Determining Half-Life from a Graph

The half-life of a radioactive isotope can be determined from a graph of activity (or count rate) against time.

Method

1. Read the initial activity (or count rate) from the y-axis at $t = 0$.
2. Calculate half of the initial activity.
3. Draw a horizontal line from this half-value to the curve.
4. Draw a vertical line from the curve down to the time axis.
5. Read the time. This is one half-life.

Checking consistency

Repeat for the next half-life:

• Find the time when the activity drops to one quarter of the initial value.
• The time between the half-value and the quarter-value should be the same as the first half-life.

If the half-lives are approximately equal, the data is consistent with exponential decay.

Worked example

A graph shows:

• At $t = 0$: activity = 800 Bq
• At $t = 3$ min: activity = 400 Bq
• At $t = 6$ min: activity = 200 Bq
• At $t = 9$ min: activity = 100 Bq

Half-life = 3 minutes (confirmed by multiple consistent readings).

Correcting for background radiation

Before plotting, subtract the background count rate from all readings. If background is not subtracted, the curve will level off at the background level instead of approaching zero, and the half-life will appear longer than it actually is.

Table method (without graph)

Time (min)	Activity (Bq)
0	1200
2	600
4	300
6	150

Each 2-minute interval halves the activity. Half-life = 2 minutes.

Common errors and how to correct them

• Not subtracting background radiation before determining the half-life.
• Trying to read the half-life when the graph has not reached half the initial value (need more data points).
• Using the time for the activity to reach zero (it never truly reaches zero due to the exponential nature of decay).

How to apply this in an exam

Read the initial activity. Find the time for it to halve. Check by finding the time for it to quarter. State the half-life clearly with units.