To find a half-life from a graph, pick a count rate, find half of that value, and read the time gap between the two points on the curve. That gap is the half-life, and it is the same wherever you start. You meet this in any question that shows a decay curve or a table of rate against time.
If your readings include background, remove it first, as in subtracting background from supplied counts. This lesson is part of half-life and background.
What does the curve show?
The vertical axis shows the corrected count rate, and the horizontal axis shows time. The curve falls steeply at first, then more gently, but never reaches zero in the plotted region. The shape comes from a fixed fraction decaying in each equal time step.
Because the fraction is fixed, the time to halve is constant along the whole curve. That is the test of a half-life graph: one halving from 200 to 100 takes the same time as one from 100 to 50.
How do I read it, step by step?
- Check the axes and units. Note whether time is in seconds, minutes, hours, days or years.
- Pick a starting value that is easy to read, for example the value at time zero.
- Halve it and mark that value on the vertical axis.
- Read across to the curve, then down to the time axis for both points.
- Subtract the two times. The gap is the half-life, not the later time itself.
- Check with a second pair of points. Both gaps should match.
Worked example
(Invented data.) A source was measured with a detector where the background was 5 counts/min. The table gives the measured readings and the corrected rates.
| Time (min) | Measured (counts/min) | Corrected (counts/min) |
|---|---|---|
| 0 | 205 | 200 |
| 8 | 105 | 100 |
| 16 | 55 | 50 |
| 24 | 30 | 25 |
Step 1: use the corrected values. Start at 200 counts/min at 0 min.
Step 2: half of 200 is 100. The curve reaches 100 counts/min at 8 min.
Step 3: the time gap is 8 − 0 = 8 min.
Check: from 100 to 50 counts/min the time goes from 8 min to 16 min, a gap of 8 min again. From 50 to 25 it goes from 16 min to 24 min, also 8 min. Three equal gaps confirm the half-life is 8 min.
On a smooth curve you could also start at 150 counts/min and halve to 75. Those points fall at about 3.3 min and 11.3 min, a gap of 8.0 min, which shows that the starting point does not matter. You can test curves like this with the graph-model and residual explorer.
The mistake to watch for
A student looked at the same graph and wrote:
Mistaken answer: half-life = 100 counts/min
The student read the half of 200 correctly, but stopped at the vertical axis and gave that value as the answer.
The half-life is a time, so the answer must come from the horizontal axis and carry a time unit.
Another slip is to give the later time (8 min) without subtracting the start, which works only when you start at zero. The correction is to read two times and take their difference. The rate and energy graph interpreter gives more practice reading axes and units.
Check yourself
1. A corrected count rate falls from 480 counts/s at time zero to 240 counts/s at 12 hours. What is the half-life?
Show answer
240 is half of 480, and the time gap is 12 − 0 = 12 hours.
2. A curve passes through 80 counts/min at 5 min and 40 counts/min at 20 min. What is the half-life?
Show answer
40 is half of 80. Time gap = 20 − 5 = 15 min.
3. A graph of measured count rate starts at 90 counts/min. The background is 10 counts/min. The measured rate reaches 50 counts/min after 6 days. What is the half-life?
Show answer
Corrected start = 90 − 10 = 80 counts/min. Corrected at 6 days = 50 − 10 = 40 counts/min, which is half of 80. The half-life is 6 days.
Where this leads next
Once you can read a half-life, use it to predict amounts in applying successive halving to a stated interval. If graph-to-answer reasoning is where marks slip, our teachers can work on it in online one-to-one Physics tuition.