FAD1022: Rapid-Fire Drill Pack — CQ7 Atomic Physics

Objective: Achieve mechanical fluency in Bohr-model calculations, photon transitions, photoelectric effect, de Broglie wavelength, and basic nuclear decay.
Target: ~2.5 min per problem. If you stall >3 minutes, skip and mark it.
Total problems: 62
Estimated time: 2.5–3 hours

Cheat Sheet (Memorize First)

Quantity Formula Key Value
Bohr radius $r_n = a_0 n^2 = (5.29 \times 10^{-11},\text{m}),n^2$ $a_0 = 5.29 \times 10^{-11},\text{m}$
Energy levels (H) $E_n = -\dfrac{13.6}{n^2},\text{eV}$ $E_1 = -13.6,\text{eV}$
Angular momentum $L_n = n\hbar = n\dfrac{h}{2\pi}$ $\hbar = 1.055 \times 10^{-34},\text{J}\cdot\text{s}$
Photon energy $E = hf = \dfrac{hc}{\lambda} = E_i - E_f$ $hc = 1240,\text{eV}\cdot\text{nm}$
Rydberg formula $\dfrac{1}{\lambda} = R_H\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)$ $R_H = 1.097 \times 10^{7},\text{m}^{-1}$
de Broglie wavelength $\lambda = \dfrac{h}{p} = \dfrac{h}{mv} = \dfrac{h}{\sqrt{2mK}}$
Photoelectric effect $hf = \phi + K_{\max}$ $K_{\max} = eV_s$
Threshold frequency $f_0 = \dfrac{\phi}{h}$ $\lambda_0 = \dfrac{hc}{\phi}$
Compton shift $\Delta\lambda = \dfrac{h}{m_e c}(1-\cos\theta)$ $\dfrac{h}{m_e c} = 2.43,\text{pm}$
Nuclear radius $R = R_0 A^{1/3}$ $R_0 = 1.2 \times 10^{-15},\text{m}$
Mass defect $\Delta m = Zm_p + Nm_n - m_{\text{nucleus}}$
Binding energy $E_B = \Delta m \cdot c^2 = \Delta m \times 931.5,\text{MeV/u}$ $1,\text{u} = 931.5,\text{MeV}/c^2$
Decay constant $\lambda = \dfrac{\ln 2}{T_{1/2}} = \dfrac{0.693}{T_{1/2}}$
Activity $A = \lambda N$ $1,\text{Bq} = 1,\text{decay/s}$
Exponential decay $N = N_0 e^{-\lambda t}$

Constants

  • $h = 6.63 \times 10^{-34},\text{J}\cdot\text{s}$
  • $c = 3.00 \times 10^{8},\text{m/s}$
  • $e = 1.60 \times 10^{-19},\text{C}$
  • $m_e = 9.11 \times 10^{-31},\text{kg}$
  • $m_p = 1.673 \times 10^{-27},\text{kg}$
  • $m_n = 1.675 \times 10^{-27},\text{kg}$
  • $1,\text{eV} = 1.60 \times 10^{-19},\text{J}$

Mnemonic — Spectral Series:

  • Lyman → Lowest (UV, $n \to 1$)
  • Balmer → Bright (Visible, $n \to 2$)
  • Paschen → Peek (IR, $n \to 3$)

Part A: Bohr Model & Orbital Properties

Target: 90 s per problem.

Set A1 — Radius, Velocity & Angular Momentum (6 problems)

Apply the Bohr quantization conditions directly. Watch your powers of 10.

  1. Calculate the radius of the $n = 4$ orbit in hydrogen.
  2. Find the angular momentum of an electron in the $n = 3$ orbit.
  3. Determine the orbital speed of an electron in the $n = 2$ state.
  4. By what factor is the $n = 5$ orbit larger than the $n = 2$ orbit?
  5. An orbit in hydrogen has radius $4.76 \times 10^{-10},\text{m}$. What is the principal quantum number $n$?
  6. The angular momentum of an electron in a hydrogen orbit is measured as $2.11 \times 10^{-34},\text{kg}\cdot\text{m}^2\cdot\text{s}^{-1}$. Identify $n$.

Score: ___/6

Set A2 — Energy Levels (6 problems)

Remember: energies are negative. Ionization means reaching $E = 0$.

  1. What is the energy of the $n = 5$ level in hydrogen?
  2. What is the energy of the first excited state?
  3. Calculate the ionization energy for an electron initially in the $n = 3$ state.
  4. An energy level in hydrogen is measured at $-0.544,\text{eV}$. What is $n$?
  5. Find the quantum number $n$ for an energy level of $-1.51,\text{eV}$.
  6. How much energy must be supplied to move an electron from the ground state to the $n = 4$ level?

