Radioactive Half-Life Calculator

How the half-life calculator works

Radioactive decay is a first-order process: the rate at which atoms decay is proportional to the number of atoms currently present. This gives an exponential form. The most intuitive way to describe it is the half-life (T½) — the time required for half the atoms to decay. After one half-life, 50% remains; after two, 25%; after three, 12.5%. After ten half-lives, less than 0.1% is left.

The formula

The remaining quantity after elapsed time t is:

N(t) = N₀ × (1/2)^(t / T½)

Here N₀ is the initial amount and T½ is the half-life. The exponent t/T½ is the number of half-lives that have elapsed — it does not need to be an integer. For a half-life of 5,730 years and an elapsed time of 10,000 years, t/T½ = 1.745, and the remaining fraction is (1/2)^1.745 ≈ 0.298 — about 29.8%.

Reverse mode: elapsed time from remaining fraction

If you know the remaining fraction f = N(t)/N₀, you can invert the formula to solve for t:

t = T½ × log(f) / log(0.5)

This is the basis of radiometric dating. A sample with 25% of its original carbon-14 has been through exactly two half-lives — about 11,460 years. A sample with 10% remaining has been through log(0.1)/log(0.5) ≈ 3.32 half-lives — about 19,025 years, which is near the upper limit of C-14 dating reliability.

Worked example

A 100-gram sample of carbon-14 (T½ = 5,730 years) after 10,000 years:

• Half-lives elapsed: 10,000 / 5,730 = 1.7452.
• Remaining fraction: (1/2)^1.7452 = 0.2981.
• Remaining mass: 100 × 0.2981 = 29.81 grams.
• Decayed fraction: 1 − 0.2981 = 70.19%.

Note that t and T½ must be in the same unit. This calculator converts both to seconds internally before computing the ratio, so you can enter a half-life in years and an elapsed time in days and get the right answer.

Unit conversions used

• 1 minute = 60 seconds
• 1 hour = 3,600 seconds
• 1 day = 86,400 seconds
• 1 year = 365.25 days = 31,557,600 seconds (Julian year, the convention used in astronomy and radiometrics)

Edge cases and gotchas

Non-exponential decay. The half-life formula assumes first-order kinetics. For radioactive isotopes this is essentially exact. For chemical processes and biological systems it is often a useful approximation, but breaks down when the concentration is very high or very low, when saturation kinetics kick in, or when multiple elimination pathways operate in parallel.

Branching decay. Some isotopes decay by more than one pathway (e.g., bismuth-212 decays by both alpha and beta emission). Each pathway has its own partial half-life, and the effective half-life combines them. If you need branch-by-branch analysis, consult a decay-chain reference.

Pharmacology caveat. Drug half-lives assume first-order elimination, which is true for most drugs at therapeutic doses. Alcohol, aspirin at high doses, and several others exhibit zero-order kinetics instead, where a fixed amount is eliminated per unit time regardless of concentration — the half-life concept does not apply cleanly.

What this calculator is not

This is a single-isotope, single-stage decay tool. It does not model decay chains (e.g., U-238 → Th-234 → Pa-234 → U-234 → …), secular equilibrium, or branching ratios. It does not convert between activity (Bq or Ci) and mass — for that you need the isotope's specific activity. And it does not apply significant-figure rules to the output; the displayed precision is for mathematical clarity, not a claim about measurement uncertainty.