Radioactive Decay Calculator
Calculate remaining quantity, elapsed time, half-life, decay constant, or activity using the exponential radioactive decay law.
Calculator is for informational purposes only. Terms and Conditions
Choose what to calculate
Select the unknown and optionally load a common radionuclide half-life.
Enter the known values
Quantity values may use any consistent unit. Time and decay-rate units are converted automatically.
Result
Primary answer first, followed by decay checks and transparent calculation steps.
Result details
- Check—
Show calculation stepsReview conversions, equations, substitutions, and reverse checks
- Enter valid values to see the complete calculation.
This calculator models a single radionuclide with a constant decay probability and no production, replenishment, or daughter-chain buildup.
Radioactive decay curve
The marker shows the calculated state on a normalized exponential-decay curve.
The curve updates after a valid decay calculation.
Method, Sources, and Assumptions
Calculation basis, radionuclide data source, activity units, and model limitations.
Half-life and decay-mode reference data are based on evaluated nuclear data available through the National Nuclear Data Center. Activity uses A = λN, and 1 Ci = 3.7 × 10¹⁰ Bq.
- The model treats one radionuclide with constant decay probability and no additional production.
- Quantity-based decay requires the same quantity unit for initial and remaining values.
- Activity in becquerels is calculated only from radioactive atom count and decay constant.
Calculator guide
Understanding Your Radioactive Decay Result
The calculator above solves the exponential radioactive decay relationship for remaining quantity, elapsed time, half-life, initial quantity, decay constant, or activity. For most decay problems, you need a starting quantity, an elapsed time, and either a half-life or decay constant; activity mode instead uses the number of radioactive atoms with a decay rate.
For a single radionuclide with a constant decay probability and no source term, the expected number of undecayed nuclei decreases exponentially. Half-life and decay constant describe the same decay rate in different forms, while activity describes how many nuclear transformations occur per unit time.
- Best for
- Single-isotope exponential decay problems and reverse-solving the same relationship.
- Core output
- The selected unknown, plus useful checks such as remaining fraction, half-lives elapsed, decay constant, and mean lifetime.
- Key assumption
- The radionuclide follows first-order decay with a constant decay constant and no production or removal process.
How to Use the Calculator
Start by choosing the quantity you want to solve for. The calculator then shows only the inputs needed for that mode and updates the result automatically when the active inputs are valid.
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Choose the unknown
Use Solve For to select remaining quantity, elapsed time, half-life, initial quantity, decay constant, or activity. This is the most important setup choice because it changes which values are required and which units are available for the answer.
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Choose how the decay rate is known
For remaining quantity, elapsed time, initial quantity, or activity, select either Half-life or Decay constant λ. These are equivalent descriptions of the same first-order decay rate.
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Use a preset only when it matches the radionuclide
The radioisotope preset can load the calculator’s reference half-life for C-14, I-131, Co-60, Cs-137, Tc-99m, or U-238. Leave the preset on Custom when you have another nuclide or a project-specific decay value.
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Enter consistent quantities and review units
When a calculation uses both initial and remaining quantity, they must represent the same physical quantity in the same unit basis. Time and decay-rate fields can use their provided unit selectors; the calculator converts those values internally without changing the represented physical quantity.
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Check the result details and steps
Use the main answer first, then use the secondary results, decay curve, and calculation steps as checks. Advanced Options lets you change the available answer units and displayed precision where those choices apply.
Radioactive Decay Equations and Method
The calculator evaluates the closed-form first-order radioactive decay equation directly. The same exponential relationship can be written using either the decay constant or the half-life, then algebraically rearranged for the selected unknown.
Exponential decay law
Plain language: the amount remaining equals the starting amount multiplied by an exponential decay factor determined by the decay constant and elapsed time.
This relationship applies to a population of identical radioactive nuclei when the decay constant is constant and there is no continuing production, transfer, or removal term in the model.
Half-life form
Plain language: every elapsed half-life multiplies the remaining amount by one-half. Fractional numbers of half-lives are handled by the same exponential relationship.
