Half-Life

1. Definition

The half-life, t_(1/2), is the time taken for:

• the number of undecayed nuclei to fall to half, or
• the activity to fall to half.

For an unchanged detector setup, the net count-rate is proportional to activity, so it also halves on average. Raw measured count-rate does not fall toward zero because background remains.

2. Key Ideas

• After each half-life, the remaining fraction halves: 1/2, 1/4, 1/8, …
• If n half-lives have passed: N_final/Nᵢₙᵢₜᵢₐₗ = (1/2)ⁿ
• If time t has passed: n = t/(t_(1/2)) (when t is a multiple of t_(1/2))
• With a GM tube, subtract background first: net count-rate = measured - background

3. Detailed Explanations

A. Half-life using repeated halving (fast method)

If the question is “after 3 half-lives”, you can do:

N → N/2 → N/4 → N/8

Same idea for activity/count-rate.

B. Finding half-life from a decay curve

1. Use the net count-rate (subtract background).
2. Choose an activity value A on the curve.
3. Find the time for the activity to drop to A/2.
4. That time difference is the half-life.

Because decay is random, real count-rate points fluctuate. Use the overall curve or repeated halving intervals rather than expecting every reading to lie exactly on a smooth line.

Finding half-life from a decay curve (with background subtraction)

A schematic decay curve showing why you subtract background first, then measure the time for the net count-rate to halve.

Scroll across the graph to read all labels.

Open full-size graph

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Values and uncertainty for Finding half-life from a decay curve (with background subtraction)

Series	Time (min)	Time uncertainty	Count-rate (cpm)	Count-rate uncertainty
Measured count-rate (source + background)	0		340
Measured count-rate (source + background)	2		180
Measured count-rate (source + background)	4		100
Measured count-rate (source + background)	6		60
Measured count-rate (source + background)	8		40
Measured count-rate (source + background)	10		30
Background count-rate	0		20
Background count-rate	10		20
Net count-rate (measured − background)	0		320
Net count-rate (measured − background)	2		160
Net count-rate (measured − background)	4		80
Net count-rate (measured − background)	6		40
Net count-rate (measured − background)	8		20
Net count-rate (measured − background)	10		10

Values and uncertainty for Finding half-life from a decay curve (with background subtraction)

Series	Time (min)	Time uncertainty	Count-rate (cpm)	Count-rate uncertainty
Measured count-rate (source + background)	0		340
Measured count-rate (source + background)	2		180
Measured count-rate (source + background)	4		100
Measured count-rate (source + background)	6		60
Measured count-rate (source + background)	8		40
Measured count-rate (source + background)	10		30
Background count-rate	0		20
Background count-rate	10		20
Net count-rate (measured − background)	0		320
Net count-rate (measured − background)	2		160
Net count-rate (measured − background)	4		80
Net count-rate (measured − background)	6		40
Net count-rate (measured − background)	8		20
Net count-rate (measured − background)	10		10

4. Common Mistakes

• Using background-included readings for half-life (subtract background first).
• Halving by a fixed amount instead of a fixed fraction.
• Mixing up “half-life” with “time for all nuclei to decay” (it never reaches zero in the ideal model).
• Halving the raw measured count-rate instead of the background-corrected rate.
• Assuming the half-life changes when the starting sample size changes.

5. Exam Tips

1. State the definition clearly (half of nuclei / half of activity).
2. Use (1/2)ⁿ for non-trivial questions.
3. In decay curve questions: show clearly where you halved the activity.

6. Worked Examples

7. Mind Stretchers

8. Practice and next step

Read half-life from both tables and curves in the Radioactivity & Half-Life Explorer, then complete the Radioactivity Structured Practice.