Welcome to Activity and Half-Life!

In the previous chapters, we learned that some atoms are unstable and release radiation to become more stable. But how fast does this happen? Can we predict when a single atom will explode? (Spoiler: We can't!) In this chapter, we will learn how to measure the "speed" of radioactive decay using activity and the half-life. Don't worry if this seems a bit "random" at first—that's because radioactivity itself is random!

1. Activity and the Becquerel

When an unstable nucleus decays, it emits radiation (like alpha, beta, or gamma). We want to know how many of these decays are happening every second. This "rate" of decay is what we call activity.

Definition: Activity is the number of decays that occur in a radioactive sample per unit of time.

The Unit: The Becquerel \( (\text{Bq}) \)

In Physics, we measure activity in a unit called the Becquerel \( (\text{Bq}) \).

  • \( 1 \text{ Bq} = 1 \text{ decay per second} \)

Example: If a wooden ancient artifact has an activity of \( 50 \text{ Bq} \), it means \( 50 \) unstable nuclei are decaying inside it every single second.

Did you know? The unit is named after Henri Becquerel, who discovered radioactivity by accident when he left some uranium salts on top of a photographic plate in a dark drawer!

2. The Random Nature of Decay

It is very important to remember that radioactive decay is a random process. This means:

  • We cannot predict exactly which nucleus will decay next.
  • We cannot predict exactly when a specific nucleus will decay.

However, if we have a huge number of atoms (which we usually do), we can use half-life to predict how the group as a whole will behave over time.

3. What is Half-Life?

The half-life is a special measurement of time. Because the activity of a source decreases over time (as there are fewer unstable atoms left to decay), we need a way to describe how quickly that decrease happens.

Definition: The half-life is the time taken for half of the unstable nuclei in a sample to decay, OR the time taken for the activity of a sample to fall to half its original value.

The "Pizza Analogy":
Imagine you have a giant pizza. Every \( 10 \) minutes, you eat exactly half of what is on the table.
- After \( 10 \) minutes (one "half-life"), you have \( \frac{1}{2} \) a pizza left.
- After \( 20 \) minutes (two "half-lives"), you have \( \frac{1}{4} \) left.
- After \( 30 \) minutes (three "half-lives"), you have \( \frac{1}{8} \) left.
The time it takes to eat half is always the same (\( 10 \) minutes), even though the actual amount of pizza you eat gets smaller each time.

4. Calculating Half-Life from a Graph

In your exam, you will often be asked to find the half-life from a graph showing how activity (on the vertical \( y \)-axis) changes with time (on the horizontal \( x \)-axis). This is called a decay curve.

Step-by-Step Guide:

1. Pick a starting activity on the \( y \)-axis (let's call this \( A \)).
2. Divide that value by two (\( \frac{A}{2} \)).
3. Draw a horizontal line from the \( \frac{A}{2} \) value until you hit the graph curve.
4. Draw a vertical line down from that point to the \( x \)-axis (time).
5. The value on the \( x \)-axis is the half-life.

Quick Tip: To be more accurate, do this twice! For example, find how long it takes to go from \( 80 \text{ Bq} \) to \( 40 \text{ Bq} \), then check how long it takes to go from \( 40 \text{ Bq} \) to \( 20 \text{ Bq} \). The time gap should be the same!

5. Simple Half-Life Calculations

You might be asked to find the activity or mass of a sample after a certain amount of time. You don't need fancy formulas—just a simple list or "arrow diagram."

Example Problem:

A radioactive isotope has a half-life of \( 10 \text{ minutes} \). Its initial activity is \( 800 \text{ Bq} \). What will its activity be after \( 30 \text{ minutes} \)?

Step 1: Calculate how many half-lives have passed.
\( \text{Number of half-lives} = \frac{30 \text{ mins}}{10 \text{ mins}} = 3 \text{ half-lives} \)

Step 2: Halve the activity three times.
- Start: \( 800 \text{ Bq} \)
- After 1st half-life: \( 400 \text{ Bq} \)
- After 2nd half-life: \( 200 \text{ Bq} \)
- After 3rd half-life: \( 100 \text{ Bq} \)

Common Mistake: Students often divide the original number by the number of half-lives (e.g., \( 800 / 3 \)). Don't do this! You must divide by \( 2 \) repeatedly.

6. Background Radiation

Before you measure the activity of a source, you must account for background radiation. This is the low-level radiation that is around us all the time from natural and man-made sources.

Sources of background radiation include:
- Cosmic rays: Radiation from space.
- Rocks and Soil: Radioactive radon gas from underground rocks.
- Food and Drink: Naturally occurring isotopes like Potassium-40.
- Medical sources: X-rays and radioactive tracers.
- Nuclear Industry: Fallout from past testing (though this is a very small percentage).

Correcting for Background Radiation:

If a Geiger-Muller (GM) detector shows a reading, it includes the background radiation. To find the corrected activity of a source:
\( \text{Corrected Activity} = \text{Total Count Rate} - \text{Background Count Rate} \)

7. Key Takeaways

  • Activity is measured in Becquerels \( (\text{Bq}) \), which means decays per second.
  • Radioactive decay is random and spontaneous.
  • Half-life is the time it takes for activity (or the number of nuclei) to halve.
  • Half-life is constant for a particular isotope; it never changes.
  • Always subtract background radiation if the question gives you a background count.

Note: For details on how we detect this radiation using photographic film or Geiger-Muller detectors, see the chapter on "Types of radiation and detection".