80. Radioactive Decay & Half-life

1. Key Definitions

Radioactive decay is a random process. While we cannot predict when an individual nucleus will decay, we can use statistical measurements for large samples:

• Activity: The overall rate of decay for all isotopes in a sample. It is measured in Becquerels (Bq), where 1 Bq = 1 decay per second.

• Count Rate: The number of decays recorded each second by a detector, such as a Geiger-Müller tube. This is used to estimate the activity.

2. What is Half-life?

Half-life is constant for a particular isotope and can be defined in two ways:

1. The time taken for the number of radioactive nuclei in a sample to half.

2. The time taken for the activity (or count rate) of a sample to half.

As time passes, there are fewer unstable nuclei left to decay, so the activity of the sample decreases over time.

3. Calculating Half-life from a Graph

On a decay graph (Activity vs. Time), the curve shows a "downward slope" that gets flatter over time:

• To find the half-life, identify the initial activity (e.g., 600 Bq).

• Find the time it takes for that value to drop by half (to 300 Bq).

• The time interval on the x-axis is the half-life.

• You can verify this by checking how long it takes to half again (e.g., from 300 Bq to 150 Bq); the time interval should be the same.

4. Example Calculation

Question: The half-life of a source is 40 hours. There are initially 3 million radioactive nuclei. How many remain after 5 days?

1. Convert time to hours: 5 days $\times$ 24 hours = 120 hours.

2. Calculate number of half-lives: 120 hours $\div$ 40 hours = 3 half-lives.

3. Half the nuclei 3 times:

   • Initial: 3,000,000

   • After 1st half-life: 1,500,000

   • After 2nd half-life: 750,000

   • After 3rd half-life: 375,000

Final Answer: 375,000 nuclei remain.

5. Summary Table