Explain why a graph of the number of undecayed nuclei N against time t for a radioactive sample produces an exponential curve, and explain what feature of the curve allows the half-life to be determined without knowing the initial number of nuclei.
Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).
Model answer (4 marks)
The rate of decay (activity) is proportional to the number of undecayed nuclei present at that instant, so dN/dt=−λN. This proportionality means the fractional rate of change of N is constant (equal to the decay constant λ), which is the defining condition for exponential decay. The half‑life is the time interval for N to halve from any chosen value on the curve, not just from the start. This constancy of the half‑life (a property of exponential functions) means N₀ need not be known; the same time interval for halving is read directly from the graph at any convenient point.
Examiner tips
- Show the proportionality dN/dt=−λN to justify exponential form.
- Explain that a constant fractional change gives an exponential curve.
- State that the half‑life is the time for N to halve from any point on the curve.
- Mention that the half‑life can be read from the graph without N₀.
Common mistakes
- Confusing the decay constant λ with the half‑life; they are related but not the same.
- Failing to recognise that the half‑life can be taken from any point, not just the start.
- Using a linear or quadratic model instead of exponential for the decay curve.
Mark scheme (4 marks)
- The rate of decay (activity) is proportional to the number of undecayed nuclei present at that instant.
- This proportionality means the fractional rate of change of N is constant (equal to the decay constant λ), which is the defining condition for exponential decay.
- The half-life is the time interval for N to halve from any chosen value on the curve, not just from the start.
- This constancy of the half-life (a property of exponential functions) means N₀ need not be known; the same time interval for halving is read directly from the graph at any convenient point.
Key terms in this question
Related
- All IB DP Physics Higher Level (2023 syllabus) revision notes →
- How to answer a "Explain" question →
- Decode the mark scheme abbreviations →
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