Half-life

CIE A-Level Physics· Unit 27: Nuclear physics, Topic 3· 10 min read

1. Definition and Key Properties of Half-life★★☆☆☆⏱ 3 min

📘 Definition

Half-life

t_{1/2}

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.

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 $8.0 \times 1 0^{18}$ 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
$1.0 \times 1 0^{18}$ 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.

2. Calculating Half-life from Data and Graphs★★★☆☆⏱ 4 min

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.

📐 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.

✓ Quick check

Test your understanding so far:

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
Reveal answer
30 Bq —
3 full half-lives pass: 240 → 120 → 60 → 30 Bq. Correct.

3. Half-life and Decay Constant Relationship★★★☆☆⏱ 3 min

🔬 Derivation

Goal:

Derive the relationship between half-life $t_{1/2}$ and decay constant $λ$

Starting from:

The radioactive decay law $N = N_{0} e^{- λ t}$

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
$-\ln 2 = -\lambda t_{1/2}$

Result:

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 $1.2 \times 1 0^{- 4}$ 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 $5.8 \times 1 0^{3}$ seconds (or ~96 minutes).

4. Common Pitfalls

Wrong move:

Assuming half-life changes with the size of the sample

Why:

Half-life is an intrinsic property of the isotope, not the sample. It is always constant for a given isotope.

Correct move:

Use the same half-life value regardless of the initial amount of the sample.

Wrong move:

Counting half-lives from time zero when starting from a non-zero starting point

Why:

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 move:

Calculate number of half-lives as $n = \frac{end time - start time}{t _{1/2}}$ .

Wrong move:

Forgetting to convert units when calculating half-life from decay constant

Why:

CIE examiners penalise incorrect or missing units even if the calculation is correct.

Correct move:

Always check the required units in the question and convert your final answer.

Wrong move:

Using base 10 logarithm instead of natural logarithm for the $t_{1/2}$ formula

Why:

The decay law uses base $e$ exponential, so the derivation requires natural logarithm.

Correct move:

Always use natural logarithm for the half-life-decay constant relationship.

5. Quick Reference 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	$t_{1/2} = ln 2/ λ \approx 0.693/ λ$
Graphical method	Average time between multiple half-value pairs
Key property	Half-life is constant for any given isotope

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · Paper 1
Half-life calculation from data
2023 · Paper 2
Graphical determination of half-life
2021 · Paper 4
Half-life decay constant relation

Going deeper

syllabusCIE 9702 2025-2027 SyllabusSee page 47 for half-life learning outcomes

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.