Half-life: the two-element definition
Half-life = the time for half the radioactive nuclei in a sample to decay (equivalently, the time for the activity to fall to half its initial value). Two required elements: a time interval AND that it is half the nuclei / activity halving. Activity = the number of nuclear decays per second, measured in becquerels (Bq); 1 Bq = 1 decay per second. Different isotopes have different half-lives, from fractions of a second to billions of years.
The half-life decay formula
With = initial nuclei (or initial activity ), = half-life, = elapsed time: The exponent is the number of half-lives. For integer , halve times: The same formula applies to activity () and to count rate. The decay curve is asymptotic — it never reaches zero.
Read half-life from an activity graph
Method: read an activity on the smooth curve, e.g. Bq at ; find when it has halved to Bq — that time is one half-life. Repeat from another point (e.g. Bq) and the interval should match. Always use two points with a 2:1 activity ratio and subtract the times. Never read the half-life from a single point, or as the time activity reaches zero — the curve is asymptotic and never touches the axis.
Drawn from real examiner reports.
Half-life: name the nuclei
"Time for activity to halve" alone omits the atomic reason — it is half the unstable (radioactive) nuclei that decay. The mark-scheme form links the time to the nuclei: "the time for the number of radioactive nuclei to fall to half its original value". "Time for radiation to halve" is rejected because it never names nuclei.
June 2024 Paper 2P — candidates who gave vague everyday definitions of half-life scored zero for omitting the reference to radioactive nuclei, even when they could do the calculation.
Non-integer half-lives need the formula
Repeated halving works ONLY for a whole number of half-lives. If min and min, then — you cannot just halve 3 times. Use . Even for integer , students often halve one too many or too few times. Always compute first, then apply the formula.
June 2024 Paper 2P — a significant minority halved the initial activity the wrong number of times, especially when the time was not an exact multiple of the half-life.
Graph: use two points, not one
Reading where the curve crosses a single activity value (e.g. 50 Bq) and calling that time the half-life only works if the initial activity is exactly 100 Bq. Instead take two readings — any activity and exactly half of it — and use the time difference. Averaging several pairs improves precision. The curve is asymptotic, so "the time the radiation stops" is always wrong.
June 2024 Paper 2P — many candidates read the half-life from a single point on an activity-time graph rather than the time difference between two points at a 2:1 activity ratio.
Half-life ≠ time to reach zero
Decay is exponential, so a sample never fully decays — the curve approaches the time axis but never touches it. The half-life is the time for half the remaining nuclei to decay, not the time until none are left. A source with a 1-minute half-life still has a tiny fraction of its nuclei after 10 minutes.
Activity (Bq) ≠ count rate
Activity is the true number of decays per second in the source, in becquerels (Bq). Count rate is what a detector records — always lower, because it captures only some emitted particles. The half-life formula applies to both, but they are not numerically equal. Use the units the specification gives, especially Bq for activity.
November 2024 Paper 2P — recall and use the specified units; weaker candidates did not distinguish activity (measured in Bq) from count rate.
Find n from the ratio, not t ÷ A
To get the number of half-lives from data, take the ratio of activities and write it as a power of 2 — e.g. , so . A common dimensional slip is dividing time by activity (e.g. ), which is meaningless. Once is known, half-life .
Bank the method marks
Show every step: (1) ; (2) state the formula ; (3) substitute; (4) evaluate; (5) give the unit (Bq for activity, counts for count rate). Each labelled step scores, even if the arithmetic slips.
Write out the halving sequence
For a whole number of half-lives, write the chain and count the arrows carefully — this earns the method mark even if you slip, and it guards against halving one time too many or too few.
Half-life from a graph: two points
Draw a horizontal line from and from across to the curve, drop verticals to the time axis, and subtract the two times. Check with a second pair (e.g. ). Never use a single point or the time the curve appears to reach zero.
Spot n as a power of 2
When activity falls by a factor of 2, 4, 8 or 16, recognise these as to read off instantly, then half-life . Example: a drop to of the start means .
| Quantity | Symbol | Formula | Unit |
|---|---|---|---|
| Number of undecayed nuclei at time | — (count) | ||
| Activity at time | becquerel, Bq | ||
| Number of half-lives elapsed | — (dimensionless) |
Where = initial number of radioactive nuclei, = initial activity (Bq), = half-life (in the same time unit as ).
Key constant: 1 Bq = 1 nuclear decay per second.
Define half-life.
A radioactive sample initially contains 1200 undecayed nuclei. The half-life of the isotope is 3 hours.
Calculate the number of undecayed nuclei remaining after 9 hours.
