Exponential Decay of Activity
Quantum and Nuclear

Exponential decay and half life

for 14-16

Using sealed sources, you can demonstrate most of the properties of alpha, beta and gamma radiation. The experiments in this collection allow students to see their ranges, penetrating powers and, in the case of beta radiation, how it is deflected in a magnetic field. They can link these properties to the nature of each type of radiation, and start to form a picture of why these types of radiation behave in the way they do.

The first experiment introduces the idea that radioactive atoms give out three distinct types of radiation, known as alpha, beta and gamma. The next four experiments allow you to investigate their properties in more detail. Each of these experiments is built around one type of radiation (and all its properties). However, you could equally choose to reshuffle these experiments and focus on each property in turn – i.e. look at the range in air for all three radiations and then move on to penetrating power.

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Measuring the half-life of protactinium

Exponential Decay of Activity
Quantum and Nuclear

Measuring the half-life of protactinium

Practical Activity for 14-16


Measuring the half-life of a radioactive isotope brings some of the wonder of radioactive decay into the school laboratory. Students can witness one element turning into another and hear (or see) the decrease in the radiation it gives out as it transmutes.

This demonstration uses a protactinium generator to show the exponential decay of protactinium-234, a grand-daughter of uranium. It has a half-life of just over a minute, which gives students the chance to measure and analyze the decay in a single lesson.

Apparatus and Materials

  • tray
  • Holder for Geiger-Müller tube
  • Geiger-Müller tube, thin window
  • Scaler
  • Stopclock
  • Retort stand, boss, and clamp
  • Ratemeter (OPTIONAL)
  • Protactinium generator

Health & Safety and Technical Notes

See the following guidance note:

Managing radioactive materials in schools

To limit the risk of radioactive liquids being spilt, there should be special instructions in the local rules for handling (and preparing) this source.

Read our standard health & safety guidance

Preparation of the protactinium generator

It is now possible to purchase the chemicals already made up in a sealed bottle. One supplier is TAAB Laboratories Equipment Ltd, 3 Minerva House, Calleva Park, Aldermaston, RG7 8NA. Tel: 0118 9817775. However, you can make your own if you prefer.

These quantities make a total volume of 20 cm3. You can scale them up if you have a larger bottle. (A '30 ml' bottle has a capacity of about 35 ml, so there is still room to shake the solution when the total volume is 30 ml.)

  1. Dissolve 1 g of uranyl nitrate in 3 cm3 of water. Wash it into a small separating funnel or beaker with 7 cm3 of concentrated hydrochloric acid.
  2. To this solution, add 10 cm3 of iso-butyl methyl ketone or amyl acetate.
  3. Shake the mixture together for about five minutes. Then run the liquid into the polypropylene bottle and firmly screw down the cap. It can help to shield the lower half of the bottle with some lead.
  4. Place the bottle in a tray lined with absorbent paper.

Once you have made the protactinium generator, you can store it with other radioactive materials, taking care to follow your school code of practice and local rules: see the Managing radioactive materials in schools guidance note:

Managing radioactive materials in schools

A polypropylene bottle is preferable to polythene because it is somewhat more resistant to attack by the acid and ketone. Nevertheless, polythene bottles can be used, provided no attempt is made to store the liquid in them for more than a few weeks.

The organic layer which separates out contains the protactinium-234. This decays with a half-life of about 70 seconds.

An alternative to protactinium: A new, effective and extremely low hazard system for measuring half-life is available from Cooknell Electronics Ltd, Weymouth, DT4 9TJ. This uses fabric gas mantles designed for camping lights. Each mantle contains a small quantity of radioactive thorium. More details are available on the Cooknell Electronics website:

