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Nuclear Physics: Half-Life and Radioactive Decay
Radioactive decay is one of the most predictable processes in physics — any given nucleus has a fixed probability of decaying in any time interval, independent of temperature, pressure, or chemical state. This predictability makes nuclear calculations both straightforward and powerful, particularly for dating and safety applications.
Half-Life: After n Half-Lives, (1/2)ⁿ Remains
The half-life (t₁/₂) is the time for half of any initial quantity to decay. After each half-life, exactly half remains. After 1 half-life: 50% remains. After 2 half-lives: 25% remains. After 3 half-lives: 12.5% remains. After 10 half-lives: (1/2)¹⁰ = 0.098% remains — less than 0.1% of the original. The general formula is: remaining fraction = (1/2)ⁿ where n = elapsed time / t₁/₂. The Half-Life Calculator handles all these calculations for any isotope.
Carbon-14 Dating
Carbon-14 has a half-life of 5,730 years, making it useful for dating organic materials up to about 50,000 years old (about 8-9 half-lives, where less than 0.4% of the original C-14 remains). After 11,460 years (two half-lives), 25% of the original C-14 remains. After 22,920 years (four half-lives), only 6.25% remains. For an organic sample showing 60% of the original C-14 activity: time elapsed = t₁/₂ × log₂(1/0.60) = 5,730 × 0.737 ≈ 4,220 years old.
Exponential Decay: N(t) = N₀ × e^(-λt)
The continuous decay equation N(t) = N₀ × e^(−λt) gives the exact quantity remaining at any time t, where λ is the decay constant (λ = ln(2) / t₁/₂). For carbon-14: λ = 0.693 / 5,730 = 0.000121 per year. After 3,000 years: N/N₀ = e^(−0.000121 × 3000) = e^(−0.363) ≈ 0.696, so about 69.6% remains. The Radioactive Decay Calculator applies this formula for any isotope given the decay constant or half-life.
Practical Applications
Common isotopes and their half-lives: Uranium-235 (703 million years) — used for geological dating; Carbon-14 (5,730 years) — archaeological dating; Caesium-137 (30.17 years) — radiation therapy and industrial gauges; Iodine-131 (8.02 days) — thyroid cancer treatment; Radon-222 (3.82 days) — home radon hazard assessment; Polonium-210 (138 days) — alpha particle source. The Half-Life Calculator and Radioactive Decay Calculator handle all these isotopes using either half-life or decay constant inputs.
Frequently Asked Questions
How do I calculate how much material remains after multiple half-lives?
Use remaining = original × (1/2)ⁿ where n = elapsed time / half-life. For Carbon-14 (t₁/₂ = 5,730 years) after 17,190 years: n = 17,190 / 5,730 = 3 half-lives, so (1/2)³ = 1/8 = 12.5% remains. You can also use N(t) = N₀ × e^(−λt) with λ = ln(2)/t₁/₂ for non-integer numbers of half-lives.
How does carbon dating work?
Living organisms maintain a constant ratio of C-14 to C-12 because they continuously exchange carbon with the environment. When they die, no new C-14 is incorporated, and existing C-14 decays with a half-life of 5,730 years. By measuring the current C-14/C-12 ratio and comparing to the known original ratio, scientists can calculate how long ago the organism died. The technique is accurate to about ±40 years for samples up to ~50,000 years old.
What is the relationship between half-life and decay constant?
The decay constant λ and half-life t₁/₂ are related by λ = ln(2) / t₁/₂ ≈ 0.693 / t₁/₂. A shorter half-life means a larger decay constant and faster decay. For iodine-131 (t₁/₂ = 8.02 days): λ = 0.693 / 8.02 = 0.0864 per day. This means about 8.64% of the remaining I-131 decays each day on average.