How to use
Solve radioactive decay forwards or backwards, compare remaining and decayed amounts, and explore a normalized half-life graph with sourced isotope presets.
Choose Remaining Amount, Elapsed Time, Half-Life or Starting Amount. Enter the known quantities using one shared amount unit. Choose Custom for your own positive half-life or select a source-verified isotope preset. The isotope search filters five presets; the current selection stays available. Half-Life mode automatically uses Custom because the half-life is being calculated.
Results update in this tab, including the primary answer, remaining and decayed quantities, percentages, half-lives elapsed, graph and milestone table. Open Show calculation for the substitution steps. Copy Result includes only accepted inputs and calculated quantities, plus source metadata for an active preset. Reset restores Carbon-14, 100 g, 11,400 years and its sourced 5,700-year half-life. Clipboard failure provides selectable text. Refresh clears edits.
Half-life formula
N = N₀ × (1/2)^(t/T½)
- N₀
- Starting amount of the parent isotope.
- N
- Expected remaining amount of that parent isotope.
- t
- Elapsed time, zero or positive.
- T½
- Positive constant half-life.
Both amounts share the selected unit. Seconds, minutes, hours, days and years convert to canonical seconds before calculating t/T½. One minute is 60 seconds, one hour 3,600 seconds, one day 86,400 seconds, and one year is exactly 365.2422 days for this calculator. This fixed elapsed-duration convention matches the year definition stated in the Cesium-137 adopted data; it does not count calendar years or leap days. Cobalt-60 is retained in its source-reported days.
How radioactive decay works
Radioactive decay changes unstable parent nuclei into other nuclei or states. Under the constant-half-life model, each half-life removes half of the expected surviving parent population. Equal time intervals therefore remove equal fractions, rather than equal absolute amounts.
An individual nucleus decays randomly; half-life does not predict its particular decay time. Fractional results for atoms or nuclei represent expected population quantities, not pieces of an atom. The continuous model is most useful for populations with many nuclei.
How many half-lives have passed
The number of half-lives elapsed is n = t/T½. At n = 0, the starting amount remains unchanged. At n = 1, half remains. At n = 2, one quarter remains. At n = 2.5, about 17.6777% remains. Fractional half-lives follow the same exponential equation; time need not be a whole multiple of T½.
Remaining vs decayed amount
Remaining amount is N. Decayed amount is N₀ − N: the amount of the original parent population that has undergone decay. Remaining percentage is 100 × N/N₀ and decayed percentage is 100 × (1 − N/N₀). Their unrounded values sum to 100% within floating-point tolerance.
“Decayed” does not mean the same quantity of a particular daughter isotope has accumulated. Daughter products, branching and subsequent decay are not tracked. Choose kg, g, mg, µg, atoms, nuclei, mol or mmol for both amounts. Changing the amount selector relabels your quantities; it does not convert them or infer mass-to-nuclei relationships. Bq is activity, not an amount unit here.
Solving backwards
Solving for time gives t = T½ × ln(N/N₀) / ln(1/2). Solving for half-life gives T½ = t × ln(1/2) / ln(N/N₀). Solving for starting amount gives N₀ = N / (1/2)^(t/T½).
Elapsed Time requires 0 < N ≤ N₀ and T½ > 0; equal amounts give t = 0. Half-Life requires 0 < N < N₀ and t > 0. If there is no represented decay, the half-life is indeterminate rather than zero. Starting Amount requires N > 0, t ≥ 0 and T½ > 0. Values implying growth receive a clear domain error. The implementation uses stable logarithms near equality.
Isotope half-life data
The presets below use central values from NNDC ENSDF adopted-level evaluations, reviewed 2026-10-01. The reported uncertainties and evaluation identifiers are retained for context; uncertainty is not propagated through the displayed results. These references are versioned evaluations, not claims of exact or timeless values.
| Isotope | Half-life | Reported uncertainty | Reference |
|---|---|---|---|
| Carbon-14 | 5700 years | ±30 years | NNDC ENSDF: ¹⁴C Adopted Levels, Gammas (opens in a new tab) 1991Aj01; half-life evaluated by V. Chechev in 1998; PDF page 1. Reviewed 2026-10-01. |
| Iodine-131 | 8.0252 days | ±0.0006 days | NNDC ENSDF: ¹³¹I Adopted Levels, Gammas (opens in a new tab) NDS 107, 2715 (2006); cutoff 2006-07-17; PDF page 1. Reviewed 2026-10-01. |
| Cobalt-60 | 1925.28 days | ±0.14 days | NNDC ENSDF: ⁶⁰Co Adopted Levels, Gammas (opens in a new tab) NDS 114, 1849 (2013); cutoff 2012-12-31; PDF page 4. Reviewed 2026-10-01. |
| Cesium-137 | 30.08 years | ±0.09 years | NNDC ENSDF: ¹³⁷Cs Adopted Levels, Gammas (opens in a new tab) NDS 108, 2173 (2007); cutoff 2006-10-01; PDF page 1. Reviewed 2026-10-01. |
| Uranium-238 | 4.468e9 years | ±0.006 × 10⁹ years | NNDC ENSDF: ²³⁸U Adopted Levels, Gammas (opens in a new tab) NDS 127, 191 (2015); cutoff 2014-06-01; PDF page 2. Reviewed 2026-10-01. |
Carbon-14 is 5,700 years in the cited adopted data, so the default uses 11,400 years for two half-lives. Cobalt-60 uses the reported 1,925.28 days rather than an independently rounded year conversion. Custom accepts a positive user-supplied half-life, with no implied isotope identity or reference endorsement.