Cooknell Electronics


  1. Support the Geiger-Muller tube holder in a clamp, so that the tube is facing downwards towards the neck of the bottle.
  2. Allow the bottle to stand for at least ten minutes. Take the background count by running the counter for at least 30 seconds. This is done with the bottle in position, because some of the count will come from the lower layer. You can do this before the experiment or some time after it has finished.
  3. Alternatively, the GM tube can be clamped horizontally with the window close to the upper layer.
  4. Shake the bottle vigorously for about 15 seconds to thoroughly mix the layers.
  5. Place the bottle in the tray.
  6. As soon as the two layers have separated, start the count and start the stop-clock.
  7. Record the time from the beginning of the experiment - i.e. the time of day for the sample.
  8. Record the count every 10 seconds. Or record it for 10 seconds every 30 seconds.
  9. Run the experiment for about five minutes, ample time to reveal the meaning of the term half-life and to illustrate the decay process.
  10. Provided you leave a few minutes between each attempt, you can repeat the experiment. In 5 minutes the activity of the protactinium in the aqueous layer grows to 15/16 of its equilibrium value.
  11. It is possible to record the growth to equilibrium. Do this by moving the GM tube so that the aqueous layer at the bottom of the bottle is immediately above the end window of the GM tube.

Teaching Notes

The chemistry of the experiment:

  • The first stages of the uranium-238 series are involved in this experiment.
  • The aqueous solution (at the bottom of the bottle) contains the uranium-238, its daughter thorium-234 and the short-lived granddaughter protactinium-234.
  • Uranium and protactinium both form anionic chloride complexes but thorium does not. At high hydrogen ion concentrations, these complexes will dissolve in the organic layer (which is floating on top of the aqueous solution).
  • When you shake the bottle, about 95% of the short-lived granddaughter (protactinium) and some of the uranium will be dissolved in the organic layer. The thorium stays in the aqueous layer.
  • Since radioactivity is a property of the innermost nucleus of the atom it is not affected by chemical combination.
  • The granddaughter (in the organic layer) decays without any more being produced by its parent (thorium) all of which is still in the aqueous layer. It emits beta particles which travel through the plastic wall of the bottle. Isolating the protactinium in the top (organic) layer allows it to decay without any top-up from its parent (thorium).
  • The radiation from the thorium and uranium should not interfere with the results, for two reasons:
    1. The counter does not detect the alpha particles from the uranium or the low energy beta particles from the thorium. It only records the high energy (2 MeV) beta particles from the granddaughter (protactinium).
    2. The uranium-238 decays with an extremely long half-life. It yields a meagre, almost constant, stream of low energy alpha particles. Its daughter, thorium-234, decays with a half-life of 24 days. During the length of this experiment the decay rate can be assumed to be constant. If these two isotopes contribute to the count at all, it will be accommodated in the background count. The stockpile of thorium is also constantly topped up in the aqueous layer as long as the protactinium is present with the thorium.

Table of count rate: Get the students to make a table of count rate against time, and correct it for background count. The first 10-second reading should be allocated to a time of zero.

Plot a graph: Get the students to plot a graph of count rate against time. They should draw a smooth curve through the points.

  • First point out the general pattern - that the count rate decreases with time. Then look for an exponential trend - that the best fit curve always takes the same amount of time to halve.
  • Get students to measure the half-life from the curve.
  • Point out the random nature of the points: although the decay follows a pattern, there is an element of randomness and it is not perfectly predictable.

How Science Works extension This experiment provides an opportunity to assess the accuracy of the measured half-life value and how the random nature of decay affects the answer.

The accepted value for the half-life of protactinium is about 70 seconds.

Explore different ways in which a half-life value can be obtained from this apparatus:

  • Amend the procedure described above so that, instead of a scaler (counter), a ratemeter is used. One student just records the time it takes for the count-rate to halve. This will provide a very approximate value.
  • Repeat the experiment with several members of the class timing how long it takes for the count-rate to halve. There is likely to be considerable spread in results across the group and the mean result may differ from the accepted value for half-life. In each case, ask students to identify errors and uncertainties in their measurement(s) and to suggest ways in which these could be reduced.

For example, ask: "how does the random nature of the decay affect the measured count-rate when the count is low, or high, compared the background count?"

  • Either you or your students may suggest a graphical method as an improvement. The procedure described in the main experiment above could then be carried out, and then the accuracy of the half life value assessed and evaluated.

