eV is needed to ionise a hydrogen atom from its ground state
- The atom has a negative mass
Show answer
13.6 eV is needed to ionise a hydrogen atom from its ground state. Raising the electron to 0 eV frees it, which takes 13.6 eV.
An electron is at -3.40 eV. The levels above it are -1.51 eV and -0.85 eV. A photon of 2.00 eV arrives. What happens?
- It is not absorbed, because no gap is exactly 2.00 eV
- It is absorbed and the electron moves to -1.51 eV
- It is absorbed and the electron moves to -1.40 eV
- The atom is ionised
Show answer
It is not absorbed, because no gap is exactly 2.00 eV. The gaps are 1.89 eV and 2.55 eV; a photon is only absorbed if its energy exactly matches a gap.

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Last updated 7 November 2026. Spotted a mistake? Tell us.