• Some atomic nuclei are very unstable and only exist for a few microseconds, seconds, minutes, hours or days before decaying (disintegrating), these are known as radioisotopes.
  • The breakdown of these unstable nuclei is called radioactive decay.
  • Decay is a random event - you can't predict which isotope nucleus will disintegrate.
  • You cannot alter the rate of the decay process - the radioactive atoms are totally unaffected by the physical or chemical state of the atoms e.g. in an element or compound and unaffected by change in temperature or pressure.
  • However, despite being a random process, there is a statistical pattern that the rate of decay follows - known as a radioactive decay curve.
  • The graph of activity follows curve with a gradually decreasing negative gradient.
  • If the gradient is steep, the more unstable is the isotope.
  • 
  • Typical radioactivity decay curves
  • The 'blue' radioisotope is more stable than the 'green' radioisotope - gradients less steep.
  • The 'green' radioisotope starts with a greater activity - a more radioactive source, but is more unstable and disintegrates (decays) much more rapidly.
• Others are very stable and take millions of years to decay away to form another atom.
• Some isotopes are completely stable and do not undergo radioactive decay at all.
• The radioactivity (emissions) of any radioactive material always decreases with time.
  • See section 3. for units and measurement of radioactivity
• A measure of the stability of a radioisotope is given by its half-life (defined below).
• Unstable nuclei disintegrate at random, you cannot predict which decays to emit alpha, beta, gamma or other nuclear/ionising radiations.
• What you can say is the radioactivity must always decrease over time but never quite reaches zero, except after a very long period of time (infinity?).
• The decay follows are particular pattern, illustrated by the graphs above and below, known as a decay curves.
  • The graph will drop steeply for very unstable nuclei but show a very small gradient if more stable.
  • Every graph shows the same mathematical feature which is that for a particular time interval the amount of
• The half-life of a radioisotope is the average time it takes for half of the remaining undecayed radioactive nuclei (atoms) to decay to a different nucleus (atom).
  • The values of half-lives can vary from a fraction of a second (highly unstable) to millions of years (relatively much more stable)
• It means in one half-life of time, on average, half of the undecayed unstable nuclei of a particular isotope disintegrate.

• Within a half-life, half of the remaining unstable nuclei decay (disintegrate), equivalent to a 50% reduction in the radioactivity.
• A short half-life means the activity (radioactivity) will fall quickly e.g. falls by 50% in a few minutes with lots of unstable nuclei decaying.
• so, a short half-life means relatively rapid decay, a long half-life means a relatively slow decay and measurable radioactivity lasts much longer e.g. 10⁹ years!
• 
• Ideal graph points for a radioisotope's decay curve - the graph is based on the 'idealised' data table below.

• time in half-lives elapsed	0	1	2	3	4	5	6
  Fraction of radioisotope left	1	1/2	1/4	1/8	1/16	1/32	1/64
  Fraction of radioisotope left	1.0	0.5	0.25	0.125	0.0625	0.03125	0.015625
  Percentage of radioisotope left	100	50	25	12.5	6.25	3.125	1.5625

• The fraction or % of the radioisotope remaining is a measure of the activity at any point in time, in fact you can measure activity in Becquerel (counts/sec, decays/s, Bq s^-1) and you might given data in any of the three ways described here - so be versatile in your thinking!

• See the decay curve graphs below representing the behaviour of relatively unstable radioactive-isotope with a half-life of 5 days.
  • This means from radioactivity measurements we can analyse the data and calculate from the graph the half-life of a radioactive-isotope or some calculation based on an initial level of activity and a later measurement of the decreased activity. Whatever method, you need accurate activity data linked to time.
  • This also means that we can make predictions of activity
• The radioactivity of any sample will decrease with time as the unstable atoms decay to more stable atoms, though sometimes by complex decay series routes e.g. ₉₂U isotopes eventually decay to ₈₂Pb isotopes.

The older a sample of a radioactive material, the less radioactive it is. The decrease in radioactivity follows a characteristic pattern shown in the graph or decay curve. The y axis can represent the % radioisotope left OR the measured radioactivity. After every half-life, in this case 5 days, working out from the graph, the % radioisotope (or radioactivity, count rate etc.) is halved, producing the initially steeply declining curve which then levels out towards zero after infinite time!

