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Half-Life & Radioactive Decay Calculator

Calculate remaining amount, initial amount, elapsed time or half-life for radioactive decay and first-order reactions. Includes decay constant and mean lifetime. Carbon-14 dating example.

Same units as N₀ (g, atoms, Bq, %…)
ResultEnter values to calculate
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    Law of Radioactive Decay formula

    N(t) = N0 (12)t/t½ = N0 e−λt
    SymbolQuantitySI unitMeaning
    NRemaining quantity—Quantity remaining after time t
    N₀Initial quantity—Initial quantity
    tElapsed timesElapsed time
    t½Half-lifesHalf-life — time for the quantity to halve

    What is Law of Radioactive Decay?

    Law of Radioactive Decay: The number of undecayed nuclei decreases exponentially with time; in each half-life, half of what remains decays. — Ernest Rutherford & Frederick Soddy, 1902

    Radioactive decay, and any first-order reaction, follows exponential decay. The half-life t½ is constant: it does not depend on how much material you start with. The decay constant is λ = ln 2 / t½, and the mean lifetime is τ = 1/λ.

    The same equation applies to drug elimination (pharmacokinetics), first-order chemical kinetics and capacitor discharge. N can be in any unit — grams, number of atoms, activity in becquerels, or a percentage.

    How to use this calculator

    1. Choose the quantity you want to find in the Calculate menu. It is highlighted as the result field.
    2. Enter the known values. Scientific notation such as 1.6e-11 is accepted.
    3. Select a unit for each value. Units are converted to SI automatically.
    4. The answer updates as you type. Open Show calculation steps to see the working, or use Copy result to share a link with your inputs.

    Worked example

    Carbon-14 has a half-life of 5730 years. What percentage remains after 10 000 years?

    1. t/t½ = 10 000 / 5730 = 1.745 half-lives.
    2. N = 100 × 0.51.745 = 29.8%.

    To date a sample, choose "Elapsed time" and enter the measured remaining fraction.

    Frequently asked questions

    How do I find the age of a sample?

    Choose "Elapsed time" in the Calculate menu. Enter N₀ = 100 and N = the percentage of the original isotope that remains, plus the half-life. The calculator returns t = t½ · log₂(N₀/N).

    What is the relationship between half-life and the rate constant?

    For first-order processes, t½ = ln 2 / k ≈ 0.693 / k. The calculator shows this decay constant as a derived result.