Score: ___/6

Part B: Transitions & Spectral Series

Target: 2 min per problem.

Set B1 — Photon Energy, Frequency & Wavelength (6 problems)

Use $\Delta E = E_i - E_f$. For emitted photons $\Delta E > 0$; for absorbed photons $\Delta E > 0$ also, but the atom gains energy.

  1. An electron drops from $n = 3$ to $n = 1$. What is the energy of the emitted photon (in eV)?
  2. An electron falls from $n = 4$ to $n = 2$. What is the photon energy in joules?
  3. For the transition $n = 5 \to n = 2$, calculate the frequency of the emitted photon.
  4. A hydrogen atom in the $n = 2$ state absorbs a photon and jumps to $n = 5$. What photon energy was absorbed?
  5. An electron in hydrogen transitions from $n = 6$ to $n = 2$. What is the wavelength of the emitted photon?
  6. What is the energy required to ionize a hydrogen atom from its ground state by photon absorption?

Score: ___/6

Set B2 — Rydberg Formula & Series Identification (6 problems)

Use $1/\lambda = R_H(1/n_f^2 - 1/n_i^2)$. Identify the series by the final level $n_f$.

  1. Calculate the wavelength of the Lyman-$\alpha$ line ($n = 2 \to n = 1$).
  2. Calculate the wavelength of the Balmer-$\alpha$ line ($n = 3 \to n = 2$).
  3. Calculate the wavelength of the Paschen-$\alpha$ line ($n = 4 \to n = 3$).
  4. A hydrogen spectral line has wavelength $97.3,\text{nm}$ and ends at $n_f = 1$. Find the initial level $n_i$.
  5. A line in the Balmer series has wavelength $486,\text{nm}$. Find the initial level $n_i$.
  6. What is the shortest wavelength in the Lyman series?

Score: ___/6

Part C: Photoelectric Effect & de Broglie Wavelength

Target: 2.5 min per problem.

Set C1 — Photoelectric Effect (6 problems)

Use $K_{\max} = hf - \phi = eV_s$. If $hf < \phi$, no photoemission occurs.

  1. A metal has work function $\phi = 2.3,\text{eV}$. Light of wavelength $400,\text{nm}$ strikes it. Find $K_{\max}$.
  2. Find the threshold frequency for a metal with $\phi = 4.5,\text{eV}$.
  3. Light of wavelength $300,\text{nm}$ ejects electrons with $K_{\max} = 1.2,\text{eV}$. Find the work function.
  4. Calculate the threshold wavelength for a metal with $\phi = 2.1,\text{eV}$.
  5. Photons of energy $5.0,\text{eV}$ strike a metal with $\phi = 2.5,\text{eV}$. Find the maximum speed of the ejected electrons.
  6. The stopping potential for a certain metal is $2.5,\text{V}$ when illuminated with $250,\text{nm}$ light. Find the work function.

Score: ___/6

Set C2 — de Broglie Wavelength & Compton Scattering (6 problems)

For non-relativistic particles: $\lambda = h/\sqrt{2mK}$. For Compton: $\Delta\lambda = \lambda_C(1-\cos\theta)$.

  1. Find the de Broglie wavelength of an electron accelerated through $100,\text{V}$.
  2. An electron travels at $2.0 \times 10^{6},\text{m/s}$. What is its de Broglie wavelength?
  3. A proton has kinetic energy $500,\text{eV}$. Calculate its de Broglie wavelength.
  4. An electron has a de Broglie wavelength of $0.10,\text{nm}$. What is its speed?
  5. An electron has de Broglie wavelength $0.50,\text{nm}$. What is its kinetic energy (in eV)?
  6. X-rays of wavelength $500,\text{pm}$ scatter off free electrons at $\theta = 90^{\circ}$. What is the scattered wavelength?

Score: ___/6

Part D: Nuclear Structure, Binding Energy & Radioactivity

Target: 2.5 min per problem.

Set D1 — Nuclear Properties & Binding Energy (6 problems)

Use $R = R_0 A^{1/3}$, $\Delta m = Zm_p + Nm_n - m_{\text{nucleus}}$, and $E_B = \Delta m \times 931.5,\text{MeV/u}$.

