Half-Life and Radioactive Decay Calculations

Decay is random for one nucleus

A radioactive material contains unstable nuclei. Each nucleus may decay, emitting ionising radiation and becoming more stable. For a particular isotope, every undecayed nucleus has a constant chance of decaying during a given interval of time.

The process is random: it is impossible to predict which particular nucleus will decay next or exactly when it will decay. Waiting for a long time does not make an individual nucleus “due” to decay.

Random does not mean that there is no overall pattern. A large sample contains enormous numbers of nuclei, so its collective behaviour is much more predictable than the behaviour of any one nucleus. Roughly the same fraction of the remaining nuclei decays during each equal time interval.

Why activity falls

The activity of a source is the number of nuclear decays occurring per second. It is measured in becquerels:

$1\ \text{Bq}=1\ \text{decay per second}$

At first, a sample has many unstable nuclei available to decay, so its activity is relatively high. As nuclei decay, fewer unstable nuclei remain. There are therefore fewer opportunities for decay per second and the activity falls.

Activity is proportional to the number of undecayed nuclei. If the number of undecayed nuclei halves, the activity also halves. This connection lets activity measurements reveal what is happening in a sample without counting its nuclei directly.

Half-life describes repeated proportional change

The half-life, written $T_{1/2}$, is the time taken for the activity to fall to half its value. It can equivalently be described as the time taken for the number of undecayed nuclei to halve.

Each isotope has its own half-life. After every complete half-life, half of the quantity present at the start of that interval remains:

This is repeated halving, not subtraction of a fixed amount. For example, the changes $800\to400$, $400\to200$ and $200\to100$ are equal proportional changes, although the amounts lost are different.

Calculating after an integral number of half-lives

For an elapsed time $t$ and half-life $T_{1/2}$, first find the number of half-lives:

$n=\frac{t}{T_{1/2}}$

In this lesson, the calculation requirement concerns an integral number of half-lives, so $n$ is a whole number. If $Q_0$ is the initial activity or emission rate and $Q$ is the value remaining, then

$Q=Q_0\left(\frac12\right)^n$

The remaining fraction and ratio are therefore

$\frac{Q}{Q_0}=\left(\frac12\right)^n$

$Q:Q_0=1:2^n$

After four half-lives, for instance, the remaining-to-original ratio is $1:16$. The net decline is the part that has disappeared, so its fraction of the original is

$1-\left(\frac12\right)^n$

After four half-lives, $1/16$ remains and $15/16$ has decayed. Always check whether a question asks for the amount remaining, the amount decayed, or a ratio in a particular order.

Reading half-life from an activity–time graph

An ideal activity–time graph is a smooth exponential curve: it slopes downwards and becomes progressively less steep. It does not lose the same activity in every equal interval; it loses the same fraction. This means its intermediate values lie on a curve, not on straight sections joining the half-life points. Random decay can make real measured readings fluctuate, but their overall trend follows this shape.

Data for Activity follows a smooth decay curve

To determine the half-life from a graph:

1. Choose an activity value on the curve, such as $A$.
2. Calculate half of it, $A/2$.
3. Read the time corresponding to $A$ and the time corresponding to $A/2$.
4. Subtract the earlier time from the later time.
5. Give the half-life with the unit shown on the time axis.

You do not have to begin at time zero. As a check, choose another pair of values, such as $A/2$ and $A/4$. The time interval should be the same, allowing for the precision of the graph.

Random events and a predictable curve

A dice model helps connect randomness with half-life. Each die can represent one undecayed nucleus, and a chosen result can represent decay. It is impossible to know which die will show that result on a particular throw, but a roughly predictable fraction of a large collection is removed each round. Repeating the model produces a decreasing trend with random fluctuations.

The model illustrates why a nucleus cannot be assigned an exact decay time while the activity of a large sample can still have a measurable half-life. Half-life is therefore a statistical property of an isotope, not a countdown for an individual nucleus.

Random decay

• An individual unstable nucleus has a constant chance of decaying in a given interval.
• You cannot predict which nucleus will decay or when it will decay.
• A large sample nevertheless shows a predictable overall decline.

Half-life

$T_{1/2}$ is the time for the activity, or number of undecayed nuclei, to halve.

Activity falls because fewer unstable nuclei remain. Each half-life removes half of what is currently present, not a fixed amount.