Find half-life when measured counts include background
Subtract a stated constant background before choosing the source halving level.
Section P5.2.4 · Half-life
Mapped to the named 0654 section using teaching from the corresponding single-science course. A 0654-specific review of every Core and Supplement objective is still in progress. Written practice is self-checked; it does not demonstrate supervised laboratory competence or award an official Cambridge grade.
Before you begin
- Calculate corrected count rates.
- Read source-only half-life curves.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which total reading marks one half-life here?
What you will learn
- Subtract a stated constant background before choosing the source halving level.
- Find background-corrected half-life and interpret the nonzero total-curve limit.
Correct before halving
A measured mean record includes a constant 20 counts/s background. Total readings at 0, 2, 4 and 6 minutes are 180, 100, 60 and 40 counts/s. Subtract 20 from each to obtain source rates 160, 80, 40 and 20 counts/s. Their half-life is 2 minutes.
Do not halve the initial total of 180 to seek 90. Half the source contribution is 80, and the background remains 20, so the correct measured target is 100. The background is not part of the decaying source.
Read the offset curve
The ideal total-reading curve approaches the background level, not zero. In general the measured half-source target is background + (initial measured rate − background) / 2. Keep the background and measured rates in matching units and under matching detector conditions.
This is a mean model with a supplied stable background. Real measurements fluctuate; short noisy intervals require repeated observations and suitable uncertainty judgement. Do not interpret an estimated negative corrected reading as physically negative source activity.
Sources for this lesson
Pause and explain
What level does the ideal total curve approach at long times?
Worked example
For background 20 counts/s and initial total 180 counts/s, find the measured reading after one source half-life.
Show the worked solution
- Initial source rate = 180 − 20 = 160.
- Half the source is 80.
- Add the unchanged background: 80 + 20 = 100.
Answer 100 counts/s.
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.
Find half-life from total rates 180, 100, 60, 40 at 0, 2, 4, 6 min with background 20 counts/s.
Give me a hint
Subtract 20 from each reading.
Compare my reasoning
- Corrected rates are 160, 80, 40, 20.
- Each halving takes 2 min.
- The source half-life is 2 min.
2 minutes.
Look for these in your work
- Correct all readings.
- Check repeated halving intervals.
Background is 10 counts/s. Total rate falls from 90 to 50 counts/s in 5 min. Find source half-life.
Give me a hint
Compare corrected rates.
Compare my reasoning
- Initial corrected rate is 80.
- Final corrected rate is 40.
- The source halves in 5 min.
5 minutes.
Look for these in your work
- Use constant matched background.
- State the interval unit.
For the 20-counts/s background curve, a learner uses total 180 → 90 to estimate half-life. Correct the target.
Give me a hint
Apply halving only to the source contribution.
Compare my reasoning
- 180 − 20 = 160.
- 160 / 2 = 80.
- The measured target is 80 + 20 = 100.
Use total 100 counts/s, not 90.
Look for these in your work
- Preserve the background.
- Distinguish total and corrected readings.
Common mistakes
- Subtract before halving.
- Add background back for a measured target.
- The total curve approaches background.
What can you explain now?
What is the corrected rate for total 180 and background 20 counts/s?
Compare with the explanation
160 counts/s
Subtract the matched background rate from the total.
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Make it stick
Find background-corrected half-life and interpret the nonzero total-curve limit.
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.