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Brainlag

Half-life calculator

Find the amount left, the time taken or the half-life from the other two, with N = N0 x (1/2)^(t / T1/2) and the working shown. For GCSE and A-level physics.

See it happen: Radioactive decay

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Lessons from the Combined and Separate Science courses on this topic: worked steps, then exam-style questions with new numbers every time.

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Formulas

Amount leftN = N₀ × (1/2)^(t / T½)
Timet = T½ × log₂(N₀ / N)
Number of half-livest / 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?

  1. 24 / 6 = 4 half-lives.
  2. Halve four times: 800, 400, 200, 100, 50.
  3. 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.

How it works

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.

  1. Enter any three of N₀ (initial amount), N (amount left), t (time) and T½ (half-life). Leave the fourth empty.
  2. Use the same time unit for t and T½, such as seconds, minutes, days or years.
  3. Use the same unit for N₀ and N: grams, numbers of atoms, counts, Bq or percentages all work.
  4. Read the answer and the working, then check the number of half-lives and the percentage remaining.
  5. Look at the decay curve and the table to see how the amount halves with each half-life.

Tips

  • Check with whole half-lives first. 80 g with a half-life of 5 years: after 15 years that is 3 half-lives, so 80 → 40 → 20 → 10 g, which is 12.5% remaining.
  • 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

How much is left after three half-lives?
One eighth. Half is left after one half-life, a quarter after two and an eighth after three. In general the fraction left after n half-lives is (1/2)^n.
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.

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