The Half-Life Calculator finds how much of a radioactive sample remains after a given time, and converts among half-life, mean lifetime and decay constant. Choose an isotope preset or enter a custom half-life; the tool applies the exponential decay law and can mark the 50%, 25% and 12.5% points on the decay curve.
Half-life is the time for half the nuclei in a sample to decay. After one half-life, half remains; after two, a quarter; after three, an eighth. The underlying process is exponential, not linear.
Calculate the quantity remaining after a time
Remaining quantity equals the initial quantity multiplied by one-half raised to the number of half-lives elapsed. Enter N₀, t½ and t; the calculator returns N(t). The same inputs also determine how many half-lives have passed: n = t / t½.
| Symbol | Quantity | Unit |
|---|---|---|
| N₀ | initial quantity | same as N (count, mass, activity units) |
| N(t) | quantity at time t | same units as N₀ |
| t½ | half-life | seconds, or any consistent time unit |
| t | elapsed time | same unit as t½ |
| λ | decay constant | 1 / time |
| τ | mean lifetime | same unit as t½ |
N₀ can be a nucleus count, a mass of isotope, or an activity figure, as long as N(t) uses the same unit. The exponential factor is dimensionless.
Time units must match: years with years, days with days. Mixing years of half-life with seconds of elapsed time without conversion produces nonsense.
Apply the half-life formula
Radioactive decay follows an exponential law: each equal time interval multiplies the remaining quantity by the same factor. When that interval is one half-life, the factor is exactly one-half, which produces the compact power-of-two form used in most isotope tables and dating problems.
N(t) = N₀ × (1/2)^(t / t½)
N(t) = N₀ × e^(−λt)With λ = ln(2) / t½, the two expressions are identical. The calculator uses the half-life form when t½ is the given nuclear data, which is how isotope tables are usually published.
After exactly one half-life, N = N₀/2. After exactly two, N = N₀/4. Non-integer numbers of half-lives are fine: t / t½ = 1.5 gives N = N₀ / (2^1.5) ≈ 0.3536 N₀.
Convert between half-life, mean lifetime and decay constant
Nuclear data sheets may list half-life, mean life or decay constant depending on the field. All three describe the same exponential rate and convert through factors of the natural logarithm of two. Mixing half-life with mean lifetime without converting is a frequent source of wrong remaining fractions.
λ = ln(2) / t½ ≈ 0.693 / t½
τ = 1 / λ = t½ / ln(2) ≈ 1.443 × t½
t½ = ln(2) / λ ≈ 0.693 / λFor carbon-14 with t½ = 5,730 years:
τ ≈ 5,730 / ln(2) ≈ 8,266.6 years
λ ≈ ln(2) / 5,730 ≈ 0.000121 / year
Enter any one of t½, τ or λ and the calculator fills the other two. The remaining-quantity path still needs N₀ and t in addition to the rate.
Calculate remaining carbon-14 after 11,460 years
Carbon-14's half-life of 5,730 years makes 11,460 years exactly two half-lives. Starting from an initial quantity of 100 units of C-14, the remaining amount after that elapsed span is 25 units, which is the engine test fixture for remaining quantity.
1. List what is known. N₀ = 100, t½ = 5,730 y, t = 11,460 y.
2. Count half-lives. t / t½ = 11,460 / 5,730 = 2
3. Apply the formula. N = 100 × (1/2)² = 100 × 1/4 = 25
Mean lifetime for the same isotope is τ ≈ 8,266.6 years. That figure is not needed for the remaining-quantity step, but it is the correct τ if a problem asks for mean life rather than half-life.
After three half-lives (17,190 years) the remaining amount would be 12.5. After four, 6.25. The calculator accepts any positive t, not only integer multiples of t½.
Select an isotope preset
Presets load published half-lives so a problem that names carbon-14, iodine-131 or uranium-238 does not require a separate data lookup. Selecting a row fills t½ from the shared reference table; initial quantity and elapsed time remain for the user to enter.
| Isotope | Half-life | Typical use |
|---|---|---|
| Carbon-14 | 5,730 years | Radiocarbon dating to ~50,000 years |
| Uranium-238 | 4.468 billion years | Oldest rocks; age of the Earth |
| Uranium-235 | 703.8 million years | Nuclear fuel; geological dating |
| Potassium-40 | 1.248 billion years | Potassium-argon dating of volcanic rock |
| Iodine-131 | 8.02 days | Thyroid imaging and treatment |
| Technetium-99m | 6.01 hours | Medical imaging tracer |
| Cobalt-60 | 5.27 years | Radiotherapy; industrial radiography |
| Caesium-137 | 30.17 years | Contaminant after nuclear accidents |
| Strontium-90 | 28.79 years | Fallout product; accumulates in bone |
| Radon-222 | 3.82 days | Indoor air hazard from soil gas |
| Plutonium-239 | 24,110 years | Nuclear fuel and weapons material |
| Tritium (H-3) | 12.32 years | Self-luminous signs; fusion research |
Short-lived medical isotopes need hours or days as the time unit. Geological isotopes need millions or billions of years. Match the unit of t to the unit of t½.