If this data was obtained in a laboratory experiment, two points should be born in mind.  (i) Before plotting the graph, all measurements should have the background radiation subtracted from them. (ii) Your graph is highly unlikely to be a 'perfect' decay curve. Radioactive disintegration is a random process, so real data will show small variations from the 'perfect' decay curve - its a statistical thing!

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• The radioactivity from a radioisotope is measured over a period of time.
  • Graphical or mathematical analysis is performed to calculate the time it takes for the radioactivity of the isotope to halve.
  • For short-lived radioactive isotopes, the radioactivity is likely to be measured in terms of the count rate.
  • Therefore the half-life will be the time it takes for the count rate to halve.
• An example of what this means is shown in the diagram below.
  • Half-life calculation example 1
  • You would use a Geiger-Marsden counter, or similar scintillation counter to make measurements of the radioactivity of a radioisotope.
    • Radioactivity, or simply 'activity' is measured in becquerels (Bq).
      • 1 becquerel = 1 disintegration or decay/second.
      • Sometimes the activity might be stated as counts per second (cps = Bq).
  • The graph shows the rapid decay of a very unstable radioactive isotope in terms of count rate per minute (cpm) versus minutes.
    • Although not shown, before plotting the graph, you should do a blank test for the background radiation and subtract this from ALL the readings.
    • You would do a blank test by taking several readings without the presence of the radioisotope and use the average to correct the readings.
    • An alternative to this is to use heavy lead shielding to protect the Geiger counter from background radiation, but should still do a blank test with the identical experiment setup.
  • From the graph you can work out the time (half-life) it takes for half of the radioactive atoms to decay from the decrease in count rate.
  • e.g. in terms of time elapsed, count rate ==> we get
  • 0 min, 400cpm ==> 10 min, 200cpm ==> 20 min, 100 cpm etc.
  • In other words, the activity halves every 10 minutes, clearly showing the half-life is 10 minutes.
  • -
• Half-life calculation example 2 not using a graph, but 'simple fraction' reasoning.
  • Suppose a sample of a radioisotope gives an initial activity of 1200 counts per minute (cpm).
  • If the activity has fallen to 150 cpm after 180 days, calculate the half-life of the radio-isotope.
  • The simple method just involves involving halving from the initial value of activity until you reach the final value.
  • In terms of activity: 1200 == ÷2 ==> 600 == ÷2 ==> 300 == ÷2 ==> 150
  • so, to get from 1200 to 150 required 3 halvings.
  • From the definition of half-life, this means 3 half-lives elapsed for the activity to drop from 1200 cpm to 150 cpm.
  • Therefore the half-life is 180÷3 = 60 days
  • -
• Half-life calculation example 3 - prediction of future activity
  • Suppose a sample of a radioisotope has an activity of 800 Bq.
  • What will be the activity after three half-lives have elapsed?
    • The rule is that activity halves over every half-life of time elapsed.
    • 800 ÷ 2 = 400,  400 ÷ 2 = 200,  200 ÷ 2 = 100.
    • Therefore the final activity is 100 Bq
    • You can also express the result as fraction or percent:
      • (i) Fraction of activity remaining: 1 ==> 1/2  ==> 1/4  ==> 1/8th (also = 100/8 = 12.5%)
      • (ii) Fraction of radioisotope decayed: 1 - 1/8 = 7/8ths (also = 100 x 7/8 = 87.5%).
      • -

• Half-life calculation example 4 - calculating half-life from one piece of data
  • Suppose the activity of a sample of a radioisotope has an activity of 8000 Bq (counts/second).
  • After 24 hours the activity had dropped to 250 Bq.
  • Calculate the half-life (t_½) of the radioisotope.
  • Now, for every half-life, the activity halves, therefore we can set out a line of 'halving' logic!
  • 8000 == t_½ ==> 4000 == t_½ ==> 2000 == t_½ ==> 1000 == t_½ ==> 500 == t_½ ==> 250
  • Therefore it took five half-lives to drop from an activity of 8000 to 250 Bq.
  • Therefore the half-life = 24 5 = 4.8 hours
  • -

