A great question.

Radioactive decay is modelled exponentially. When a nucleus decays, emitting a gamma ray, an alpha particle or a beta particle, it changes into a different isotope. Every nucleus has the same probability that it will decay.

But which ones do decay is a random or stochastic process. When an isotope has a long half life the probability of an individual nucleus decaying at a given time point is very low. Isotopes with a short half life have a much higher probability of decay.

Physicists have derived a formula which allows you to tell how many (but not precisely which) nuclei in a sample will have decayed after a certain period of time has elapsed. This formula is as follows :

...where N is the number of nuclei that decay, N

o is the number of nuclei that your started with, e is the natural (Naperian) logarithm, lambda is a mathematical value called the decay constant (given by ln2/half-life), and t is time.

The half life is the time taken for half of the radioactive nuclei to decay.

Therefore, if you take a known amount of the isotope you are interested in you can work out how many atoms (and hence nuclei) it contains. Then, by measuring how many 'counts' or decays it undergoes over a known period of time you can use the above formula to work out the half life.

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