Free Half-Life Calculator
Calculate how much of a radioactive (or any exponentially decaying) substance is left after a given time using the half-life decay law N = N₀·(½)^(t/t½).
Enter the initial amount, the half-life, and the elapsed time (using the same time unit for both times) to find how much remains.
N = N₀·(½)^(t/t½). Time t and half-life t½ must share the same unit (seconds, days, years, …) so they cancel in the exponent. Decay is exponential, so the amount halves every t½ and approaches — but never reaches — zero.
Quick answer
Radioactive decay follows N = N₀·(½)^(t/t½), where N₀ is the starting amount, t is the elapsed time, and t½ is the half-life. For example, with N₀ = 100, t½ = 5730 years, and t = 5730 years, exactly one half-life has passed, so N = 100·(½)¹ = 50 — half the sample remains. Keep t and t½ in the same time unit and the result is independent of those units.
Formula & method
Remaining amount after time t
N = N₀ · (½)^(t / t½)
- N — amount (mass, atoms, or activity) remaining after time t
- N₀ — initial amount at t = 0
- t — elapsed time
- t½ — half-life — time for the amount to fall to one half
t and t½ must use the same time unit (seconds, days, years, …); the units cancel in the exponent. Equivalent forms: N = N₀·2^(−t/t½) and N = N₀·e^(−λt) with decay constant λ = ln(2)/t½ ≈ 0.693/t½.
Fraction remaining and number of half-lives
fraction = N / N₀ = (½)^n, n = t / t½
n is the number of half-lives elapsed. Percent remaining = 100 · (½)^n. After n = 1, 2, 3, 10 half-lives, 50%, 25%, 12.5%, and ≈0.098% remain.
Examples
- Input
- N₀ = 100, t½ = 5730 years, t = 5730 years
- Result
- N = 50 (50% remaining, 1 half-life)
- Why
- n = t/t½ = 5730/5730 = 1, so N = 100·(½)¹ = 100·0.5 = 50. Exactly half the carbon-14 is left, which is the meaning of a single half-life.
- Input
- N₀ = 100, t½ = 5730 years, t = 11460 years
- Result
- N = 25 (25% remaining, 2 half-lives)
- Why
- n = 11460/5730 = 2, so N = 100·(½)² = 100·0.25 = 25. Each half-life halves the previous amount: 100 → 50 → 25.
- Input
- N₀ = 200 mg, t½ = 14 days, t = 30 days
- Result
- N ≈ 45.2862 mg (≈22.6431% remaining, ≈2.142857 half-lives)
- Why
- n = 30/14 ≈ 2.142857, so N = 200·(½)^2.142857 = 200·0.226431 ≈ 45.2862 mg. The exponent need not be a whole number — decay is continuous.
- Input
- N/N₀ = 0.25 (25% of carbon-14 left), t½ = 5730 years
- Result
- t = 11460 years
- Why
- Rearrange to t = t½·log₍½₎(N/N₀) = t½·ln(0.25)/ln(0.5) = 5730·2 = 11460 years. Because 25% = (½)², the sample is two half-lives old.
When to use this tool
- Estimating how much of a radioactive isotope (carbon-14, iodine-131, uranium-238, etc.) remains after a known elapsed time.
- Radiocarbon or radiometric dating — solving for the age of a sample from the fraction of parent isotope still present.
- Pharmacokinetics and clinical dosing — gauging how much of a drug with a known elimination half-life is left in the body between doses.
- Any first-order / exponential decay problem: capacitor discharge, reaction kinetics, or concentration decline that halves over a fixed interval.
Common mistakes
- Mixing time units — t in days but t½ in years (or vice versa). Both must be in the same unit so they cancel in t/t½; convert one first.
- Treating decay as linear: assuming half is gone after t½ and all of it after 2·t½. After two half-lives 25% still remains, not 0%; decay is exponential and never reaches exactly zero.
- Confusing half-life t½ with the mean lifetime τ = 1/λ. They are not equal: τ = t½/ln(2) ≈ 1.4427·t½.
- Rounding the number of half-lives to a whole number. The exponent t/t½ can be fractional (e.g. 2.14), and you must keep it fractional for an accurate remaining amount.
Frequently asked questions
What is the half-life formula?
The remaining amount is N = N₀·(½)^(t/t½), where N₀ is the starting amount, t is the elapsed time, and t½ is the half-life. Equivalent forms are N = N₀·2^(−t/t½) and N = N₀·e^(−λt) with λ = ln(2)/t½. The fraction remaining is simply (½) raised to the number of half-lives, t/t½.
How do I calculate how much is left after a certain time?
Divide the elapsed time by the half-life to get the number of half-lives n = t/t½, then multiply the initial amount by (½)^n. For example, after 3 half-lives N = N₀·(½)³ = N₀·0.125, so 12.5% remains. This calculator does the arithmetic for any t and t½ instantly.
Does the substance ever fully disappear?
No. Mathematically, exponential decay approaches zero but never reaches it — each half-life removes only half of what is currently present. In practice an amount becomes negligible after about 10 half-lives (≈0.098% remaining), but it is never exactly zero.
How do I find the half-life if I know the decay constant λ?
Use t½ = ln(2)/λ ≈ 0.693/λ. Conversely, λ = ln(2)/t½. The decay constant λ is the probability per unit time that a given atom decays, while the half-life is the time for half of a large sample to decay.
How is half-life used in carbon dating?
Carbon-14 has a half-life of about 5730 years. By measuring the fraction of carbon-14 remaining in a sample relative to a living reference, you solve N/N₀ = (½)^(t/t½) for t: t = t½·ln(N/N₀)/ln(0.5). If 25% remains, that is two half-lives, so the sample is about 11,460 years old.
Can I use this calculator for drug half-lives?
Yes. Drug elimination is usually first-order, so the same formula applies. Enter the dose as N₀, the drug's elimination half-life as t½, and the time since the dose as t (in the same unit) to estimate the amount still in the body. After about 4–5 half-lives roughly 94–97% has cleared.
Sources & references
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Provided “as is” for general information only — results may be inaccurate, so verify before you rely on them. No warranty; use at your own risk.
Built and reviewed by HIFreeTools against the formula shown above and any authoritative references cited on this page. See our methodology and editorial standards.
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