  1. For gold-197 ($^{197}_{79}\text{Au}$), state $A$, $Z$, $N$ and calculate the nuclear radius.
  2. Calculate the nuclear radius of carbon-12.
  3. Given $m_p = 1.007276,\text{u}$, $m_n = 1.008665,\text{u}$ and $m(^{4}\text{He}) = 4.00260,\text{u}$, find the binding energy of helium-4.
  4. The atomic mass of $^{16}\text{O}$ is $15.994915,\text{u}$. Calculate its binding energy per nucleon.
  5. The mass of a lithium-7 nucleus is $7.016004,\text{u}$. Find its mass defect and binding energy.
  6. Carbon-12 has $E_B/A \approx 7.68,\text{MeV/nucleon}$; uranium-235 has $E_B/A \approx 7.59,\text{MeV/nucleon}$. Which nucleus is more stable, and by how much per nucleon?

Score: ___/6

Set D2 — Radioactivity & Half-Life (6 problems)

Use $\lambda = 0.693/T_{1/2}$, $N = N_0 e^{-\lambda t}$, and $A = \lambda N$.

  1. Technetium-99m has a half-life of $6.0,\text{h}$. Calculate its decay constant in $\text{h}^{-1}$.
  2. What fraction of a radioactive sample remains after 3 half-lives?
  3. How long does it take for a Tc-99m sample to decay to $10%$ of its initial activity?
  4. Carbon-14 has $T_{1/2} = 5730,\text{yr}$. If a sample has $25%$ of its original C-14 remaining, how old is it?
  5. How many atoms are in $1.0,\text{g}$ of carbon-14? ($M \approx 14,\text{g/mol}$)
  6. A nuclide $^{226}_{88}\text{Ra}$ undergoes 3 alpha decays followed by 2 beta-minus decays. What are $A$ and $Z$ of the final daughter nuclide?

Score: ___/6

Part E: Mixed & Reverse Problems

Target: 3–4 min per problem.

These problems combine multiple skills or require you to work backwards from a given result.

  1. A hydrogen electron drops from an unknown level $n$ to $n = 2$, emitting light of wavelength $486,\text{nm}$. Identify $n$.
  2. In a photoelectric experiment, the stopping potential is $1.5,\text{V}$ for a metal with $\phi = 1.8,\text{eV}$. What is the incident wavelength?
  3. A hydrogen atom in the ground state absorbs a photon of wavelength $97.3,\text{nm}$. What is the final quantum number $n$?
  4. A deuteron ($^{2}_1\text{H}$) has atomic mass $2.014102,\text{u}$. Calculate its binding energy.
  5. An electron has de Broglie wavelength $0.122,\text{nm}$. Through what accelerating voltage was it produced?
  6. A parent nuclide $^{234}_{90}\text{X}$ undergoes 2 alpha decays and 3 positron emissions. Identify the daughter nuclide ($A$ and $Z$).
  7. A He-Ne laser emits at $\lambda = 632.8,\text{nm}$ with power $5.0,\text{mW}$. How many photons does it emit per second?
  8. For a hydrogen-like ion, the $n = 2$ orbit has radius $1.06 \times 10^{-10},\text{m}$. Determine the atomic number $Z$.

Score: ___/8

Part F: Common Traps & Exam Tricks

Target: 2 min per problem.

These problems are designed to catch the most frequent exam mistakes. Read carefully.

  1. A student writes: "The energy of the $n = 2$ level is $+3.4,\text{eV}$." Calculate the correct energy difference for a transition $n = 2 \to n = 1$ and show why the sign matters.
  2. Compare the magnitude of energy change for $n = 2 \to 5$ (absorption) versus $n = 5 \to 2$ (emission). Are they the same or different?
  3. A metal has threshold wavelength $620,\text{nm}$. Will it exhibit photoemission under red light of wavelength $700,\text{nm}$? Explain in one sentence.
  4. In a binding-energy calculation, a student uses the mass number $A = 4$ instead of the atomic mass $4.00260,\text{u}$ for helium-4. By how much is their mass defect in error (in u)?
  5. A radioactive sample loses half its nuclei in one half-life. True or false: its activity also drops to exactly one-half? Justify with the formula.
  6. After 2 half-lives, what fraction of a radioactive sample has decayed (not remained)?
  7. An isotope undergoes one alpha decay and one beta-minus decay. What is the net change in $Z$ and $A$?
  8. Convert $5.0,\text{eV}$ to joules. A common error is to multiply by $1.6 \times 10^{-19}$ instead of using the correct operation. What is the correct value in joules?