For iodine-131, λ ≈ 0.0864 /day when t½ = 8.02 days, which matches the conversion fixture in the engine tests.
Read the decay curve
Exponential decay falls quickly at first and then flattens toward zero without reaching it in finite time. Markers at one, two and three half-lives sit at 50%, 25% and 12.5% of N₀. The calculator can place those markers on the curve drawn from the current N₀ and t½.
| Half-lives elapsed | Fraction remaining |
|---|---|
| 0 | 100% |
| 1 | 50% |
| 2 | 25% |
| 3 | 12.5% |
| 4 | 6.25% |
| 5 | 3.125% |
The curve never hits exactly zero on the plot; at ten half-lives about 0.1% remains. Activity (decays per second) follows the same exponential factor when the detection efficiency is constant.
A linear plot of N versus t curves downward. A plot of ln N versus t is a straight line with slope −λ, which is how half-lives are often extracted from count data.
Understand radiocarbon dating
Radiocarbon dating estimates the time since an organism stopped exchanging carbon with the atmosphere by measuring how much carbon-14 remains relative to stable carbon. The physical decay step uses the 5,730-year half-life; full archaeological ages also apply calibration curves for past atmospheric C-14 levels.
The practical limit is about 50,000 years, roughly eight or nine half-lives, beyond which so little C-14 remains that measurement uncertainty dominates. Calibration against tree rings and other records adjusts for past changes in atmospheric C-14; the simple exponential on this page is the physical decay step, not a full archaeological age report.
Enter N₀ as 100% (or as an initial activity) and t as the age to see the remaining percentage. Solving for t from a measured remaining fraction is the dating direction: t = t½ × log₂(N₀/N).
Uranium and potassium systems date rocks on geological timescales where carbon-14 is long gone.
Frequently asked questions
What is half-life?
Half-life is the time required for half the radioactive nuclei in a sample to decay. After one half-life, half remain; after two, a quarter remain. It is a fixed property of each isotope under ordinary conditions.
How do I calculate remaining quantity?
Use N(t) = N₀ × (1/2)^(t/t½). For N₀ = 100, t½ = 5,730 years and t = 11,460 years, exactly two half-lives elapse and N = 25.
What is the difference between half-life and mean lifetime?
Mean lifetime τ = t½ / ln(2) ≈ 1.443 × t½. It is the average life of a nucleus before decay. Half-life is shorter. For C-14, t½ = 5,730 y and τ ≈ 8,266.6 y.
How are half-life and decay constant related?
λ = ln(2) / t½. A short half-life means a large decay constant and a steep exponential. The calculator converts either way.
Which isotope preset should I choose?
Pick the isotope named in the problem, or the one that matches the application: carbon-14 for organic dating, technetium-99m or iodine-131 for nuclear medicine, uranium isotopes for deep time. Custom half-lives are available when the nuclide is not listed.
Why does the remaining amount never reach zero?
Exponential decay multiplies by one-half each half-life. That product approaches zero as t grows but does not hit zero at a finite time. In practice, after enough half-lives the remaining activity falls below detection.
How does radiocarbon dating use half-life?
After death, C-14 in organic material decays with t½ = 5,730 years while stable carbon does not. Comparing the remaining C-14 fraction to the initial fraction estimates the time since death, up to about 50,000 years, with calibration for past atmospheric levels.
Can I solve for the time elapsed?
Yes. From N and N₀, t = t½ × log₂(N₀/N) = t½ × ln(N₀/N) / ln(2). Enter N₀, N and t½ to recover t when the calculator mode allows solving for time.
Do chemical bonds change nuclear half-life?
For practical purposes in this calculator, no. Nuclear half-lives are set by the nucleus. Ordinary temperature and chemistry do not change t½ for the isotopes listed here.
What does activity mean?
Activity is the number of decays per unit time, often in becquerels (one decay per second). For a given isotope, activity is proportional to the number of undecayed nuclei, so it follows the same exponential factor as N(t). Halving N halves the activity when detection efficiency is unchanged.
How many half-lives until almost nothing is left?
After ten half-lives about 0.1% of the original quantity remains (1/1024). After twenty, about one part in a million remains. "Almost nothing" depends on the detector and the starting amount; the mathematics never reaches exact zero at finite time.
Summary
The Half-Life Calculator applies N(t) = N₀ × (1/2)^(t/t½) and converts among half-life, mean lifetime and decay constant with factors of ln(2). Carbon-14 at t½ = 5,730 years leaves 25 of an initial 100 after 11,460 years, and has mean lifetime about 8,266.6 years.
Isotope presets fill half-lives from the shared reference table for dating, medicine and teaching problems. The decay curve marks successive halvings at 50%, 25% and 12.5%.