Half-life and Uses of Radiation

Because radioactive decay is random, we can never say when a particular nucleus will decay. Yet a large sample behaves in a remarkably predictable way: in a fixed period of time, a fixed fraction of the unstable nuclei always decays. This regularity is captured by the half-life — the single most useful number for describing how quickly a source decays. Half-life lets us predict how much of a source is left after a given time and, together with the type of radiation, decide which isotope suits a job. Radiation is genuinely double-edged: the very property that makes it dangerous — its ability to ionise atoms and damage cells — is also what makes it useful, from medical scans to smoke detectors. This lesson, part of Topic P4 (Waves and radioactivity) of OCR Gateway Combined Science A, defines half-life, works through decay-curve calculations, distinguishes irradiation from contamination, and surveys the uses, hazards and safety of radiation.

By the end of this lesson you should be able to define half-life, find the amount or count-rate of a source remaining after a whole number of half-lives, read a half-life from a decay curve, distinguish irradiation from contamination, describe the main uses and hazards of radiation, and describe the precautions taken to reduce exposure.

This lesson builds AO1 understanding of half-life and radiation safety, AO2 application when you calculate the fraction of a source remaining after a whole number of half-lives, and AO3 analysis when you read a half-life from a decay curve.

Radioactive Decay Is Random

Radioactive decay is a completely random process, which has two parts: we cannot predict which individual nucleus in a sample will decay next, and we cannot predict when any particular nucleus will decay. It is also spontaneous — not triggered by anything outside the nucleus. Heating, cooling, crushing or chemically reacting a source makes no difference to whether or when a nucleus decays.

Even though we cannot predict a single nucleus, a large sample contains an enormous number of nuclei, and on average a fixed fraction decays each second. This is why radioactivity is predictable on average — which is what makes half-life a reliable measure. The activity of a source is the number of decays per second, measured in becquerel (Bq), where $1 \text{ Bq} = 1$ decay per second; as the unstable nuclei are used up, the activity decreases over time.

Exam Tip: "Random" means you cannot predict which nucleus will decay or when; "spontaneous" means the decay is not affected by temperature, pressure or chemistry. Both phrases score marks.

What Is Half-Life?

The half-life of a radioactive isotope is the time taken for half the undecayed nuclei in a sample to decay. Equivalently — and just as usefully — it is the time taken for the activity (or count-rate) of the sample to fall to half its value. These two definitions are really the same statement, because the activity is proportional to the number of undecayed nuclei present: halve the number of unstable nuclei and you halve the rate at which they decay.

The half-life is a fixed property of a given isotope: it is always the same, no matter how big the sample is or how old it is. Half-lives vary enormously between isotopes — from tiny fractions of a second to billions of years.

The key feature is repeated halving. After one half-life, half the original nuclei remain. After two half-lives, half of that half — a quarter — remains. After three half-lives, an eighth remains, and so on. The amount never reaches exactly zero; it just keeps halving.

Number of half-lives	Fraction of original remaining
0	$1$ (all)
1	$\dfrac{1}{2}$
2	$\dfrac{1}{4}$
3	$\dfrac{1}{8}$
4	$\dfrac{1}{16}$
$n$	$\left(\dfrac{1}{2}\right)^{n}$

Exam Tip: A frequent misconception is that after two half-lives the source has "all gone". Not so — after two half-lives a quarter is still left, and the amount never reaches exactly zero. Define half-life as the time for half the undecayed nuclei to decay (or for the count-rate to halve), and remember it is fixed for each isotope.

The Decay Curve

If you plot the count-rate of a source against time, you get a decay curve — a smooth curve that falls steeply at first and then more gently, getting closer and closer to zero without ever quite reaching it. This shape is called exponential decay.

To find the half-life from a decay curve: pick a starting value on the count-rate axis, find the time at which the count-rate has fallen to half of it, and read off the time interval. A good check is to do it twice — from the start value to its half, and from that half to a quarter — and confirm you get the same time both times, because the half-life is constant. In the curve above, the count-rate falls from $800$ to $400$ in one half-life, from $400$ to $200$ in the next, and from $200$ to $100$ in the next: each step takes the same time.

