Half-Life Calculator

How to use this half-life calculator

What this calculator does

This calculator estimates how much of an exponentially decaying quantity remains after a chosen time. It also reports the decay constant, mean lifetime, elapsed number of half-lives, and the amount already decayed. The model applies to radioactive nuclei and to other processes that follow first-order exponential decay. It does not identify an isotope, predict radiation dose, or determine whether a material is safe to handle.

When to use it

Use it to check laboratory exercises, compare isotopes with different half-lives, estimate how activity changes during storage, or explore first-order decay in chemistry and pharmacokinetics. The U.S. Environmental Protection Agency's explanation of radioactive decay and half-life provides useful background for interpreting the calculation.

How to calculate

1. The calculator opens with a complete demonstration: 1,000 units, a 5-minute half-life, and 10 minutes elapsed. Results and a validated Excel workbook are immediately available.
2. Replace Initial quantity (N₀) with the amount present at time zero and select a descriptive quantity unit.
3. Enter the Half-life time (T), select the time unit, and enter Total time (t) in that same unit.
4. Read the remaining quantity first, then use the supporting values, chart, and decay schedule to understand the result.
5. Select Download Excel to export the current typed values and results. Reset clears the demonstration and disables export until a complete valid state is entered again.

Input guide

Initial quantity (N₀) is required and accepts a positive decimal in a period-decimal format, such as 1000 or 2.5. Choose units, g, kg, mg, mol, or Bq from Quantity unit. The unit is a label only; the exponential fraction is unchanged. A larger initial quantity raises both the remaining and decayed quantities proportionally. Do not mix commas as decimal separators or type unit symbols into the number field.

Half-life time (T) is required and must be greater than zero. A value such as 5 with minutes selected means the quantity halves every five minutes. A longer half-life slows decay. Time unit applies to the half-life, total time, decay constant, and mean lifetime, so do not enter 5 minutes and 10 hours without converting one value first.

Total time (t) is required and may be zero or positive. Zero returns the full initial quantity. A larger elapsed time decreases the remainder but never makes it negative. Scientific notation and decimal commas are intentionally rejected to avoid ambiguous pasted values.

Output guide

Remaining quantity (N(t)) is the modeled amount after the selected interval. Decayed quantity is the initial amount minus the remainder. Elapsed half-lives equals total time divided by half-life; an integer value shows how many complete halving intervals have passed. The summary pills report the same count plus the percentages remaining and decayed.

Decay constant (λ) is the first-order rate constant, expressed per selected time unit. A larger λ means faster decay. Mean lifetime (τ) is the reciprocal of λ and is longer than the half-life by a factor of 1/ln(2). The line chart shows the modeled quantity at nine times from zero through the selected total time. The Decay schedule table lists the step, elapsed time, elapsed half-lives, remaining quantity, and remaining percentage for those same points.

Worked example

With N₀ = 1,000 units, T = 5 minutes, and t = 10 minutes, the elapsed count is 10 ÷ 5 = 2 half-lives. The model gives N(t) = 1,000 × 0.5² = 250 units. Therefore 25.00% remains and 750 units, or 75.00%, have decayed. The decay constant is ln(2) ÷ 5 = 0.138629 min⁻¹, and the mean lifetime is 1 ÷ λ = 7.2135 minutes.