| Date | May 2013 | Marks available | 6 | Reference code | 13M.3.SL.TZ1.5 |
| Level | Standard level | Paper | Paper 3 | Time zone | Time zone 1 |
| Command term | Explain, Predict, Sketch, and Suggest | Question number | 5 | Adapted from | N/A |
Question
This question is about atomic energy levels.
Explain how atomic spectra provide evidence for the quantization of energy in atoms.
Outline how the de Broglie hypothesis explains the existence of a discrete set of wavefunctions for electrons confined in a box of length L.
The diagram below shows the shape of two allowed wavefunctions ѱA and ѱB for an electron confined in a one-dimensional box of length L.
(i) With reference to the de Broglie hypothesis, suggest which wavefunction corresponds to the larger electron energy.
(ii) Predict and explain which wavefunction indicates a larger probability of finding the electron near the position \(\frac{L}{2}\) in the box.
(iii) On the graph in (c) on page 7, sketch a possible wavefunction for the lowest energy state of the electron.
Markscheme
atomic spectra have discrete line structures / only discrete frequencies/wavelengths;
photon energy is related to frequency/wavelength;
photons have discrete energies;
photons arise from electron transitions between energy levels;
which must have discrete values of energy;
de Broglie suggests that electrons/all particles have an associated wavelength; this wave will be a stationary wave which meets the boundary conditions of the box; the stationary wave has wavelength \(\frac{{2L}}{n}\) (where L is the length of the box and where n is an integer);
(i) wavelength of ψA larger than ψB;
therefore momentum of ψB larger than ψA (from de Broglie hypothesis); therefore ψB has larger energy;
Award [1 max] for a bald correct answer.
or
ψB has n=3, ψA has n=2;
EK ∝ n2;
so ψB corresponds to the larger energy;
(ii) ψA=0, ψB≠0 in the middle of the box/at \(\frac{L}{2}\);
so ψB corresponds to the larger probability since probability ∝ΙψΙ2;
Accept ∝ ψ2.
or
the probability (of finding the electron) is related to the amplitude;
amplitude of ψB is greater than amplitude of ψA so ψB is more likely to be found;
Award [1 max] for a bald correct answer.
(iii)
correct sketch; (accept -ψ)
Accept wavefunction with any amplitude.
Examiners report
(a) Candidates struggled with this question. Although they demonstrated some familiarity with the idea, they could not clearly describe the connection between atomic structure and the emission spectra, usually discussing electrons without photons. The arguments leading from atomic spectra to energy levels were not logically organised.
There were very few correct answers to (b).
(i) was reasonably well done by many, although many did not refer to the de Broglie hypothesis explicitly and thus relate wavelength to momentum and so to energy.
(ii) was poorly answered. Not many candidates understood the relation between amplitude and probability of locating the particle.
(iii) was well done by most.