Score: ___/6

Final Scorecard

Part Sets Problems Raw Score
A — Bohr Model & Orbitals A1, A2 12 ___/12
B — Transitions & Spectra B1, B2 12 ___/12
C — Photoelectric & de Broglie C1, C2 12 ___/12
D — Nuclear & Radioactivity D1, D2 12 ___/12
E — Mixed & Reverse E1–E8 8 ___/8
F — Traps & Tricks F1–F6 6 ___/6
TOTAL 62 ___/62

Proficiency Benchmarks

  • 43/62 (70%) — Proficient. You can handle standard exam problems.
  • 53/62 (85%) — Solid. Fast and accurate on most archetypes.
  • 58/62 (93%) — Exam-ready. Any mistake is a careless slip.

Speed Benchmarks

  • <2.5 h: Excellent mechanical fluency.
  • 2.5–3.5 h: Good. Review the sets you scored lowest on.
  • >3.5 h: Drill the specific parts you scored lowest on again tomorrow.

Error Log Template

After grading, list every wrong problem number with a one-word reason:

Problem Reason
e.g. A2.3 sign error

Re-solve all wrong problems immediately with notes, then again in 24 hours without notes.

Answer Key

Set A1

  1. $8.46 \times 10^{-10},\text{m}$
  2. $3.17 \times 10^{-34},\text{kg}\cdot\text{m}^2\cdot\text{s}^{-1}$
  3. $1.09 \times 10^{6},\text{m/s}$
  4. $6.25$
  5. $n = 3$
  6. $n = 2$

Set A2

  1. $-0.544,\text{eV}$
  2. $-3.40,\text{eV}$
  3. $1.51,\text{eV}$
  4. $n = 5$
  5. $n = 3$
  6. $12.75,\text{eV}$

Set B1

  1. $12.1,\text{eV}$
  2. $4.08 \times 10^{-19},\text{J}$
  3. $6.89 \times 10^{14},\text{Hz}$
  4. $2.86,\text{eV}$ (absorbed)
  5. $411,\text{nm}$
  6. $13.6,\text{eV}$

Set B2

  1. $121.5,\text{nm}$
  2. $656,\text{nm}$
  3. $1875,\text{nm}$
  4. $n_i = 4$
  5. $n_i = 4$
  6. $91.2,\text{nm}$

Set C1

  1. $0.80,\text{eV}$
  2. $1.09 \times 10^{15},\text{Hz}$
  3. $2.93,\text{eV}$
  4. $590,\text{nm}$
  5. $9.4 \times 10^{5},\text{m/s}$
  6. $2.46,\text{eV}$

Set C2

  1. $0.123,\text{nm}$
  2. $0.364,\text{nm}$
  3. $1.28 \times 10^{-12},\text{m}$
  4. $7.28 \times 10^{6},\text{m/s}$
  5. $6.0,\text{eV}$
  6. $502.4,\text{pm}$

Set D1

  1. $A = 197$, $Z = 79$, $N = 118$, $R = 6.98,\text{fm}$
  2. $2.75,\text{fm}$
  3. $27.3,\text{MeV}$
  4. $7.72,\text{MeV/nucleon}$
  5. $\Delta m = 0.0405,\text{u}$, $E_B = 37.7,\text{MeV}$
  6. C-12 is more stable by $0.09,\text{MeV/nucleon}$

Set D2

  1. $0.116,\text{h}^{-1}$
  2. $1/8$
  3. $19.9,\text{h}$ ($\approx 20,\text{h}$)
  4. $11,460,\text{yr}$
  5. $4.30 \times 10^{22}$
  6. $A = 214$, $Z = 84$ ($^{214}_{84}\text{Po}$)

Part E

  1. $n = 4$
  2. $376,\text{nm}$
  3. $n = 4$
  4. $2.22,\text{MeV}$
  5. $101,\text{V}$
  6. $A = 226$, $Z = 83$ ($^{226}_{83}\text{Bi}$)
  7. $1.6 \times 10^{16},\text{photons/s}$
  8. $Z = 2$ (He$^+$)

Part F

  1. $\Delta E = (-3.40) - (-13.6) = 10.2,\text{eV}$. The sign determines whether energy is released or required.
  2. Same magnitude ($2.86,\text{eV}$); opposite direction (absorption vs emission).
  3. No, because $700,\text{nm} > 620,\text{nm}$, so photon energy is below the work function.
  4. Error $= 4.00260 - 4 = 0.00260,\text{u}$.
  5. True; $A = \lambda N$, so if $N \to N/2$, then $A \to A/2$.
  6. $3/4$ has decayed (only $1/4$ remains).
  7. Net: $\Delta Z = -2 + 1 = -1$; $\Delta A = -4 + 0 = -4$.
  8. $8.0 \times 10^{-19},\text{J}$ (multiply $5.0$ by $1.6 \times 10^{-19}$).

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