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Half-Life Calculator | Solve half-life time or quantity

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Models radioactive decay and exponential decay problems. Enter any three of the initial quantity, remaining quantity, elapsed time, and half-life, and the tool solves for the missing value using N = N0 × (1/2)^(t/T), with the working shown. Used for radiometric dating, medication dosing, and decay homework.

Last updated: June 2026 · Free · No sign-up required

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Results

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Quick reference

Half-lives elapsedFraction remaining
150%
225%
312.5%
46.25%
10≈ 0.1%

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Interpretation guide

IsotopeHalf-life
Carbon-145,730 years
Iodine-1318.02 days
Uranium-2384.47 billion years
Technetium-99m6.01 hours

lightbulb Iodine-131 decaying over 24 days

A thyroid treatment dose contains 500 mg of iodine-131, which has a half-life of 8.02 days. After 24 days, how much remains? The elapsed time covers 24 ÷ 8.02 ≈ 2.99 half-lives.

Result: N = 500 × (1/2)^(24/8.02) = 500 × (1/2)^2.9925 ≈ 500 × 0.1256 ≈ 62.8 mg.

What this means: Just under one-eighth remains — exactly what you'd expect after nearly three half-lives (1/2³ = 1/8). This is why iodine-131 is useful medically: it's potent when administered and essentially gone within weeks, and the same math schedules safe follow-up doses.

Formula & methodology

Formula: N = N0 x (1/2)^(t / T); T = half-life; decay constant lambda = ln(2) / T

How half-life is calculated

The tool applies the exponential decay law N = N₀ × (1/2)^(t ÷ T), where N₀ is the initial amount, t the elapsed time, and T the half-life — the time for half the substance to decay. The decay constant λ = ln(2) ÷ T links the half-life to the exponential rate. Enter any three of the four variables and the tool solves for the missing one.

Example: a 100 mg sample of a substance with an 8-day half-life has 100 × (1/2)^(16÷8) = 25 mg remaining after 16 days — exactly two half-lives.

Reading the result: after n half-lives, 1/2ⁿ of the original remains, which is why the curve halves repeatedly rather than declining at a constant rate. Half-life math underlies radioactive dating, medical dosing, and drug elimination in pharmacology.

Authoritative source: Wolfram MathWorld

Frequently asked questions

What is half-life?
Half-life is the time it takes for a quantity to decrease to half its original value. After one half-life, 50% remains; after two, 25%; after three, 12.5%, and so on. It is a constant for a given substance, which is why it is used to characterize radioactive isotopes and drug elimination.
How do you calculate the remaining amount after several half-lives?
Use N = N0 x (1/2)^(t/T), where N0 is the starting amount, t is elapsed time, and T is the half-life. Divide the elapsed time by the half-life to get the number of half-lives, then halve the starting amount that many times. 80 g after 3 half-lives is 80 x 0.125 = 10 g.
How do you find the half-life from the decay constant?
The half-life T equals the natural log of 2 divided by the decay constant lambda: T = ln(2) / lambda ≈ 0.693 / lambda. Conversely, lambda = ln(2) / T. The decay constant is the probability per unit time that a given atom decays, so a larger lambda means a shorter half-life.
Does half-life depend on the starting amount?
No. Half-life is independent of how much you start with — it always takes the same time to halve, whether you begin with 1 gram or 1 kilogram. This is the defining feature of exponential (first-order) decay and is what makes carbon dating and drug dosing predictable.
How is half-life used in carbon dating?
Living things absorb carbon-14, which stops replenishing at death and decays with a 5,730-year half-life. By measuring how much carbon-14 remains compared to a living sample and applying N = N0 (1/2)^(t/T), scientists solve for t — the time since death. This reliably dates organic material up to roughly 50,000 years old.
Can the page solve for the half-life itself?
Yes — give it any three of the four quantities and it finds the fourth. Enter starting amount, remaining amount, and elapsed time, and it inverts N = N₀(1/2)^(t/T) to isolate T. That's how you'd determine an unknown isotope's half-life from two activity measurements.
What is the decay constant λ, and why are there two formulas?
The decay constant λ = ln(2)/T ≈ 0.693/T is the per-unit-time probability of decay — the exponential rate. N = N₀e^(−λt) and N = N₀(1/2)^(t/T) are the same law in different clothes: substitute λ and the exponents match exactly. Physicists prefer λ; the half-life form is more intuitive.
Does half-life depend on how much you start with?
No — that's the defining feature of exponential decay. Whether you begin with a gram or a kilogram, half is gone after one half-life and a quarter remains after two. The page's results scale linearly with the starting amount for exactly this reason.
Why do real measurements scatter around the decay curve?
The math describes the expected value for a large sample; real counting has statistical noise, background radiation, and possible impurities. A Geiger counter reading near the predicted 62.8 mg might show 60 or 66 — the curve is exact, the measurement isn't. Average multiple readings to converge on it.

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