Half-Life Calculator
Calculate radioactive decay, remaining quantity, half-life, or elapsed time using the half-life formula.
Model radioactive or exponential decay — find how much of a substance remains after a given time, solve for the half-life itself, or work out how much time has elapsed.
How Half-Life Calculator Works
The core relationship is N(t) = N₀ × (1/2)^(t / t½), where N₀ is the starting amount, t½ is the half-life, and t is the elapsed time — every time one full half-life passes, exactly half of the remaining amount decays away. Rearranging this same formula using logarithms lets the calculator instead solve for the half-life itself (given a known remaining amount) or for the elapsed time (given a known half-life).
The decay constant λ = ln(2) / half-life and the mean lifetime τ = half-life / ln(2) describe the exact same decay process using two other common conventions from nuclear physics and chemistry — λ representing the fractional decay rate per unit time, and τ representing the average time a single atom survives before decaying.
"Half-lives elapsed" is simply t divided by the half-life, and doesn't need to be a whole number — a value of 1.5, for instance, means one and a half half-life periods have passed, corresponding to just over a third of the original material remaining.
See It In Action
Who Uses Half-Life Calculator and Why
- Estimating how much of a radioactive isotope remains after a given number of years for a chemistry or physics problem.
- Working through a carbon-14 dating style calculation to estimate an artifact's age from remaining material.
- Modeling how a medication or caffeine level decreases in the body over time using the same exponential decay pattern.
- Solving for an unknown half-life itself when the starting amount, remaining amount, and elapsed time are all known.
Mistakes to Avoid
- Assuming the remaining amount hits exactly zero after a whole number of half-lives — it only ever approaches zero, halving indefinitely without ever fully reaching it.
- Confusing the decay constant (λ) with the half-life itself — they're related but not interchangeable, and λ = ln(2) ÷ half-life specifically.
- Applying this exponential decay formula to a process that doesn't actually follow a fixed 'halves every set interval' pattern.
Tips for Best Results
- Use the 'half-lives elapsed' figure to sanity check a fractional decay period — a value of 1.5 means one and a half half-life periods have passed.
- Remember mean lifetime (τ) is always longer than the half-life by a factor of about 1.4427 (1/ln 2), since some material persists well beyond the median decay point.
Fixing Common Problems
Solving for half-life gives an unexpected number. — Double-check the starting amount, remaining amount, and elapsed time were all entered in consistent units — the rearranged formula (half-life = t × ln(2) ÷ ln(N₀/Nt)) is sensitive to inconsistent time units or a swapped N₀/Nt pair.
Terms Explained
Decay constant (λ): The fraction of remaining material that decays per unit time, calculated as λ = ln(2) ÷ half-life.
Mean lifetime (τ): The average time a single particle survives before decaying, always longer than the half-life by a factor of 1/ln(2).