Half-Life

IGCSE Edexcel Physics
7.11–7.13 Radioactive decay, activity and half-life calculations
Key Concepts: Half-life ($t_{1/2}$) is the time taken for the activity (or number of undecayed nuclei) of a radioactive sample to halve. Activity is measured in becquerels (Bq). Radioactive decay is random and spontaneous — it cannot be changed by temperature, pressure or chemical state.

Section A — Concepts

1. Define half-life. [2]
2. State what is meant by the activity of a radioactive source and give its unit. [2]
3. State two properties of radioactive decay that show it is random and spontaneous. [2]

Section B — Half-Life Calculations

4. A sample contains 400 undecayed nuclei and has a half-life of 5 years. How many undecayed nuclei remain after 10 years? [2]
5. A radioactive source has an activity of 800 Bq and a half-life of 2 hours. Calculate the activity after 6 hours. [2]
6. A sample starts with 6400 Bq. After 40 days the activity is 200 Bq. Calculate the half-life. [3]
7. An isotope has a half-life of 3 hours. A sample currently has an activity of 960 Bq. Calculate the activity 12 hours ago. [3]

Section C — Decay Graphs

8. Describe the shape of a graph of activity (y-axis) against time (x-axis) for a radioactive source. [2]
9. A student measures the activity of a sample every 10 minutes and records: 480 Bq, 240 Bq, 120 Bq, 60 Bq. State the half-life and justify your answer. [2]
10. Explain why the activity of a radioactive source never reaches exactly zero. [2]

Total marks: 22

Mark Scheme

1. The time taken for the activity of a radioactive source [1] to fall to half its initial value (or for the number of undecayed nuclei to halve) [1] [2]
2. Activity is the number of nuclear decays per second [1]; unit: becquerel (Bq) [1] [2]
3. It cannot be predicted when a particular nucleus will decay [1]; it is not affected by external factors such as temperature, pressure or chemical state [1] [2]
4. 10 years = 2 half-lives: $400 \rightarrow 200 \rightarrow 100$ nuclei [2]
5. 6 hours = 3 half-lives: $800 \rightarrow 400 \rightarrow 200 \rightarrow 100\,\text{Bq}$ [2]
6. $6400 \rightarrow 3200 \rightarrow 1600 \rightarrow 800 \rightarrow 400 \rightarrow 200$ — 5 half-lives [1]; $t_{1/2} = 40/5 = 8\,\text{days}$ [2] [3]
7. 12 hours ago = 4 half-lives before now; activity then was $960 \times 2^4 = 960 \times 16 = 15\,360\,\text{Bq}$ [3]
8. Exponential decay curve [1]; starting at a high value, decreasing rapidly at first then more slowly, approaching but never reaching zero [1] [2]
9. Activity halves every 10 minutes [1]; so the half-life = 10 minutes [1] [2]
10. There will always be some undecayed nuclei remaining [1]; since each nucleus decays independently and randomly, there is always a small (but non-zero) chance a nucleus has not yet decayed [1] [2]