• Half-life calculation example 5 - half-life calculation - comparing the activity of two sources after a period of time
  • After the reprocessing of nuclear fuels rods two fractions of nuclear waste were separated out.
  • Radioactive source A had an initial activity of 50 MBq and a half-life of 100 days.
  • Source B has an initial activity of 40 MBq and a half-life of 200 days.
  • (a) Which source is the greatest hazard initially?
    •  Source A is more radioactive than B, 50 MBq > 40 MBq
  • (b) Calculate which source is the greatest hazard after 400 days?
    • For source A: 400 days = 400/100 = 4 half-lives
    • Fraction remaining: 1 => 1/2 => 1/4 => 1/8 => 1/16
    • So activity of A after 400 days = 50/16 = 3.13 MBq (3 sf)
    • For source B: 400 days = 400/200 = 2 half-lives
    • Fraction remaining: 1 => 1/2 => 1/4
    • So activity of B after 400 days = 40/4 = 10 MBq
    • Therefore B is more radioactive, and therefore more hazardous than A after 400 days.
  • (c) Sketch a graph to illustrate the given data.
    • It should look something like the graph on the right.
  • (d) Suggest some implications for the storage of radioactive waste.
    • For highly radioactive waste materials, with a short half-life, safe storage for a relatively short time might be sufficient until it reaches a safe level of activity.
    • However, if a less radioactive source has a much longer half-life, it will remain radioactive for much longer and need a long-term secure storage facility.
    • Radioactive materials with long half-lives are more of a hazard in the long-run than highly active radioisotopes with short half lives.

• Half-life calculation example 6 - half-life calculation of an isotope used an industrial process
  • When first used, the initial activity of this was found to be 384 kBq.
  • After 24 years the activity was found to be 6 kBq.
  • What was the half-life of the radioisotope?
  • 384 => 192 => 96 => 48 => 24 => 12 => 6
  • From an activity of 384 to 6 takes 6 half-lives.
  • So the half-life is 384/6 = 4 years
  • -
• Example 7. How long can an radioisotope be useful
  • A radioisotope has an initial activity of 6000 Bq, and a half-life of 10 days.
  • In order to be useful, its activity must exceed 700 Bq.
  • For how long, approximately, is the radioisotope useful?
  • Working through the half-life decay sequence
  • 6000 ==> 3000 ==> 1500 ==> 750 ==> 375
  • For 3 half-lives the activity exceeds 700 Bq, so the useful life is a little over 3 x 10 = 30 days.
• You need to practice these sort of calculations of half-life determination, radioactive residue left, and dating calculations (see below) using the multiple choice QUIZ (higher GCSE = AS GCE in this case)
• You can do a class experiment to illustrate the random nature of radioactive decay and half-life e.g.
  • Use say 50 normal dice numbered 1-6 shaken in a container.
  • Make zero time that before the first 'throw' (tip the lot out of the box), so t = 0, d = 50
  • Throw the dice and remove all sixes, pretending they were the ones to disintegrate (decay).
  • Make this t = 1, d = dice left.
  • Just repeat a few times, removing all the sixes each time and counting the dice left.
  • After, say six goes, plot the number of dice left versus the time interval 0, 1, 2, 3 etc. and a decay curve graph will emerge.
  • You can then estimate the 'half-life' in time intervals.
  • The time interval could represents, seconds, minutes, hours ... millions of years, it doesn't matter!

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• From the half-life you can calculate how much of the radio-active atoms are left e.g. after one half-life, ¹/₂ is left, after two half-lives, ¹/₄ is left, after three half-lives, ¹/₈ is left in other words its a 'halving pattern' etc.
  • Example Q: The half-life of a radioisotope is 10 hours. Starting with 2.5g, how much is left after 30 hours?
    • 2.5g =10h=> 1.25g =10h=> 0.625g =10h=> 0.3125g (after total time of 30h)
    • Another way to think - if the time elapsed is equal to a whole number of half-lives you can just divide the 30 h by 10 h, giving 3 half-lives.
    • Therefore you just have to halve the amount three times!
      • e.g. 2.5 ==> 1.25 ==> 0.625 ==>  0.3125g
• The half-life of a radioisotope has implications about its use and storage and disposal.
  • If the half-life is known then the radioactivity of a source can be predicted in the future (see (1) above).
  • Plutonium-244 produced in the nuclear power industry has a half-life of 40 000 years!
  • Even after 80 000 years there is still a 1/4 of the dangerously radioactive material left.
  • Quite simply, the storage of high level nuclear reactor radioactive waste is going to be quite a costly problem for many (thousands?) of years!
  • Storage of waste containing these harmful substances must be stable for hundreds of thousands of years! So we have quite a storage problem for the 'geological time' future! see also dangers and background radiation.
• Radioisotopes used as tracers must have short half-lives, particularly those used in medicine to avoid the patient being dangerously over exposed to the harmful radiation, but a long enough half-life to enable accurate measurement and monitoring of the tracer.

