Half-Life Calculator
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The Exponential Decay Formula
Half-life describes any quantity that decreases at a rate proportional to its current amount — most famously radioactive isotopes, but also drug concentration in the bloodstream or capacitor discharge. The governing equation is N(t) = N₀ × (1/2)t / T, where N₀ is the starting quantity, T is the half-life, and t is the elapsed time in the same units as T. This calculator also reports the decay constant λ = ln(2) / T and the mean lifetime τ = 1 / λ = T / ln(2), which is the average time a single atom or particle survives before decaying — always about 44% longer than the half-life itself.
A Common Misconception: Half-Lives Don't "Run Out"
Decay is exponential, not linear — a substance never mathematically reaches exactly zero, it just gets arbitrarily small. After 1 half-life, 50% remains; after 2, it's 25%, not 0%; after 10 half-lives, roughly 0.1% remains, which is why 10 half-lives is often treated as "effectively gone" in practice. Each half-life removes half of whatever is left at that moment, not half of the original amount.
Related Tools
Half-life problems are really exponent and logarithm problems in disguise — if you want to see the underlying algebra worked out step by step, try the exponent calculator or the log calculator.
Frequently Asked Questions
What is the formula for half-life decay?
The remaining quantity is N(t) = N0 x (1/2)^(t/T), where N0 is the starting amount, T is the half-life, and t is the elapsed time measured in the same units as T. This calculator applies that formula directly, along with the decay constant (lambda = ln(2)/T) and mean lifetime (tau = 1/lambda), which are the two other standard ways scientists express the same decay rate.
Why is the mean lifetime longer than the half-life?
Half-life is the median survival time (50% of a sample has decayed), while mean lifetime (tau) is the average survival time of an individual atom, which is pulled higher by the small fraction that survives much longer than average. Mathematically tau = T / ln(2), so the mean lifetime is always about 44% longer than the half-life, regardless of the substance.