Radioactive materials raise significant safety issues, providing an opportunity to discuss the value and use of secondary data sources.

This experiment was safety-tested in February 2007

Up next

Simple model of exponential decay

Exponential Decay of Activity
Quantum and Nuclear

Simple model of exponential decay

Practical Activity for 14-16

Class practical

In this activity, students model radioactive decay using coins and dice. By relating the results from the model to the experimental results in...

Measuring the half-life of protactinium

...students can see that the model helps to explain the way in which a radioactive substance decays. The model provides an insight into what might be happening within radioactive atoms.

This activity is a good analogy of radioactive decay as it is based on probability. The decaying trend will be noticeable and so too will the random nature.

Apparatus and Materials

  • Pennies or other coins, plentiful supply
  • Dice, plentiful supply (OPTIONAL)

Health & Safety and Technical Notes

Read our standard health & safety guidance

The more coins each student has, the better the analogy of radioactive decay. You could use as few as one per student to keep it simple. Any more than four is quite difficult to manage.

Small coins will turn around more in their cupped hands.

A canvas bag containing 500 plastic cubes (each side 10 mm), each with one face identified, is available in the UK from Lascells, order code 60-010.



  1. Explain the procedure (as follows) to the class.
  2. Each student has a number of coins. This could be between one and four. They hold them in their cupped hands.
  3. On your instruction "shake", the students shake their coins for at least 5 seconds (they should ensure that the coins are moving around inside their cupped hands). On the instruction "stop", they stop shaking and open their hands with one hand flat and facing upwards so that they can see their coins.
  4. If any coins come down heads, they take them out of their palm and place them on the desk.
  5. On your instruction "show", they put up a number of fingers corresponding to the number of coins they took out of their palm.
  6. Record this number on the board.
  7. They keep the remaining coins in their hands and repeat from step 3. If you can arrange it that you take a reading once every minute, then you can record the readings against time. It will then give results very similar to protactinium.
  8. Analyze the result by plotting a graph.

Teaching Notes

  • You might want to appoint a counter and a scribe to count the coins and record the results.
  • Take care with how you ask students to signal the numbers. They may be tempted to add their own (rude) gestures.
  • Draw out the similarities with the protactinium experiment. The trend is the same and there is also some randomness. The close match between the results from this model and the results from

    Measuring the half-life of protactinium

    show that radioactive atoms have a chance of decaying in any fixed time.
  • Use the activity to explain the downward trend of the decay curve. Only coins that are left can decay. As there are fewer of them each time, fewer will decay.
  • The activity raises the interesting question about how long a radioactive source will last and what happens to the last atom.
  • An alternative to shaking the coins in students' palms is to flick them. But this takes longer.
  • You could repeat the experiment with small dice to give a longer half-life. Combining results (as outlined here) makes for a smoother curve.

This experiment was safety-tested in May 2007

Up next

Developing a model of the atom: radioactive atoms

Exponential Decay of Activity
Quantum and Nuclear

Developing a model of the atom: radioactive atoms

Teaching Guidance for 14-16

Initially, students may regard atoms as the fundamental chemical particles. True, electrons can be chipped off an atom, and possibly all an atom’s electrons stripped off to leave a bare nucleus; yet according to the simple story, the nucleus is still fixed and determines the element by its charge, Ze.

Therefore, to change one element into another, the alchemist’s dream of lead into gold, would require a change of nuclear charge. At first sight this seems impossible because the nucleus is buried deep in the atom bound together by tremendous forces. But it does happen in radioactive elements.

Soon after the discovery of radioactivity in 1896 by Becquerel, Marie Curie and her husband Pierre discovered a new element which they named radium. They extracted dangerously large samples of radium from vast quantities of rock and experimented on its radioactive behaviour.

You could say: Radioactive atoms do not just stay there as atoms of ordinary copper do; they are completely different: they are unstable, they suddenly break up, flinging out a particle such as an alpha particle, becoming an atom of a different element.