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• Carbon-14 is formed at a constant rate in the upper atmosphere by high energy nuclear processes.
  • Cosmic rays are very high energy charged particles e.g. from the Sun (or even outside of our solar system) that collide with atomic nuclei to release neutrons.
  • Nitrogen-14 atoms collide with, and capture neutrons to form carbon-14 and a proton.
    • 
    • A neutron smashes out, and replaces, a proton in the nitrogen-14 nucleus, so carbon-14 is all nicely prepared for the future archaeologist!
  • The carbon-14 atoms, like any other carbon atoms, become part of carbon dioxide in the atmosphere, and carbonates and organic compounds in aquatic or land-based organisms.
  • Carbon-14 is constantly radioactively decaying, but is also being constantly replaced and balanced by the nuclear reactions in the upper atmosphere.
• Most carbon atoms are of the stable isotope carbon-12.
• A very small % of them are radioactive due to carbon-14 with a half-life of 5700 years.
• It decays by beta emission to stable nitrogen-14.
• 
• Archaeologists can use any material containing carbon of 'organic living' origin to determine its age.
• This can be bone, wood, leather etc. and the technique is sometimes called radiocarbon-14 dating.
• When the 'carbon containing' material is in a living organism there is a constant interchange of carbon with the environment e.g. as absorbed food from photosynthesis or carbon dioxide given out in respiration - all parts of the carbon cycle.
• This means the carbon-14 % remains constant as long as the plant or animal organism is alive!
• When the organism is dead the exchange stops and the carbon-14 content of the material begins to fall as it radioactively decays.
• Compared to when it was 'alive' ...
  • if an object has ¹/₂ (¹/₂ of 1, 50%) of the expected carbon-14 it must be 5700 years old,
  • if it only has ¹/₄ (¹/₂ of a ¹/₂, 25%) of the expected ¹⁴C left, the object it must be 11400 years old (5 700 + 5 700),
  • and if only ¹/₈ (¹/₂ of ¹/₄, 12.5%) of the ¹⁴C left it is 17100 years old (11 400 + 5700) etc. etc.

  • time in years	% 14C left
    0	100.00
    5730	50.00
    11460	25.00
    17190	12.50
    22920	6.25
    28650	3.13
    34380	1.56
    40110	0.78
    45840	0.39

  • Decay curve for carbon-14 disintegration
  • Example of a simple dating calculation.
  • An archaeologist had a sample of bone from a prehistoric skeleton analysed for its carbon-14 content.
  • The bone sample was found to contain 6.25% of the original carbon-14, calculate the age of the skeleton.
  • Just using a simple halving calculation technique you get ...
    • 100% ==> 50% ==> 25% ==> 12.5% ==> 6.25%
    • So to get to 6.25% takes four half-lives
    • therefore the age of the skeleton is 4 x 5700 =  22800 years
  • All the analysis of rock samples is done with a mass spectrometer.

  • APPENDIX 1: Four 'slides' summarising how radiocarbon dating is done (for advanced level students only)

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• Certain elements with very long half-lives can be used to date the geological age of igneous rocks and even the age of the Earth.
  • has a half-life of 1.3 x 10⁹ years. It decays to form .
• If the argon gas is trapped in the rock, the ratio of potassium-40 to argon-40 decreases over time and the ratio can be used to date the age of rock formation i.e. from the time the argon gas first became trapped in the rock.
  • The method is more reliable for igneous rocks, rather than sedimentary rocks because the argon will tend to diffuse out of porous sedimentary rocks but would be well trapped in harder and denser igneous rocks.
  • If the ⁴⁰Ar/⁴⁰K ratio is 1.0 (50% of ⁴⁰K decayed, 50% left )  the rocks are 1.3 x 10⁹ years old
  • If the ⁴⁰Ar/⁴⁰K ratio is 3.0 (75% of ⁴⁰K decayed, 25% left)  the rocks are 2.6 x 10⁹ years old
  • If the ⁴⁰Ar/⁴⁰K ratio is 7.0 (87.5% of ⁴⁰K decayed, 12.5% left)  the rocks are 3.9 x 10⁹ years old
    • These are worked out on the basis of 100% =half-life=> 50% =half-life=> 25% =half-life=> 12.5% etc.
• Long lived isotopes of uranium (element 92) decay via a complicated series of relatively short-lived radioisotopes to produce stable isotopes of lead (element 82).
  • The uranium isotope/lead isotope ratio decreases with time and so the ratio can be used to calculate the age of igneous  rocks containing uranium compounds.
• All the analysis of rock samples is done with a mass spectrometer.

