How to use it
- Choose the quantity you want in Quantity to find.
- Fill in the other boxes that it needs: the Amount at the start, the Amount remaining, the Half-life and the Elapsed time.
- Use the same time unit, such as years or days, for the half-life and the elapsed time.
- Read the Result, the table of the amount after each half-life and the steps.
The decay of a radioactive substance
Radioactive decay follows first-order kinetics, so the number of atoms that decay in a moment is proportional to the number that are left. The amount left after a time t is N = N0 e^(−λt), where λ is the decay constant. The half-life T is the time in which half of the atoms decay, and it fixes λ (OpenStax, Chemistry 2e, section 21.3).
N = N0 e^(−λt) = N0 × 2^(−t ÷ T)
λ = ln 2 ÷ T = 0.693 ÷ T
N0 = amount at the start
N = amount remaining after the time t
T = half-life
t = elapsed time (the same unit as T)- After one half-life half is left, after two a quarter, after three an eighth, and so on.
- Solved for the time, t = −ln(N ÷ N0) ÷ λ; solved for the half-life, T = ln 2 × t ÷ ln(N0 ÷ N).
Rounding in the textbook
The OpenStax example uses λ = 0.132 per year for cobalt-60, which has a half-life of 5.27 years. The exact value is 0.131527, so the calculator gives 13.9 % after 15 years and 29.7 years for 2.0 % to remain. The book's 13.8 % and 29.6 years come from the rounded λ.
What this calculator does not include
- Chains of decays, where a product is itself radioactive, and the build-up of a daughter isotope.
- The radiation dose, the activity in becquerels and the shielding of the radiation.
- A half-life that is very short compared with the time you count, where the measurement itself limits the result.
Worked example: cobalt-60 after 15 years
Cobalt-60 has a half-life of 5.27 years, so λ = 0.693 ÷ 5.27 = 0.1315 per year. After 15 years the fraction left is e^(−0.1315 × 15) = 0.1391, which is 13.9 % of the start. The time for 2.0 % to remain is ln(100 ÷ 2) ÷ 0.1315 = 29.7 years.
| Question | Answer |
|---|---|
| Decay constant λ | 0.1315 per year |
| Fraction left after 15 years | 13.9 % |
| Time for 2.0 % to remain | 29.7 years |
| Half-lives in 15 years | 2.85 |
Frequently asked questions
What is the half-life of a substance?
The half-life is the time in which half of the radioactive atoms decay. It is a fixed property of each isotope, and it does not depend on how much of the substance there is.
Why does the time unit not matter?
The result is a ratio of times, t ÷ T, so it does not depend on the unit as long as the half-life and the elapsed time use the same one. A result of time comes out in that unit.
Why can the amount never reach zero?
Each half-life halves the amount, so it gets smaller and smaller without reaching zero. The calculator shows the fraction that is left, and the table lists it after each half-life.
Why does my textbook give a slightly different answer?
Textbooks often round the decay constant before the next step. The OpenStax cobalt-60 example rounds λ to 0.132, which changes the second significant figure of the answer.
Is my data sent to a server?
No. The calculation runs in your browser, and nothing you type is sent to us or stored.
Sources
Figures and text last checked on Tuesday, November 10, 2026.