The Memory Trick
💡 Half Remains Each Half-Life
A radioactive sample's half-life is the time it takes for exactly half of the remaining radioactive nuclei to decay. Crucially, this isn't a one-time event — after one half-life, 50% of the original sample remains; after two half-lives, 25% remains (half of the remaining half); after three, 12.5% remains — the sample keeps losing half of whatever is LEFT with each successive half-life interval.
Why It Works
Radioactive decay is a purely statistical, random process at the level of individual nuclei — each nucleus has a fixed probability of decaying in any given time interval, independent of how long it's already existed. This memoryless randomness is exactly what produces the exponential decay pattern, where a CONSTANT FRACTION (not a constant amount) decays in each equal time interval.
Step by Step
Working Through Half-Life Problems
1
Track the halving pattern directly
Starting with a known amount, repeatedly halve it for each half-life that passes — this is often faster and more intuitive than plugging into the full exponential decay formula for simple whole-number half-life counts.
Starting with 100 g and a half-life of 1 hour: after 1 hr → 50 g, after 2 hr → 25 g, after 3 hr → 12.5 g, after 4 hr → 6.25 g.
2
Half-life is a property of the specific isotope
Every radioactive isotope has its own characteristic, fixed half-life — ranging from fractions of a second to billions of years, depending on the specific nuclear species.
Carbon-14 has a half-life of about 5,730 years, which is exactly why it's useful for dating organic materials up to roughly 50,000 years old.
3
Real-world applications rely on this predictability
Because half-life is so precisely predictable and constant for a given isotope, it's used practically in carbon dating (archaeology/geology), nuclear medicine (diagnostic and treatment isotopes), and nuclear reactor design (fuel and waste management).
Carbon dating measures the remaining fraction of Carbon-14 in an organic sample and works backward using the known half-life to estimate the sample's age.
🏥 Worked Example
A radioactive sample starts at 80 g with a half-life of 2 hours. How much of the sample remains after 8 hours?
1
Find the number of half-lives elapsed: 8 hours ÷ 2 hours per half-life = 4 half-lives.
2
Halve the sample four times: 80 → 40 → 20 → 10 → 5 g.
3
Conclusion: 5 g of the original 80 g sample remains after 8 hours — exactly (½)⁴ = 1/16 of the original amount.
📌 Exam Application
Exams test correctly calculating the number of half-lives elapsed and applying repeated halving (or the exponential decay formula for non-whole-number half-life counts), and understanding that half-life describes a CONSTANT FRACTION decaying, not a constant amount.
⚠️ Most Common Radioactive Decay and Half-Life Mistakes
The most common trap is assuming a sample fully decays after two half-lives (mistakenly thinking 50%+50%=100%) — in reality, the sample never mathematically reaches exactly zero; it just keeps approaching zero, losing half of whatever remains with each successive half-life, indefinitely.
✓ Quick Self-Test
1) What is the definition of half-life? The time it takes for half of a radioactive sample's remaining nuclei to decay. 2) After two half-lives, what percentage of the original sample remains? 25%. 3) After three half-lives, what percentage remains? 12.5%. 4) Is half-life the same for every radioactive isotope, or does it vary? It varies — each isotope has its own fixed, characteristic half-life. 5) Name a real-world application of half-life. Carbon dating (also acceptable: nuclear medicine, reactor design).