RADIOACTIVE DECAY CHAINS
An unstable nucleus may emit an alpha or beta particle to form another element, or a gamma ray
and become less excited nucleus. This daughter product as it is called may also be radioactive
and so a whole chain of decay products can be formed.
One of the most well known of these is the decay chain of uranium 238 (shown below).
Isotope
Uranium 238
Thorium 234
Protactinium 234
Uranium 234
Thorium 230
Radium 226
Radon 222
Polonium 218
Lead 214
Bismuth 214
Polonium 214
Lead 210
Bismuth 210
Polonium 210
Lead 206
Half life
4.5 x 109 years
24 days
1.2 minutes
2.5 x 105 years
8.0 x 104 years
1620 years
3.8 days
3.1 minutes
27 minutes
20 minutes
1.6 x 10-4 s
19 years
5.0 days
138 days
stable
Lead 206 is stable and is therefore the end of the
decay chain.
Since all the nuclei are in dynamic equilibrium in a decay series - that is there can be no build up
of any particular element the rate of decay of all the components in the series must be the same in other words dN/dt for every component is the same.
Therefore dN1/dt = -λ1N1 = dN2/dt = -λ2N2 = dN3/dt = -λ3N3 = dN4/dt = -λ4N4 = etc. so:One consequence of this is that the elements with the long half lives will be present in larger
quantities than those with short half lives because; λ1N1 = λ2N2 etc. or λN = a constant
Because N is a constant and the half life (T) of a radioactive isotope in the decay chain is
proportional to 1/
we have for a decay chain
Decay chains:
N1/N2 = T1/T2
The number of nuclei of a particular radioactive isotope in a decay chain will be inversely
proportional to the half-life of that isotope when equilibrium conditions have been reached.
Example problem
Find the ratio of the anmount of lead 214 to bismuth 214 in the above series.
Ratio of numbers of nuclei = TPb/TBi = 27/20 = 1.35 times as much Lead 214 as there will be Bismuth 214.
Notice that we have to make an adjustment if we want to deal with the relative masses of
components in the series.
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Since N = (m/M)L
T1/T2 = N1/N2 = (m1/M1)/(m2/M2) = (m1/m2)x(M2/M1)
Uses of radioisotopes
Radioactive isotopes can be very useful. They are used in:
1. Medicine for both treatment and diagnosis
2. Archaeological and geological dating using carbon 14 or uranium
3. Fluid flow measurement - water, blood, mud, sewage etc.
4. Thickness testing of materials such as polythene
5. Radiographs of metal castings
6. Sterilisation of food and insects
7. Tracers in fertilisers used in agriculture
8. Smoke alarms in houses
9. Tracing phosphate fertilisers using phosphorus 32
10. Checking the silver content of coins
11. Atomic lights using krypton 85
12. Testing for leaks in pipes
Proof of A = Ao/2n
This can be proved as follows.
Start with the standard radioactive decay law and take logs to the base e:
A = Aoe-t
ln A = ln Ao - t = ln Ao – ln(2t/T) where T is the half life.
Therefore:
ln A = ln[Ao/2n) where n = t/T and so A = Ao/2n
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