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	Half life
Uranium 238	4.5x109 years
Thorium 234	24 days
Protactinium 234	1.2 minutes
Uranium 234	2.5x105 years
Thorium 230	8x104 years
Radium 226	1620 years
Radon 222	3.8 days
Polonium 218	3.1 minutes
Lead 214	27 minutes
Bismuth 214	20 minutes
Polonium 214	1.6x10-4 s
Lead 210	19 years
Bismuth 210	5.0 days
Polonium 210	138 days
Lead 206	STABLE

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:



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

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)

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^-λt
ln A = ln A_o -λt = 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