Find remaining amount, initial amount, elapsed time, half-life, decay constant, and mean lifetime for exponential or radioactive decay. Works for any consistent amount unit such as g, mg, mol, atoms, or Bq, and keeps all calculations in your browser.
Inputs & Options
Required: initial quantity N0, half-life T1/2, and time elapsed t.
Display-only label. Initial and remaining amounts must use the same unit.
Choose a common setup, then use the selected example to fill the required fields.
Choose a solve mode, enter the known values, then calculate.
Results
Ready.
This calculator assumes constant half-life and first-order exponential decay.
Primary result
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Fraction remaining
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Percent remaining
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Percent decayed
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Elapsed half-lives
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Decay constant λ
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Mean lifetime τ
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Decay Curve
Range: waiting for inputsCurrent point: not calculated
Half-Life, Decay Constant, and Mean Lifetime Converter
Enter any one value to calculate the other two using λ = ln(2) / T1/2 and τ = 1 / λ.
Enter half-life, λ, or τ to convert units and constants.
These relations are valid for first-order exponential decay with a single constant half-life. That model is standard for ideal radioactive decay and many idealized first-order processes.
How to Use It
Select the unknown: initial quantity N0, remaining quantity Nt, time elapsed t, or half-life T1/2.
Enter the three known values using the same amount unit.
Enter the known time value and choose its unit.
Calculate to get the result, fraction remaining, λ, and τ.
Use copy or download if you need a quick lab note or calculation record.
In practice, the safest workflow is to keep the amount units consistent and focus on whether the time relationship is sensible. If one half-life passes, half the amount should remain. If two half-lives pass, one quarter should remain. If three pass, one eighth should remain. Those checkpoints are useful for spotting input mistakes quickly before you rely on the detailed result. The calculator exposes the decay constant and milestone table for that reason: they give you an immediate sanity check on whether the entered half-life matches the story told by the remaining amount.
Initial quantity and remaining quantity must be greater than zero for ratio-based solves.
Remaining quantity cannot exceed initial quantity for forward radioactive decay with nonnegative time.
If remaining quantity equals initial quantity after positive time, the implied half-life is infinite.
For very old carbon-14 samples, this calculation is only the exponential decay step; calibration and contamination controls still matter.
This tool is for educational and general scientific calculation use. It does not replace isotope-specific reference data, radiation safety procedures, or medical/pharmacokinetic advice.
Worked Examples
Carbon-14 dating
Inputs:N0 = 100, Nt = 25, T1/2 = 5730 years.
Substitution:t = 5730 × log2(100 / 25).
Result:t = 11,460 years.
Check: 25% remains, which is two half-lives, so the age should be about 2 × 5730 years.
Carbon-14 examples use a half-life of 5,730 years; iodine-131 examples use 8.02 days; uranium-238 FAQ uses 4.47 billion years; radium-226 FAQ uses 1,600 years.
Precision note: calculations use JavaScript double-precision floating point arithmetic and display rounded values. Input uncertainty, isotope data uncertainty, detector background, calibration curves, daughter products, and non-exponential processes can dominate the numerical rounding shown here.
5 quick facts about half-life
1
Half-life is a fraction rule
Each half-life cuts the remaining amount in half, regardless of whether you started with grams, atoms, moles, or activity.
2
Equal time steps do not remove equal amounts
Exponential decay removes the same proportion each interval, so the absolute drop gets smaller over time.
3
After 10 half-lives, very little remains
Only about 0.098% of the original amount is left after ten half-lives, which is why the quantity often becomes negligible in practice.
4
Decay constant and half-life carry the same information
They are linked by λ = ln(2) / T1/2, so knowing either one lets you compute the other immediately.
5
The model reaches zero only asymptotically
In ideal continuous exponential decay, the amount approaches zero but never becomes exactly zero at a finite time.
FAQ
What is half-life?
Half-life is the time required for an exponentially decaying quantity to fall to one half of its current amount. In radioactive decay this is a statistical average for many atoms, not a promise that every small sample loses exactly half its atoms at that instant.
How do I calculate half-life from measured data?
Use T1/2 = t / log2(N0 / Nt), where N0 is the initial quantity, Nt is the remaining quantity, and t is elapsed time. The measured remaining quantity must be less than the initial quantity for a finite half-life.
What is the carbon-14 half-life?
Carbon-14 has a half-life of about 5,730 years. Radiocarbon dating also needs calibration and sample context because atmospheric carbon-14 has varied over time.
What is the uranium-238 half-life?
Uranium-238 has a half-life of about 4.47 billion years.
What is the radium half-life?
Radium has several radioactive isotopes. Radium-226, the commonly cited isotope, has a half-life of about 1,600 years.
How is decay constant different from half-life?
The decay constant λ is a rate per unit time, while half-life T1/2 is a duration. They carry the same information through λ = ln(2) / T1/2.
Does half-life mean exactly half the atoms decay?
For a large sample, about half the radioactive nuclei remain after one half-life. For very small samples, decay is random, so the observed count can differ from exactly half.
Why does exponential decay never reach zero?
The continuous exponential model multiplies the remaining amount by a fraction over each interval. That makes the amount approach zero asymptotically, but it never becomes exactly zero at a finite time in the model.
Can I use grams, moles, atoms, or becquerels?
Yes. The calculator treats amount as a generic quantity. Initial and remaining quantities must use the same amount unit.
Does this handle changing decay rates?
No. It assumes a single constant half-life and first-order exponential decay with no replenishment, daughter buildup, biological clearance, or multi-step decay chain effects.
