Quantum Mechanics 101 · Unit 1: The Classical World Cracks

Quantum Academy · Chapter 5: Atoms Get Weird: Bohr's Model & Spectral Lines

Barcodes of light, and electrons that only live on certain floors

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Instructions. Work through these on paper before checking the answers. Use Eₙ = −13.6 eV/n², photon energy equal to the difference between levels, the shortcut λ ≈ 1240/ΔE (nanometres and electron-volts), the Rydberg formula 1/λ = R(1/n₁² − 1/n₂²) with R = 1.097×10⁷ per metre, and rₙ = n²a₀ with a₀ = 0.0529 nm. Take 1 eV = 1.602×10⁻¹⁹ joules and g = 9.81 m/s². Show units at every step. One point for each vocabulary match and each question; the challenge is a bonus.

Part A Vocabulary matching

Write the letter of the matching definition in the blank beside each term.

1.Emission spectrum
2.Absorption spectrum
3.Spectroscopy
4.Nucleus
5.Stationary state
6.Ground state
7.Excited state
8.Ionization energy
9.Quantum leap
10.Spectral series
11.Bohr radius
12.Rydberg constant

A.A family of lines produced by jumps that all end on the same level, such as the Lyman (n = 1), Balmer (n = 2) and Paschen (n = 3) series.
B.In Bohr's model, an allowed state of the atom with a fixed energy, in which the electron does not radiate.
C.Any state of an atom above the ground state, reached by absorbing energy and usually lasting only billionths of a second.
D.The lowest-energy state of an atom; for hydrogen, n = 1 at −13.6 eV.
E.The radius of the smallest orbit in Bohr's hydrogen atom, a₀ ≈ 0.0529 nm; in modern theory, the most likely distance of a ground-state electron.
F.The constant R ≈ 1.097×10⁷ per metre in the formula for hydrogen's spectral lines; Bohr derived it from fundamental constants.
G.A discrete jump of an atom between two allowed states, with no states in between; the smallest possible change, not a giant one.
H.Dark lines on a continuous spectrum where a cooler gas has absorbed light, at the same wavelengths as its emission lines.
I.The measurement and interpretation of spectra, used to identify elements and measure temperature, motion and magnetic fields.
J.The tiny, dense centre of an atom, holding all its positive charge and nearly all its mass; discovered by Rutherford in 1911.
K.The bright lines at specific wavelengths given off by a hot, thin gas; unique to each element.
L.The minimum energy needed to remove an electron from an atom; 13.6 eV for hydrogen in its ground state.

Part B Questions

1.
Rutherford's 1911 conclusion from the Geiger–Marsden scattering experiments was that:Circle one
1. AElectrons can move only in quantized orbits
2. BAn atom's positive charge and nearly all its mass are concentrated in a tiny nucleus
3. CPositive charge is spread evenly through the atom, with electrons embedded in it
4. DAtoms emit light only at certain wavelengths
2.
Which transition in hydrogen emits the photon with the shortest wavelength?Circle one
1. An = 3 to n = 2
2. Bn = 6 to n = 2
3. Cn = 2 to n = 1
4. Dn = 5 to n = 3
3.
A cloud of cool atomic hydrogen at about 100 K, like the thin gas between the stars, sits between a bright source of continuous light and a spectrometer. Which lines appear dark?Circle one
1. AThe four visible Balmer lines
2. BThe infrared Paschen lines
3. CThe ultraviolet Lyman lines
4. DNo lines at all, because cool gas cannot absorb light
4.
According to Bohr, an electron in a stationary state radiates light continuously as it circles the nucleus.Circle one
TrueFalse
5.
Less energy is needed to ionize a hydrogen atom from its n = 2 level than from its ground state.Circle one
TrueFalse
6.
A ground-state hydrogen atom can absorb a photon of any energy between 10.2 eV and 13.6 eV.Circle one
TrueFalse
7.
The lowest-energy state of an atom, in which an undisturbed atom remains indefinitely, is called the state.Fill in the blank
8.
The family of hydrogen lines produced by jumps down to n = 2, which includes all of hydrogen's visible lines, is the series.Fill in the blank
9.
Explain why an element produces bright lines in its emission spectrum and dark lines at exactly the same wavelengths in its absorption spectrum.Short answer
10.
Give two things Bohr's model of hydrogen gets right and two things it gets wrong.Short answer
11.
Use the Rydberg formula to find the wavelength of the Balmer line produced by a jump from n = 4 to n = 2, and the energy of its photon in electron-volts.Show your working
Working
Answer:
12.
Find the series limit of the Balmer series, the shortest wavelength any Balmer line can have, and the photon energy that corresponds to it. Which part of the spectrum is it in?Show your working
Working
Answer:
13.
Neutral-atom quantum computers excite atoms into Rydberg states with n of several tens or more. In Bohr's model of hydrogen, find the orbit radius and the energy for n = 100, and the energy needed to ionize the atom from that state.Show your working
Working
Answer:
14.
Express the energy of hydrogen's red quantum leap, 1.89 eV, in joules. How many such leaps would supply the energy needed to lift a 1.0-gram paperclip by 1.0 millimetre?Show your working
Working
Answer:

Part C Mad scientist challenge bonus

The star that broke the hydrogen code. In 1896 Edward Pickering reported, in the star ζ Puppis, a series of lines (spotted on Harvard's photographic plates by Williamina Fleming, whom he credited) resembling hydrogen's but with extra lines between the Balmer lines; in 1912 Alfred Fowler produced them in a tube of hydrogen mixed with helium. (a) Bohr's model for a single electron bound to a nucleus of charge Ze gives Eₙ = −13.6 eV × Z²/n². Write the levels of singly ionized helium, He⁺ with Z = 2, for n = 1 to 6. (b) Find the wavelengths of the He⁺ lines from n = 5, 6, 7 and 8 down to n = 4. (c) Which of these coincide with hydrogen's Balmer lines, which are new, and why were astronomers fooled into blaming hydrogen? (d) Bohr's reply to Fowler: the electron and nucleus both circle their common centre of mass, which multiplies each Rydberg constant by 1/(1 + m/M), where M is the nuclear mass. Using M/m = 1836 for the proton and 7294 for the helium nucleus, find the ratio of helium's effective constant (including Z² = 4) to hydrogen's, and estimate how far apart hydrogen's 656 nm line and helium's line from 6 to 4 really lie.

Working