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After n half-lives, the fraction remaining is 1/2ⁿ, where n = total time ÷ half-life time. Free GCSE AQA Physics notes with key definitions and 13 FAQs.
Half-Life Calculations (HT) is part of Atoms and nuclear radiation in GCSE AQA Higher Physics.
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After n half-lives, the fraction remaining is 1/2ⁿ, where n = total time ÷ half-life time. Free GCSE AQA Physics notes with key definitions and 13 FAQs.
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Use this lesson to stay inside Atoms and nuclear radiation while connecting the detail back to the wider Atomic structure topic.
Read the core explanations from Half-Life Calculations (HT) before testing yourself from memory.
Imagine you have 1000 radioactive atoms. After one half-life, 500 remain. After another half-life, 250 remain. Then 125, then 62.5...
This halving pattern is completely predictable, even though individual atoms decay randomly. The half-life is like a countdown timer that never changes for a particular isotope.
Here's the key insight: after n half-lives, the fraction remaining is always $\frac{1}{2^n}$. This gives us a powerful calculation tool.
To find the number of half-lives, divide the total time by the half-life time: $n = \frac{\text{total time}}{\text{half-life time}}$
Worked example: Half-life calculation with ratios
Carbon-14 has a half-life of 5730 years. A fossil contains 1/8 of its original carbon-14. How old is the fossil?
Answer: 17,190 years
Activity measures how fast nuclei are decaying right now. It follows the same halving pattern as the number of nuclei.
If you start with 1000 Bq, after one half-life you have 500 Bq. After two half-lives, 250 Bq. The same $\frac{1}{2^n}$ rule applies.
Worked example: Activity decline calculation
A radioactive source has initial activity 2400 Bq. Its half-life is 8 hours. What is the activity after 24 hours?
Answer: 300 Bq (ratio 1:8 compared to original)
Exam questions often ask for the 'net decline' expressed as a ratio. This means comparing what's left to what you started with.
If you start with 1000 nuclei and end with 125 nuclei, the ratio is 125:1000, which simplifies to 1:8. The net decline is the difference: 875 nuclei have decayed.
Worked example: Net decline ratio calculation
A sample starts with 3200 radioactive nuclei. After 15 minutes, 200 nuclei remain. Express the net decline as a ratio.
Answer: 1:16 (or 1/16 of original remains)
Half-life questions are mark-heavy in exams because they test both mathematical skills and scientific understanding. Here's how to maximise your marks.
The biggest trap is mixing up 'fraction remaining' with 'fraction decayed'. If 1/4 remains, then 3/4 has decayed. Always check which one the question asks for. Also, be careful with your fraction calculations – for example, an expression like 8/8 equals 1, not 0.125.
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Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
The half-life of carbon-14 is 5730 years. A fossil containing 1/8 of its original carbon-14 has been through 3 half-lives, so it is 3 × 5730 = 17,190 years old.
Activity is the rate at which radioactive nuclei decay in a sample, measured in becquerels (Bq). It follows the same halving pattern as the number of nuclei, so the 1/2ⁿ rule applies.
After n half-lives, the fraction remaining is 1/2ⁿ, where n = total time ÷ half-life time. After 3 half-lives the fraction remaining is 1/2³ = 1/8, so a 2400 Bq source with an 8 hour half-life has 300 Bq after 24 hours.
Write the fraction remaining as a power of 2 to find the number of half-lives, then multiply by the half-life. Carbon-14 has a half-life of 5730 years, so a fossil with 1/8 of its original carbon-14 has passed 3 half-lives and is 3 × 5730 = 17,190 years old.
Net decline is the total reduction in radioactive nuclei or activity over a given time, often expressed as a ratio of final to initial values. A sample that starts with 3200 nuclei and ends with 200 has a ratio of 200 : 3200, which simplifies to 1:16.
The fraction remaining is the part of the radioactive nuclei still left, while the fraction decayed is the part that has gone. If 1/4 remains, then 3/4 has decayed; starting with 1000 nuclei and ending with 125, 875 nuclei have decayed.
The number of half-lives is found by dividing the total time by the half-life: n = total time ÷ half-life time. For a half-life of 8 hours over 24 hours, n = 24 ÷ 8 = 3 half-lives.
A source with initial activity 2400 Bq and a half-life of 8 hours has an activity of 300 Bq after 24 hours. That is 3 half-lives, so the fraction remaining is 1/2³ = 1/8, and 2400 × 1/8 = 300 Bq.
Radioactive decay is expressed as a ratio by comparing the final amount to the initial amount and simplifying. A sample that starts with 3200 nuclei and has 200 left gives 200 : 3200, which simplifies to 1 : 16.
A fraction of 1/16 remaining means 4 half-lives have passed, because after n half-lives the fraction remaining is 1/2ⁿ and 16 = 2⁴. In the same way, 1/8 means 3 half-lives, since 8 = 2³.
If 125 of 1000 nuclei remain, 875 have decayed; this difference is the net decline. The ratio of final to initial is 125 : 1000, which simplifies to 1 : 8.
The fraction remaining after n half-lives is 1/2ⁿ. The same rule applies to the number of nuclei, the activity and the count rate, so 1000 Bq becomes 500 Bq after one half-life and 250 Bq after two.
A common fraction mistake in half-life calculations is writing 8/8 as 0.125 when it equals 1. Another trap is mixing up fraction remaining with fraction decayed: if 1/4 remains, then 3/4 has decayed.