Half-Life Calculation GCSE: Formula and Examples

A GCSE half-life calculator works out how much of a radioactive sample remains after a given time, or how long it takes for the sample to fall to a stated amount. Enter the initial and remaining quantities and the elapsed time, and it returns the number of half-lives and the half-life duration.

Formula

The GCSE relationship is that the amount halves once per half-life. If the initial and remaining quantities are known, first find the number of half-lives, then divide the elapsed time by that number:

Number of half-lives: n = log₂(N₀ / N) = log₁₀(N₀ / N) / log₁₀(2)

Half-life: T = t / n

• N₀ is the initial amount, in grams, number of nuclei, or another suitable quantity.
• N is the remaining amount, measured in the same units as N₀.
• n is the number of half-lives elapsed; it has no unit.
• t is the total elapsed time, in seconds, minutes, years, or another time unit.
• T is the half-life, in the same time unit as t.

The calculator uses base-10 logarithms to calculate log₂; these are equivalent ways of expressing the same formula. The calculation assumes the remaining amount is positive and no greater than the initial amount.

Worked Example

Using the default values, an initial amount of 80 g falls to 10 g over 18 seconds.

1. Find the fraction remaining: 10 / 80 = 1/8.
2. Recognise that 1/8 = (1/2)³, so n = 3 half-lives. The calculator’s logarithm gives log₂(80 / 10) = log₂(8) = 3.00.
3. Divide the total time by the number of half-lives: T = 18 s / 3 = 6.00 s.

The calculated half-life is 6.00 seconds. The amount would halve from 80 g to 40 g after 6 seconds, to 20 g after 12 seconds, and to 10 g after 18 seconds.

What Half-Life Means at GCSE

Half-life is the time taken for the number of undecayed radioactive nuclei in a sample to halve. It can also be described using the sample’s activity: after one half-life, its activity is half its original value, assuming the measurement reflects the same sample under comparable conditions.

Radioactive decay is random. You cannot predict exactly when a particular nucleus will decay, but a large collection of nuclei follows a predictable statistical pattern. As a result, half-life describes a sample or population, not a countdown for each individual atom. The material does not become non-radioactive after one half-life; half the original undecayed nuclei remain on average.

How to Use the Calculator

• Enter the starting quantity and the later quantity in matching units. Both may be mass, number of nuclei, or another comparable measure.
• Enter the elapsed time in the unit you want the answer expressed in. If time is in minutes, the calculated half-life is in minutes; if it is in years, the answer is in years.
• Read the first result as the number of half-lives, then read the second as the half-life duration.

The initial and remaining quantities do not need to be measured in grams specifically: the ratio is what matters. For example, comparing 80 nuclei with 10 nuclei gives the same number of half-lives as comparing 80 g with 10 g, provided both values describe the same kind of quantity and are measured consistently.

Finding the Remaining Amount Instead

Many GCSE questions give the half-life and ask how much remains after a particular time. Use:

N = N₀ × (1/2)^t/T

Here, t is elapsed time and T is half-life, and both must use the same time unit. For example, if a sample begins with 160 nuclei and undergoes four half-lives, the successive amounts are 80, 40, 20, then 10 nuclei. The formula gives 160 × (1/2)⁴ = 10.

If the number of half-lives is a whole number, repeatedly halving is often the quickest method in an exam. If it is not a whole number, the formula or a graph can help. For instance, after 1.5 half-lives, the remaining fraction is (1/2)^1.5, about 0.354, or roughly 35.4% of the starting amount.

Common Mistakes and Limits

• Mixing units: Do not divide a time in minutes by a number intended to represent elapsed seconds. Convert first, or consistently use one time unit throughout.
• Using mismatched quantities: The initial and remaining amounts must be comparable. A mass cannot be divided directly by a count of nuclei.
• Confusing a half-life with half the elapsed time: The sample halves after each interval T; the interval stays constant even as the amount gets smaller.
• Rounding too early: Keep extra digits during intermediate calculations, then round the final result to a sensible number of significant figures. Experimental measurements limit meaningful precision.
• Expecting an exact count in every sample: Decay is random, so actual observations may differ from the smooth predicted pattern, particularly for small numbers of nuclei.

This calculator finds a half-life from two amounts and an elapsed time using the standard exponential-decay model. It cannot establish whether real measurements are accurate, account for background radiation, or identify a substance’s half-life from its name. For practical experiments, measurement uncertainty and any background count should be considered separately.

Frequently Asked Questions

What is the half-life formula for GCSE?
For the amount remaining, use N = N₀ × (1/2)^t/T. To find the number of half-lives from two amounts, use n = log₂(N₀/N), then calculate T = t/n if elapsed time is known.

How do I calculate half-life without a calculator?
Look for repeated halvings. If a sample falls from 64 g to 8 g, it halves three times: 64 to 32 to 16 to 8. If that takes 15 minutes, each half-life is 15/3 = 5 minutes.

What does one half-life mean?
It means that, on average, half the undecayed radioactive nuclei in the sample remain after that interval. It does not mean all the radioactivity disappears or that every nucleus decays during that time.

Can half-life be measured in years or seconds?
Yes. A half-life can use any suitable time unit. The elapsed time and half-life must use the same unit when applying the remaining-amount formula.

Why is the remaining amount not always a whole number?
The mathematical model describes an expected proportion. A calculation can therefore produce a fractional number of nuclei, while an actual sample contains whole nuclei and may fluctuate because decay is random.

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in initial | Initial amount | g | 80 | 0
in remaining | Remaining amount | g | 10 | 0
in elapsed | Elapsed time | s | 18 | 0
out half_lives | Number of half-lives | - | log10(initial / remaining) / log10(2) | 2
out half_life | Half-life | s | elapsed / half_lives | 2