Half-Life Calculator for Radioactive Decay

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

Solve for Required: initial quantity N0, half-life T1/2, and time elapsed t.

Amount label Display-only label. Initial and remaining amounts must use the same unit.

Initial quantity N0

Remaining quantity Nt

Half-life T1/2

Time elapsed t

Example 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 inputs Current 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.

Half-life T1/2

Decay constant λ

Mean lifetime τ

Enter half-life, λ, or τ to convert units and constants.

Half-Life Milestones

Half-lives	Elapsed time	Fraction remaining	Percent remaining	Amount remaining
0	—	1	100%	—

Half-Life Formulas

Equivalent decay equations

Nt = N0 × (1/2)^{t / T1/2}

Nt = N0 × e^-λt

Nt = N0 × e^{-t / τ}

N0: initial quantity

Nt: quantity remaining after time t

t: elapsed time

T1/2: half-life

λ: decay constant per unit time

τ: mean lifetime

Rearranged solver forms

Remaining quantity: Nt = N0 × 2^{-t / T1/2}
Initial quantity: N0 = Nt / 2^{-t / T1/2}
Time elapsed: t = T1/2 × log2(N0 / Nt)
Half-life: T1/2 = t / log2(N0 / Nt)
Constants: λ = ln(2) / T1/2, T1/2 = ln(2) / λ, τ = 1 / λ = T1/2 / ln(2)

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.

Remaining sample after several half-lives

Inputs: N0 = 100 g, T1/2 = 5.27 days, t = 15.81 days.

Substitution: Nt = 100 × 2^{-15.81 / 5.27}.

Result: Nt = 12.5 g.

Check: 15.81 days is 3 half-lives, so one eighth of the starting sample remains.

Solving half-life from measured data

Inputs: N0 = 80 counts/s, Nt = 10 counts/s, t = 21 hours.

Substitution: T1/2 = 21 / log2(80 / 10).

Result: T1/2 = 7 hours.

Check: The activity dropped by a factor of 8, or three halvings, so the half-life is one third of 21 hours.

Converting half-life to λ and τ

Inputs: T1/2 = 5730 years.

Substitution: λ = ln(2) / 5730, τ = 5730 / ln(2).

Result: λ ≈ 0.000121 per year, τ ≈ 8267 years.

Check: Mean lifetime is longer than half-life because 1/e remaining, about 36.8%, occurs after one τ.

Sources and Accuracy

Last reviewed: June 29, 2026

Model and formulas

The calculator uses the standard first-order radioactive decay model Nt = N0e^-λt, the half-life relationship λ = ln(2) / T1/2, and τ = 1 / λ.

Lumen/OpenStax physics: Half-Life and Activity

Reference half-lives and limits

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.

EPA iodine radionuclide basics · EPA uranium radionuclide basics · EPA radium radionuclide basics

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

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.

Equal time steps do not remove equal amounts

Exponential decay removes the same proportion each interval, so the absolute drop gets smaller over time.

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.

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.

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.