How to use this calculator
- Enter or choose initial quantity (N₀).
- Enter or choose remaining quantity (N).
- Enter or choose half-life (T½).
- Enter or choose time elapsed (t).
- The answer updates as you type. Use Copy result or Share to keep it.
Method
N = N₀ × (½)^(t ÷ T½); T½ = t × ln 2 ÷ ln(N₀ ÷ N); t = T½ × log₂(N₀ ÷ N); λ = ln 2 ÷ T½; τ = 1 ÷ λ
- N₀
- starting quantity
- N
- quantity remaining after time t
- T½
- half-life
- λ
- decay constant
- τ
- mean lifetime
Worked example
- t ÷ T½ = 11,460 ÷ 5,730 = 2 half-lives.
- N = 100 × (½)² = 100 × ¼ = 25.
- λ = ln 2 ÷ 5,730 ≈ 0.000120968 per year.
After 11,460 years, two carbon-14 half-lives, 25 of the original 100 units remain.
What the result means
After 7 half-lives less than 1% remains, and after 10 about 0.1%, which is why “ten half-lives” is a common rule of thumb for “essentially gone”.
Assumptions and limits
- Simple first-order (exponential) decay with a constant half-life.
- All quantities use the same unit, and the half-life and time use the same time unit.
- Real samples can contain several isotopes or decay chains, and drug elimination often departs from a single half-life; those need more detailed models.
Questions
Does the amount ever reach zero?
In the model it never quite does: each half-life halves what is left. In practice a sample becomes undetectable, or for atoms, the last one decays at a random moment.