Half-life is a population statistic, not a countdown timer for an individual nucleus. The decay time of one particular nucleus cannot be predicted; the exponential law describes the expected behavior of a sufficiently large population.
Half-life, decay constant, and activity
A shorter half-life corresponds to a larger decay constant. Mean lifetime τ is the reciprocal of the decay constant, and activity equals the decay constant multiplied by the number of radioactive atoms present.
Mass, atoms, and activity are not interchangeable
For fractional decay, any consistent proportional quantity can be used for the starting and remaining values, including atom count, moles, radionuclide mass, activity, or a relative amount. The initial and remaining entries must use the same basis.
Absolute activity is different: the calculator uses \(A=\lambda N\), so it needs the number of radioactive atoms. If you start from radionuclide mass, convert mass to moles using the nuclide molar mass and then to atoms using Avogadro’s constant before calculating Bq or Ci.
- \(N_0\)
- Initial number of radioactive nuclei or another consistently proportional starting quantity.
- \(N(t)\)
- Radioactive nuclei or proportional quantity remaining after elapsed time \(t\).
- \(t\)
- Elapsed time. Its units must be compatible with the units used to express the decay rate.
- \(t_{1/2}\)
- Half-life: the time for the expected undecayed population, and therefore its activity, to fall to one-half.
- \(\lambda\)
- Decay constant, with units of inverse time.
- \(A\)
- Activity. When \(\lambda\) is expressed in s\(^{-1}\) and \(N\) is a number of atoms, \(A\) is in becquerels.
Worked Radioactive Decay Example
Suppose a sample starts at 100 arbitrary units, has a 10-year half-life, and is allowed to decay for 20 years. This matches the calculator’s default remaining-quantity example and provides a simple two-half-life verification.
Substitute the values
Result
25 units remain
Two half-lives have elapsed, so the sample is at 25% of its starting quantity and 75% has decayed.
How to Interpret the Results
The primary answer is only one view of the same exponential process. The remaining fraction, half-lives elapsed, decay constant, mean lifetime, and decay curve are especially useful for checking whether the answer is physically and mathematically reasonable.
The decay curve is exponential rather than linear. Equal time intervals do not remove equal absolute amounts; each half-life removes half of what remains at the start of that interval. The curve therefore drops more steeply in absolute quantity early on and progressively flattens as it approaches zero.
Amount and activity fall by the same fraction
For one radionuclide with constant \(\lambda\), \(A=\lambda N\). Therefore \(A/A_0=N/N_0=e^{-\lambda t}\). If 25% of the radioactive nuclei remain, 25% of the original activity remains as well.
Half-life changes the time scale
Holding elapsed time and initial quantity constant, a shorter half-life produces a smaller remaining fraction. Doubling elapsed time does not subtract a fixed amount; it multiplies the current amount by another exponential decay factor.
Use half-lives as a mental check
After 1, 2, 3, and 4 half-lives, the expected remaining fractions are 50%, 25%, 12.5%, and 6.25%. A result far from these landmarks deserves a unit, input, or solve-mode check.
Half-Life Fraction Reference
For a pure single-isotope decay process, each elapsed half-life leaves one-half of the quantity that was present at the start of that interval. The table below is a quick manual check for the calculator.
| Half-lives elapsed | Fraction remaining | Percent remaining |
|---|---|---|
| 0 | 1 | 100% |
| 1 | 1/2 | 50% |
| 2 | 1/4 | 25% |
| 3 | 1/8 | 12.5% |
| 4 | 1/16 | 6.25% |
| 5 | 1/32 | 3.125% |
| 10 | 1/1024 | 0.09765625% |
These values follow directly from \((1/2)^n\), where \(n=t/t_{1/2}\). Fractional values of \(n\) are valid too; the calculator evaluates the full exponential relationship rather than rounding to a whole number of half-lives.
Assumptions and Limits
The equations are exact within the ideal single-isotope model, but real measurements and applications can introduce uncertainty that the model does not represent. The key question is whether your physical system can actually be represented by a single constant-\(\lambda\) exponential decay term.
Single radionuclide
The basic equation tracks one radionuclide at a time. A mixture of isotopes has multiple decay constants, so each component must be modeled separately before quantities or activities are combined.
