Measuring radioactive half-life

Practical method

Aim

Measure the half-life of protactinium-234m using a protactinium generator.

Variables

Independent variable: time
Dependent variable: corrected count rate
Control variables: position of the GM tube; counting interval.

Apparatus

• Sealed protactinium generator bottle
• GM tube and counter (or data logger)
• Stopwatch
• Drip tray

Method

1. Measure the background count rate.
2. An authorised teacher shakes the sealed generator so that protactinium-234m collects in the upper organic layer, then places it by the GM tube.
3. Record the count in each 10 s interval (or use a data logger) for about 5 minutes.
4. Subtract background and convert to count rate.
5. Plot ln C against t: gradient −λ. Half-life t½ = \(\dfrac{\ln 2}{\lambda }\).

Sample results

Background 0.50 counts s⁻¹ subtracted.

Analysis

ln C falls linearly with gradient −0.0099 s⁻¹, so λ = 9.9 × 10⁻³ s⁻¹ and t½ = \(\dfrac{\ln 2}{0.0099}\) = 70 s, close to the accepted 70 s (1.17 min) for protactinium-234m.

Uncertainty

Counts in each interval follow √N statistics; later intervals have few counts and large percentage uncertainty. Using the gradient of ln C weights all points and averages the scatter.

Errors and improvements

Risk assessment

Common mistakes

• Not subtracting background.
• Reading the half-life from the first two points only.
• Students handling the source.

Worked analysis

Gradient = \(\dfrac{2.208 - 3.689}{150}\) = −0.0099 s⁻¹. t½ = \(\dfrac{\ln 2}{0.0099}\) = 70 s.

Common misconception

The half-life is the same whatever the starting count rate; it is the time for any amount of the isotope to halve.