Solved Examples (Dedicated Problem Set)

How to Use This Problem Set

Your full workout for Nuclei, grouped by theme: mass conversions & counting, size & density, binding energy, Q-values, fission & fusion energy budgets, and decay-law problems.

Keep these handy:

• 1 u = $1.66 \times 10^{-27}$ kg = 931.5 MeV/$c^2$; 1 MeV = $1.6 \times 10^{-13}$ J; 1 year = $3.154 \times 10^7$ s
• Radius/density: $R = 1.2A^{1/3}$ fm; $V \propto A$; $\rho \approx 2.3 \times 10^{17}$ kg/m³ (all nuclei)
• Binding: $\Delta M = [Zm_H + (A-Z)m_n] - M_{atom}$; $E_b = \Delta M \times 931.5$ MeV; $E_{bn} = E_b/A$
• Q-value: $Q = [\Sigma m_i - \Sigma m_f] \times 931.5$ MeV; Q > 0 exothermic
• Decay: $N = N_0e^{-\lambda t}$; $T_{1/2} = 0.693/\lambda$; $\tau = 1/\lambda$; $R = \lambda N$; after n half-lives: $N_0/2^n$
• Masses: $m_H$ = 1.007825 u, $m_n$ = 1.008665 u, $m(^4He)$ = 4.002603 u

State the condition, then substitute.

Solved Examples - Conversions & Counting

Example 1. Convert 0.5 u into MeV.

Solution: $0.5 \times 931.5 = 465.8$ MeV.

Example 2. Energy equivalent of an electron's mass (0.000548 u)?

Solution: $0.000548 \times 931.5 \approx 0.511$ MeV — the famous electron rest energy.

Example 3. Atoms in 10 g of $^{235}$U?

Solution: $N = \dfrac{10}{235} \times 6.023 \times 10^{23} = 2.56 \times 10^{22}$.

Example 4. Express $3.2 \times 10^{-11}$ J in MeV.

Solution: $\dfrac{3.2 \times 10^{-11}}{1.6 \times 10^{-13}} = 200$ MeV — one fission's worth.

Example 5. An element's isotopes of mass 6.015 u (7.5%) and 7.016 u (92.5%): mean atomic mass?

Solution: $0.075 \times 6.015 + 0.925 \times 7.016 = 6.94$ u — lithium.

Solved Examples - Size & Density

Example 6. Radius of $^{27}$Al?

Solution: $R = 1.2 \times 27^{1/3} = 1.2 \times 3 = 3.6$ fm.

Example 7. Ratio of radii of $^{208}$Pb and $^{26}$Al… take A = 208, 26.

Solution: $\left(\dfrac{208}{26}\right)^{1/3} = 8^{1/3} = 2$ — lead's nucleus is exactly twice aluminium's radius.

Example 8. Gold vs silver radii (NCERT 13.4)?

Solution: $\left(\dfrac{197}{107}\right)^{1/3} \approx 1.23$.

Example 9. Nuclear density of $^{56}$Fe (mass 55.85 u)? (NCERT Example 13.1)

Solution: $\rho = \dfrac{9.27 \times 10^{-26}}{\frac{4\pi}{3}(1.2 \times 10^{-15})^3 \times 56} = 2.29 \times 10^{17}$ kg/m³.

Example 10. Mass number of a nucleus with radius 6.0 fm?

Solution: $A = \left(\dfrac{6.0}{1.2}\right)^3 = 5^3 = 125$.

Solved Examples - Binding Energy

Example 11. Binding energy of $^{14}_{7}$N (14.00307 u)? (NCERT 13.1)

Solution: $\Delta M = [7(1.007825) + 7(1.008665)] - 14.00307 = 0.11236$ u; $E_b = 104.7$ MeV.

Example 12. $E_{bn}$ of nitrogen-14 from Example 11?

Solution: $104.7/14 = 7.48$ MeV per nucleon.

