## The Rate of Decay

The rate at which these unstable isotopes undergo decay varies greatly between the different isotopes. The process is random for each atom. However there is a fixed probability that an atom will disintegrate over a fixed time scale.

It is rather like throwing a dice – on an individual throw then you cannot be certain what number will show up. However, you know that with a large number of throws the probability of throwing any number is one in six.

If we have a lump of pure Polonium 210 which undergoes alpha decay to produce stable Lead 206. For every alpha particle that is produced there is one less Polonium atom. Therefore the amount of radiation detected will slowly decrease as the Polonium is converted to lead.

The time it takes for the radiation to decay to half of what is was previously is constant and is called the half life.

It is usually assumed that all significant quantities of the isotope have disappeared after about 10 halflives.

The longer the halflife the fewer the number of atoms that decay in a given length of time. Therefore an isotope with a long halflife emits less radiation than an isotope with a short halflife.

## The Mathematics

You do not need this for the workshop but I have put this in because I have been asked questions about it. However, you will need to understand this to do any calculations on *secular equilibrium*.

The rate of change of the number of atoms with time is equal to the number of atoms times some constant.

Where N is the number of atoms and ? is the decay constant.

If we integrate both sides and take N_{0} to be the initial number of atoms

If we want to find the half life then all we have to do is set N equal to half N_{0}

Cancel the N_{0} and take logs of both sides

When doing calculations I usually convert the halflife to the decay constant and then use that. I find it easier to do it this way rather than working out where all the ln(2)s go in the formula.

## See Also

Tags: decay rate, halflife

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