Exponential Decay of Activity
Quantum and Nuclear

Episode 514: Patterns of decay

Lesson for 16-19 IOP TAP

This episode assumes that students already have an idea of half-life, and links it to the empirical decay curve.

Lesson Summary

  • Discussion and demonstration: Measuring half life (40 minutes)
  • Worked examples: Involving whole numbers of half lives (30 minutes)
  • Student example and discussion: Plotting a graph (30 minutes)
  • Student questions: Calculations (30 minutes)

Discussion and demonstrations: Measuring half life

Each radioactive nuclide has its own unique half-life. Values range from millions of years (e.g. uranium-238 at 4.47 × 109 year) to minute fractions of a second

(e.g. beryllium-8 at 7 × 10-17 s).

Sealed school/college sources have half-lives chosen to ensure that they will remain radioactive over a period of years (though Co-60 will become significantly less active year-by-year):

RadiationSourceNuclideNotesT1/2
αamericium-24124195AmAlso emits gammas458 year
βstrontium-909038SrThe energetic betas come from the daughter Ys-9028.1 year
γcobalt-606027CoAlso emits betas, but these may be absorbed internally5.26 year

You need something else if you are to measure half-life in the course of a lesson. A short half-life source commonly available in schools and colleges is protactinium-234, T ½  = 72 s. Another is radon-220, T ½  = 55 s.

Demonstrate the measurement of half-life for one of these. (There is a data logging opportunity here.) It is vital to correct for background radiation. Explain how to find T ½ from the graph.

(Students could repeat this experiment for themselves later.)

Demonstration of half life of protactinium: Measuring half life

Episode 514-1: Half life of protactinium (Word, 64 KB)


Worked examples: Involving whole numbers of half-lives

Here is a set of questions involving integral numbers of half lives. You could set them as examples for your students, or work through them to explain some basic ideas.

Question: The nuclear industry considers that after 20 half lives, any radioactive substance will no longer present a significant radiological hazard. The half life of the fission product from a nuclear reactor, caesium-137, is 30 years. What fraction will still be active after 20 half lives?

Answer: Use the long method. Calculate ½ of ½ and so on for twenty steps:

1/contents4 after 10 half lives

1/2048, …, 1/1048576 after 20 half lives

i.e. less than one millionth of the original quantity remains radioactive.

A quicker method is to calculate ½ 20. Show how this is done with a calculator, using the yx key.

Question: How many years into the future will Cs-137 be safe?

Answer: 20  ×  30 years = 600 years

Question: If after ten half lives the activity of a substance is reduced to one thousandth of its original value, how many more half lives must elapse so that the original activity is reduced to one millionth of its original value?

Answer: Ten half lives reduces activity by a factor of 11000. One millionth is 11000  ×  11000, so ten more half lives are needed.

Student example and discussion: Plotting a graph

Provide your students with the following data.

timeactivity / Bq
010057
56253
103648
152296
201403
25798
30508
35307
40201

Ask them to inspect the data and estimate the half-life. (It lies between 5 and 10 seconds.)

Now ask them to plot a graph and thereby determine the half life of the substance. They should deduce a total time of 35 seconds for five halvings of the original activity, gives a half life of 7 seconds.

Emphasise the definition of half life T ½ . The half life of a radioactive substance is the time taken on average for half of any quantity of the substance to have decayed. (Check the precise wording of your specification.)

Introduce the term exponential to describe this behaviour, in which a quantity decreases by a constant factor in equal intervals of time.

Episode 514-2: Half life (Word, 25 KB)


Exponential behaviour in nature is very common. It appears in the post-16 specification several times, so this may not be the first time your students have met it. (The equalisation of electric charge on a capacitor whose plates are connected via a resistor; the reduction in amplitude of a damped simple harmonic oscillator; the absorption of electromagnetic radiation passing through matter (e.g. γ rays by lead, visible light by glass); Newton’s Law of Cooling. So either refer back to previous situations and make the analogy explicit, or flag forward to other examples to come.

Exponential Decay of Activity
can be analysed using the quantity Activity
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