Explain why two samples of the same radioactive isotope, one containing twice as many atoms as the other, will have the same half-life but different activities.

IB DP Physics Higher Level (2023 syllabus) — E.3 Radioactive decay · Explain · 4 marks · View as Markdown

Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).

Model answer (4 marks)

Activity is proportional to the number of undecayed nuclei (A = λN). The larger sample has twice as many nuclei, so its activity is twice that of the smaller sample.
The decay constant λ depends only on the isotope, not on sample size, so both samples share the same λ.
Half‑life is t½ = ln2/λ. Because λ is identical for both, t½ is the same.
After one half‑life each sample has lost half its nuclei; the larger still has twice as many undecayed nuclei and therefore twice the activity at every stage.

Examiner tips

  • Use the formula A = λN to link activity to N.
  • State that λ is a property of the isotope only.
  • Show the relationship t½ = ln2/λ to explain equal half‑lives.
  • Mention that the larger sample always has twice the activity.

Mark scheme (4 marks)

  1. Activity is proportional to the number of undecayed nuclei present (A = λN), so the larger sample has greater activity.
  2. The decay constant λ depends only on the nature of the nucleus (the isotope), not on the size of the sample, so both samples share the same λ.
  3. Half-life is related to the decay constant by t½ = ln2 / λ, so since λ is the same for both samples, the half-life is the same.
  4. After one half-life each sample retains half of its original nuclei, so the larger sample still has twice as many undecayed nuclei and twice the activity as the smaller sample throughout the decay process.

Key terms in this question

half-life

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