Radioactive decay follows a fixed, substance-specific half-life. This works out how much of a sample remains after a given time, or how much time has passed given a known remaining amount.
How it works
Exponential decay
N(t) = N₀ × (½)^(t / half-life)
N₀ is the initial amount, t is elapsed time, and half-life is the substance’s own characteristic constant.
A half-life is not a countdown timer on any single atom
Any one radioactive atom has no “memory” of how long it has already existed — its chance of decaying in the next hour is identical whether it formed a second ago or a billion years ago. The half-life describes the population’s average behaviour, which is why this equation works cleanly for a bulk sample (billions of atoms averaging out) and says nothing about when any one specific atom will decay.
After one half-life, half remains — not zero, and never quite zero
This corrects the everyday misconception the name invites. After ten half-lives, roughly 0.1% of the original amount is still there, not a clean zero — the decay curve approaches zero asymptotically and mathematically never actually reaches it.
How to use this calculator
- Enter the initial amount and the substance’s half-life.
- Choose whether you know the elapsed time or the amount remaining.
Frequently asked questions
Does the half-life change as the sample decays?
No — it’s a fixed property of the specific radioactive isotope, unaffected by how much of the sample remains, its temperature, or its chemical form. This is what makes the exponential formula valid throughout the entire decay process.
How is half-life used in radiocarbon dating?
Carbon-14’s known half-life (about 5,730 years) lets scientists work backward from how much C-14 remains in an organic sample to estimate how long ago the organism died — solving this exact equation for elapsed time.
Can the remaining amount ever be zero?
Mathematically, never exactly — the exponential decay curve approaches zero but never reaches it. In physical reality, once you’re down to a single atom, decay becomes a discrete probabilistic event rather than a smooth curve, but the formula itself never outputs a literal zero.
Why does this calculator decline an amount remaining greater than the initial amount?
Because decay only ever reduces the amount present — a “remaining” figure larger than what you started with isn’t a valid decay scenario for this equation.
Does this apply to non-radioactive decay processes too?
The same exponential mathematics describes any process with a constant proportional decay rate — some drug elimination in the body, for instance — though the term “half-life” and this specific formula are most commonly associated with radioactivity.
