Half-Life Data With Too Few Counts
A school uses a weak source with a half-life of about 5 minutes and a Geiger–Müller tube to record count rate every minute for 12 minutes. Background is measured as 14 counts per minute. The corrected count rates are 210, 168, 141, 118, 101, 86, 74, 64, 55, 48, 42, and 36 counts/min. A student plots the data on linear axes, gets a curve, and concludes "the source doesn't have a constant half-life because the rate doesn't halve evenly." The teacher has one lesson to correct this and link it to the meaning of random decay.
1. Test the student's conclusion quantitatively. Does the data set support a constant half-life? Show the check.
Read the worked answer
The right test: for exponential decay, the ratio of successive rates at equal time intervals should be constant - not the difference.
Successive ratios (1-minute intervals):
168/210 = 0.800; 141/168 = 0.839; 118/141 = 0.837; 101/118 = 0.856; 86/101 = 0.851; 74/86 = 0.860; 64/74 = 0.865; 55/64 = 0.859; 48/55 = 0.873; 42/48 = 0.875; 36/42 = 0.857.
Mean ratio ≈ 0.852 per minute.
Half-life: 0.852^n = 0.5 → n = ln0.5/ln0.852 = (−0.693)/(−0.160) = 4.33 min.
So the data are consistent with a constant half-life of about 4.3 min, close to the stated ~5 min.
Where the student went wrong: checking whether the rate halves over equal intervals is the wrong criterion. The rate falls by a constant fraction per interval, which produces a curve on linear axes. The differences shrink (210−168 = 42, then 36, 23, 17…) precisely because the decay is exponential - the student read this as evidence against constancy when it is evidence for it.
Statistical caveat: near the end, 36 counts/min over one minute has a Poisson uncertainty of √36 ≈ 6, i.e. ~17%, so the late ratios scatter more. The early ratios are tighter and support the constant-fraction model.
2. Design the analysis you would have students do, and explain how to handle the random nature of decay honestly.
Read the worked answer
Analysis design:
- Correct for background first and state it explicitly - subtracting 14 counts/min is a systematic correction that must be applied before any ratio is taken.
- Plot ln(rate) against time. A straight line is the signature of exponential decay; the gradient gives the decay constant λ, and t½ = ln2/λ. This turns a curve-fitting judgement into a straight-line fit students can do by hand.
- Compare two methods: the ratio method above and the log-plot gradient. Agreement between independent methods is a genuine check.
- Quantify uncertainty from counting statistics (√N) and show error bars - students should see that the late points are noisier, not that the physics changed.
Handling randomness honestly:
- State plainly that we cannot predict when an individual nucleus decays; we predict the statistical behaviour of large numbers.
- Distinguish "the model is wrong" from "the sample is small." With ~200 counts/min, fluctuations of ±14 are expected.
- Have students pool repeated runs to show the curve smooths as statistics improve.
Closing question: "If you doubled the counting time per point, what would happen to the scatter, and what would not change?" This separates statistical uncertainty from the underlying decay constant.