Section 10.6: Radioactivity

Radioactivity is the spontaneous emission of particles or energy from unstable atomic nuclei. It is a random process but follows statistical laws.

  • Types of Radiation:
    • Alpha (α) particles: He nuclei, low penetration, stopped by paper.
    • Beta (β) particles: Electrons or positrons, moderate penetration, stopped by thin metal.
    • Gamma (γ) rays: Electromagnetic radiation, high penetration, requires dense shielding.
  • Decay Law: \( N = N_0 e^{-\lambda t} \) where \( N \) = remaining nuclei, \( N_0 \) = initial nuclei, \( \lambda \) = decay constant, \( t \) = time.
  • Half-Life: \( t_{1/2} = \frac{\ln 2}{\lambda} \)

Example: Half-Life

A sample of a radioactive isotope has a half-life of 4 hours. If the initial mass is 80 g, find the remaining mass after 12 hours.

Number of half-lives: \( n = \frac{12}{4} = 3 \)
Remaining mass: \( m = 80 \times (1/2)^3 = 80 \times 1/8 = 10 \, \text{g} \)

Practice Problems

  1. An isotope has a decay constant \( \lambda = 0.693/5 \, \text{days}^{-1} \). Find its half-life.
  2. A 100 g sample decays to 12.5 g. How many half-lives have passed?
  3. Classify the type of radiation emitted in: \( {}^{210}_84Po \to {}^{206}_82Pb + \alpha \).
  4. A radioactive substance has 5,000 nuclei initially. After 3 hours, 2,500 nuclei remain. Find the half-life.
  5. Explain why gamma radiation is more penetrating than alpha radiation.
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