6.
How long does material
remain radioactive? halflife, some uses and implications
6a.
What is a radioisotope and what is the halflife of a radioisotope?
 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.
 A measure of the stability of a radioisotope is given by its halflife
(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 halflife 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 halflives can vary from a fraction of a
second (highly unstable) to millions of years (relatively much more stable)

It means in one halflife of time, on average, half of the undecayed unstable
nuclei of a particular isotope disintegrate.
 Within a halflife, half of the remaining
unstable nuclei decay (disintegrate), equivalent to a 50% reduction in
the radioactivity.
 A short halflife 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 halflife means relatively rapid
decay, a long halflife means a relatively slow decay and measurable
radioactivity lasts much longer e.g. 10^{9} years!

Ideal graph points for a radioisotope's decay curve 
the graph is based on the 'idealised' data table below.

time in halflives 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 radioactiveisotope with a
halflife of 5 days.
 This means from radioactivity measurements we can analyse
the data and calculate from the graph the halflife of a radioactiveisotope 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. _{92}U isotopes
eventually decay to _{82}Pb
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
halflife, 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!

(i)
6b.
Four Uses of radioactive decay data and halflife
values
(1) Determination of the halflife 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 shortlived radioactive isotopes, the radioactivity is likely to be measured in terms of the count rate.
 Therefore the halflife will be the time it takes for the count rate to halve.

An example of what this means is shown in the diagram below.

Halflife calculation example 1

You would use a GeigerMarsden 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
(halflife) 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
halflife is 10
minutes.
 

Halflife 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 halflife of the radioisotope.
 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 halflife, this means 3 halflives
elapsed for the activity to drop from 1200 cpm to 150 cpm.
 Therefore the halflife is 180÷3
= 60 days
 

Halflife 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 halflives have
elapsed?
 The rule is that activity halves over every halflife
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%).
 
 Halflife calculation example 4
 calculating halflife 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 halflife (t_{½}) of
the radioisotope.
 Now, for every halflife, 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 halflives to drop from an
activity of 8000 to 250 Bq.
 Therefore the halflife = 24 5 =
4.8
hours
 

Halflife calculation example
5
 halflife 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 halflife of 100 days.
 Source B has an initial activity of 40 MBq and a
halflife 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
halflives
 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
halflives
 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 halflife, 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 halflife, it will remain radioactive for much longer
and need a longterm secure storage facility.
 Radioactive materials with long halflives
are more of a hazard in the longrun than highly active
radioisotopes with short half lives.

Halflife calculation example 6  halflife 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 halflife of the radioisotope?
 384 => 192 => 96 => 48 => 24 => 12 => 6
 From an activity of 384 to 6 takes 6 halflives.
 So the halflife 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 halflife 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 halflife decay sequence
 6000 ==> 3000 ==> 1500 ==> 750 ==> 375
 For 3 halflives the activity exceeds 700 Bq, so the
useful life is a little over 3 x 10 = 30 days.
 
 7.
 You need to practice these sort of calculations
of halflife
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 halflife e.g.
 Use say 50 normal dice numbered 16 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 'halflife' in time intervals.
 The time interval could represents, seconds, minutes,
hours ... millions of years, it doesn't matter!
(2) Using halflife data in
hazard analysis and more on prediction of radioisotope residues
 From the halflife you can calculate how
much of the radioactive atoms are left e.g. after
one halflife, ^{1}/_{2} is left, after two
halflives, ^{1}/_{4} is left, after three
halflives, ^{1}/_{8} is left in other words its a
'halving pattern' etc.
 Example Q: The halflife 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 halflives you can just divide the
30 h by 10 h, giving 3 halflives.
 Therefore you just have to halve the
amount three times!
 e.g. 2.5 ==> 1.25 ==> 0.625 ==>
0.3125g
 The halflife of a radioisotope has implications about its use and storage and
disposal.
 If the halflife is known then the radioactivity of a source can be predicted in the
future (see (1) above).
 Plutonium244 produced in the nuclear power industry has a halflife 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 halflives, particularly those used in medicine to avoid the patient being dangerously over exposed to the harmful radiation,
but a long enough halflife to enable accurate measurement and monitoring of the
tracer.
(3)
Archaeological dating with the isotope carbon14
 Carbon14 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.
 Nitrogen14 atoms collide with, and capture neutrons
to form carbon14 and a proton.

 A neutron smashes out, and replaces, a proton in the
nitrogen14 nucleus, so carbon14 is all nicely prepared for the future archaeologist!
 The carbon14 atoms, like any other carbon atoms,
become part of carbon dioxide in the atmosphere, and carbonates and organic
compounds in aquatic or landbased organisms.
 Carbon14 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 carbon12.
 A very small % of them are radioactive due to
carbon14 with a halflife of 5700 years.
 It decays by beta emission to stable nitrogen14.