Example 13. Binding energy of $^{56}_{26}$Fe (55.934939 u)? (NCERT 13.2)

Solution: $\Delta M = [26(1.007825) + 30(1.008665)] - 55.934939 = 0.528461$ u; $E_b = 492.3$ MeV; $E_{bn} = 8.79$ MeV.

Example 14. Binding energy of $^{209}_{83}$Bi (208.980388 u)? (NCERT 13.2)

Solution: $\Delta M = [83(1.007825) + 126(1.008665)] - 208.980388 = 1.760877$ u; $E_b = 1640$ MeV; $E_{bn} = 7.85$ MeV.

Example 15. Energy to dismantle all nuclei in a 3.0 g copper coin ($^{63}$Cu, 62.92960 u)? (NCERT 13.3)

Solution: per-nucleus $\Delta M = [29(1.007825) + 34(1.008665)] - 62.92960 = 0.591935$ u → 551.4 MeV; atoms $= 2.87 \times 10^{22}$; total ≈ $1.58 \times 10^{25}$ MeV $= 2.5 \times 10^{12}$ J.

Example 16. $E_{bn}$ of $^{4}$He (4.002603 u)?

Solution: $\Delta M = [2(1.007825) + 2(1.008665)] - 4.002603 = 0.030377$ u; $E_b = 28.3$ MeV; $E_{bn} = 7.07$ MeV.

Solved Examples - Q-values

Example 17. Q of $^{1}H + {}^{3}H \to {}^{2}H + {}^{2}H$? (NCERT 13.5i)

Solution: $Q = [1.007825 + 3.016049 - 2(2.014102)] \times 931.5 = -0.00433 \times 931.5 \approx -4.03$ MeV — endothermic.

Example 18. Q of $^{12}C + {}^{12}C \to {}^{20}Ne + {}^{4}He$? (NCERT 13.5ii)

Solution: $Q = [24.000000 - 19.992439 - 4.002603] \times 931.5 = 0.004958 \times 931.5 \approx +4.62$ MeV — exothermic.

Example 19. Is $^{56}Fe \to 2\,^{28}Al$ possible? (NCERT 13.6)

Solution: $Q = [55.93494 - 2(27.98191)] \times 931.5 = -26.9$ MeV < 0 — not energetically possible.

Example 20. Q of the deuteron-deuteron → He-3 + n reaction, from masses m($^2$H) = 2.014102 u, m($^3$He) = 3.016029 u, $m_n$ = 1.008665 u?

Solution: $Q = [2(2.014102) - 3.016029 - 1.008665] \times 931.5 = 0.003510 \times 931.5 \approx 3.27$ MeV. ✔ (Matches NCERT Eq. 13.13b.)

Example 21. Complete and identify: $^{1}_{0}n + {}^{235}_{92}U \to {}^{140}_{54}Xe + X + 2\,^{1}_{0}n$.

Solution: A = 236 - 140 - 2 = 94; Z = 92 - 54 = 38 → $X = {}^{94}_{38}$Sr.

Solved Examples - Fission & Fusion Budgets

Example 22. Energy from complete fission of 1 kg of $^{239}$Pu at 180 MeV each? (NCERT 13.7)

Solution: $N = \dfrac{1000}{239} \times 6.023 \times 10^{23} = 2.52 \times 10^{24}$; $E = 4.53 \times 10^{26}$ MeV $\approx 7.3 \times 10^{13}$ J.

Example 23. How long does 2.0 kg of deuterium power a 100 W lamp (3.27 MeV per d-d reaction)? (NCERT 13.8)

Solution: reactions $= 3.01 \times 10^{26}$; E ≈ $1.58 \times 10^{14}$ J; $t = 1.58 \times 10^{12}$ s ≈ $5 \times 10^4$ years.

Example 24. Coulomb barrier of two touching deuterons (r = 2.0 fm)? (NCERT 13.9)

Solution: $U = \dfrac{9 \times 10^9 (1.6 \times 10^{-19})^2}{4 \times 10^{-15}} = 5.76 \times 10^{-14}$ J ≈ 360 keV.