Decay graph
The normalized SVG curve plots expected remaining fraction N/N₀ = 2⁻ⁿ against elapsed half-lives. It starts at 100%, passes through 50% at one half-life, 25% at two, 12.5% at three and 6.25% at four. The current-result dot and adjacent text describe the same calculated point.
The x-axis shows at least five half-lives and expands up to twenty. A result beyond twenty is stated using its actual half-life count, and its marker is omitted rather than falsely clamped to the chart edge. The graph has a fixed aspect ratio and a text equivalent.
Half-life milestones
| Half-lives | Remaining | Decayed |
|---|---|---|
| 0 | 100% | 0% |
| 1 | 50% | 50% |
| 2 | 25% | 75% |
| 3 | 12.5% | 87.5% |
| 4 | 6.25% | 93.75% |
| 5 | 3.125% | 96.875% |
Each additional half-life halves what remains. The interactive milestone table also applies these fractions to your current starting amount.
Worked examples
Carbon-14: two half-lives
Starting with 100 g and using the sourced T½ = 5,700 years, elapsed time 11,400 years gives n = 2. N = 100 × (1/2)² = 25 g; 25% remains and 75 g, or 75%, has decayed.
Reverse time: 80 g to 10 g
With a Custom half-life of 6 hours, 80 → 40 → 20 → 10 g represents three half-lives. t = 6 × ln(10/80)/ln(1/2) = 18 hours.
A fractional half-life count
Starting at 200 mg, with a Custom 10-day half-life and 15 days elapsed, n = 1.5. N = 200 × 2⁻¹·⁵ = 70.71067811865476 mg, displayed as 70.7107 mg. About 35.3553% remains.
Recover the starting amount
With 25 g remaining, a Custom 10-year half-life and 20 years elapsed, n = 2. N₀ = 25/(1/2)² = 100 g.
For these examples, you can review significant figures in the reported decay result separately. A chosen display precision does not improve the accuracy of a measured amount or sourced half-life, or establish radiological safety.
Assumptions and limitations
This calculator models ideal exponential radioactive decay for a population of unstable parent nuclei using a constant half-life. It does not model decay chains, daughter-product buildup, branching ratios, changing environmental chemistry, radiation dose, shielding, biological half-life, effective half-life, detector response, safety exposure, criticality or nuclear reactions.
The tool is educational and mathematical. It provides no radioactive-material handling, acquisition, disposal, source-strength planning, dose-limit or shielding-design guidance.
Known amounts must be between 1e-100 and 1e100 in the shared unit. Time values normalize to 1e-12 through 1e25 seconds; elapsed time may also be zero. Inputs allow at most 64 characters, and the elapsed half-life count is capped at 1,000. Computed amounts must remain positive and finite; out-of-range or underflowing calculations receive errors. Derived remaining amounts can be smaller than the input floor if representable. IEEE-754 floating-point arithmetic and rounding apply; numerical checks use relative tolerance 1e-12. Display precision does not claim greater scientific accuracy than a preset measurement.
FAQ
What is radioactive half-life?
The half-life is the time over which the expected number of undecayed parent nuclei falls to half its initial value. It describes a population, not a scheduled decay time for one atom.
What formula does the calculator use?
N = N₀ × (1/2)^(t/T½), with a constant positive half-life and nonnegative elapsed time. The two time quantities are converted to the same unit before division.
How do I calculate how much remains after several half-lives?
Divide elapsed time by the half-life to find n, then multiply the starting amount by 2⁻ⁿ. For 100 g after three half-lives, the expected remaining amount is 12.5 g.
What does 2 half-lives mean?
Two successive halvings have passed. Half of a half is one quarter, so 25% remains and 75% of the original parent population has decayed.
Can I solve for elapsed time instead?
Yes. Choose Elapsed Time and enter starting amount, remaining amount and half-life. The remaining amount must be positive and cannot exceed the starting amount; equal amounts return zero elapsed time.
Can I calculate the half-life from starting and remaining amounts?
Yes. Choose Half-Life and enter positive elapsed time, positive starting amount and a positive remaining amount smaller than the starting amount. This mode uses Custom; unchanged amounts do not determine a finite half-life.
Why does radioactive decay follow an exponential curve?
In this ideal model, each surviving nucleus has a constant probability per unit time of decaying. The expected rate of loss is therefore proportional to the number still present, producing an exponential decline.
Are isotope half-life values exact?
No. They are evaluated scientific measurements with uncertainties and reference versions. The five presets retain their cited significant digits and reported uncertainty. This calculator uses the central values and does not propagate uncertainty.
What is the difference between half-life and decay rate?
Half-life is a duration. The decay constant λ = ln(2)/T½ has units of inverse time, while activity is the expected number of decays per unit time, λN for a parent-nucleus count. This calculator does not calculate activity or accept Bq as an amount.
Does this calculator handle decay chains or radiation dose?
No. It models a single parent population with a constant half-life. Daughter buildup, decay chains, radiation dose, shielding and biological or effective half-life are outside its scope.
Sources
Reviewed October 1, 2026. NNDC adopted-level evaluations supply the preset half-lives and reported uncertainties. OpenStax supports the explanatory exponential-decay model. Reference editions and data cutoffs appear beside each isotope. Sources do not endorse BuzCalculator.
- NNDC ENSDF: ¹⁴C Adopted Levels, Gammas (opens in a new tab)
- NNDC ENSDF: ¹³¹I Adopted Levels, Gammas (opens in a new tab)
- NNDC ENSDF: ⁶⁰Co Adopted Levels, Gammas (opens in a new tab)
- NNDC ENSDF: ¹³⁷Cs Adopted Levels, Gammas (opens in a new tab)
- NNDC ENSDF: ²³⁸U Adopted Levels, Gammas (opens in a new tab)
- OpenStax College Physics 2e: 31.5 Half-Life and Activity (opens in a new tab)