Translation between representations (TBR)

Hydrogen energy levels and spectra

Unit 15
12 points
About 28 minutes

You can use a calculator on this question, just like on exam day.

A multipart problem about one situation shown in several ways. You draw a diagram, derive equations, sketch or draw graphs, and then explain whether your answers agree with each other or use them to predict what happens when the situation changes. On the exam: Question 2 of 4. New for May 2027: the free-response section is 95 minutes, down from 100, for all 4 questions (50% of your score), one of each type in this order. Calculator and equation sheet allowed. The CED suggests 25–30 minutes. This question type started in May 2025, so Physics 2 free-response questions from 2024 and earlier are built differently.

The question

The energy levels of a hydrogen atom are E₁ = −13.6 eV, E₂ = −3.40 eV, E₃ = −1.51 eV and E₄ = −0.85 eV, and an electron with energy 0 eV is no longer bound to the atom. A tube of hydrogen gas is excited by an electric discharge, so some atoms are raised as high as the n = 4 level. Use h = 4.14 × 10⁻¹⁵ eV·s and c = 3.00 × 10⁸ m/s.

Suggested time: 28 minutes

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Part (a)

3 points

Describe an energy-level diagram for hydrogen up to n = 4, showing every downward transition that can produce a photon from an atom that starts in the n = 4 level. Identify the transition that produces the photon with the shortest wavelength and the one that produces the photon with the longest wavelength, and justify your choices.

Your answer to part (a)

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Part (b)

3 points

Derive an expression for the wavelength λ of the photon emitted when an atom drops from a level with energy $E_{\text{upper}}$ to a level with energy $E_{\text{lower}}$ . Then calculate the wavelengths of the photons from the 3→2 and 4→2 transitions.

Your answer to part (b)

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Part (c)

3 points

(i) Light from the tube passes through a spectrometer. Describe the graph of intensity versus wavelength you would observe for visible wavelengths (400 nm to 700 nm), and explain why no other lines appear in that range. (ii) Describe how the result changes if, instead, white light passes through a container of cool hydrogen gas, in which all the atoms are in the ground state, before entering the spectrometer.

Your answer to part (c)

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Part (d)

3 points

(i) A photon with energy 11.0 eV strikes a hydrogen atom in the ground state. Is the photon absorbed? (ii) A photon with energy 15.0 eV strikes a hydrogen atom in the ground state. Describe the result, and calculate any kinetic energy given to the electron. (iii) Explain how both results are consistent with your energy-level diagram.

Your answer to part (d)

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