No production term
The equation assumes the radionuclide is not being created while it decays. Ongoing activation, breeding, or another source requires an added production term rather than the simple decay-only equation.
No parent-daughter chain model
When a daughter radionuclide is produced by decay of a parent and its buildup matters, coupled decay equations are required. A single exponential for the parent does not describe the daughter inventory.
Activity needs an atom count
The calculator’s activity mode uses \(A=\lambda N\) with the number of radioactive atoms. Generic mass or arbitrary quantity units can be used for fractional decay, but they are not enough by themselves to produce an absolute activity in Bq.
Very small theoretical remainders
After many half-lives, the mathematical model may return an extremely small nonzero value. Whether that value is practically measurable can depend on detector sensitivity, background, sampling, and the specific application.
Radiometric dating needs more than the decay equation
The exponential law is the mathematical core of radiometric dating, but real age determination can require assumptions or corrections for initial conditions, contamination, system closure, calibration, and measurement uncertainty.
Sources and Calculation Basis
The calculation uses standard first-order radioactive decay relationships. The worked example was checked both with the half-life form and by repeated halving, while activity-unit definitions are tied to SI/NIST references.
- OpenStax Physics — Half Life and Radiometric Dating — supports the exponential decay law, the half-life relation, activity as decay rate, and the becquerel definition.
- OpenStax University Physics Volume 3 — Radioactive Decay — supports mean lifetime, \(A=\lambda N\), and the fact that activity follows the same exponential decay factor as the radioactive population.
- NIST Guide to the SI — Units Outside the SI — gives the curie conversion \(1\ \mathrm{Ci}=3.7\times10^{10}\ \mathrm{Bq}\).
- National Nuclear Data Center — NuDat 3 — authoritative evaluated nuclear-data resource used for radionuclide properties such as half-life and decay information.
The isotope preset values belong to the calculator interface and should be treated as reference data for the selected nuclide, not as a substitute for checking the current evaluated data source when a consequential decision depends on a precise half-life value.
Radioactive Decay Calculator FAQs
These questions cover the most common interpretation and unit issues that are not already answered by the calculator controls.
What is the radioactive decay formula?
The standard single-isotope decay equation is \(N(t)=N_0e^{-\lambda t}\). Using half-life instead of decay constant gives the equivalent form \(N(t)=N_0(1/2)^{t/t_{1/2} }\).
How are half-life and decay constant related?
They describe the same decay rate: \(\lambda=\ln(2)/t_{1/2}\). A larger decay constant means a shorter half-life, while a smaller decay constant means a longer half-life.
How much remains after three half-lives?
One-eighth of the starting radioactive quantity remains: \((1/2)^3=1/8=12.5\%\). The activity of that same single radionuclide is also 12.5% of its initial activity.
What is the difference between Bq and Ci?
The becquerel is the SI unit of radioactive activity, with 1 Bq corresponding to one decay per second. The curie is a non-SI activity unit; \(1\ \mathrm{Ci}=3.7\times10^{10}\ \mathrm{Bq}\).
Is radioactive activity the same as radiation dose?
No. Activity describes nuclear transformations per unit time. Absorbed dose and dose-related protection quantities depend on additional factors such as radiation energy and type, geometry, distance, shielding, exposure pathway, and biological context.
Does a radioactive sample ever reach exactly zero?
The continuous exponential model approaches zero asymptotically rather than reaching it at a finite time. Real samples contain a discrete number of nuclei, so at very small populations the deterministic continuous model is no longer a literal count of individual remaining atoms.
Can I use the calculator for carbon-14 dating?
You can use the C-14 preset and the elapsed-time solve mode to calculate the exponential-decay age implied by consistent starting and remaining quantities or activities. That is not the same as a calibrated radiocarbon age determination, which can require calibration, contamination control, reservoir corrections, sample preparation, and measurement uncertainty.
Can I use one calculation for a mixture of isotopes?
Not as a single exponential unless every component has the same decay constant. For a mixture, calculate each radionuclide with its own half-life or decay constant, then combine quantities or activities only when that combination is physically meaningful for the task.