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).





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 = -l₁N₁ = dN₂/dt = -l₂N₂ = dN₃/dt = -l₃N₃ = dN₄/dt = -l₄N₄ = 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; -



Because lN is a constant and the half life (T) of a radioactive isotope in the decay chain is proportional to 1/l

we have for a decay chain:



The number of nuclei of a particular radioactive isotope in a decay chain will be proportional to the half-life of that isotope when equilibrium conditions have been reached.




Notice that we have to make an adjustment if we want to deal with the relative masses of components in the series.
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 = A_o/2ⁿ

This can be proved as follows.
Start with the standard radioactive decay law and take logs to the base e:

A = A_oe^-lt
ln A = ln A_o -lt = ln A_o – ln(2t/T) where T is the half life.

Therefore:

ln A = ln[A_o/2ⁿ) where n = t/T and so A = A_o/2ⁿ