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2.6.2Forces, Space and Radioactivity

Half-Life — WJEC GCSE Science (Double Award) - Physics Revision Notes & Topical Questions

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📚 WJEC GCSE Science (Double Award) - Physics Hub / Topical Revision
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Section 1 of 9What half-life means

Because decay is random, we cannot say when a given nucleus will decay, but for a large number of identical nuclei the pattern is very regular.

Half-life
The half-life of a radioactive isotope is the time taken for the number of undecayed nuclei in a sample to fall to half of its original value.
Activity
Equivalent definitions are the time for the activity (or the corrected count rate) to fall to half of its original value.

Half-lives vary enormously

  • some isotopes have half-lives of fractions of a second, others thousands of millions of years.
  • Each isotope has its own characteristic half-life, and it does not depend on how big the sample is, nor on the temperature or chemical state.

Section 2 of 9Halving again and again

After one half-life half the nuclei remain. After two half-lives, half of that half remains, which is one quarter. The fraction left after n half-lives is (½)n.

Half-lives passed01234
Fraction remaining11/21/41/81/16

The same halving applies to the activity. A sample with an initial activity of 800 Bq has an activity of 400 Bq after one half-life, 200 Bq after two, and 100 Bq after three. Note that it is only the undecayed atoms that halve; the decayed atoms still exist, as atoms of the daughter product.

Section 3 of 9Using a decay curve

A graph of activity (or count rate) against time is a smooth curve that falls steeply at first and then flattens, approaching zero without touching it. To find the half-life from the graph:

1
Read the initial value at time zero.
2
Find half of that value on the vertical axis.
3
Read across to the curve and then down to the time axis: that time is one half-life.
4
Repeat from a different starting point on the curve (for example from the value at 2 hours to half of it) to check that the time taken is the same.
  • It always is.

If the readings include background, subtract it first, otherwise the curve will level off at the background value instead of falling towards zero and the halving will be wrong.

Section 4 of 9Calculations

A typical method for a calculation is:

Calculations
Work out how many half-lives fit into the time: n = total time ÷ half-life.
Halve the starting value n times (or multiply by (½)n).

When the time is not a whole number of half-lives the exact answer needs a logarithm, which is beyond this level; read an approximate value from a graph instead. You may also be asked for the fraction or percentage remaining. Percentage remaining = fraction × 100%.

You can also reason backwardsif a sample of activity 12 kBq is measured now, and its half-life is 3 days, then 6 days ago (two half-lives) it had 48 kBq.

Section 5 of 9Why half-life matters

The half-life of an isotope decides what it can be used for and how it must be stored.

1
A medical tracer needs a short half-life (hours) so that it does not stay active in the patient.
2
A source used in industry to monitor thickness needs a long half-life so that it does not have to be replaced frequently.
3
Radioactive waste with a very long half-life stays dangerous for a very long time and has to be stored securely.
4
Isotopes with a half-life of a fraction of a second are not generally useful because they disappear before they can be used.
5
You will see these applications in the next lessons.

Section 6 of 9Worked reasoning: reading a graph

Suppose a graph shows corrected count rate falling from 160 counts/s at 0 h.

1
Half of 160 is 80, and the curve passes through 80 at 4 h, so the half-life is 4 h.
2
Check: from 80 the next halving to 40 should happen at 8 h, and from 40 to 20 at 12 h.
3
If your readings do not agree, the curve has been misread or background has not been subtracted.
4
Because decay is random, real data points scatter slightly around the smooth curve,
  • so a best-fit curve is drawn through them before reading values, and the average of several half-life readings is the most reliable result.
5
Larger counts give less scatter, so counting for a longer time improves accuracy.

Linka source used up quickly (short half-life) has high initial activity for the same number of nuclei, while a long half-life source has a low but steady activity.

Section 7 of 9Worked examples

  1. An isotope has a half-life of 6 hours. A sample has an initial activity of 640 Bq. What is its activity after 24 hours?

  2. The corrected count rate from a source falls from 96 counts/s to 12 counts/s in 15 days. Calculate the half-life.

  3. A radioactive sample contains 2.4 × 1020 undecayed nuclei. Its half-life is 20 minutes. How many undecayed nuclei remain after 1 hour, and what percentage of the original is that?

Section 8 of 9Key terms

TermMeaning
Half-lifeTime for the number of undecayed nuclei, or the activity, of a sample to fall to half its original value.
IsotopeAtoms of an element with the same number of protons but different numbers of neutrons.
Decay curveGraph of activity or count rate against time, falling steeply then more gently.
Daughter nucleusThe nucleus produced when a parent nucleus decays.
Corrected count rateCount rate with the background count rate subtracted.

Section 9 of 9Exam technique

1
Thinking the sample has all decayed after two half-lives. After two half-lives one quarter remains, not zero.
2
Dividing the activity by the number of half-lives rather than halving that many times.
3
Using the half-life as the time for the sample's mass to halve. The nuclei that decay still exist as a daughter element, so the total mass changes little.
4
Forgetting to subtract background before using count rate to find a half-life.
5
Believing a bigger sample has a longer half-life. Half-life is a property of the isotope, independent of sample size.

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