Half-Life Calculator
Calculate the remaining amount of a radioactive substance after a given time using the half-life decay formula. Also find elapsed time or half-life.
Remaining Amount
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Percent Remaining —
Half-Lives Elapsed —
Elapsed Time —
Half-Life —
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Remaining Amount
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Percent Remaining —
Half-Lives Elapsed —
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Remaining Amount
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Percent Remaining —
Percent Decayed —
Half-Lives Elapsed —
Elapsed Time —
Half-Life —
Decay Constant λ —
Mean Lifetime (τ) —
How to Use This Calculator
- Select what to solve for: Remaining Amount, Elapsed Time, or Half-Life.
- Enter the initial amount, half-life, and elapsed time (or the two known values).
- Results show remaining amount, percentage remaining, and number of half-lives elapsed.
Formula
N = N₀ × (½)^(t ÷ t½)
t = t½ × log(N₀ ÷ N) ÷ log(2)
t½ = t × log(2) ÷ log(N₀ ÷ N)
Example
Example: N₀ = 100 g, t½ = 5,730 yr, t = 11,460 yr → N = 25 g (25% remaining, 2 half-lives).
Frequently Asked Questions
- A half-life (t½) is the time it takes for exactly half of a radioactive substance (or unstable quantity) to decay or transform into another substance. After one half-life, 50% of the original amount remains. After two half-lives, 25% (50% of 50%) remains. After three, 12.5% remains. The pattern follows: Remaining % = (1/2)ⁿ × 100%, where n is the number of half-lives elapsed. Half-life is constant for a given isotope — it does not depend on the initial amount, temperature, pressure, or chemical state of the substance.
- Carbon-14 (¹⁴C) has a half-life of 5,730 years. It is formed in the upper atmosphere when cosmic ray neutrons strike nitrogen-14, and it is incorporated into living organisms through the food chain at a constant ratio. When an organism dies, it stops absorbing new ¹⁴C, and the existing amount decays. By measuring the ratio of ¹⁴C to stable ¹²C in a sample and comparing to the known atmospheric ratio, scientists can date organic materials up to about 50,000 years old. This technique, called radiocarbon dating, was developed by Willard Libby in the 1940s.
- Use the half-life decay formula: N = N₀ × (1/2)^(t ÷ t½), where N is the remaining amount, N₀ is the initial amount, t is the elapsed time, and t½ is the half-life. For example, starting with 100 g of Carbon-14 (t½ = 5,730 years), after 11,460 years (two half-lives): N = 100 × (1/2)^(11460 ÷ 5730) = 100 × (1/2)² = 100 × 0.25 = 25 g. Equivalently, use N = N₀ × e^(−λt) where λ = ln(2) ÷ t½ is the decay constant. Both formulas give identical results.
- A radioactive substance never fully disappears — the amount approaches zero asymptotically. However, for practical purposes: after 7 half-lives, 0.78% remains; after 10 half-lives, 0.098% remains; after 20 half-lives, 0.0001% remains. Nuclear safety guidelines for radioactive waste typically use 10 half-lives as a benchmark for when activity becomes negligible. Medical radioactive tracers are chosen with short half-lives (hours to days) so they clear the body quickly. For long-lived isotopes like Uranium-238 (half-life = 4.47 billion years), decay is practically imperceptible in human timeframes.
- Yes — rearrange the half-life formula to solve for time: t = t½ × log(N₀ ÷ N) ÷ log(2). This is equivalent to t = t½ × ln(N₀ ÷ N) ÷ ln(2). For example, if an initial 100 g sample is now 10 g, and the half-life is 5,730 years: t = 5730 × log(100÷10) ÷ log(2) = 5730 × 1 ÷ 0.30103 = 19,035 years (approximately 3.32 half-lives). In this calculator, select "Solve for: Elapsed Time (t)", enter the initial amount, remaining amount, and half-life, and the time is calculated automatically.