Example 25. Fissions per second in a 500 MW reactor (200 MeV each)?

Solution: $\dfrac{5 \times 10^8}{3.2 \times 10^{-11}} \approx 1.6 \times 10^{19}$ per second.

Example 26. Verify the p-p cycle total.

Solution: 2(0.42) + 2(1.02) + 2(5.49) + 12.86 = 26.7 MeV. ✔

Example 27. Hydrogen the Sun burns per second ($3.8 \times 10^{26}$ W)?

Solution: per proton 6.68 MeV = $1.07 \times 10^{-12}$ J → $3.6 \times 10^{38}$ protons/s → ≈ $6 \times 10^{11}$ kg/s.

Solved Examples - Decay Law

Example 28. $T_{1/2}$ = 20 min. Fraction left after 1 hour?

Solution: 3 half-lives → $\dfrac{1}{8}$.

Example 29. Decay constant for $T_{1/2}$ = 1386 s?

Solution: $\lambda = \dfrac{0.693}{1386} = 5 \times 10^{-4}$ s⁻¹.

Example 30. Activity of $10^{18}$ atoms with $\lambda = 10^{-4}$ s⁻¹?

Solution: $R = \lambda N = 10^{14}$ Bq.

Example 31. Activity falls from 8000 to 500 Bq in 36 hours: half-life?

Solution: ratio 16 = $2^4$ → 4 half-lives → $T_{1/2}$ = 9 hours.

Example 32. Mean life of a nuclide with $T_{1/2}$ = 693 years?

Solution: $\tau = \dfrac{T_{1/2}}{0.693} = 1000$ years.

Example 33. What fraction decays during one mean life?

Solution: remaining $= e^{-1} \approx 37\%$; decayed ≈ 63%.

Example 34. A sample is 1/64 of its original after 30 days. Find $T_{1/2}$.

Solution: $1/64 = (1/2)^6$ → 6 half-lives → $T_{1/2}$ = 5 days.

Solved Examples - Mixed Mastery

Example 35. $^{238}_{92}$U alpha-decays. Identify the daughter and compute Q given m(U) = 238.05079 u, m(Th) = 234.04363 u, m(He) = 4.002603 u.

Solution: daughter $^{234}_{90}$Th; $Q = [238.05079 - 234.04363 - 4.002603] \times 931.5 = 0.004557 \times 931.5 \approx 4.25$ MeV.

Example 36. Mass converted to energy per second by a 1000 MW source?

Solution: $\dot{m} = \dfrac{P}{c^2} = \dfrac{10^9}{9 \times 10^{16}} \approx 1.1 \times 10^{-8}$ kg/s ≈ 1 gram per day.

Example 37. The Sun loses how much mass per second ($3.8 \times 10^{26}$ W)?

Solution: $\dfrac{3.8 \times 10^{26}}{9 \times 10^{16}} \approx 4.2 \times 10^{9}$ kg/s — four million tonnes a second.

Example 38. Two nuclei have volumes in ratio 8 : 1. Ratio of mass numbers and radii?

Solution: A ratio = 8 : 1 (V ∝ A); radii ratio = 2 : 1.

Example 39. A nucleus at rest emits a 5 MeV alpha. Why is the daughter's recoil energy small, and roughly how big for A = 234?

Solution: momentum conservation: $p_\alpha = p_{daughter}$; $K = \dfrac{p^2}{2m}$ → $K_d = K_\alpha \times \dfrac{m_\alpha}{m_d} \approx 5 \times \dfrac{4}{234} \approx 0.085$ MeV — the heavy partner takes the momentum but little energy.

Example 40. How much $^{235}$U (200 MeV/fission) equals the chemical energy of 1000 kg of coal ($10^7$ J/kg)?

Solution: need $10^{10}$ J; per kg U ≈ $8.2 \times 10^{13}$ J → $m = \dfrac{10^{10}}{8.2 \times 10^{13}}$ kg ≈ 0.12 g. A coal truck versus a dust grain.