Sketch and use exponential radioactive decay
Sketch the decay curve and calculate activity, undecayed nuclei or source count rate with x = x₀e⁻λᵗ.
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
What happens to an exponential decay curve on linear axes at long times?
What you will learn
- Describe the exponential nature of radioactive decay
- Sketch and use x = x₀e⁻λᵗ
- Use the relation for activity, undecayed nuclei or source count rate
Exponential decay is a constant fractional decrease
The relation x = x₀e⁻λᵗ models radioactive decay, where x₀ is the initial quantity and x is its value after time t. The quantity x may be activity, number of undecayed nuclei or source count rate.
On ordinary linear axes, the curve falls steeply at first and then flattens toward zero. It does not become negative and does not reach exactly zero after a finite time in this model.
Substitute time and decay constant with compatible units
The product λt must be dimensionless. If λ is in min⁻¹, use t in minutes; if λ is in s⁻¹, use seconds. Keep the initial and final quantities in the same type and unit.
When interpreting measured count rate, subtract a stated constant background first if the model is for the source alone. A background offset should not be allowed to decay with the source.
Pause and explain
For x = x₀e⁻λᵗ, what must be true of the product λt?
Worked example
A source has initial activity 800 Bq and decay constant 2.00 × 10⁻³ min⁻¹. Find its activity after 500 minutes.
Show the worked solution
- Use A = A₀e⁻λᵗ.
- λt = 2.00 × 10⁻³ × 500 = 1.00.
- A = 800e⁻¹ = 294 Bq to three significant figures.
Answer 294 Bq (3 s.f.)
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
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 sample has N₀ = 1.2 × 10⁶ nuclei and λ = 0.010 s⁻¹. Find N after 100 s.
Give me a hint
Find λt, then multiply N₀ by e raised to the negative result.
Compare my reasoning
- λt = 0.010 × 100 = 1.0.
- N = 1.2 × 10⁶e⁻¹.
- e⁻¹ ≈ 0.368, so N ≈ 4.4 × 10⁵.
4.4 × 10⁵ nuclei (2 s.f.).
Look for these in your work
- Calculates λt
- Uses the negative exponent
- Keeps the same nuclei quantity
Sketch activity against time for a radioactive source and mark one half-life.
Give me a hint
Show a smooth curve, not a sequence of equal absolute drops.
Compare my reasoning
- Start at A₀ when t = 0.
- Mark A₀/2 at t = t₁/₂.
- Draw a decreasing curve that flattens toward zero.
A decreasing exponential curve through (t₁/₂, A₀/2), approaching zero.
Look for these in your work
- Correct initial value
- Half-life point marked
- Curve approaches zero without crossing it
A detector records 150 counts per minute initially and 60 counts per minute after 5 minutes. A constant background is 10 counts per minute. What fraction of the source count rate remains?
Give me a hint
Subtract background from both readings before forming the ratio.
Compare my reasoning
- Initial source rate = 150 − 10 = 140 counts per minute.
- Later source rate = 60 − 10 = 50 counts per minute.
- Fraction remaining = 50/140 = 0.357 to three significant figures.
0.357 of the initial source count rate (about 35.7%).
Look for these in your work
- Subtracts background from each reading
- Compares source-only rates
- Calculates a dimensionless fraction
Common mistakes
- Using e⁺λᵗ instead of e⁻λᵗ
- Mixing hours and seconds between λ and t
- Applying the exponential relation to a total count rate with an unremoved constant background
What can you explain now?
What happens to an exponential decay curve on linear axes at long times?
Compare with the explanation
It approaches zero and becomes less steep
The rate of decrease slows as the remaining quantity falls.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
Why must λ and t use compatible units in x = x₀e⁻λᵗ?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.