Energy levels and spectra

Electrons in atoms can only have certain energies, called energy levels. When an electron drops to a lower level it emits a photon whose energy exactly matches the gap; when it absorbs a photon of exactly the right energy it jumps up. Each element has its own pattern of lines – a fingerprint that lets astronomers find out what distant stars are made of.

Key facts

  • Photon energy = difference between levels: hf = E_upper − E_lower.
  • Emission spectrum: bright lines on a dark background (hot gas). Absorption spectrum: dark lines on a continuous spectrum (cool gas in front of a hot source).
  • A hot dense solid gives a continuous spectrum.
  • Energy levels are negative; 0 eV means the electron has just escaped (ionisation).
  • Hydrogen: E_n = −13.6 ÷ n² eV. From level n there are n(n − 1)/2 possible lines.
  • Use λ = hc ÷ ΔE (ΔE in joules) to find the wavelength of a line.

Worked examples

Example 1

An electron drops from n = 3 (−1.51 eV) to n = 2 (−3.4 eV) in hydrogen. Find the photon energy and wavelength.

  1. ΔE −1.51 − (−3.4) = 1.89 eV = 3.02 × 10⁻¹⁹ J
  2. λ = hc ÷ ΔE λ = 6.63 × 10⁻³⁴ × 3.0 × 10⁸ ÷ 3.02 × 10⁻¹⁹ = 6.58 × 10⁻⁷ m
  3. About 658 nm: the red hydrogen line.

Answer: 1.89 eV; 658 nm (red)

Example 2

How much energy is needed to ionise hydrogen from its ground state, and what is the longest wavelength of light that can do it?

  1. Energy 0 − (−13.6) = 13.6 eV = 2.18 × 10⁻¹⁸ J
  2. λ = hc ÷ E λ = 1.989 × 10⁻²⁵ ÷ 2.18 × 10⁻¹⁸ = 9.14 × 10⁻⁸ m = 91.4 nm

Answer: 13.6 eV; 91.4 nm

Example 3

Gas atoms are excited up to level 4. How many different lines can they emit?

  1. From level 4 the electron can drop to 3, 2 or 1: three lines.
  2. From level 3 it can drop to 2 or 1: two more lines.
  3. From level 2 it can only drop to 1: one more line.
  4. 3 + 2 + 1 = 6 lines.

Answer: 6