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1. The isotopes of carbon: Note the tiny amount of carbon-14 in the first place

2. How is carbon-14 formed: Its a bit of upper atmosphere nuclear physics

3. How a mass spectrometer works: The diagram briefly describes how a 'time of flight' mass spectrometer works

The archaeological organic material is converted into carbon dioxide to obtain the ¹²CO₂:¹⁴CO₂ ratio

4. The carbon-14 decay curve: You get the date of the carbon containing material from the % of ¹⁴C left or the ¹⁴C;¹²C ratio.

Atomic structure, history, definitions, examples and explanations including isotopes

1. Atomic structure and fundamental particle knowledge needed to understand radioactivity

2. What is Radioactivity? Why does it happen? Three types of atomic-nuclear-ionising radiation

3. Detection of radioactivity, its measurement and radiation dose units, ionising radiation sources - radioactive materials, background radiation

4. Alpha, beta & gamma radiation - properties of 3 types of radioactive nuclear emission & symbols ,dangers of radioactive emissions - health and safety issues and ionising radiation

5. Uses of radioactive isotopes emitting alpha, beta (+/–) or gamma radiation in industry and medicine

6. The half-life of a radioisotope - how long does material remain radioactive? implications!, uses of decay data and half-life values - archaeological radiocarbon dating, dating ancient rocks

7. What actually happens to the nucleus in alpha and beta radioactive decay and why? nuclear equations!, the production of radioisotopes - artificial sources of radioactive-isotopes, cyclotron

8. Nuclear fusion reactions and the formation of 'heavy elements' by bombardment techniques

9. Nuclear Fission Reactions, nuclear power as an energy resource

Easier Foundation Tier Radioactivity multiple choice QUIZ

Harder Higher Tier Radioactivity multiple choice QUIZ

Worksheet QUIZ Question 1 on RADIOACTIVITY - absorption of alpha, beta and gamma radiation

Worksheet QUIZ Question 2 on RADIOACTIVITY - dangers & monitoring ionising radiation levels

Worksheet QUIZ Question 3 on RADIOACTIVITY - revision of atomic structure

Worksheet QUIZ Question 4 on RADIOACTIVITY - what happens to atoms in radioactive decay?

Worksheet QUIZ Question 5 on RADIOACTIVITY - uses of radioisotope and half-life data

ANSWERS to the WORD-FILL WORKSHEET QUIZZES

Crossword puzzle on radioactivity and ANSWERS!

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OTHER CHEMICAL CALCULATION PAGES

1. What is relative atomic mass?, relative isotopic mass and calculating relative atomic mass

2. Calculating relative formula/molecular mass of a compound or element molecule

3. Law of Conservation of Mass and simple reacting mass calculations

4. Composition by percentage mass of elements in a compound

5. Empirical formula and formula mass of a compound from reacting masses (easy start, not using moles)

6. Reacting mass ratio calculations of reactants and products from equations (NOT using moles) and brief mention of actual percent % yield and theoretical yield, atom economy and formula mass determination

   • Reacting masses, concentration of solution and volumetric titration calculations (NOT using moles)

7. Introducing moles: The connection between moles, mass and formula mass - the basis of reacting mole ratio calculations (relating reacting masses and formula mass)

8. Using moles to calculate empirical formula and deduce molecular formula of a compound/molecule (starting with reacting masses or % composition)

9. Moles and the molar volume of a gas, Avogadro's Law

10. Reacting gas volume ratios, Avogadro's Law and Gay-Lussac's Law (ratio of gaseous reactants-products)

11. Molarity, volumes and solution concentrations (and diagrams of apparatus)

12. How to do volumetric titration calculations e.g. acid-alkali titrations (and diagrams of apparatus)

13. Electrolysis products calculations (negative cathode and positive anode products)

14. Other calculations e.g. % purity, % percentage & theoretical yield, volumetric titration apparatus, dilution of solutions (and diagrams of apparatus), water of crystallisation, quantity of reactants required, atom economy

15. Energy transfers in physical/chemical changes, exothermic/endothermic reactions

16. Gas calculations involving PVT relationships, Boyle's and Charles Laws

17. Radioactivity & half-life calculations including dating materials (this page)