Mastering Radioactive Decay and Half-Life for GCSE Physics

Radioactive decay is a fundamental concept in nuclear physics that describes how unstable isotopes become stable over time. For your GCSE Physics exams, you need to understand not just the definitions, but the mathematical behaviour of these substances.

In this guide, we will explore why radioactive decay is considered a random process, how we define half-life, and how to solve common numerical problems. Mastering these concepts is essential for understanding everything from medical tracers to carbon dating.

The Random Nature of Radioactive Decay

Radioactive decay is inherently random. If you have a single unstable nucleus, it is impossible to predict exactly when it will decay. You cannot know which nucleus will decay next, nor can you predict the precise moment a specific atom will emit radiation.

However, when dealing with a large sample containing millions of nuclei, we can observe a predictable pattern. While individual events are random, the overall rate of decay for a large population follows a statistical trend. This rate is unaffected by external conditions such as temperature, pressure, or chemical state. The activity of a sample is measured in Becquerels (Bq), where 1 Bq equals one decay per second.

Defining Half-Life

Because the number of radioactive nuclei decreases over time, the activity of a sample also drops. We use the term half-life to quantify how quickly this happens. The half-life of an isotope is defined as:

1. The time it takes for the number of radioactive nuclei in a sample to halve.
2. The time it takes for the count rate (or activity) from a sample to fall to half its initial level.

It is important to note that the activity never technically reaches zero; it simply becomes smaller and smaller as time progresses.

Calculating Half-Life: Worked Example 1

A sample of a radioactive isotope has an initial activity of 800 Bq. Its half-life is 2 hours. Calculate the activity of the sample after 6 hours.

Step 1: Determine the number of half-lives that have passed. $$n = \frac{\text{Total time}}{\text{Half-life}} = \frac{6 \text{ hours}}{2 \text{ hours}} = 3 \text{ half-lives}$$

Step 2: Halve the initial activity for each half-life.

• After 1 half-life: $800 \div 2 = 400 \text{ Bq}$
• After 2 half-lives: $400 \div 2 = 200 \text{ Bq}$
• After 3 half-lives: $200 \div 2 = 100 \text{ Bq}$

Answer: The activity after 6 hours is 100 Bq.

Calculating Remaining Nuclei: Worked Example 2

A sample starts with 40,000 radioactive nuclei. After 12 days, only 5,000 nuclei remain. Calculate the half-life of the isotope.

Step 1: Determine how many times the sample has halved.

• Start: 40,000
• 1st half-life: 20,000
• 2nd half-life: 10,000
• 3rd half-life: 5,000

It took 3 half-lives to reach 5,000 nuclei.

Step 2: Calculate the half-life. $$\text{Half-life} = \frac{\text{Total time}}{\text{Number of half-lives}} = \frac{12 \text{ days}}{3} = 4 \text{ days}$$

Answer: The half-life of the isotope is 4 days.

Interpreting Decay Curves

In an exam, you may be asked to interpret a decay curve graph. The y-axis typically represents activity or the number of nuclei, while the x-axis represents time. To find the half-life from a graph:

1. Identify the initial activity at $t = 0$.
2. Find the value that is exactly half of the initial activity on the y-axis.
3. Draw a horizontal line from this value to the curve, then drop a vertical line down to the x-axis.
4. The value on the x-axis is the half-life.

Common Mistakes

• Confusing activity with time: Students often try to divide the activity by the number of half-lives instead of halving the value repeatedly.
• Ignoring background radiation: When calculating activity from a Geiger-Müller tube, remember to subtract the background radiation count rate from your total readings before performing half-life calculations.
• Miscounting half-lives: Always write out the steps (e.g., 100 -> 50 -> 25) to ensure you do not miss a step in the sequence.

Frequently Asked Questions

Does the half-life change as the sample decays? No, the half-life is a constant property of a specific radioactive isotope.

Can we predict when a single atom will decay? No, radioactive decay is a random process; we can only predict the behaviour of large numbers of atoms.

What is the unit for activity? Activity is measured in Becquerels (Bq), where 1 Bq is one decay per second.

Why does the activity never reach zero? Because you are always halving the remaining amount, you will mathematically approach zero but never reach it.

Conclusion

Understanding radioactive decay and half-life is a core requirement for your GCSE Physics success. By mastering these calculations and the underlying theory, you are well-prepared for your assessments. To reinforce your learning with interactive visuals, head over to MathInstructor AI and generate a free animated lesson on this topic today.