Radioactive Decay Calculator

Find the Remaining Amount of a Radioactive Isotope

Radioactive Decay Formula

Formula

\[ N = N_0 \cdot 2^{-t/t_{1/2}} \]

Where:

\(N\) — remaining amount or activity after time \(t\), in the same unit as \(N_0\)
\(N_0\) — initial amount or activity (g, mol, Bq, Ci, or number of atoms)
\(t\) — elapsed time
\(t_{1/2}\) — half-life of the isotope, in the same time unit as \(t\)

The formula expresses exponential decay: every time one half-life passes, the remaining amount is halved. An equivalent form uses the decay constant \(\lambda = \ln 2 / t_{1/2}\), giving \(N = N_0 e^{-\lambda t}\).

Radioactive Decay – Calculation Example

Example: a sample initially contains 100 g of carbon-14, whose half-life is 5730 years. How much remains after 11460 years?

\[ n = \frac{t}{t_{1/2}} = \frac{11460}{5730} = 2 \text{ half-lives} \]

\[ N = 100 \cdot 2^{-2} = 100 \cdot 0.25 = 25 \text{ g} \]

Result: 25 g of carbon-14 remains; 75% has decayed, 25% remains, and 2 half-lives have passed. The decay constant is \(\lambda = 1.2097 \times 10^{-4}\) per year (about \(3.833 \times 10^{-12}\) per second).

How to Use the Calculator

Enter the Initial Amount or Activity (N₀) — a mass, number of atoms, moles, or activity — and choose its unit (g, mg, kg, mol, Bq, MBq, or Ci).
Enter the Half-life (t½) of the isotope and select its time unit (seconds, minutes, hours, days, or years). The default value, 5730 years, is the half-life of carbon-14.
Enter the Elapsed Time (t) since the initial measurement and its time unit. The half-life and elapsed time may use different units — the calculator converts them automatically.
Read the results: the Remaining Amount or Activity (N) in the same unit as your input, the Decayed Fraction and Remaining Fraction in percent, the Number of Half-Lives (n), and the Decay Constant \(\lambda\) per second and per your chosen half-life unit.

What Is Radioactive Decay?

Radioactive decay is the spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation — alpha particles, beta particles, or gamma rays — and transforms into a different, more stable nuclide. Because decay is a random, statistical process at the level of individual atoms, a large sample of a pure isotope decays at a rate that is perfectly predictable: the number of decay events per unit time is always proportional to the number of atoms present.

This proportionality leads to exponential decay. The half-life is the characteristic time in which half of the atoms in a sample decay. After one half-life, 50% of the isotope remains; after two half-lives, 25%; after three, 12.5%, and so on. The amount never reaches exactly zero, but it becomes negligibly small after roughly 7–10 half-lives.

The result \(N\) tells you how much of the original isotope is still present after the elapsed time. Note that the "lost" material has not vanished — it has been converted into a daughter product. For example, carbon-14 decays into nitrogen-14, and uranium-238 eventually becomes stable lead-206 through a decay chain. If you work with amounts of substance rather than masses, the atoms-to-moles converter can help you translate between atoms and moles.

Because the decay equation works for any quantity proportional to the number of atoms — mass, moles, atom count, or activity in becquerels or curies — the calculator accepts all of these. The remaining fraction in percent is the same regardless of the unit you choose.

Assumptions, Limitations, and Practical Tips

Pure isotope: the calculation assumes the starting material contains only the isotope of interest. In a real sample, daughter products and other contaminants add mass that is not accounted for by the decay equation.
Closed system: no isotope is added or removed during the elapsed time. Leaching, contamination, or chemical exchange will distort the result — a key concern in radiometric dating.
Activity inputs: if you enter an activity (Bq, MBq, Ci), the result is the remaining activity, not the remaining mass. Activities of different isotopes cannot be compared directly by mass without knowing their half-lives and molar masses.
Year definition: the calculator uses the Julian year of 365.25 days, the standard convention in nuclear and geochemistry data tables. For very short-lived isotopes, use seconds or minutes to avoid rounding issues.
Many half-lives: after about 10 half-lives the remaining fraction is under 0.1%. Reported results for such old samples reflect mathematical precision, not measurement precision — real detectors and mass spectrometers have practical detection limits.

A quick sanity check: the number of half-lives \(n\) alone tells you the answer. If \(n\) is a whole number, the remaining fraction is simply \(1/2^n\). For other chemistry calculations, such as solution concentrations or reaction yields, see the molarity calculator or the percent yield calculator.

Frequently Asked Questions

How do I calculate radioactive decay by hand without a calculator?

Divide the elapsed time by the half-life to get the number of half-lives \(n\), then multiply the starting amount by \(2^{-n}\). If \(n\) is a whole number, simply halve the amount \(n\) times: after 1 half-life 50% remains, after 2 half-lives 25%, after 3 half-lives 12.5%. For fractional half-lives, convert \(2^{-n}\) to \(e^{-0.693n}\) or use a scientific calculator's exponent key.

What is the difference between half-life, decay constant, and mean lifetime?

The half-life \(t_{1/2}\) is the time for half the atoms to decay. The decay constant \(\lambda = \ln 2 / t_{1/2}\) is the probability per unit time that one atom decays. The mean lifetime \(\tau = 1/\lambda = t_{1/2}/\ln 2 \approx 1.443\,t_{1/2}\) is the average time an individual atom survives. All three describe the same exponential decay, just parameterized differently.

How do I convert between becquerels and curies?

One curie equals \(3.7 \times 10^{10}\) becquerels, and 1 Bq is one decay per second. So to convert Ci to Bq, multiply by 37 billion; to convert Bq to Ci, divide by \(3.7 \times 10^{10}\). For example, a 1 µCi source has an activity of 37,000 Bq. Note that activity depends on both the amount of substance and the half-life, so 1 g of two different isotopes can have wildly different activities.

Why does the remaining amount never reach zero?

Exponential decay is asymptotic: each half-life halves what is left, so mathematically some fraction always remains. Practically, after 7 half-lives less than 1% remains and after 10 half-lives less than 0.1%, which falls below the detection limits of most instruments. At that point the isotope is effectively gone for experimental purposes, even though the equation gives a tiny nonzero value.

Can I use this calculator for carbon dating?

Yes, as a first approximation. Enter the half-life of carbon-14 (5730 years) and the time since death to find how much C-14 remains. Real radiocarbon dating works in reverse: scientists measure the remaining C-14 fraction and solve for \(t\). Keep in mind that accurate dating requires corrections for atmospheric C-14 variation and contamination, so treat simple results as estimates rather than definitive ages.

What happens if the elapsed time and half-life are in different units?

No problem — the calculator converts both to seconds internally before comparing them. You can enter a half-life in years and an elapsed time in days, or any other combination. If you compute by hand, you must convert both times to the same unit yourself first; mixing units is one of the most common errors in decay calculations and can throw the answer off by orders of magnitude.

Does the decayed material disappear, and what about decay chains?

The decayed atoms do not vanish — they become daughter nuclides. For example, C-14 turns into N-14, and U-238 passes through a chain of intermediates before ending as stable Pb-206. This calculator tracks only the parent isotope. In decay chains, the mass of the original element decreases, but total sample mass stays nearly constant, since daughters remain in the material.