Define half-life and calculate the decay constant
Use half-life as a time interval and calculate λ = 0.693/t₁/₂ with matching time units.
A Level · Cambridge International Physics 9702, 2025–2027 syllabus. The Topic 23 outcomes were also checked against Cambridge’s 2028–2030 syllabus, which reports no significant teaching changes. Online practice is self-checked and cannot replace supervised practical work, full timed assessment or human subject-teacher review.
Before you begin
- Use scientific notation, SI units, proportional reasoning and the particle numbers in nuclide notation.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
After three half-lives, what fraction of the original nuclei remains?
What you will learn
- Define half-life
- Use λ = 0.693/t₁/₂
- Keep the units of decay constant consistent with the half-life
Half-life describes a repeated halving
The half-life is the time taken for the activity, number of undecayed nuclei or source count rate to decrease to half its initial value. It is a characteristic time for a particular radioactive nuclide.
After one half-life, half remains; after two, one quarter remains; after three, one eighth remains. The elapsed time must be measured from the stated starting value.
Calculate λ with matching units
For exponential decay, λ = 0.693/t₁/₂. The decay constant has the reciprocal unit of the half-life: a half-life in minutes gives λ in min⁻¹, while seconds give s⁻¹.
Convert the half-life before substituting only when the required λ unit is specified. Activity in Bq requires λ in s⁻¹ in A = λN.
Pause and explain
A half-life is 8.0 s. What is λ in s⁻¹?
Worked example
A nuclide has a half-life of 20.0 minutes. Find its decay constant in min⁻¹.
Show the worked solution
- Use λ = 0.693/t₁/₂.
- Substitute 20.0 min: λ = 0.693/20.0.
- Keep the reciprocal minute unit.
Answer λ = 0.0347 min⁻¹ (3 s.f.)
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A source’s activity halves in 12.0 days. Find λ in day⁻¹.
Give me a hint
Divide 0.693 by the half-life in days.
Compare my reasoning
- λ = 0.693/12.0 days.
- λ = 0.05775 day⁻¹.
- Round to three significant figures.
0.0578 day⁻¹.
Look for these in your work
- Uses the stated half-life
- Divides 0.693 by time
- Uses reciprocal-day units
A half-life is 2.0 hours. Calculate λ in s⁻¹.
Give me a hint
Convert 2.0 hours to seconds first.
Compare my reasoning
- t₁/₂ = 2.0 × 3600 = 7200 s.
- λ = 0.693/7200 s.
- λ = 9.63 × 10⁻⁵ s⁻¹.
9.6 × 10⁻⁵ s⁻¹ (2 s.f.).
Look for these in your work
- Converts hours to seconds
- Uses λ = 0.693/t₁/₂
- Reports s⁻¹
An activity falls from 640 Bq to 80 Bq. How many half-lives have elapsed?
Give me a hint
Count repeated factors of two from 640 to 80.
Compare my reasoning
- 640 → 320 after one half-life.
- 320 → 160 after two; 160 → 80 after three.
- Three halvings are required.
Three half-lives.
Look for these in your work
- Halves the value repeatedly
- Counts each interval
- Does not confuse the final fraction with elapsed half-lives
Common mistakes
- Defining half-life as the time for every nucleus to decay
- Giving λ the same unit as time instead of the reciprocal
- Substituting a half-life in hours when the question asks for s⁻¹
What can you explain now?
After three half-lives, what fraction of the original nuclei remains?
Compare with the explanation
One eighth
Each half-life halves the amount: 1 → 1/2 → 1/4 → 1/8.
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Make it stick
What unit should λ have if the half-life is given in minutes?
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