Radioactive Decay Formula Calculator and Examples

A radioactive decay calculator estimates how much of a radionuclide remains after a chosen time, using its initial amount and half-life. Enter those values and the elapsed time to calculate the remaining amount, the amount that has decayed, and the fraction left.

Formula

The exponential radioactive-decay law is N = N₀e^−λt = N₀(1/2)^t/T_1/2. Here, N is the amount remaining, N₀ is the initial amount, λ is the decay constant, t is elapsed time, and T_1/2 is the half-life. The calculator uses λ = ln(2)/T_1/2, with ln(2) represented by 0.69314718056. The half-life and elapsed time must use the same time unit; this calculator uses years for both. The input amount may be a mass of one radionuclide, provided it is expressed in the same unit as the output.

“Amount decayed” is calculated as N₀ − N. “Fraction remaining” is N/N₀, a unitless proportion between 0 and 1. The amount remaining is the headline result; the other outputs help interpret it.

Worked Example

Using the default inputs, the initial amount is 10 g, the half-life is 5 years, and the elapsed time is 12 years.

1. Calculate the elapsed time in half-lives: 12 y / 5 y = 2.4.
2. Calculate the fraction remaining: (1/2)^2.4 = 0.18946457.
3. Calculate the remaining amount: 10 g × 0.18946457 = 1.8946457 g, which rounds to 1.8946 g.
4. Calculate the amount decayed: 10 g − 1.8946457 g = 8.1053543 g, which rounds to 8.1054 g.

The fraction remaining, rounded to six decimal places, is 0.189465. The outputs are rounded for display, so subtracting the rounded amounts may differ slightly from the rounded decay result.

Assumptions

• The sample contains one radionuclide with a known half-life.
• Decay follows the exponential law; the calculation predicts an expected amount, not the exact number of individual atoms in a small sample.
• The half-life is constant over the period considered, and elapsed time is nonnegative.
• The starting amount is the amount of radioactive material at the time defined as zero. No additional material is added or removed.

When to Use the Calculator

Use this calculation for a simple estimate of how a radionuclide’s quantity changes over time. It is useful in classroom problems, interpreting a stated half-life, and comparing the expected remaining fractions at different dates. It does not identify an unknown isotope, determine a half-life from measurements, or assess whether a sample is safe.

For example, if a material has a half-life of 5 years, after 5 years half the original amount remains; after 10 years, one-quarter remains. At 12 years, 2.4 half-lives have passed, so the remaining fraction is about 0.1895. A half-life is not the time for all the material to disappear: each successive half-life halves the amount then present.

If the known starting quantity is activity rather than mass, the same equation can calculate remaining activity, as long as the starting activity and result use the same activity unit, such as becquerels. In that case, enter activity as the initial amount and interpret the headline output as activity, not mass. Activity measures nuclear decays per second; it is not the same as material mass or radiation dose.

How Radioactive Decay Works

Radioactive decay is a random process for individual unstable nuclei. A particular nucleus cannot generally be assigned a predictable decay time, but a large population has a statistically predictable decay rate. The half-life describes the time over which the expected number of undecayed nuclei—and therefore the activity of a single radionuclide—falls to half its starting value.

The decay constant λ is the probability per unit time that a nucleus decays, expressed with inverse-time units. For a half-life of 5 years, λ = 0.69314718056 / 5 y = 0.1386294361 y⁻¹. It is not necessary to enter λ in this calculator because the half-life determines it.

The model concerns the parent radionuclide. Its decay may produce a radioactive daughter isotope. If the daughter also decays, or if decay products build up, the amount of each isotope requires a decay-chain calculation rather than this single-isotope formula.

Choosing Inputs and Reading Results

• Initial amount: Use the quantity present at the start of the interval, not the amount currently remaining unless the current time is your chosen starting point. Use a nonnegative value and record whether it represents mass, activity, or another amount.
• Half-life: Use a value for the specific radionuclide, not a generic half-life for an element. Half-lives can range from fractions of a second to billions of years.
• Elapsed time: Enter time since the starting amount was defined. This calculator expects years, so convert other time units before entering them.

For a time measured in days, convert it to years using the convention appropriate to the problem. For example, using 365 days per year, 730 days is 2 years. Do not enter 730 as years when the half-life is entered in years: matching time units is essential because the ratio t/T_1/2 controls the result.

Results are numerical estimates and should not be interpreted as a measurement. Rounding can hide small differences, especially after many half-lives. If only a tiny fraction remains, the calculator may display zero at the selected precision even though the mathematical value is positive.

Limits, Safety, and Common Misconceptions

This equation does not calculate radiation dose, exposure risk, shielding needs, contamination, or legal disposal requirements. A small remaining mass does not automatically imply low hazard: the isotope, radiation type, activity, chemical form, exposure route, and circumstances matter. Conversely, an amount calculation alone cannot establish that a sample is dangerous. Do not use this page to make handling or safety decisions. For radioactive materials, follow applicable radiation-safety rules and guidance from qualified radiation-safety personnel.

A common mistake is to confuse half-life with the time required for complete decay. For any finite time in this idealized model, a positive fraction remains; the fraction becomes very small but does not reach exactly zero. Another mistake is treating an isotope’s half-life as the half-life of every isotope of that element. Half-life is specific to the radionuclide.

Measurement uncertainty also matters. The initial amount and half-life may be known only approximately, while real instruments have detection limits and measurement uncertainty. A more detailed analysis should account for those uncertainties and, where relevant, daughter products and decay chains.

Frequently Asked Questions

What is the radioactive decay formula?
The remaining amount is N = N₀e^−λt, or equivalently N = N₀(1/2)^t/T_1/2. N₀ is the initial amount, t is elapsed time, and T_1/2 is the half-life.

How do I calculate how much radioactive material remains?
Divide elapsed time by the half-life to find the number of half-lives, then multiply the starting amount by (1/2) raised to that number. Make sure both times use the same unit.

How many half-lives until a sample is gone?
The ideal exponential formula never reaches exactly zero in a finite number of half-lives. In practice, a sample may become too small to measure or fall below a relevant detection threshold.

Can I use the formula for radioactive activity?
Yes, for a single radionuclide with no additions or removals, activity decreases by the same exponential factor as the number of parent nuclei. Enter activity as the initial quantity and keep its units consistent.

Does radioactive decay calculate radiation dose?
No. It estimates the amount or activity remaining. Dose depends on additional factors such as radiation type, energy, exposure conditions, distance, shielding, and how radiation interacts with matter.

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in n0 | Initial amount | g | 10 | 0
in h | Half-life | y | 5 | 0.000001
in t | Elapsed time | y | 12 | 0
out n | Amount remaining | g | n0 * exp(-0.69314718056 * t / h) | 4
out d | Amount decayed | g | n0 - n | 4
out f | Fraction remaining | - | exp(-0.69314718056 * t / h) | 6