Lessons 11–15 cover Rutherford's nuclear model, Bohr's quantised atom, quantum mechanics and wave-particle duality, the strong nuclear force and binding energy, and radioactive decay. This checkpoint assesses your understanding of atomic and nuclear structure and your ability to apply quantum and nuclear physics concepts.
Rutherford scattering, alpha particles, closest approach, nuclear size
Quantised orbits, energy levels, Rydberg formula, spectral series
de Broglie wavelength, Schrödinger equation, quantum numbers, Pauli exclusion
Binding energy, mass defect, nuclear stability, fission/fusion
Alpha, beta, gamma, exponential decay, half-life, carbon dating
15 questions — instant feedback
Q1. In Rutherford's scattering experiment, large-angle deflection of alpha particles indicated:
Correct: B. Large-angle scattering requires close approach to a concentrated positive charge.
Q2. According to Bohr, electrons in allowed orbits:
Correct: A. Bohr postulated that electrons in stationary states do not radiate, contradicting classical electromagnetism.
Q3. The wavelength of the Lyman-$\\alpha$ line ($n=2 \\rightarrow n=1$) in hydrogen is approximately:
Correct: C. $1/\\lambda = 1.097\\times10^7(1 - 1/4) = 8.23\\times10^6$. $\\lambda = 122$ nm (ultraviolet).
Q4. The de Broglie wavelength of a particle is inversely proportional to its:
Correct: B. $\\lambda = h/p$. Higher momentum means shorter wavelength.
Q5. The Pauli exclusion principle explains:
Correct: D. No two electrons can share the same four quantum numbers, limiting each subshell to a specific capacity.
Q6. The strong nuclear force has a range of approximately:
Correct: A. ~1-3 fm ($10^{-15}$ m), comparable to nuclear dimensions.
Q7. The binding energy per nucleon is highest for:
Correct: C. Fe-56 has the maximum binding energy per nucleon (~8.8 MeV).
Q8. In beta-minus decay, the atomic number $Z$:
Correct: B. A neutron becomes a proton, so $Z$ increases by 1 while $A$ stays the same.
Q9. A sample has half-life 12 hours. After 36 hours, the fraction remaining is:
Correct: D. 36 hours = 3 half-lives. $(1/2)^3 = 1/8$ remains.
Q10. Gamma decay involves:
Correct: A. Gamma decay releases excess energy as a photon; $A$ and $Z$ remain unchanged.
Q11. The Heisenberg uncertainty principle states that:
Correct: C. $\\Delta x \\Delta p \\geq \\hbar/2$ is a fundamental quantum limit.
Q12. An electron has de Broglie wavelength $5.0\\times10^{-10}$ m. Its momentum is approximately:
Correct: B. $p = 6.63\\times10^{-34}/5.0\\times10^{-10} \\approx 1.33\\times10^{-24}$ kg·m/s.
Q13. Rutherford's model could not explain why:
Correct: D. Classical physics predicts accelerating charges radiate energy. Bohr's quantisation postulate solved this.
Q14. Nuclear fusion releases energy because:
Correct: A. Fusion of light nuclei moves up the binding energy per nucleon curve toward Fe-56.
Q15. In quantum mechanics, an orbital (as opposed to a Bohr orbit) represents:
Correct: C. $||\\psi||^2$ gives the probability density; orbitals are probability clouds, not trajectories.
5 questions — model answers revealed
SAQ 1. (a) Describe the Geiger-Marsden experiment and explain how it led to Rutherford's nuclear model. (b) Calculate the distance of closest approach for a 5.0 MeV alpha particle approaching a copper nucleus ($Z = 29$) head-on. (c) Explain why this distance is an upper limit for the nuclear radius. (d) State one limitation of Rutherford's model. (4 marks)
Model answer (4 marks):
(a) Alpha particles fired at thin gold foil. Most passed straight through (atoms mostly empty space), some deflected (concentrated positive charge), very few bounced back (small dense nucleus) (1.5 marks).
(b) $r = (8.99\\times10^9)(2)(29)(1.6\\times10^{-19})^2/(5.0\\times1.6\\times10^{-13}) = 1.67\\times10^{-14}$ m (1 mark).
(c) The alpha particle turns around before reaching the nuclear surface due to Coulomb repulsion; actual radius is smaller (0.5 mark).
(d) Cannot explain why electrons don't spiral in / only works for hydrogen / no explanation for quantisation (1 mark).
SAQ 2. (a) Calculate the wavelength of light emitted when an electron transitions from $n = 5$ to $n = 2$ in hydrogen. Identify the spectral series. (b) Calculate the de Broglie wavelength of an electron with kinetic energy 50 eV. (c) Explain how de Broglie's hypothesis accounts for Bohr's angular momentum quantisation condition. (d) Distinguish between a Bohr orbit and a quantum orbital. (4 marks)
Model answer (4 marks):
(a) $1/\\lambda = 1.097\\times10^7(1/4 - 1/25) = 2.30\\times10^6$. $\\lambda = 435$ nm (Balmer series, visible) (1.5 marks).
