What is Nuclear Decay and Half-Life?
Nuclear decay is the spontaneous breakdown of an unstable atomic nucleus, releasing energy and particles. Half-life is the time it takes for half of the original nuclei to decay — the key measure of radioactivity.
Nuclear decay follows exponential decay: N(t) = N₀ × (1/2)^(t/t₁/₂), where N₀ is initial nuclei, t is elapsed time, and t₁/₂ is the half-life.
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Step-by-step worked examples
Carbon-14 has a half-life of 5,730 years. If a fossil has 25,000 C-14 atoms initially, how many remain after 11,460 years?
t = 11,460 years = 2 half-lives (11,460 ÷ 5,730) N(t) = 25,000 × (1/2)² = 25,000 × 0.25 = 6,250 atoms
Iodine-131 has a half-life of 8 days. Start with 1 million atoms. How many remain after 24 days?
24 days = 3 half-lives (24 ÷ 8) N(t) = 1,000,000 × (1/2)³ = 1,000,000 × 0.125 = 125,000 atoms
Uranium-238 has a half-life of 4.5 billion years. Start with 80 kg. How much remains after 13.5 billion years?
13.5 billion = 3 half-lives (13.5 ÷ 4.5) Remaining = 80 × (1/2)³ = 80 × 0.125 = 10 kg
Flashcards
Quick quiz
Q1.After 2 half-lives, what percentage of nuclei remain?
Q2.The half-life of Strontium-90 is 29 years. True or false: after 58 years, all nuclei are gone?
Q3.Which type of radiation is least penetrating?
Q4.N(t) = 1000 × (1/2)^(t/10), where t is in hours. After 30 hours, N(t) = ?
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Common mistakes
After 2 half-lives, no nuclei remain. — Correct: After 2 half-lives, 25% remain. Decay is exponential, never reaching zero.
Half-life is the time for complete decay. — Correct: Half-life is the time for half the nuclei to decay. Complete decay takes infinite time.
All isotopes decay at the same rate. — Correct: Each isotope has its own unique half-life.
If you double the initial amount, the half-life doubles. — Correct: Half-life is constant; only the number of nuclei changes.
FAQ
What is the nuclear decay formula?
N(t) = N₀ × (1/2)^(t/t₁/₂), where N(t) is remaining nuclei, N₀ is initial nuclei, t is time, and t₁/₂ is half-life.
What happens after one half-life?
Exactly half of the original radioactive nuclei have decayed; the other half remain.
Is radioactive decay dangerous?
Yes — high-energy radiation can damage cells. Lead and concrete provide shielding.
How is half-life used in medicine?
Radioactive tracers with short half-lives are used in PET scans and other diagnostics.