A radium atom remains a radium atom, with the chemical behaviour of a heavy metal, until it suddenly hurls out this alpha particle. (The alpha particle has such a huge energy that it must come from the nucleus.) The remainder of the radium atom is no longer a heavy metal, but a quite different element. This ‘daughter’ of radium is an atom of a heavy inert gas, the end of the helium, neon, argon, krypton, xenon series. It is called radon. The atomic masses have been measured directly, radium-226, radon-222 (a difference of 4 suggesting that the lost alpha particle is a helium nucleus). Separate measurements confirm this.

When you have a mixture of a parent element and a daughter element which have different chemical properties, then they can be separated by ordinary chemical methods.

Radon gas is itself unstable and radioactive. Each of its atoms suddenly, at an unpredictable moment, hurls out an alpha particle. The remainder is a new atom, very unstable, which is called polonium, the ‘daughter’ of radon and the ‘granddaughter’ of radium. The series continues through several more radioactive elements and stops at a stable form of lead. The series does not begin with radium: it begins with uranium several stages earlier. Radioactive uranium (Z=92) has turned into lead (Z=82).

Making unstable atoms

A century ago, radioactivity was a peculiarity of a few mostly heavy, elements: the last few at the end of the Periodic Table. Nowadays scientists can bombard samples of lighter elements with high speed, high energy protons or neutrons, provided directly or indirectly by an accelerator. They can make unstable isotopes of every element in the periodic table. This has opened up the field of nuclear chemistry. Radioactive isotopes behave chemically like their stable isotopes and can be mixed with them. Their progress as radioactive tags can be traced, like luggage labels, following the progress of a ‘labelled’ isotope through the human body or an industrial process.


All the unstable members of these strange families have a constant, reliable characteristic: the atoms show no signs of ageing, or growing weaker, however long they last. Each radioactive element has a constant chance of breaking up in each succeeding second. This is described by a useful length of time, the ‘half-life’ of the radioactive element. For each individual atom the betting is 50:50 for and against its breaking up at any time during one half-life from now. The break-up seems to be controlled by pure chance. That chance does not change and make the break-up more likely for atoms that need to survive longer.

For radium the half-life is 1650 years. Start with 1000 mg of radium now and 1650 years later you will have only 500 mg left. After a further 1650 years only 250 mg will be left and so on. For radium’s daughter, radon -222, the half-life is 3.8 days. In less than four days half the radon gas will have disappeared. You will find helium gas there instead, with the solid products.

The instability appears to be something inherent in the nuclear structure. Nowadays, taking a wave view of the behaviour of nuclear particles, you can picture a stationary wave pattern defining the life of an alpha particle inside the nucleus. But the wave is not completely confined, it leaks through the potential barrier round the nucleus and runs on as a faint wave outside. The wave is interpreted as describing probabilities of locations. It is not a mechanical wave carrying energy and momentum.

While the alpha particle is expected to be found inside the nucleus, there is a chance of finding it one day outside, despite what would seem an insurmountable potential wall. That chance of the alpha particle being outside, being emitted, is definite and constant, a part of the defining wave property, as long as the nucleus lasts. It suggests that high energy alpha particles go with a short half-life of the parent nucleus.

Up next

Exponential decay of a radioactive substance

Exponential Decay of Activity
Quantum and Nuclear

Exponential decay of a radioactive substance

Teaching Guidance for 14-16

One of the most important characteristics of radioactivity is that it decays exponentially. This has two basic mathematical implications at this level.

  1. The rate falls by a constant ratio in a given time interval. The time it takes to fall by a half is always the same. It also falls to a tenth in equally regular, but longer, time intervals.
  2. The rate of decay is proportional to the amount that is left. This can be seen in the experiment to model radioactive decay. The number of coins that decay in any ‘shake’ is proportional to the number that is left.
  3. Radioactive decay experiment

From these features, you can argue, respectively, the following points.

  • The chance of an atom disintegrating is constant in time. Radioactive decay is a series of many chance events, all with an unalterable chance.
  • The rate of disintegrations is proportional to the total number of unchanged radioactive atoms at that moment. Both the rate and the stockpile itself die away exponentially with the same characteristic half-life.

