AHL Nuclear decay

Radioactive decay is a random process. This leads to an interesting link with mathematics.

If the rate of change of a quantity is proportional to the quantity itself then the change is exponential.


Key Concepts

Random processes

Nuclear decay is a random process. In a sample of unstable nuclei:

  • We do not know which nucleus will be the next to decay
  • We do not know when the next nucleus will decay
  • There is a fixed probability that a given proportion of decays will occur in a given period of time. For a large sample of dice, one sixth would land on a six in a given roll.

The probability of decay is related to the amount of energy that will be released when it happens.

The consequence of the random nature of decay can be demonstrated with a more everyday substance: beer foam. Since the bubbles pop randomly, the amount of beer foam in a glass decays exponentially.

Exponential decay graphs are an example of a Non-linear graph.


The number of decays per second in a sample of material is directly proportional to the number of nuclei. This is referred to as the activity of the sample.

\(A\propto N\)

\(A=\lambda N\)

  • \(A\) is the activity of the sample (Bq, equivalent to s-1)
  • \(\lambda\) is the decay constant (s-1)
  • \(N\) is the number of undecayed nuclei

Half life

Half life gives a measure of the rate of decay of a sample. Protactinium is an isotope with a half life of about 1 minute. This makes it convenient for school experiments.

There are methods for measuring short and long half lives:

  • If the half life is short, a Geiger-Muller tube and counter can collect measurements for how count rate varies with time. The background count rate must be measured (without the presence of the sample) and deducted from each reading. A graph can be plotted of count rate against time, and the half life measured and confirmed using intervals on the count axis.
  • The activity of isotopes with half lives of hundreds or thousands of years does not noticeably decrease in the time over which observations can be made. Instead, to find half life, a pure sample of the isotope can be isolated. The mass can be used to find the number of nuclei, which is combined with the activity to find decay constant.

Half life is often the quantity that is quoted for a given isotope in exam questions. A good tip for commencing further calculations is to use the half life to find the decay constant:

\(\lambda={\ln2\over t_{1\over 2}}\)

  • \(\lambda\) is the decay constant
  • \(t_{1\over2}\) is the half life

The half life of a isotope can be used to determine whether the isotope may have uses for mankind:

  • The long half life of carbon-14 (5730 years) means it can be used to date ancient objects
  • The short half life of technetium-99m (6 hours) permits the medical use as a tracer in the human body 


Exponential decay

If \({\mathrm{d} N\over \mathrm{d}t} ∝ N\) then the decay is exponential:

\(N =N_0\mathrm{e}^{-\lambda t}\)

  • \(N\) is the number of undecayed nuclei remaining
  • \(N_0\) is the number of undecayed nuclei when \(t=0\)
  • \(\lambda\) is the decay constant, a fixed probability of a given proportion of nuclei decaying per unit time (units correspond to \(t\) e.g. s-1)
  • \(t\) is time (e.g. s)

Since activity is proportional to the number of undecayed nuclei, it is also exponential, leading to equations that can often be more useful than the number of particles. After all, it's difficult to count individual nuclei!

\(A=\lambda N_0\mathrm{e}^{-\lambda t}\)

\(A =A_0\mathrm{e}^{-\lambda t}\)

  • \(A\) is the activity of the source (s-1)
  • \(A_0\) is the activity when \(t=0\)

The exponential decay equation enables the calculation of the number of nuclei remaining at any time.

A linear relationship can be produced by taking natural logarithms of each side of the equation:

\(\ln A=-\lambda t+\ln{A_0}\)

This is the kind of mathematical manipulation that may required for your Investigation.

Test Yourself

Use flashcards to practise your recall.

Use quizzes to practise application of theory.