Half-life Calculator
Find the amount left, the time or the half-life from the radioactive decay formula, with the working.
See it happen: Radioactive decay
Formulas
- Amount left N = N₀ × (1/2)^(t / T½)
- Time t = T½ × log₂(N₀ / N)
- Number of half-lives t / T½
Worked example
A sample gives 800 counts per minute and has a half-life of 6 hours. What is the count rate after 24 hours?
- 24 / 6 = 4 half-lives.
- Halve four times: 800, 400, 200, 100, 50.
- Or with the formula: N = 800 × (1/2)⁴ = 800 / 16 = 50.
Answer: 50 counts per minute.
Common mistakes
- Taking away half of the original amount each time instead of halving what is left.
- Mixing time units: the time and the half-life must be in the same unit.
- Forgetting the background count: take it away from each reading before using the formula.
About this tool
This half-life calculator works with the radioactive decay formula N = N₀ × (1/2)^(t / T½). Fill in any three of the initial amount, the amount left, the time and the half-life, and it finds the fourth with the working. You also get the number of half-lives, the percentage remaining, a decay curve with each half-life marked and a table of the amount after each one.
Tips
- When t is not a whole number of half-lives, the formula still works. Solving for t or T½ uses logarithms: t = T½ × log(N₀/N) / log(2).
- Half-life is the time for half of the remaining nuclei to decay, not half of the original amount each time. That is why the curve never quite reaches zero.
- Activity in Bq follows the same pattern as the number of nuclei, so you can enter activities as N₀ and N.
FAQ
- Which units should I use?
- Any units, as long as they are consistent. t and T½ must be in the same time unit, and N₀ and N must be in the same unit as each other. The answer comes out in the unit you used.
- How does it find the time or the half-life?
- It rearranges N = N₀ × (1/2)^(t / T½) using logarithms, for example t = T½ × log(N₀ / N) / log(2). The working shows each step, so you can follow the same method by hand.
- What does the number of half-lives mean?
- It is t divided by T½. After 1 half-life, 50% is left. After 2, 25%. After 3, 12.5%. A number that is not whole, such as 2.5 half-lives, is fine: about 17.7% remains.