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A2 · Practice questions · Energy levels and line spectra

Every line is a jump between two levels.

Six original Cambridge-style questions on discrete energy levels, emission and absorption of photons with hf = E₁ − E₂, converting level gaps to wavelengths, and why atoms give line spectra.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.
Keep these straight

The gap between two levels is one photon.

01
Analysis
[2 marks]

The energy levels of an atom are quoted as negative values, with the ground state the most negative. Explain why the values are negative, and state what is meant by the ground state.

  • The zero of energy is taken as an electron just free of the atom (at rest, infinitely far away); a bound electron must be given energy to reach this, so its energy is below zero, that is negative. ✓
  • The ground state is the lowest energy level available to the electron, the state it normally occupies and from which most energy must be supplied to remove it. ✓
02
Calculation
[3 marks]

Two energy levels of an atom are at −1.6 eV and −5.5 eV. An electron falls from the higher level to the lower one. Calculate the energy of the emitted photon in joules and its wavelength, and state the region of the electromagnetic spectrum in which the line appears.

  • ΔE = −1.6 − (−5.5) = 3.9 eV = 3.9 × 1.60 × 10⁻¹⁹ = 6.24 × 10⁻¹⁹ J. ✓
  • λ = hc / ΔE = (6.63 × 10⁻³⁴ × 3.00 × 10⁸) / 6.24 × 10⁻¹⁹ = 3.2 × 10⁻⁷ m (320 nm). ✓
  • This is shorter than visible light (about 400 nm to 700 nm), so the line is in the ultraviolet. ✓
03
Analysis
[3 marks]

The light from a low-pressure gas discharge lamp is passed through a diffraction grating and gives a set of sharp, separate coloured lines. Explain how this line spectrum arises, and explain why the same gas, when cool and placed in front of a white-light source, produces dark lines at exactly the same wavelengths.

  • Electrons in the excited atoms drop between discrete energy levels, and each drop emits a photon of energy equal to the gap, hf = E₁ − E₂. ✓
  • Only certain gaps exist, so only certain photon energies, and hence only certain wavelengths, are emitted: each allowed transition gives one line. ✓
  • A cool atom can absorb a photon only if its energy exactly matches a gap, lifting the electron to a higher level; those wavelengths are removed from the white light, and since the gaps are the same the dark lines coincide with the bright emission lines. ✓
04
Calculation
[3 marks]

A green spectral line of wavelength 546 nm is emitted when an electron in an atom falls to a level of energy −5.77 eV. Calculate the energy of the photon in electronvolts and hence the energy of the level from which the electron fell.

  • E = hc / λ = (6.63 × 10⁻³⁴ × 3.00 × 10⁸) / (546 × 10⁻⁹) = 3.64 × 10⁻¹⁹ J. ✓
  • E = 3.64 × 10⁻¹⁹ / 1.60 × 10⁻¹⁹ = 2.28 eV. ✓
  • The upper level is higher by the photon energy: E₁ = −5.77 + 2.28 = −3.49 eV. ✓
05
Calculation
[3 marks]

An atom has four energy levels: −10.4 eV, −5.5 eV, −3.7 eV and −1.6 eV. Electrons in the highest level are able to make every possible downward transition. State the number of different spectral lines that can be produced, identify the transition that gives the longest wavelength, and calculate that wavelength.

  • Each pair of levels gives one line: 3 + 2 + 1 = 6 lines. ✓
  • The longest wavelength comes from the smallest gap, −3.7 eV to −5.5 eV, which is 1.8 eV = 2.88 × 10⁻¹⁹ J. ✓
  • λ = hc / ΔE = (6.63 × 10⁻³⁴ × 3.00 × 10⁸) / 2.88 × 10⁻¹⁹ = 6.9 × 10⁻⁷ m (690 nm, red). ✓
06
Calculation
[4 marks]

In the hydrogen atom the n = 2 level has energy −3.40 eV, the n = 3 level −1.51 eV and the n = 4 level −0.85 eV. An electron in the n = 2 level absorbs a photon of energy 2.55 eV.

(a) State the level to which the electron is raised. (b) Calculate the wavelength of the absorbed photon. (c) Explain why a photon of energy 3.0 eV would not be absorbed by this electron.

  • (a) −3.40 + 2.55 = −0.85 eV, so the electron is raised to the n = 4 level. ✓
  • (b) E = 2.55 × 1.60 × 10⁻¹⁹ = 4.08 × 10⁻¹⁹ J. ✓
  • λ = hc / E = (6.63 × 10⁻³⁴ × 3.00 × 10⁸) / 4.08 × 10⁻¹⁹ = 4.9 × 10⁻⁷ m (490 nm, blue-green). ✓
  • (c) −3.40 + 3.0 = −0.40 eV, and there is no energy level at −0.40 eV; a photon can be absorbed only if its energy exactly matches the gap to an allowed level. ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.