Radioactive Tracer Dilution in a Water Tank
A laboratory technician prepares a short-lived tracer for a leak test on a ship's fresh-water tank. A stock solution is labelled 8.0 × 10⁴ Bq/mL. The technician withdraws 0.50 mL, dilutes it to 250 mL with distilled water, then injects the whole 250 mL into a 2.0 m³ tank of water and stirs thoroughly. A 1.0 mL sample of the tank water is counted and gives 60 counts per minute above background. The counter's efficiency is 20%, and 1 Bq corresponds to 60 decays per minute. The half-life of the isotope is 6.0 hours, and the measurement is made 3.0 hours after injection. The supervisor questions whether the result is consistent with the expected activity.
1. Calculate the expected activity concentration in the tank at the moment of injection, then correct for decay to the time of sampling.
Read the worked answer
Activity withdrawn: 8.0 × 10⁴ Bq/mL × 0.50 mL = 4.0 × 10⁴ Bq.
After dilution to 250 mL, concentration = 4.0 × 10⁴ / 250 = 160 Bq/mL.
All 4.0 × 10⁴ Bq enters 2.0 m³ = 2000 L = 2.0 × 10⁶ mL.
Concentration in tank = 4.0 × 10⁴ / 2.0 × 10⁶ = 0.020 Bq/mL at injection.
After 3.0 h (half of 6.0 h), activity halves: 0.020 / 2 = 0.010 Bq/mL at sampling.
2. Convert the measured count rate to an activity concentration and decide whether the measurement agrees with the prediction.
Read the worked answer
Measured 60 cpm at 20% efficiency means true disintegrations = 60 / 0.20 = 300 dpm in 1.0 mL.
Convert to Bq: 300 / 60 = 5.0 Bq per mL.
Predicted at sampling was 0.010 Bq/mL, so the measured value is about 500 times higher - a gross disagreement, not a rounding issue.
The judgement: the measurement is not consistent with the dilution model. Likely causes are a mislabelled stock activity, incomplete mixing (sampling a concentrated plume), or a units error in the dilution step. The technician should not report the tracer as validated; the tank should be resampled after further mixing and the stock certificate rechecked.