Nuclei Class 12 Notes

• Nuclear radius:
• $R=R_0{A}^{1/3}$
• with
• $R_0\approx 1.2\text{fm}$
• .
• Nuclear density is nearly constant for all nuclei, about
• $2.3\times 10^{17}{\text{kg/m}}^3$
• — far denser than ordinary matter.
• The strong nuclear force binds nucleons and is short-range, stronger than electric repulsion between protons.

• 1 atomic mass unit (u) = 1.66 × 10⁻²⁷ kg ≈ 931.5 MeV of energy.
• Binding energy per nucleon peaks near iron-56 (~8.8 MeV), the most stable nucleus.
• Nuclei lighter than iron gain energy by fusing; nuclei heavier than iron gain energy by fission.

• Decay law:
• $N=N_0{e}^{-\lambda t}$
• where λ is the decay constant.
• Activity:
• $A=\lambda N$
• , in becquerel (1 Bq = 1 decay/s).
• Mean life:
• $\tau =\frac{1}{\lambda }$
• .

• Carbon-14 dating uses the 5730-year half-life to date once-living material.
• Mean life is the average lifetime of a nucleus:
• $\tau =1.443T_{1/2}$
• .
• If the fraction remaining is f, the age is
• $t=\frac{\ln (1/f)}{\lambda }$
• .

• Fusion joins light nuclei (e.g. deuterium + tritium → helium + neutron) releasing enormous energy — the source of the Sun's power.
• Fusion needs very high temperatures (thermonuclear) and is not yet harnessed for power plants.
• Energy released:
• $Q=(m_{\text{initial}}-m_{\text{final}})c^2$
• .

Example: The half-life of a radioactive substance is 5 years. What fraction of a sample remains after 15 years?

Solution: 15 years = 3 half-lives, so the fraction remaining is ${\left(\frac{1}{2}\right)}^3=\frac{1}{8}$. Equivalently, using $N=N_0{e}^{-\lambda t}$ with $\lambda =0.693/5$ gives the same result.

Exercises (NCERT Q13.1–Q13.10)

• 13.1) Obtain the binding energy (in MeV) of a nitrogen nucleus ¹⁴₇N, given m(¹⁴₇N) = 14.00307 u.
• 13.2) Obtain the binding energy of the nuclei Fe (⁵⁶₂₆) and Bi (²⁰⁹₈₃) in units of MeV from the following data: m(⁵⁶₂₆Fe) = 55.934939 u; m(²⁰⁹₈₃Bi) = 208.980388 u.
• 13.3) A given coin has a mass of 3.0 g. Calculate the nuclear energy that would be required to separate all the neutrons and protons from each other. For simplicity assume that the coin is entirely made of ⁶³₂₉Cu atoms (of mass 62.92960 u).
• 13.4) Obtain approximately the ratio of the nuclear radii of the gold isotope ¹⁹⁷₇₉Au and the silver isotope ¹⁰⁷₄₇Ag.
• 13.5) The Q value of a nuclear reaction A + B → C + D is defined by Q = [m_A + m_B − m_C − m_D]c² where the masses refer to the respective nuclei. Determine from the given data the Q-value of the following reactions and state whether the reactions are exothermic or endothermic. (i) ¹₁H + ³₁H → ²₁H + ²₁H (ii) ¹²₆C + ¹²₆C → ²⁰₁₀Ne + ⁴₂He. Atomic masses are given to be m(²₁H) = 2.014102 u, m(³₁H) = 3.016049 u, m(¹²₆C) = 12.000000 u, m(²⁰₁₀Ne) = 19.992439 u.
• 13.6) Suppose, we think of fission of a ⁵⁶₂₆Fe nucleus into two equal fragments, ²⁸₁₃Al. Is the fission energetically possible? Argue by working out Q of the process. Given m(⁵⁶₂₆Fe) = 55.93494 u and m(²⁸₁₃Al) = 27.98191 u.
• 13.7) The fission properties of ²³⁹₉₄Pu are very similar to those of ²³⁵₉₂U. The average energy released per fission is 180 MeV. How much energy, in MeV, is released if all the atoms in 1 kg of pure ²³⁹₉₄Pu undergo fission?
• 13.8) How long can an electric lamp of 100 W be kept glowing by fusion of 2.0 kg of deuterium? Take the fusion reaction as ²₁H + ²₁H → ³₂He + n + 3.27 MeV.
• 13.9) Calculate the height of the potential barrier for a head on collision of two deuterons. (Hint: The height of the potential barrier is given by the Coulomb repulsion between the two deuterons when they just touch each other. Assume that they can be taken as hard spheres of radius 2.0 fm.)
• 13.10) From the relation R = R₀A^(1/3), where R₀ is a constant and A is the mass number of a nucleus, show that the nuclear matter density is nearly constant (i.e. independent of A).

Answers at a glance

1. 1
   Q13.1 → B.E. = 104.7 MeV.
2. 2
   Q13.2 → B.E.(Fe) = 492.3 MeV (8.79 MeV/nucleon); B.E.(Bi) = 1640.3 MeV (7.85 MeV/nucleon).
3. 3
   Q13.3 → E = 1.58 × 10²⁵ MeV = 2.53 × 10¹² J.
4. 4
   Q13.4 → R(Au)/R(Ag) = 1.23.
5. 5
   Q13.5 → (i) Q = −4.03 MeV, endothermic. (ii) Q = +4.62 MeV, exothermic.
6. 6
   Q13.6 → Q = −26.9 MeV — the fission of ⁵⁶Fe into two ²⁸Al nuclei is not energetically possible.
7. 7
   Q13.7 → E ≈ 4.5 × 10²⁶ MeV (4.54 × 10²⁶ MeV).
8. 8
   Q13.8 → t ≈ 5.0 × 10⁴ years (1.58 × 10¹² s).
9. 9
   Q13.9 → Barrier height = 360 keV.
10. 10
    Q13.10 → ρ = 3m_p/(4πR₀³), independent of A — nuclear density is a constant.

MCQ Questions

1 Mark Questions

1. 1
   Define the half-life of a radioactive nucleus.
2. 2
   State the mass–energy equivalence relation.
3. 3
   Define the mass defect of a nucleus.
4. 4
   Give the SI unit of activity of a radioactive sample.

2 Mark Questions

1. 1
   Calculate the energy equivalent of 1 amu in MeV. (Ans: 931.5 MeV)
2. 2
   A radioactive sample has a half-life of 8 days. What fraction of the original nuclei remains after 16 days? (Ans: 1/4)
3. 3
   The activity of a sample is 6 × 10⁶ Bq and its decay constant is 2 × 10⁻⁴ s⁻¹. Find the number of undecayed nuclei. (Ans: N = A/λ = 3 × 10¹⁰)

3 Mark Questions

1. 1
   Explain mass defect and binding energy. Sketch the binding energy per nucleon versus mass number curve and state the region of maximum stability.
2. 2
   Give the nature, charge, mass and penetrating power of α, β and γ radiations emitted by a radioactive nucleus.
3. 3
   State the law of radioactive decay and obtain the relation N = N₀e^(−λt) from it.

5 Mark Questions

1. 1
   Compare nuclear fission and nuclear fusion with suitable examples (e.g. U-235 fission and deuterium–tritium fusion), and explain how the release of energy follows from the binding energy per nucleon curve.
2. 2
   Derive the radioactive decay law N = N₀e^(−λt). Define the decay constant, half-life and activity, and obtain the relation between half-life and decay constant.