Up next

Some useful equations for half-lives

Exponential Decay of Activity
Quantum and Nuclear

Some useful equations for half-lives

Teaching Guidance for 14-16

The rate of decay of a radioactive source is proportional to the number of radioactive atoms (N) which are present.

dNdt = -λN

is the decay constant, which is the chance that an atom will decay in unit time. It is constant for a given isotope.

The solution of this equation is an exponential one where N0 is the initial number of atoms present.

N = N0e -λt (Equation 1)

Constant ratio

This equation shows one of the properties of an exponential curve: the constant ratio property.

The ratio of the value, N1, at a time t1 to the value, N2, at a time t2 is given by:

N1N2 = e -λt1e -λt2

N1N2 = e -λ(t1-t2)

In a fixed time interval, t2t1 is a constant. Therefore the ratio

N1N2 = a constant

So, in a fixed time interval, the value will drop by a constant ratio, wherever that time interval is measured.

Straight line log graph

Another test for exponential decay is to plot a log graph, which should be a straight line.


N = N0e -λt

Taking natural logs of both sides:


Therefore a graph of N against t will be a straight line with a slope of QuantitySymbol{-λ}.

Half-life and decay constant

The half-life is related to the decay constant. A higher probability of decaying (bigger λ) will lead to a shorter half-life.

This can be shown mathematically.

After one half life, the number, N of particles drops to half of N0 (the starting value). So:

N = N02 when t=T½

By substituting this expression in equation (see above),

N02 = N0e -λT½

Taking natural logs of both sides gives:


ln2 = +λT½

T½ = 0.693λ

Up next

Managing radioactive materials in schools

Ionising Radiation
Quantum and Nuclear

Managing radioactive materials in schools

Teaching Guidance for 14-16

Countries have national laws to control how radioactive materials are acquired, used and disposed of. These laws follow internationally agreed principles of radiological protection.

The following principles apply to schools:

  • There should be a person designated to be responsible for the security, safety and proper use of radioactive sources.
  • Sealed radioactive sources should be of a safe design and type suitable for school science.
  • Sealed sources should be used whenever possible in preference to unsealed sources. Unsealed sources can only be justified when the scientific demonstrations would not be practicable using sealed sources.
  • Records of all radioactive sources should be properly kept, showing what they are, when they were bought, when and by whom they have been used, and eventually, how they were disposed of.
  • Radioactive sources should be used only when there is an educational benefit.
  • Radioactive sources should be handled in ways that minimize both staff and student exposures.
  • Sealed sources should be carefully checked periodically to make sure they remain in a safe condition.
  • The school should have a suitable radioactivity detector in good working order.

UK regulation & guidance

Generally, school employers will insist you obtain their permission before acquiring new radioactive sources.

You must follow your employer’s safety guidance relating to the use the radioactive sources. Most school employers will require you to use either SSERC or CLEAPSS safety guidance, as follows:

In Scotland, safety guidance for use of radioactive sources in schools is issued by the Scottish Schools Equipment Research Centre (SSERC) and is available to members through their website.

In the rest of the UK and British Isles Crown Dependencies, guidance is available from CLEAPSS, the School Science Service. Their guidance document, L93, is freely available from their website, even to non-members.

In the UK...

  • In classes where children are under the age of 16, the use of radioactive material shall be restricted to demonstrations by qualified science teachers, (which includes newly qualified teachers). However, closer inspection of devices containing low-activity sources such as diffusion cloud chambers is permitted provided the sources are fully enclosed within the devices and not removed during the inspection.
  • Young persons aged 16 and over may use radioactive sources under supervision. Although the use of radioactive material is regulated, it should not be used as an excuse to avoid practical work. As the ASE points out, "Using the small sources designed for school science gives a good opportunity to show the properties of radioactive emissions directly, and to discuss the radiation risks. Just as importantly, it is an opportunity to review pupils' perception of risks, as they are likely to have constructed their own understanding from a variety of sources, including science fiction films and internet sites. If the work is restricted just to simulations, it may reinforce exaggerated perceptions of risk from low-level radiation.”

