Energy Levels in Atoms and Line Spectra

Section: Quantum Physics | Syllabus: Cambridge A Level Physics (9702)

Energy Levels in Atoms and Line Spectra

Discrete Energy Levels

Electrons in an atom can only occupy specific, defined energy states (energy levels). The energy is quantised.

Energies are typically negative, with the ground state (n=1) being the most negative. Zero energy represents a free electron (ionised atom).

Line Spectra

Emission Line Spectrum: A series of bright coloured lines on a dark background. Produced when hot gas atoms de-excite, emitting photons of specific frequencies.

Absorption Line Spectrum: Dark lines on a continuous spectrum background. Produced when white light passes through a cool gas; photons of specific frequencies are absorbed to excite electrons.

Energy Transitions

When an electron moves between energy levels, a photon is emitted or absorbed.

hf = E_1 - E_2 = Δ E
(hc/λ) = Δ E

De-excitation (Emission): Electron falls from higher level (E_1) to lower level (E_2). Photon emitted with energy hf = E_1 - E_2.
Excitation (Absorption): Electron absorbs a photon and moves from lower to higher level. Photon energy must exactly match the energy difference.

Worked Examples

Worked Example: Hydrogen Transition

Question: An electron in a hydrogen atom falls from E_3 = -1.51 eV to E_2 = -3.40 eV. Calculate the wavelength of the emitted photon.

Solution

1. Find energy difference in eV: Δ E = -1.51 - (-3.40) = 1.89 eV

2. Convert to Joules: 1.89 × 1.60 × 10^-19 = 3.024 × 10^-19 J

3. Use λ = hc/Δ E:

λ = (6.63 × 10^-34)(3.00 × 10^8)3.024 × 10^-19 = 6.58 × 10^-7 m

Answer: 658 nm (Red light)

Worked Example: Hydrogen Energy Level Transitions

Question: Refer to the electron energy levels of the hydrogen atom (Ground: -13.6 eV, n=2: -3.40 eV, n=3: -1.51 eV, n=4: -0.85 eV).
(a) Calculate the energy lost by an electron moving between n = 4 and n = 3.
(b) An isolated hydrogen atom is given sufficient energy for the electron to reach the n = 4 level. Show that there are six possible ways in which the electron can lose energy.

Solution

(a) Energy lost = E_4 - E_3 = -0.85 - (-1.51) = 0.66 eV

(b) From n = 4, the electron can transition through any combination of levels down to n = 1:

Direct: 4→1
Two-step: 4→2→1, 4→3→1
Three-step: 4→3→2→1

Counting individual transitions: 4→3, 4→2, 4→1, 3→2, 3→1, 2→1 = 6 possible transitions

Worked Example: Absorption Spectrum

Question: When a continuous spectrum of white light is passed through helium gas, 12 black lines are seen.
(a) Explain what causes these black lines.
(b) One line is at 686.7 nm. Calculate the energy of the transition in eV.

Solution

(a) Electrons in helium atoms absorb photons with energies that exactly match the energy difference between discrete energy levels. These specific wavelengths are removed from the continuous spectrum, leaving dark lines at those positions. The 12 lines correspond to 12 possible electron transitions in helium.

(b) Using E = (hc/λ):

E = (6.63 × 10^-34)(3.00 × 10^8)686.7 × 10^-9 = 2.90 × 10^-19 J

Convert to eV: E = 2.90 × 10^-191.60 × 10^-19 = 1.81 eV

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Energy Levels in Atoms and Line Spectra