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Lesson 11.5 • Competency 11.5Complete Lesson
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Radioactivity and Nuclear Reactions

Master nuclear composition (A, Z, N), nuclear radius R = R₀ A^(1/3), constant nuclear density, mass defect Δm, binding energy per nucleon curve, radioactive decay law N = N₀ e^(-λt), half-life T_½ = ln 2 / λ, activity A = λN, and fission/fusion energy release.

Marking a lesson complete represents finishing your study of the topic; it does not denote tested mastery.

Learning Objectives

  • Calculate nuclear radius and explain why nuclear density is approximately constant across all stable nuclei (~2.3 × 10¹⁷ kg/m³).
  • Determine mass defect Δm and calculate nuclear binding energy per nucleon using Einstein’s equivalence ΔE = Δm c² (1 u = 931.5 MeV).
  • Solve radioactive decay, remaining count N(t), half-life T_½, and activity problems using exponential decay equations.

Core Theory & Concepts

The atomic nucleus is held together against intense proton-proton electrostatic repulsion by the short-range, charge-independent strong nuclear force. When nucleons bind into a nucleus, the total mass is strictly less than the sum of the individual free nucleon masses. This difference, known as the mass defect, represents binding energy through Einstein’s relation ΔE = Δm c². Unstable nuclei undergo spontaneous radioactive decay to achieve greater nuclear stability.

Mass Defect & Binding Energy

Mass defect Δm = [Z m_p + (A - Z) m_n] - M_nucleus. Binding Energy BE = Δm (in u) × 931.5 MeV. The BE/A curve peaks near Fe-56 (~8.8 MeV/nucleon).

\Delta m = [Z m_p + (A - Z) m_n] - M_{\text{nucleus}}, \quad BE = \Delta m \cdot c^2

Radioactive Decay Law & Half-Life

Activity is A = -dN/dt = λN. Number of remaining nuclei follows exponential decay: N(t) = N₀ e^(-λt). Half-life is T_½ = ln(2) / λ ≈ 0.693 / λ.

N(t) = N_0 e^{-\lambda t}, \quad T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}

Common Misconceptions

Misconception

Half-life decreases as a radioactive sample gets older and decays.

Physical Clarification:Half-life T_½ is an intrinsic, time-invariant property of the specific nuclear isotope that is completely independent of the age, mass, temperature, pressure, or chemical combination of the sample.

Syllabus Topic Outline

Competency 11.5
  • 1Nuclear composition: atomic number Z, neutron number N, mass number A
  • 2Nuclear radius R = R₀ A^(1/3) and constant nuclear matter density
  • 3Strong nuclear force characteristics (short range, charge independent, attractive)
  • 4Mass defect Δm and nuclear binding energy: BE = Δm c² (1 u = 931.5 MeV)
  • 5Binding energy per nucleon curve: peak at Iron-56 and stability analysis
  • 6Radioactive decay law: N(t) = N₀ e^(-λt) and Activity A = λN
  • 7Half-life relation: T_½ = ln 2 / λ ≈ 0.693 / λ
  • 8Alpha, beta, and gamma radiation properties and penetration hazards
  • 9Nuclear fission in reactors and nuclear fusion in the Sun