# Half-life

> CIE A-Level Physics · Nuclear Physics
> Source: https://www.owlsprep.com/study/cie-9702-u27-half-life/

This module explains the definition of half-life, how to calculate it from experimental data and graphs, and its relationship with the decay constant. You will learn to solve common CIE exam problems on radioactive decay.

**Prerequisites:** [Radioactive decay law and decay constant](https://www.owlsprep.com/study/cie-9702-u27-radioactive-decay-laws/); Exponential functions and natural logarithms

## Learning objectives

- Define half-life of a radioactive nuclide
- Calculate half-life from decay data and graphs
- Relate half-life to decay constant using the standard formula
- Solve quantitative problems involving multiple half-lives

## Definition and Key Properties of Half-life

**Half-life** — The half-life of a radioactive isotope is the time taken for either: (1) the number of undecayed nuclei in a sample to decrease to half its original value, or (2) the activity of a sample to decrease to half its original activity. Half-life is constant for any given isotope, regardless of sample size.

*Notation:* t_{1/2}

*Example:* Carbon-14 has a half-life of ~5730 years, while Radon-220 has a half-life of 55.6 seconds.

Half-life describes the statistical behavior of large numbers of nuclei, since it is impossible to predict the decay of an individual nucleus. The constant value of half-life for a given isotope is a direct consequence of the random nature of radioactive decay.

**Worked example:** A sample initially contains $8.0 \times 10^{18}$ undecayed nuclei of an isotope with half-life 2.0 minutes. How many undecayed nuclei remain after 6.0 minutes?

1. First, calculate the number of half-lives that have passed:
2. $$n = \frac{\text{total time}}{t_{1/2}} = \frac{6.0}{2.0} = 3$$
3. After each half-life, the number of undecayed nuclei is halved, so the remaining number is:
4. $$N = N_0 \times \left(\frac{1}{2}\right)^n$$
5. Substitute the values into the formula:
6. $$N = 8.0 \times 10^{18} \times \left(\frac{1}{2}\right)^3 = 8.0 \times 10^{18} \times \frac{1}{8} = 1.0 \times 10^{18}$$
7. $1.0 \times 10^{18}$ undecayed nuclei remain after 6.0 minutes.

> **Exam tip:** CIE examiners accept either the number of nuclei or activity definition for full marks. Always mention half-life is constant for a given isotope to get all marks.

## Calculating Half-life from Data and Graphs

Half-life is most commonly found experimentally by plotting a graph of activity (or number of undecayed nuclei) against time. For exponential decay, the time interval between any activity value and half that value is constant.

> **tip**
>
> To reduce error when reading half-life from a graph, measure the interval between multiple pairs of half-values and take the average of your results.

**Worked example:** The table below shows activity of a radioactive sample over time:
| Time (s) | 0 | 20 | 40 | 60 | 80 |
|----------|---|----|----|----|----|
| Activity (Bq) | 120 | 61 | 30 | 15 | 7 |
What is the half-life of the isotope?

1. Start from the initial activity of 120 Bq. Half of this is 60 Bq. From the table, activity reaches 60 Bq at ~20 s.
2. Check by taking the next half-value: half of 60 Bq is 30 Bq, which occurs at ~40 s. The time interval between these two points is $40 - 20 = 20$ s.
3. Check again: 30 Bq drops to 15 Bq at 60 s, interval is $60 - 40 = 20$ s.
4. All intervals give approximately 20 s, so the half-life is 20 s.

**Check your understanding**

Test your understanding so far:

1. A sample has an initial activity of 240 Bq, half-life 10 minutes. What is the activity after 30 minutes?

   - 30 Bq
   - 80 Bq
   - 120 Bq
   - 60 Bq

   *Why:* 3 full half-lives pass: 240 → 120 → 60 → 30 Bq. Correct.

