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Radioactive Half-Life Remaining Amount Calculator

Calculate how much of a radioactive sample remains after a given time, based on its half-life.

Result

Amount Remaining
18.9465 g
Percent Remaining
18.95%
Half-Lives Elapsed
2.4

Uses the general exponential decay formula N(t) = N₀ × (1/2)^(t/half-life). Our Carbon-14 Dating Calculator applies this same math specifically for age-dating an organic sample; this version works for any radioactive isotope and any known half-life.

About the Half-Life Remaining Amount

This calculator estimates how much of a radioactive sample remains after a specified amount of time, based on the sample's half-life. It applies the general exponential decay formula to any radioactive isotope, given an initial amount, a known half-life, and an elapsed time, all in whatever consistent units you choose to enter.

How It Works

Enter the initial amount of the sample, its half-life, and how much time has elapsed, using the same time unit for half-life and elapsed time. The calculator divides elapsed time by half-life to get the number of half-lives that have passed, then multiplies the initial amount by one-half raised to that power to find the remaining amount, and also reports that figure as a percentage of the original.

N(t) = N0 x (1/2)^(t / half-life), where N0 is the initial amount, t is elapsed time, and half-life is the isotope's half-life in the same time units.

Formula & Methodology

With an initial amount of 100 g, a half-life of 5 years, and 12 years elapsed: half-lives elapsed = 12 / 5 = 2.4. Remaining amount = 100 x 0.5^2.4, which works out to about 18.95 g, or 18.95 percent of the original sample.

Examples

Default 5-year half-life sample

A 100 g sample with a 5-year half-life, after 12 years elapsed, has gone through 2.40 half-lives and has about 18.95 g (18.95 percent) remaining.

Longer half-life over 20 years

A 50 g sample with an 8-year half-life, after 20 years elapsed, has gone through 2.50 half-lives, leaving about 8.84 g, or roughly 17.68 percent of the original amount.

Advantages

  • Works for any radioactive isotope and any half-life value, unlike a calculator built around one specific element's fixed decay rate.
  • Reports the result three ways, remaining amount, percent remaining, and number of half-lives elapsed, giving both an absolute and a relative sense of how far decay has progressed.
  • Guards against a zero or negative half-life entry internally, avoiding a division-by-zero error that would otherwise produce an undefined result.

Common Mistakes

  • Entering elapsed time and half-life in different units, say years for one and days for the other, which throws off the half-lives-elapsed calculation entirely.
  • Assuming the sample reaches exactly zero after some fixed number of half-lives, when decay is asymptotic and technically never reaches zero, even though it becomes negligible after roughly 7 to 10 half-lives.
  • Applying this general calculation to a decay chain scenario, where the decay product is also radioactive, without accounting for the daughter isotope's own separate half-life.

Edge Cases to Watch For

  • Half-life is clamped to a minimum of 0.0001 in the underlying calculation, preventing a division by zero if a user enters a half-life of zero or a negative number.
  • The remaining amount approaches zero but mathematically never reaches it exactly, since each half-life only removes half of whatever is currently left; after enough half-lives the displayed figure may round to 0.0000 g even though a nonzero trace theoretically remains.
  • The formula assumes a single isotope decaying at a constant, known half-life, and doesn't account for decay chains where the daughter product is itself radioactive and decaying at a different rate.

Common Use Cases

  • Students working through radioactive decay problems in a chemistry or physics course.
  • Anyone estimating the residual activity of a radioactive material or tracer after a known storage or elapsed period.
  • Educators building examples that show how percent remaining and half-lives elapsed relate to each other across different isotopes.
Written & fact-checked by the Calculateus TeamLast updated August 5, 2026How we verify our formulas

Frequently asked questions

Does a sample ever reach exactly zero?

Mathematically, no - exponential decay approaches zero asymptotically but never reaches it exactly, since each half-life only removes half of whatever remains. In practice, after roughly 7-10 half-lives, the remaining amount becomes negligible (under 1%) for most measurement and safety purposes, even though a nonzero trace theoretically persists.

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Conclusion

Radioactive decay follows the same exponential pattern regardless of which isotope is involved, only the half-life value changes. This calculator applies that general formula so remaining amount, percent remaining, and half-lives elapsed can all be checked in one step instead of computed by hand.