How to calculate half-life and the amount left (worked examples)

How to · Updated 2026-11-10

Amount left = start × (1/2)^(t ÷ half-life). How to use it both ways, a worked iodine-131 example, finding a half-life from two readings, and quick checks.

The amount of a radioactive sample left after a time t is the starting amount × (1/2)^(t ÷ T½), where T½ is the half-life. To find the half-life from two readings, turn it round: T½ = t × ln 2 ÷ ln(start ÷ end). The decay calculator does the first sum from dates and times and tells you when a sample will drop to any level you type.

What you need

• A starting amount. This can be a mass, a number of atoms, a count rate or an activity in becquerels (Bq). The formula works the same for all of them.
• The half-life, the time it takes for half of the sample to decay. Or, if that is the unknown, a second reading taken later.
• The time that has passed, in the same unit as the half-life.

Step by step

Each half-life, whatever is left is cut in half. So after 1 half-life you have 1/2, after 2 you have 1/4, after 3 you have 1/8. In general:

N = N₀ × (1/2)^n, where n = t ÷ T½

• N₀ is the amount at the start, N the amount left.
• t is the time passed and T½ the half-life, in the same units.
• n is the number of half-lives. It does not have to be a whole number.

To find the half-life when you know the start, the end and the time, take logs of both sides:

T½ = t × ln 2 ÷ ln(N₀ ÷ N)

You may also meet the decay constant λ (lambda), the fraction of nuclei that decay per second: λ = ln 2 ÷ T½ = 0.693 ÷ T½.

With the tool:

1. Pick the Nuclide from the list, or choose “Your own half-life…” and type the half-life and its unit.
2. Type the Activity at the reference time and pick its unit.
3. Set the Reference date and time and the Wanted date and time. The large result shows the activity at the wanted time and the percentage left.
4. Under the result, “Time between” also tells you how many half-lives have passed.
5. Optionally type a value in When will it fall to to get the date and time it reaches that level.
6. Set the step of the Decay table to the half-life to see the halving row by row.

A worked example

A lab receives iodine-131 with an activity of 200 MBq. Its half-life is 8.0252 days. How much is left 24 days later?

Number of half-lives. n = 24 ÷ 8.0252 = 2.991.

Amount left. N = 200 × (1/2)^2.991 = 25.16 MBq, which is 12.58 % of the start.

A rough check: 24 days is about 3 half-lives, and 200 → 100 → 50 → 25. Our answer is just above 25, because 2.991 is slightly less than 3.

In the decay calculator, choose I-131, type 200 with MBq, and set the wanted date 24 days after the reference. It gives 25.16 MBq (12.58 % of the reference) and 2.991 half-lives. Type 10 in “When will it fall to” and it says 34.68 days after the reference.

Finding a half-life from two readings. A detector reads 640 counts per second from a source. Ten hours later it reads 200. What is the half-life?

T½ = 10 × ln 2 ÷ ln(640 ÷ 200) = 10 × 0.6931 ÷ 1.1632 = 5.96 hours

To check it on the page, choose “Your own half-life…”, type 5.96 hours, set 640 Bq at the reference time and the wanted time 10 hours later. The result comes out at 200 Bq.

Check your answer

• Count the halvings. If the time is close to a whole number of half-lives, halve the start that many times in your head. Your answer should be near that.
• The answer is never zero. A sample halves forever. After 10 half-lives about 0.1 % is still there.
• Less time, more left. If less than one half-life has passed, more than half must remain.
• Units must match. A half-life in days with a time in hours gives nonsense. Convert one of them first.
• Plug the half-life back in. For the second example, 640 × (1/2)^(10 ÷ 5.96) is about 200, so the answer holds.

Questions people ask

Why does the same fraction decay each half-life, no matter how much there is? Each nucleus has the same fixed chance of decaying in a given second, and it has no memory of how long it has waited. The radioactive decay dice page shows this with hundreds of dice: remove every six on each throw and a steady half-life appears.

Is half-life only for radioactivity? No. Anything that falls by a fixed fraction in each equal time has one, such as a drug leaving the blood or a first-order chemical reaction. The same formula applies.

How do I find the half-life from a graph? Read the starting value, find half of it on the vertical axis, go across to the curve and down to the time axis. That time is one half-life. Repeat from there and you should get the same gap.