Exponential Law of Decay
- Radioactive decay is completely random — you can't predict which atom's nucleus will decay when.
- But although you can't predict the decay of an individual nucleus, if you take a very large number of nuclei, their overall behaviour shows a pattern.
- Isotopes of an element have the same number of protons, but different numbers of neutrons in their nuclei.
- Any sample of a particular isotope has the same rate of decay — i.e. the same proportion of atomic nuclei will decay in a given time.
- Each unstable nucleus within the isotope will also have a constant decay probability.
\(\(A=\lambda N\)\)
- where \(A\) is the activity in Becquerels
- where \(\lambda\) is the decay constant in \(s^{-1}\)
- where \(N\) is the number of unstable nuclei in the sample
Also can be written as:
\(\(A=-\frac{\Delta N}{\Delta t}\)\)
Combining these equations:
\(\(\frac{\Delta N}{\Delta t}=-\lambda N\)\)
- where \(\frac{\Delta N}{\Delta t}\) is the rate of change of unstable nuclei
- where \(\lambda\) is the decay constant in \(s^{-1}\)
- where \(N\) is the number of unstable nuclei in the sample
\(\(N=N_{0}e^{-\lambda t}\)\)
- where \(N\) is the number of unstable nuclei remaining
- where \(N_{0}\) is the original amount of unstable nuclei
- where \(\lambda\) is the decay constant
- and where \(t\) is time in seconds
\(\(N=nN_{A}\)\)
- \(N\) is the number of atoms in a sample
- \(n\) is the number of moles in the sample
- \(N_{A}\) is Avogadro's Constant