Summary of legislation (UK)

Updated October 2008

The following summarizes the somewhat complicated legislative framework in which schools are expected to work with radioactive sources in the UK. However, teachers do not need to obtain and study this legislation; this has been done by CLEAPSS and SSERC, and it is incorporated into their guidance in plain English.

In the European Union, member states have implemented the 1996 EU Basic Safety Standards Directive (as amended) that in turn reflects the 1990 International Commission on Radiological Protection recommendations. In the UK, this has been done through the Radioactive Substances Act 1993 (RSA93), which controls the security, acquisition and disposal of radioactive material, and the Ionising Radiations Regulations 1999 (IRR99) which controls the use of radioactive material by employers. Transport of radioactive material is controlled by The Carriage of Dangerous Goods and Use of Transportable Pressure Equipment Regulations 2007.

There are exemptions from parts of the RSA93 and schools can make use of The Radioactive Substances (Schools etc.) Exemption Order 1963, The Radioactive Substances (Prepared Uranium and Thorium Compounds) Exemption Order 1962, and others. These exemption orders are conditional and to make use of them and avoid costly registration with the Environment Agency (or SEPA in Scotland, or the Environment and Heritage Service in Northern Ireland) you must adhere to the conditions. Note that currently, these exemption orders are being reviewed.

The way in which these laws are implemented in England, Wales, Northern Ireland and Scotland varies. The Department for Children, Schools and Families (DCSF) has withdrawn its guidance AM 1/92, and the associated regulations requiring this have been repealed. Consequently, purchase of radioactive sources by maintained schools in England is no longer regulated by the DCSF. The DCSF commissioned CLEAPSS to prepare and issue ‘Managing Ionising Radiations and Radioactive Substances in Schools, etc L93’ (September 2008) and has commended it to schools in England. Similar regulations relating to other educational institutions in the UK have not changed; English institutions for further education remain regulated through the Department for Innovation, Universities and Skills. Similarly, schools in Wales should follow the guidance from the Welsh Assembly Government Department for Children, Education, Lifelong Learning and Skills. Schools in Scotland should follow the guidance from the Scottish Government Education Directorate and associated guidance issued by SSERC. Schools in Northern Ireland should follow the guidance from the Department of Education Northern Ireland (DENI). The Crown Dependencies Jersey, Guernsey and Isle of Man are not part of the UK and schools and colleges should follow the guidance from their own internal government departments responsible for education.

In the UK, if an employer carries out a practice with sources of ionising radiations, including work with radionuclides that exceed specified activities (which is 100 kBq for Co-60, and 10 kBq for Sr-90, Ra-226, Th-232, Am-241 and Pu-239), the practice must be regulated according to the IRR99 and the employer must consult with a Radiation Protection Adviser (RPA). Since 2005, the RPA must hold a certificate of competence recognized by the Health and Safety Executive. Education employers are unlikely to have staff with this qualification, so the RPA will usually be an external consultant. Education employers need to notify the HSE 28 days before first starting work with radioactive sources. This is centralized at the HSE’s East Grinstead office.

Note: For higher risk work with radioactive material, the IRR99 requires designated areas, called controlled areas and supervised areas, to be set up if special procedures are needed to restrict significant exposure – special means more than normal laboratory good practice. It should never be necessary for a school to designate an area as controlled, and only in special circumstances would it be necessary to designate an area as supervised. The normal use of school science radioactive sources, including the use of school science half-life sources, does not need a supervised or controlled area.

Disposal of sources in the UK

Sources that become waste because they are no longer in a safe condition, or are no longer working satisfactorily, or are of a type unsuitable for school science, should be disposed of. In England and Wales, the Environment Agency has produced a guidance document through CLEAPSS that explains the available disposal routes. Similarly, SSERC has produced guidance for schools in Scotland. Schools in Northern Ireland should refer to DENI.

Health and safety statement

See the health and safety notes in each experiment. This is general guidance.

Health and safety in school and college science affects all concerned: teachers and technicians, their employers, students, their parents or guardians, and authors and publishers. These guidelines refer to procedures in the UK. If you are working in another country you may need to make alternative provision.

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