 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 radiocarbon14 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 carbon14 % remains constant as
long as the plant or animal organism is alive!
 When the organism is dead the exchange stops and the carbon14 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 carbon14 it must be 5700 years old,
 if it only has ^{1}/_{4 } (^{1}/_{2} of a
^{1}/_{2}, 25%) of the expected ^{14}C 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 ^{14}C left it is 17100 years old (11 400 + 5700)
etc. etc.

time in years 
% ^{14}C 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 carbon14 disintegration
 Example of a simple dating calculation.
 An archaeologist had a sample of bone from a
prehistoric skeleton analysed for its carbon14 content.
 The bone sample was found to contain 6.25% of
the original carbon14, 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 halflives
 therefore the age of the skeleton is 4 x
5700 =
22800
years


(4) Geological dating of
igneous
rocks
 Certain elements with very long halflives can be used to
date the geological age of igneous rocks and even the age of the Earth.

has a halflife of 1.3 x 10^{9} years. It decays to form
.
 If the argon gas is trapped in the rock, the ratio of potassium40 to
argon40 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 ^{40}Ar/^{40}K ratio
is 1.0 (50% of ^{40}K decayed, 50% left ) the rocks are 1.3
x 10^{9} years old
 If the ^{40}Ar/^{40}K ratio
is 3.0 (75% of ^{40}K decayed, 25% left) the rocks are 2.6 x
10^{9} years old
 If the ^{40}Ar/^{40}K ratio
is 7.0 (87.5% of ^{40}K decayed, 12.5% left) the rocks are
3.9 x 10^{9} years old
 These are worked out on the basis of
100% =halflife=> 50% =halflife=> 25% =halflife=> 12.5%
etc.
 Long
lived isotopes of uranium (element 92) decay via a complicated series of relatively
shortlived 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.
TOP OF PAGE
Atomic structure, radioactivity and
nuclear physics revision notes index
Atomic structure, history, definitions,
examples and explanations including isotopes gcse chemistry
notes
1. Atomic
structure and fundamental particle knowledge needed to understand radioactivity gcse physics
revision
2.
What
is Radioactivity? Why does it happen? Three types of atomicnuclearionising radiation
gcse physics notes
3. Detection of
radioactivity, its measurement
and radiation dose units,
ionising
radiation sources
 radioactive materials, background radiation gcse physics revision
notes
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
gcse physics revision
5.
Uses of radioactive isotopes emitting alpha, beta (+/–) or gamma radiation in
industry and medicine gcse notes
6. The halflife of a radioisotope  how
long does material remain radioactive? implications!, uses of decay data and halflife values

archaeological radiocarbon dating, dating ancient rocks
gcse physics revision
7. What
actually happens to the nucleus in alpha and beta radioactive decay and why? nuclear
equations!, the
production of radioisotopes  artificial sources of radioactiveisotopes,
cyclotron gcse physics revision notes
8.
Nuclear
fusion reactions and the formation of 'heavy elements' by bombardment techniques
gcse physics notes
9. Nuclear Fission Reactions, nuclear power
as an energy resource gcse physics revision
notes
RADIOACTIVITY
multiple choice QUIZZES and WORKSHEETS
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 halflife data
ANSWERS to the WORDFILL WORKSHEET QUIZZES
Crossword
puzzle on radioactivity
and
ANSWERS!
OTHER CHEMICAL CALCULATION PAGES

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

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

Law of Conservation of Mass and simple reacting mass calculations

Composition by percentage mass of elements
in a compound

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

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

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

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

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

Reacting gas volume
ratios, Avogadro's Law
and GayLussac's Law (ratio of gaseous
reactantsproducts)

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

How to
do volumetric titration calculations e.g. acidalkali titrations
(and diagrams of apparatus)

Electrolysis products calculations (negative cathode and positive anode products)

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

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

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

Radioactivity & halflife calculations including
dating materials
(this page)
TOP OF PAGE
Doc Brown's
Chemistry Advanced Level PreUniversity Chemistry Revision Study Notes for UK IB
KS5 A/AS GCE advanced level physical theoretical chemistry students US K12 grade 11 grade 12
nuclear chemistry radioisotopes and halflives and calculations
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& ~US grades 8, 9 and 10 school science courses or equivalent for ~1416 year old
students of radioactivity halflife calculations uses Doc Brown's
school physics revision notes: GCSE physics, IGCSE physics, O level
physics, ~US grades 8, 9 and 10 school science courses or equivalent for ~1416 year old
students of radioactivity halflife calculations uses
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halflife of radioisotopes should help with IGCSE/GCSE/ chemistry or physics
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