Describe radioactive decay as a random spontaneous process, apply the exponential decay law and the decay constant, and relate it to half-life and activity

What this dot point is asking

SEAB wants you to describe radioactive decay as a random and spontaneous process, to apply the exponential decay law with the decay constant, and to relate the decay constant to half-life and activity. The randomness at the level of individual nuclei produces a precise statistical law for large numbers.

The answer

Random and spontaneous decay

Radioactive decay is random: it is impossible to predict which nucleus will decay next or exactly when a given nucleus will decay. Every undecayed nucleus has the same constant probability of decaying per unit time. Decay is also spontaneous: it is unaffected by external conditions such as temperature, pressure or chemical state. Despite this randomness, the behaviour of a very large number of nuclei is highly predictable on average.

The decay constant and activity

The decay constant $\lambda$ is the probability per unit time that a given nucleus decays. The activity $A$ of a source is the rate at which nuclei decay, measured in becquerels (Bq), one decay per second. Because each nucleus has the same probability of decaying:

where $N$ is the number of undecayed nuclei. Since the rate of decay is proportional to the number remaining, the population falls exponentially.

The exponential decay law

The number of undecayed nuclei follows:

and the activity, being proportional to $N$, follows the same form:

A graph of $\ln N$ (or $\ln A$) against time is a straight line of gradient $-\lambda$, which is the standard way to measure the decay constant from data.

Half-life

The half-life $t_{1/2}$ is the time for half the undecayed nuclei (and so half the activity) to decay. Setting $N = \tfrac{1}{2}N_0$ in the decay law gives:

Each half-life reduces the quantity to half its previous value, so after $n$ half-lives a fraction $(\tfrac{1}{2})^n$ remains. The half-life is constant for a given isotope, independent of how much material is present.

Examples in context

Example 1. Carbon dating. Living things maintain a fixed proportion of radioactive carbon-14. After death, the carbon-14 decays with a half-life of about $5700$ years. Measuring the remaining activity and applying $A = A_0 e^{-\lambda t}$ gives the age of organic remains, a direct use of the exponential decay law.

Example 2. Medical tracers. A radioactive tracer used in diagnosis is chosen with a short half-life so it decays quickly and limits the patient's dose. Technetium-99m, with a half-life of about $6$ hours, gives a strong signal during a scan but falls to a negligible activity within a day, a deliberate use of the half-life relation.

Try this

Q1. State what is meant by the activity of a radioactive source and its unit. [2 marks]

• Cue. The rate at which nuclei decay, $A = \lambda N$; unit becquerel (Bq), one decay per second.

Q2. An isotope has a decay constant of $0.050\ \text{s}^{-1}$. Find its half-life. [2 marks]

• Cue. $t_{1/2} = \dfrac{\ln 2}{\lambda} = \dfrac{0.693}{0.050} = 13.9\ \text{s}$.

Q3. A source has an initial activity of $1.6 \times 10^5\ \text{Bq}$ and a half-life of $2.0$ hours. Find its activity after $6.0$ hours. [2 marks]

• Cue. $6.0$ hours is $3$ half-lives, so $A = 1.6 \times 10^5 \times (\tfrac{1}{2})^3 = 2.0 \times 10^4\ \text{Bq}$.