Learning outcomes
- Define half-life accurately.
- Find half-life from a decay curve or table.
- Perform repeated-halving calculations.
- Use corrected count rate or number of undecayed nuclei.
- Explain why activity never reaches exactly zero in the model.
9.1 Definition of half-life
The half-life of a particular isotope is the time taken for half the nuclei of that isotope in any sample to decay. It is also the time for the corrected activity or corrected count rate to fall to half its value, provided detection conditions remain constant.
Half-life is characteristic of the isotope and is not changed by the initial sample size, ordinary temperature, pressure or chemical state. A larger sample has a greater initial activity but the same half-life.
9.2 Repeated halving
After one half-life, 1/2 remains; after two, 1/4; after three, 1/8; after n half-lives, the fraction remaining is (1/2)ⁿ. The same factors apply to number of undecayed nuclei, mass of a pure radionuclide, activity and corrected count rate.
If a question asks how much has decayed, first find what remains, then subtract from the original. After three half-lives, 1/8 remains and 7/8 has decayed.
9.3 Finding half-life from a graph
Choose a corrected count rate on the curve, find half that value, and read the time difference between the two points. Repeat with another pair to check consistency. Do not assume the half-life is the time for the curve to reach zero.
If the graph shows raw measured count including background, subtract background first. A decay curve with a non-zero horizontal asymptote often indicates background has not been removed.
9.4 Tables and non-integer cases
In a table, identify times where the corrected rate halves. If values do not fall exactly at half, interpolate between nearby times when appropriate. Most O Level calculations use a whole number of half-lives, but the graph may require careful reading.
Calculate number of half-lives = elapsed time / half-life. Then halve the initial amount that many times. Keep units consistent.
9.5 Random fluctuations and trend
Individual count-rate readings may lie above or below a smooth exponential curve because decay is random. Draw a best-fit smooth curve rather than joining every point with straight segments.
The mathematical curve approaches zero but does not reach exactly zero. A real finite sample eventually may contain no unstable nuclei, but the ideal statistical model is continuous.
Worked examples
Remaining activity
Initial corrected rate is 960 counts/min and half-life is 6 h. After 18 h, three half-lives have passed: 960 → 480 → 240 → 120 counts/min.
Finding elapsed time
A sample falls from 8000 to 500 nuclei. The sequence 8000 → 4000 → 2000 → 1000 → 500 is four half-lives. If half-life is 3 days, elapsed time is 12 days.
Amount decayed
After two half-lives, 1/4 remains, so 3/4 or 75% has decayed.
Practical focus
Investigation
Plot supplied count-rate data against time. Subtract background, draw a smooth curve, estimate half-life from at least two different starting values and compare results. Add error bars if uncertainty information is supplied.
Examination guidance
- Use corrected count rate when background is given.
- Count the number of halvings before multiplying by half-life.
- Distinguish fraction remaining from fraction decayed.
- Half-life is not the time for all nuclei to decay.
- Read a smooth best-fit curve, not isolated fluctuations.
Check your understanding
- A 1600-count/min sample has half-life 5 min. What is its rate after 20 min?
- What fraction has decayed after three half-lives?
- Why should background be removed before finding half-life?
Answers
- 100 counts/min.
- 7/8 has decayed.
- Background does not decay with the source and would prevent the measured rate from showing true halving.