## Half-life and Decay Constant Relationship

**Derivation:** Derive the relationship between half-life $t_{1/2}$ and decay constant $\lambda$

*Starting from:* The radioactive decay law $N = N_0 e^{-\lambda t}$

1. When $t = t_{1/2}$, the number of undecayed nuclei is $N = \frac{N_0}{2}$. Substitute into the decay law:
2. $$\frac{N_0}{2} = N_0 e^{-\lambda t_{1/2}}$$
3. Cancel $N_0$ from both sides, then take the natural logarithm of both sides:
4. $$\ln\left(\frac{1}{2}\right) = -\lambda t_{1/2}$$
5. Use the log rule $\ln(1/2) = -\ln 2$ to simplify:
6. $$-\ln 2 = -\lambda t_{1/2}$$

*Conclusion:* Cancel the negative signs and rearrange to get the standard relationship:

$$t_{1/2} = \frac{\ln 2}{\lambda} \quad \text{or} \quad \lambda = \frac{\ln 2}{t_{1/2}}$$

**Worked example:** An isotope has a decay constant of $1.2 \times 10^{-4}$ s⁻¹. Calculate its half-life.

1. Recall $\ln 2 \approx 0.693$, substitute into the formula:
2. $$t_{1/2} = \frac{0.693}{1.2 \times 10^{-4} \text{ s}^{-1}} = 5775 \text{ s} \approx 5.8 \times 10^3 \text{ s}$$
3. The half-life is approximately $5.8 \times 10^3$ seconds (or ~96 minutes).

**Exam command terms**

Common CIE command terms for this topic:

- **Show that** — Verify the half-life value or relationship with decay constant. You must show all working steps. *(Always start from the decay law when asked to derive the $t_{1/2}$ relationship.)*

- **Determine** — Calculate half-life from data or a graph. You must show your method, not just give the final answer.

## Common pitfalls

- **Wrong:** Assuming half-life changes with the size of the sample
  - Why it fails: Half-life is an intrinsic property of the isotope, not the sample. It is always constant for a given isotope.
  - Correct: Use the same half-life value regardless of the initial amount of the sample.
- **Wrong:** Counting half-lives from time zero when starting from a non-zero starting point
  - Why it fails: The number of half-lives depends on the time elapsed since your starting activity, not the absolute time from the start of the experiment.
  - Correct: Calculate number of half-lives as $n = \frac{\text{end time} - \text{start time}}{t_{1/2}}$.
- **Wrong:** Forgetting to convert units when calculating half-life from decay constant
  - Why it fails: CIE examiners penalise incorrect or missing units even if the calculation is correct.
  - Correct: Always check the required units in the question and convert your final answer.
- **Wrong:** Using base 10 logarithm instead of natural logarithm for the $t_{1/2}$ formula
  - Why it fails: The decay law uses base $e$ exponential, so the derivation requires natural logarithm.
  - Correct: Always use natural logarithm for the half-life-decay constant relationship.

## Cheatsheet

| Concept | Formula / Rule |
| --- | --- |
| Half-life definition | Time for activity/nuclei to decrease by half |
| Number of half-lives | $n = t / t_{1/2}$ |
| Remaining activity/nuclei | $A = A_0 (1/2)^n, N = N_0 (1/2)^n$ |
| Relation to decay constant | $t_{1/2} = \ln 2 / \lambda \approx 0.693 / \lambda$ |
| Graphical method | Average time between multiple half-value pairs |
| Key property | Half-life is constant for any given isotope |

## What's next

Half-life is a core foundational concept for all nuclear physics topics in CIE A-Level. You will use half-life to solve problems in radioactive dating, which is a common application of exponential decay. Half-life also underpins the study of decay series, where multiple sequential decays occur, and is used to calculate activity levels for nuclear medicine and safety applications. Mastery of half-life is required for more advanced topics including nuclear binding energy and fission/fusion reactions that are assessed in Paper 4.

- [Nuclear fission and fusion](https://www.owlsprep.com/study/cie-9702-u27-nuclear-fission-and-fusion/)
- [Radiation safety](https://www.owlsprep.com/study/cie-9702-u27-radiation-safety/)
- [Medical imaging](https://www.owlsprep.com/study/cie-9702-u28-overview/)

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