Exam Tip: To read a half-life off a graph, find the time for the count-rate to drop from any value to half of that value (e.g. $800 \to 400$ , or $600 \to 300$ ). Reading it at two different places and getting the same answer confirms you have done it correctly.

Calculating the Amount Remaining

The repeated-halving pattern lets you calculate how much of a source remains after a whole number of half-lives. The method is simple: work out how many half-lives have passed, then halve the starting amount that many times.

Worked Example 1 — Mass remaining

A sample contains $80 \text{ g}$ of a radioactive isotope with a half-life of $6$ years. What mass of the isotope remains after $18$ years?

Step 1 — find the number of half-lives: $18 \div 6 = 3$ half-lives.

Step 2 — halve the mass three times: $80 \to 40 \to 20 \to 10 \text{ g}$ .

Answer: after $18$ years, $10 \text{ g}$ of the isotope remains. (Check: $80 \times \left(\tfrac{1}{2}\right)^{3} = 80 \times \tfrac{1}{8} = 10 \text{ g}$ .)

Worked Example 2 — Count-rate remaining

A source has an initial count-rate of $2400 \text{ Bq}$ . Its half-life is $15$ minutes. What is the count-rate after $1$ hour?

Step 1 — find the number of half-lives in $1$ hour: $60 \div 15 = 4$ half-lives.

Step 2 — halve the count-rate four times: $2400 \to 1200 \to 600 \to 300 \to 150 \text{ Bq}$ .

Answer: after $1$ hour the count-rate is $150 \text{ Bq}$ . (Check: $2400 \times \left(\tfrac{1}{2}\right)^{4} = 2400 \times \tfrac{1}{16} = 150 \text{ Bq}$ .)

Worked Example 3 — Finding the number of half-lives

A radioactive source has an activity of $640 \text{ Bq}$ . After some time its activity has fallen to $40 \text{ Bq}$ . How many half-lives have passed?

Step 1 — halve repeatedly until you reach $40$ : $640 \to 320 \to 160 \to 80 \to 40$ .

Step 2 — count the halvings: that is 4 halvings.

Answer: $4$ half-lives have passed (since $640 \times \tfrac{1}{16} = 40 \text{ Bq}$ ).

Worked Example 4 — Finding the time taken

A radioactive isotope has a half-life of $8$ days. How long does it take for its activity to fall to one-eighth of its original value?

Step 1 — work out how many half-lives give one-eighth: $\tfrac{1}{8} = \left(\tfrac{1}{2}\right)^{3}$ , so 3 half-lives.

Step 2 — multiply by the half-life: $3 \times 8 = 24$ days.

Answer: it takes $24$ days for the activity to fall to one-eighth.

Exam Tip: The reliable method is: (1) find how many half-lives have passed (time ÷ half-life), then (2) halve the starting amount that many times. For a time, first express the fraction as a power of a half ( $\tfrac{1}{8} = (\tfrac{1}{2})^3$ ) to get the number of half-lives, then multiply by the half-life.

Irradiation Versus Contamination

This distinction is one of the most heavily tested ideas in the whole topic, and the two terms are easy to confuse, so it is worth getting absolutely clear.

Irradiation is being exposed to radiation from a source that is outside your body. The source itself does not touch you or get onto or into you — you are simply in the path of the radiation it emits. Once you move away from the source (or it is removed), the irradiation stops. Irradiation does not make you radioactive.
Contamination is getting the radioactive material itself onto your skin, your clothing, or inside your body (by swallowing, breathing it in, or through a wound). Because the radioactive atoms are now on or in you, they keep on irradiating you from close range for as long as they remain — and you can spread them to others.

Half-life and Uses of Radiation

Half-life and Uses of Radiation

Radioactive Decay Is Random

What Is Half-Life?

The Decay Curve

Calculating the Amount Remaining

Worked Example 1 — Mass remaining

Worked Example 2 — Count-rate remaining

Worked Example 3 — Finding the number of half-lives

Worked Example 4 — Finding the time taken

Irradiation Versus Contamination

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