Line spectra of the hydrogen atom
An electron falling to a lower level emits a photon with hν = E(initial) − E(final). Fixed levels allow only fixed wavelengths, so hydrogen gives a line spectrum.
Line spectra: the fingerprint of an element
When a gas is heated or an electric discharge is passed through it at low pressure, it glows. Seen through a prism, its light is not a full rainbow. It is a set of bright lines at fixed wavelengths: an emission line spectrum. Each element has its own set of lines, like a fingerprint. When white light passes through the cool gas, dark lines appear at the same wavelengths: an absorption spectrum. Hydrogen, the simplest atom, has the simplest spectrum.
How Bohr's model explains the lines
- 1Heat or an electric discharge gives energy to the atoms and lifts electrons to higher energy levels (excited states).
- 2An excited atom stays in that state only for a very short time.
- 3The electron falls to a lower level and emits a photon: hν = E(initial) − E(final).
- 4The levels are fixed, so only certain energy gaps exist. So only certain wavelengths appear: a line spectrum.
- 5In absorption, an electron takes in a photon only if its energy exactly matches a gap, which gives dark lines at the same wavelengths.
Extra reading (not in your textbook): names of the spectral series
| Series | Electron falls to | Falls from | Region |
|---|---|---|---|
| Lyman | n = 1 | n = 2, 3, 4, ... | Ultraviolet |
| Balmer | n = 2 | n = 3, 4, 5, ... | Visible (its first lines) |
| Paschen | n = 3 | n = 4, 5, 6, ... | Infrared |
| Brackett | n = 4 | n = 5, 6, 7, ... | Infrared |
| Pfund | n = 5 | n = 6, 7, 8, ... | Infrared |
Worked example: wavelength of the red line
Find the wavelength of the photon emitted when the electron in a hydrogen atom falls from n = 3 to n = 2. (Eₙ = −13.6/n² eV, hc = 1240 eV nm)
- E₃ = −13.6/9 eV = −1.51 eV; E₂ = −13.6/4 eV = −3.40 eV.
- Photon energy = E₃ − E₂ = −1.51 − (−3.40) = 1.89 eV.
- Formula: λ = hc/(photon energy).
- λ = 1240 eV nm ÷ 1.89 eV ≈ 656 nm.
λ ≈ 656 nm: the red line of the hydrogen spectrum
Worked example: longest and shortest wavelengths for falls to n = 1
Electrons in hydrogen atoms fall to the ground state, n = 1. Find the longest and the shortest wavelengths emitted. (hc = 1240 eV nm)
- Longest wavelength comes from the smallest energy gap: n = 2 → n = 1.
- Energy = −3.40 − (−13.6) = 10.2 eV; λ = 1240 ÷ 10.2 ≈ 122 nm.
- Shortest wavelength comes from the largest gap: n = ∞ → n = 1.
- Energy = 0 − (−13.6) = 13.6 eV; λ = 1240 ÷ 13.6 ≈ 91.2 nm.
Longest ≈ 122 nm; shortest ≈ 91.2 nm; both in the ultraviolet
How to draw the energy level diagram with transitions
- Draw a vertical energy axis in eV. Draw horizontal lines for n = 1 (−13.6), 2 (−3.40), 3 (−1.51), 4 (−0.85), 5 (−0.54) and ∞ (0), getting closer together going up.
- Draw downward arrows ending on n = 1: these photons are in the ultraviolet.
- Draw downward arrows ending on n = 2; mark the 3 → 2 arrow red line, 656 nm. The first few of these are visible.
- Draw arrows ending on n = 3: these photons are in the infrared.
- In each group, the longest arrow gives the shortest wavelength.
- Extra reading (not in your textbook): the groups ending on n = 1, 2 and 3 are called the Lyman, Balmer and Paschen series, and atoms excited to level n can give up to n(n − 1)/2 different lines.
The red hydrogen line at 656 nm gives glowing gas clouds in space their colour in photographs. Telescopes such as the one at Hanle in Ladakh use such lines to find hydrogen in distant stars.
2 marks: “A hydrogen atom in its ground state absorbs a photon, and its electron moves to n = 3. Find the energy and the wavelength of the photon. (Eₙ = −13.6/n² eV, hc = 1240 eV nm)” Energy = E₃ − E₁ = −1.51 − (−13.6) = 12.09 eV. λ = 1240 ÷ 12.09 ≈ 103 nm, in the ultraviolet.
3 marks: “Distinguish between an emission line spectrum and an absorption line spectrum. Use Bohr's model to explain why the dark lines of hydrogen lie at the same wavelengths as its bright lines.” Emission: bright lines on a dark background, as electrons fall to lower levels. Absorption: dark lines in a continuous spectrum, as electrons take in photons to rise. Same energy gaps, so same wavelengths.
2 marks: “Find the shortest wavelength a hydrogen atom can emit when its electron falls to the ground state. (hc = 1240 eV nm)” Largest gap: n = ∞ to n = 1, energy 13.6 eV. λ = 1240 ÷ 13.6 ≈ 91.2 nm (ultraviolet).
Fixed levels, fixed gaps, fixed lines: a bigger energy gap gives a shorter wavelength (λ = hc/ΔE).