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6. How long does material remain radioactive?

6a half-life of radioisotopes, uses of decay data & implications

6b uses of decay data and radioisotope half-life values

Doc Brown's Chemistry - KS4 science GCSE Physics Revision Notes

Practice revision questions on half-life calculations and radioactive decay. Problem solving using the half-life of a radioisotope or radioactive emission data to work out the half-life of a radioactive isotope. What is the half-life of radioactive isotopes? What is a radionuclide decay curve? How long are radioactive materials dangerous for? Are half-lives of radioisotopes useful? How do archaeologists use half-lives to date prehistoric materials? How do geologists use very long half-live values to date rocks? All of these questions are answered and explained with examples of how half-life data is so useful. These revision notes on the half-life of radioisotopes should help with GCSE/IGCSE physics courses and A/AS level physics courses

6. How long does material remain radioactive? half-life, some uses and implications

6a. The half-life of a radioisotope

• Some atomic nuclei are very unstable and only exist for a few microseconds, seconds, minutes, hours or days.
• 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.
• 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 graph below, known as a decay curve.
• 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.
• See the decay curve graph 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!

6b. Four Uses of decay data and half-life values

(1) Determination of the half-life of a Radioisotope and using it to predict future activity

• 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
• 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
• 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 (t½) of the radioisotope.
• Now, for every half-life, the activity halves, therefore we can set out a line of 'halving' logic!
• 8000 == t½ ==> 4000 == t½ ==> 2000 == t½ ==> 1000 == t½ ==> 500 == t½ ==> 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
• -
• 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)
• 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!

(2) Using half-life data in hazard analysis and more on prediction of radioisotope residue

• 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.

(3) Archaeological dating with the isotope carbon-14

• 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 as food or carbon dioxide.
• This means the carbon-14 % remains constant as long as the 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.
• 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
• For more details The decay curve for carbon-14 is shown in an Excel file webpage

(4) Geological dating of igneous rocks

• 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. 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.

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RADIOACTIVITY multiple choice QUIZZES and WORKSHEETS

five word-fills on radioactivity * Q2 * Q3 * Q4 * Q5and ANSWERS!