Half-Life Calculator
Calculate radioactive or exponential decay using half-life. Enter the initial amount, half-life and elapsed time to get remaining amount, fraction left, number of half-lives and a chart.
How to use this tool
- Enter initial amount, half-life (time units) and elapsed time (same units) in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your remaining amount and the full breakdown beneath it.
The half-life t½ is the time needed for a quantity to reduce to half its value. Using A(t) = A₀ · e^(−λt) where λ = ln(2)/t½, this calculator finds the remaining and decayed amounts at any elapsed time.
Formula
Remaining = A0 × e−kt
where decay constant k = ln(2) / t1/2, A0 = initial amount, t = elapsed time, t1/2 = half-life.
Fraction remaining = e−kt | Number of half-lives = t ÷ t1/2
How it works
This calculator uses the continuous exponential decay law, deriving the decay constant k from the half-life via k = ln(2) / t1/2. The remaining amount at any elapsed time is then A0 × e−kt. This model is exact for first-order processes such as radioactive decay and is a good approximation for many biological and chemical decay scenarios. The time units for half-life and elapsed time must match for the result to be correct.
Worked example
- Initial amount A₀ = 1000, half-life t½ = 10 time units, elapsed time t = 20 time units.
- Decay constant k = ln(2) / 10 = 0.06931 per time unit.
- Number of half-lives elapsed = 20 / 10 = 2.
- Remaining = 1000 × e^(−0.06931 × 20) = 1000 × e^(−1.3863) = 1000 × 0.25 = 250.
- Amount decayed = 1000 − 250 = 750.
Remaining amount: 250.0; Amount decayed: 750.0; Half-lives elapsed: 2.0.
Common mistakes to avoid
- Entering half-life and elapsed time in different units (e.g. half-life in days, elapsed time in hours) without converting first — the decay constant k = ln(2)/t_half, so mismatched units give wrong remaining amounts.
- Confusing biological half-life (drug elimination by the body) with radiological half-life (nuclear decay) — the formula is identical but the safe-exposure implications differ entirely.
- Expecting the remaining amount to reach exactly zero — exponential decay is asymptotic; the quantity approaches zero but never mathematically reaches it.
Key terms
- Half-life (t½)
- The time required for a quantity undergoing exponential decay to reduce to half its initial value.
- Decay constant (k)
- The rate parameter in continuous decay, related to half-life by k = ln(2) / t½. Larger k means faster decay.
- Fraction remaining
- The proportion of the original amount still present at time t, equal to e^(−kt) or equivalently (0.5)^(t/t½).
- Number of half-lives
- The elapsed time expressed as a multiple of the half-life (t / t½). After n half-lives, 1/2^n of the original amount remains.
- First-order decay
- A decay process where the rate of loss is proportional to the current amount — the mathematical basis for radioactive decay and the half-life formula.
Frequently asked questions
- What substances have known half-lives?
- Carbon-14 has a half-life of about 5,730 years (used in carbon dating). Uranium-238 has ~4.5 billion years. Iodine-131 used in medicine has ~8 days.
- Can I use this for non-radioactive decay?
- Yes — any process that halves on a fixed schedule (drug concentration, population loss, etc.) can be modelled this way.