An air sample is contaminated with ¹⁵O,which has a half-life of 2.03 minutes.You can pass it through a long pipe to allow it to decay until it can be safely released into the atmosphere.If the speed of the gas in the pipe is 1.1 m/s,how long must the pipeline be for the sample to decay to 3.0% of its original activity?

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To find the length of the pipeline, we first calculate the time required for the substance to decay to 3.0% of its original activity. Using the half-life formula and the fact that after $n$ half-lives, the fraction remaining is $(1/2)^n$, we can find $n$ such that $(1/2)^n = 0.03$. Solving for $n$, we get:$$n = \frac{\log(0.03)}{\log(0.5)} \approx 5.05$$Since each half-life is 2.03 minutes, the total time for the decay to reach 3.0% is:$$5.05 \times 2.03 \text{ minutes} \approx 10.25 \text{ minutes}$$Converting time to seconds (since 1 minute = 60 seconds):$$10.25 \text{ minutes} \times 60 \text{ seconds/minute} = 615 \text{ seconds}$$Given the speed of the gas is 1.1 m/s, the length of the pipeline ($L$) required is:$$L = \text{speed} \times \text{time} = 1.1 \text{ m/s} \times 615 \text{ s} = 676.5 \text{ m}$$Rounding to the nearest whole number, the length of the pipeline should be approximately 680 meters.

An Air Sample Is Contaminated with 15O,which Has a Half-Life

An Air Sample Is Contaminated with ¹⁵O,which Has a Half-Life