Nuclear & Radiochemistry

Q1. (4 marks) (a) Magic numbers — numbers of protons or neutrons (2, 8, 20, 28, 50, 82, 126) at which nuclei show extra stability, corresponding to closed nuclear shells. (1) (b) Binding energy per nucleon — the total nuclear binding energy divided by the mass number $A$; measures average energy needed to remove one nucleon. (2) Peaks near $^{56}$Fe because attractive nuclear force saturates while Coulomb repulsion grows with $Z$; the balance gives maximum stability (~8.8 MeV/nucleon) around iron. (1)

Q2. (4 marks) (1 each) (a) $\alpha$: $Z \to Z-2$, $A \to A-4$. (b) $\beta^-$: $Z \to Z+1$, $A$ unchanged. (c) $\beta^+$: $Z \to Z-1$, $A$ unchanged. (d) EC: $Z \to Z-1$, $A$ unchanged.

Q3. (5 marks) (a) $\lambda = \dfrac{\ln 2}{t_{1/2}} = \dfrac{0.693}{8.0} = 0.0866\ \text{day}^{-1}$. (2) (b) $\tau = \dfrac{1}{\lambda} = \dfrac{t_{1/2}}{0.693} = 11.5\ \text{days}$. (1) (c) After 24 days = 3 half-lives: fraction $= (1/2)^3 = 1/8 = 0.125$ (12.5%). (2)

Q4. (4 marks) Law: rate of decay $-\dfrac{dN}{dt} = \lambda N$ (first order). (1) Integrate: $\dfrac{dN}{N} = -\lambda\,dt \Rightarrow \ln N = -\lambda t + C$, so $N = N_0 e^{-\lambda t}$. (2) At $t=t_{1/2}$, $N = N_0/2$: $\tfrac12 = e^{-\lambda t_{1/2}} \Rightarrow \ln 2 = \lambda t_{1/2}$, hence $t_{1/2} = \dfrac{\ln 2}{\lambda} = \dfrac{0.693}{\lambda}$. (1)

Q5. (4 marks) (a) Mass change $= 238 - 206 = 32 = 4 \times n_\alpha \Rightarrow n_\alpha = 8$. (1.5) Charge: $\alpha$ decays reduce $Z$ by $2\times8 = 16$, from 92 to $92-16 = 76$; actual $Z=82$, so $\beta^-$ needed $= 82 - 76 = 6$. (1.5) So 8 α and 6 β⁻. (b) The $4n+2$ series is the uranium (radium) series. (1)

Q6. (5 marks) Definition: Q-value = energy released (or absorbed) in a nuclear reaction = $(\Sigma m_{\text{reactants}} - \Sigma m_{\text{products}})c^2$; positive Q means exothermic. (1) $\Delta m = (2.01410 + 3.01605) - (4.00260 + 1.00867)$ $= 5.03015 - 5.01127 = 0.01888\ \text{u}$. (2) $Q = 0.01888 \times 931.5 = 17.59\ \text{MeV}$. (2)

Q7. (4 marks) (a) Critical mass: minimum mass of fissile material needed to sustain a self-propagating chain reaction (neutron multiplication factor $k=1$). (1) (b) Any two: thermal reactor uses slow (thermal) neutrons + moderator, uses $^{235}$U-enriched fuel; fast reactor uses fast neutrons, no moderator, uses higher-enrichment fuel / can breed $^{239}$Pu. (2) (c) Moderator slows fast neutrons to thermal energies to increase fission probability in $^{235}$U. (1)

Q8. (4 marks) $25\% = (1/2)^n \Rightarrow n = 2$ half-lives. (2) Age $= 2 \times 5730 = 11460$ years. (2)

Q9. (4 marks) (1 each) (a) Bq — activity; 1 Bq = 1 decay per second. (b) Gy — absorbed dose; 1 Gy = 1 J kg$^{-1}$. (c) Sv — equivalent (biological) dose; = absorbed dose × radiation weighting factor, unit J kg$^{-1}$. Shielding for γ-rays: lead (or concrete). (1)

Q10. (2 marks) (a) Tc-99m: γ-emitting tracer for imaging (e.g. bone/organ scans). (0.5) (b) I-131: treatment/diagnosis of thyroid disorders (β/γ emitter). (0.5) Pu-238: long half-life (~88 yr) α-emitter producing steady heat converted to electricity by thermocouples — reliable, long-lived power where solar is impractical. (1)

[
  {"claim":"Half-life 8 days gives decay constant 0.0866/day",
   "code":"lam=ln(2)/8; result = abs(float(lam)-0.0866)<0.001"},
  {"claim":"Fraction remaining after 24 days (3 half-lives) is 0.125",
   "code":"f=(Rational(1,2))**3; result = f==Rational(1,8)"},
  {"claim":"U-238 to Pb-206 emits 8 alpha and 6 beta particles",
   "code":"na=(238-206)/4; nb=82-(92-2*na); result = (na==8 and nb==6)"},
  {"claim":"D-T fusion Q-value is 17.59 MeV",
   "code":"dm=(2.01410+3.01605)-(4.00260+1.00867); Q=dm*931.5; result = abs(Q-17.59)<0.05"},
  {"claim":"C-14 activity 25% corresponds to 11460 years",
   "code":"n=log(4)/log(2); age=n*5730; result = abs(float(age)-11460)<1"}
]