Calculate successive half-lives and remaining fractions
Find remaining parent nuclei and distinguish them from nuclei that have already decayed.
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
- Divide elapsed time by a half-life.
- Distinguish parent isotope and total material.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which fraction has decayed after three half-lives?
What you will learn
- Find remaining parent nuclei and distinguish them from nuclei that have already decayed.
- Calculate remaining and decayed fractions and reverse complete half-life steps.
Count intervals then halve
For a fixed half-life, divide elapsed time by half-life to count complete intervals. With 800 parent nuclei and a half-life of 3 hours, 9 hours contains three half-lives: 800 → 400 → 200 → 100. One eighth remains and seven eighths have decayed.
The stated parent population is a mean model. The individual events are random, so the small integer values in a real sample need not exactly match every predicted mean.
Interpret mass and inverse questions
If a problem supplies the mass of the radioactive parent isotope, halve that parent-isotope mass in the same model. Decayed parent atoms become products; the statement does not mean all material has disappeared. A 24 g parent-isotope amount with a 2-day half-life has 3 g of parent remaining after 6 days.
For an inverse question, work backward by doubling for each completed interval. If 50 parent nuclei remain on the model after four half-lives, the initial mean population was 50 × 2 × 2 × 2 × 2 = 800.
Sources for this lesson
Pause and explain
When parent-isotope mass falls, must all that material disappear?
Worked example
For 800 parent nuclei, half-life 3 hours and elapsed time 9 hours, find remaining and decayed populations.
Show the worked solution
- 9 / 3 = 3 intervals.
- 800 / 8 = 100 remain.
- 800 − 100 = 700 have decayed.
Answer 100 remain; 700 have decayed.
Try a topic test
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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.
Find remaining parent nuclei from 800 after 9 hours with a 3-hour half-life.
Give me a hint
Count three intervals.
Compare my reasoning
- 9 / 3 = 3.
- Halve three times.
- 800 / 8 = 100.
100 on the mean model.
Look for these in your work
- Count the intervals.
- Give the remaining quantity.
A supplied 24 g parent-isotope amount has half-life 2 days. Find parent mass after 6 days.
Give me a hint
Use three half-lives and name the isotope account.
Compare my reasoning
- 6 / 2 = 3.
- 24 / 8 = 3.
- This is remaining parent mass, not total sample mass.
3 g of parent isotope.
Look for these in your work
- State grams.
- Distinguish parents from products.
A model has 50 parent nuclei after four half-lives. Find the initial population.
Give me a hint
Reverse four halvings.
Compare my reasoning
- Double four times.
- 50 × 16 = 800.
- Check 800 / 16 = 50.
800 initially on the model.
Look for these in your work
- Use inverse halving.
- Check the forward calculation.
Common mistakes
- Remaining and decayed fractions differ.
- Count intervals with consistent units.
- Parent isotope loss is not disappearance of all matter.
What can you explain now?
What remains after three half-lives?
Compare with the explanation
One eighth
The three successive factors are 1/2 × 1/2 × 1/2.
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Make it stick
Calculate remaining and decayed fractions and reverse complete half-life steps.
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.