Half-Life Calculations: Formula and GCSE Examples

A half-life calculator works out how much of a radioactive substance remains after a given time, or how much time has passed, using its half-life. Enter the starting and remaining amounts plus the half-life or elapsed time to calculate the missing quantity.

Formula

For GCSE questions, the key relationship is: amount remaining = starting amount × (1/2)^{number of half-lives}. In symbols, N = N₀(1/2)^t/T. Here, N is the amount remaining, N₀ is the starting amount, t is the elapsed time, and T is the half-life. Amounts can be measured in grams, kilograms, or number of nuclei; t and T must use the same time unit.

This calculator instead finds elapsed time from two amounts. It uses t = T × log₂(N₀/N). The number of half-lives is n = log₂(N₀/N), so t = nT. The calculator outputs elapsed time in minutes and the number of half-lives, both rounded to two decimal places. Its inputs must be positive, and the remaining amount must not exceed the starting amount.

Worked Example

Using the default values, a sample starts at 80 g, 10 g remains, and its half-life is 6 minutes.

1. Find the ratio: 80 ÷ 10 = 8.
2. Find how many halvings produce that ratio: log₂(8) = 3 half-lives.
3. Multiply by the half-life: 3 × 6 min = 18 min.

The calculator therefore gives an elapsed time of 18.00 minutes and 3.00 half-lives. This checks against successive halving: 80 g becomes 40 g after 6 minutes, 20 g after 12 minutes, and 10 g after 18 minutes.

What Half-Life Means

Half-life is the time taken for the number of unstable nuclei in a radioactive sample to decrease to half its value. It is a property of the isotope: changing the sample’s mass does not change its half-life. For example, if an isotope has a half-life of 6 minutes, half of its radioactive nuclei decay during each 6-minute interval on average.

Radioactive decay is random for an individual nucleus: there is no way to predict exactly when one particular nucleus will decay. But a sufficiently large collection follows a predictable pattern. Half-life describes that statistical pattern; it does not mean that every nucleus lasts precisely one half-life, or that all the material disappears after two half-lives.

How to Calculate Half-Life Questions

A common GCSE method is to halve the starting amount repeatedly and count the intervals. This works especially neatly when the final amount is a simple fraction of the initial amount.

• Write down the starting amount and the amount remaining, keeping their units consistent.
• Halve the starting amount until you reach the remaining amount, counting each halving.
• Multiply the number of halvings by the half-life to get elapsed time.
• If the question gives elapsed time instead, divide by the half-life to find the number of intervals, then halve the amount once per interval.

For example, a 160 g sample with a half-life of 4 hours has 40 g after two half-lives: 160 → 80 → 40 g. Two intervals take 2 × 4 = 8 hours. If the time is not a whole number of half-lives, use the formula rather than forcing the amount into whole-number halvings.

Using the Exponential Formula

Real measurements need not land exactly on a sequence such as 160, 80, 40, and 20. The exponential formula handles any positive amount, including a fraction of a half-life. If the elapsed time is half of T, then the remaining amount is N₀/√2, approximately 0.707N₀, not half the starting amount. One full half-life is required for the amount to halve.

When a question gives a graph, read two points carefully and compare the amount at the start with later values. A graph may show activity or count rate rather than mass. If the measured activity is proportional to the number of undecayed nuclei, it falls by the same factor over each half-life, so the same halving approach applies. Background radiation can affect measured count rates, so classroom experiments may require subtracting a stated background count before comparing readings.

Common Mistakes and Limits

• Mixing time units: If the half-life is in seconds, convert elapsed time to seconds before dividing; alternatively convert both to minutes or hours.
• Counting intervals instead of endpoints: Going from 80 g to 40 g is one half-life, not two. Count each change from one value to the next.
• Confusing half-life with complete decay: In the mathematical model, the amount approaches zero but does not become exactly zero after a fixed number of half-lives.
• Rounding too early: Keep calculator values unrounded through intermediate steps, then round the final result to the precision requested.
• Using incompatible quantities: Do not compare a starting mass with a remaining count unless the question establishes a valid relationship between them.

The formula models radioactive decay for an isotope with a constant half-life. A measured sample may not show exact values because decay is random, measurements have uncertainty, and background radiation can contribute to readings. The calculator also assumes the remaining amount is positive; zero cannot be used in the logarithm.

Frequently Asked Questions

What is the half-life formula for GCSE physics?
N = N₀(1/2)^t/T, where N is the amount remaining, N₀ the starting amount, t the elapsed time, and T the half-life. Keep t and T in the same units.

How do I find the number of half-lives?
Divide elapsed time by the half-life: n = t/T. If you know the starting and remaining amounts instead, use n = log₂(N₀/N).

Does half-life depend on how much radioactive material there is?
No. The isotope’s half-life is unchanged by the sample size. A larger sample generally contains more unstable nuclei, but the fraction expected to decay during a given interval is the same.

What happens after three half-lives?
One eighth of the initial amount remains, because it is halved three times: (1/2)³ = 1/8. That is 12.5% of the starting amount.

Can half-life be measured from a graph?
Yes. Find a starting value, then read the time taken for the graph to fall to half that value. Repeat at other points if possible; readings may vary because of measurement uncertainty and random decay.

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in initial | Starting amount | g | 80 | 0
in remaining | Amount remaining | g | 10 | 0
in half | Half-life | min | 6 | 0
out elapsed | Elapsed time | min | half * log2(initial / remaining) | 2
out halves | Number of half-lives | - | log2(initial / remaining) | 2