Half-Life Measurement with a Noisy Counter
A student measures the activity of a short-lived radioactive sample with a Geiger counter, recording counts over successive 2-minute intervals: 480, 340, 240, 170, 120, 85 (counts per 2 min). The background count rate, measured with no source, is 20 counts per 2 min. The student plots the raw counts on log-linear graph paper and estimates a half-life, but forgets to subtract background. A classmate argues the background is negligible because it is small compared with the early readings.
1. Explain why background must be subtracted before extracting a half-life, and show the effect on the estimate.
Read the worked answer
Radioactive decay is exponential only for the source's own activity. The counter records source + background, so the measured rate is $N_{meas} = N_0 e^{-\lambda t} + B$. This is an exponential plus a constant, which is not a straight line on a log plot - it flattens at late times.
Subtracting $B = 20$: corrected counts are 460, 320, 220, 150, 100, 65.
Successive ratios (corrected): $320/460 = 0.696$, $220/320 = 0.688$, $150/220 = 0.682$, $100/150 = 0.667$, $65/100 = 0.65$ - roughly constant ≈ 0.68 per 2 min, consistent with exponential decay. The raw ratios (480→340→240…) are 0.708, 0.706, 0.708, 0.706, 0.708 - deceptively constant early but the background inflates the late points. The classmate is wrong: background is a fixed additive term that matters most precisely when the source is weak (late times), which is where the half-life fit is most sensitive.
2. Estimate the half-life from the corrected data and comment on the reliability of the estimate.
Read the worked answer
Mean ratio per 2-min interval ≈ 0.68, so the decay constant per interval satisfies $e^{-\lambda \Delta t} = 0.68$, giving $\lambda\Delta t = -\ln 0.68 = 0.386$.
Half-life in intervals: $t_{1/2}/\Delta t = \dfrac{\ln 2}{0.386} = \dfrac{0.693}{0.386} \approx 1.80$ intervals.
Each interval is 2 min, so $t_{1/2} \approx 3.6$ min.
Reliability: Poisson counting statistics give a fractional uncertainty $\approx 1/\sqrt{N}$ per point (e.g. $1/\sqrt{460} \approx 4.7%$ early, $1/\sqrt{65} \approx 12%$ late). The late points dominate the error, and the background subtraction (20 counts) adds a systematic uncertainty of about $\sqrt{20}\approx4.5$ counts. A single 3.6 min figure should be quoted with a range (roughly 3.4–3.8 min) and ideally confirmed by a least-squares fit to the corrected log data.