(b) $v = \\sqrt{2E_k/m} = \\sqrt{2(50\\times1.6\\times10^{-19})/(9.11\\times10^{-31})} = 4.19\\times10^6$ m/s. $\\lambda = h/(m_e v) = 6.63\\times10^{-34}/(9.11\\times10^{-31}\\times4.19\\times10^6) = 1.74\\times10^{-10}$ m (1.5 marks).
(c) Standing wave condition: $2\\pi r = n\\lambda = nh/(mv)$, giving $mvr = n\\hbar$ (0.5 mark).
(d) Bohr orbit = precise circular path; orbital = probability cloud with no definite trajectory (0.5 mark).
SAQ 3. (a) Outline the properties of the strong nuclear force. (b) Define binding energy and mass defect. (c) Calculate the binding energy per nucleon for oxygen-16 ($m_p = 1.007276$ u, $m_n = 1.008665$ u, $m_O = 15.994915$ u, 1 u = 931.5 MeV/c²). (d) Explain why very heavy nuclei are unstable despite the strong force. (4 marks)
Model answer (4 marks):
(a) Short range (~1-3 fm), attractive between all nucleons, saturated, repulsive at very short distances (1 mark).
(b) Mass defect = difference between mass of separated nucleons and nucleus mass. Binding energy = $\\Delta m \\cdot c^2$ (0.5 mark).
(c) $\\Delta m = 8(1.007276) + 8(1.008665) - 15.994915 = 0.137005$ u. $E_b = 0.137005 \\times 931.5 = 127.6$ MeV. $E_b/A = 127.6/16 = 7.98$ MeV/nucleon (1.5 marks).
(d) Coulomb repulsion is long-range and cumulative; strong force is short-range and saturates. For large $Z$, Coulomb wins (1 mark).
SAQ 4. (a) Distinguish between alpha, beta-minus, and gamma decay. (b) Write the complete decay equation for $^{226}_{88}\\text{Ra}$ undergoing alpha decay. (c) A radioactive sample has initial activity 2,400 Bq and half-life 8 hours. Calculate its activity after 24 hours. (d) Explain why individual nuclear decays are random but large samples follow an exponential law. (e) A sample of ancient wood has $^{14}$C activity 25% of that in living wood. Estimate its age ($t_{1/2}$ of $^{14}$C = 5,730 years). (5 marks)
Model answer (5 marks):
(a) Alpha: emits $^4_2$He, $A$ decreases by 4, $Z$ by 2. Beta-minus: neutron → proton + e⁻ + $\\bar{\\nu}_e$, $Z$ increases by 1, $A$ unchanged. Gamma: emits photon, no change in $A$ or $Z$ (1.5 marks).
(b) $^{226}_{88}\\text{Ra} \\rightarrow \\; ^{222}_{86}\\text{Rn} + \\; ^4_2\\text{He}$ (1 mark).
(c) 24 h = 3 half-lives. $A = 2400 \\times (1/2)^3 = 300$ Bq (1 mark).
(d) Individual decays are quantum mechanical and inherently unpredictable. For large $N$, statistics produce the smooth exponential law (1 mark).
(e) 25% = 2 half-lives. Age = $2 \\times 5,730 = 11,460$ years (0.5 mark).
SAQ 5. (a) Explain how Bohr's model accounts for the line spectrum of hydrogen. (b) Calculate the energy required to ionise a hydrogen atom from its ground state. (c) Explain why Bohr's model fails for multi-electron atoms. (d) Describe how quantum mechanics (Schrödinger equation) improves upon Bohr's model. (e) State the Pauli exclusion principle and explain how it determines the electron capacity of the $n = 2$ shell. (5 marks)
Model answer (5 marks):
(a) Electrons occupy discrete energy levels. Photons are emitted/absorbed with energy equal to level differences, producing discrete wavelengths (1 mark).
(b) $E_{ionisation} = 0 - (-13.6) = 13.6$ eV $= 13.6 \\times 1.6\\times10^{-19} = 2.18\\times10^{-18}$ J (1 mark).
(c) No account for electron-electron repulsion; assumes circular orbits; quantisation is postulated not derived (1 mark).
(d) Schrödinger equation derives energy levels from fundamental principles; gives probability distributions (orbitals) not trajectories; works for multi-electron atoms (approx.) (1 mark).
(e) No two electrons share all four quantum numbers. $n = 2$: $l = 0$ (2s, 2 electrons) and $l = 1$ (2p, 6 electrons). Total = 8 electrons (1 mark).