What are Nuclear Reactions and Decay?
Nuclear decay occurs when unstable atomic nuclei lose energy by emitting particles or radiation, transforming into new elements. This spontaneous process is the basis of radioactivity and is described by the exponential decay law.
Radioactive decay follows N(t) = N₀ × (1/2)^(t/t₁/₂), where N₀ is initial nuclei, t is time, and t₁/₂ is half-life. Each decay type (α, β, γ) involves different particles and conserves mass and charge numbers.
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Step-by-step worked examples
Carbon-14 has a half-life of 5,730 years. A fossil contains 12.5% of its original ¹⁴C. How old is it?
12.5% = 1/8 of original → (1/2)³ = 1/8 So the sample has gone through 3 half-lives. Age = 3 × 5,730 = 17,190 years.
A sample starts with 8,000 nuclei. After 20 days, 500 remain. Find the half-life.
Use N(t) = N₀ × (1/2)^(t/t₁/₂) 500 = 8000 × (1/2)^(20/t₁/₂) 0.0625 = (1/2)^(20/t₁/₂) (1/2)⁴ = (1/2)^(20/t₁/₂) 4 = 20/t₁/₂ t₁/₂ = 5 days
Uranium-235 has a half-life of 704 million years. What fraction decays in 1 million years?
N(t)/N₀ = (1/2)^(1,000,000 / 704,000,000) = (1/2)^(0.00142) ≈ 0.9990 About 99.9% remains; only 0.1% (1/1000) decays in 1 million years.
Flashcards
Quick quiz
Q1.After 3 half-lives, what percentage of the original sample remains?
Q2.In alpha decay, what is emitted?
Q3.If a sample's half-life is 10 days and you start with 1000 nuclei, how many remain after 30 days?
Q4.What is conserved in all nuclear reactions?
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Common mistakes
Half-life means the sample disappears after one half-life. — Correct: After one half-life, 50% remains; after two, 25%; after three, 12.5%. Theoretically, a tiny amount always remains.
Alpha decay produces a new element with fewer protons but same neutrons. — Correct: Alpha decay reduces both protons (atomic number) by 2 and nucleons (mass number) by 4, creating a different element.
Decay rate is constant (linear). — Correct: Decay is exponential: the rate depends on the current amount. Fewer nuclei → slower decay.
Only heavy nuclei decay; light elements are stable forever. — Correct: Most nuclei with Z > 83 (bismuth) are radioactive, but some light nuclei (C-14, K-40) are also radioactive.
FAQ
How is half-life used in carbon dating?
Organic matter absorbs C-14 during life. After death, C-14 decays at a known rate. Measuring remaining C-14 relative to C-12 reveals age, accurate to ~50,000 years.
Why do some nuclei decay and others don't?
Nuclei with too many neutrons or protons relative to the stable configuration (nuclear binding) become unstable and decay to reach a more stable state.
What is the difference between alpha and beta decay?
Alpha: a helium nucleus (⁴₂He) is ejected; mass number drops by 4, atomic number by 2. Beta: an electron (or positron) is ejected; atomic number changes by 1, mass number stays same.
Can we predict when a specific nucleus will decay?
No, individual decay is random and unpredictable. Only the average behavior of many nuclei follows the half-life law.