Retrieval Practice

Radioactive Decay & Half-Life — AQA A-Level Physics

Q1. Define radioactive decay and state its two key properties.
  • The spontaneous disintegration of a nucleus to form a more stable nucleus by emitting radiation.
  • It is random (equal probability for each nucleus, unpredictable) and spontaneous (unaffected by external conditions).
Q2. Define the decay constant λ and state its unit.
  • The probability that an individual nucleus will decay per unit of time.
  • Unit: s⁻¹.
Q3. State the equation relating activity, decay constant and number of nuclei.
  • A = λN.
  • Activity (Bq) = decay constant (s⁻¹) × number of undecayed nuclei.
Q4. Write the exponential decay equation for the number of undecayed nuclei.
N = N₀\(e^{−λt}\), where N₀ is the initial number of nuclei, λ is the decay constant, and t is time.
Q5. State two other quantities that follow the same exponential decay law as N.
  • Activity: A = A₀\(e^{−λt}\).
  • Count rate: C = C₀\(e^{−λt}\).
  • Both are proportional to N.
Q6. Define half-life.
  • The average time taken for a given number of nuclei of a particular isotope to halve.
  • Also the time for the activity to halve.
Q7. Derive the half-life equation from N = N₀\(e^{−λt}\).
  • Set N = ½N₀: ½ = \(e^{−λt₁/₂}\).
  • Take ln: −ln 2 = −λt₁/₂.
  • Therefore t₁/₂ = ln 2 / λ ≈ 0.693 / λ.
Q8. How do you find half-life from a decay curve?
  • Read the initial activity from the y-axis.
  • Draw a horizontal line at half this value to the curve, then draw vertically down to the time axis.
  • The time read off is the half-life.
Q9. On a log graph of ln N against t, what does the gradient represent?
  • The gradient is −λ (negative of the decay constant).
  • The y-intercept is ln N₀.
Q10. Explain why carbon dating is unreliable for samples older than 60,000 years.
After ~10 half-lives (5730 × 10 ≈ 57,000 years), the remaining C-14 activity is too low to distinguish from background radiation.
Q11. What isotope is used for uranium-lead dating and what is its half-life?
  • Uranium-238, with a half-life of 4.5 billion years.
  • It decays through a chain to stable lead-206.
Q12. State the equation to find the number of nuclei in a sample of known mass.
N = (m × Nₐ) / M, where m = mass in grams, Nₐ = 6.02 × 10²³ mol⁻¹, M = molar mass in g mol⁻¹.
Q13. How do you rearrange N = N₀\(e^{−λt}\) to find the time t?
  • Divide both sides by N₀: N/N₀ = \(e^{−λt}\).
  • Take ln: ln(N/N₀) = −λt.
  • Therefore t = −(1/λ) ln(N/N₀).
Q14. What fraction of the original nuclei remain after 3 half-lives?
(½)³ = 1/8 = 12.5% of the original nuclei remain.
Q15. Why is Pu-238 used to power space probes?
  • It is an alpha emitter with a long half-life (87.7 years), providing steady heat output for decades. 1 g generates ~500 mW.
  • Alpha radiation is easily shielded.
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