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Chapter 19: Problem 68

The principle behind the home smoke detector is described on page 516. Americium- 241 is present in such detectors. It has a decay constant of \(1.51 \times 10^{-3} \mathrm{y}^{-1}\). You are urged to check the battery in the detector at least once a year. You are, however, never encouraged to check how much Am-241 remains undecayed. Explain why.

Short Answer

Expert verified
One is encouraged to check the battery in a smoke detector because the battery's lifespan is much shorter than the half-life of Am-241, which is approximately 459 years. This means that the Am-241 remains effective for a long time and does not lose its functionality significantly over a human's typical lifespan. Therefore, it is important to check the battery annually to ensure the proper functionality of the smoke detector, while there is no need to check the remaining amount of Am-241 due to its long half-life.

Step by step solution

01

Find the half-life of Am-241

To find the half-life of Am-241, we can use the decay constant given in the problem: \(\lambda = 1.51 \times 10^{-3} \mathrm{y}^{-1}\). The relation between half-life, T, and decay constant is given by the formula: \(T = \frac{\ln 2}{\lambda}\). Substituting the given decay constant, we can calculate the half-life of Am-241. \(T = \frac{\ln 2}{1.51 \times 10^{-3}} = \frac{0.693}{1.51 \times 10^{-3}} \approx 459 \,\text{years}\)
02

Compare the half-life to the advised checking frequency

The half-life of Am-241 is approximately 459 years, which is significantly larger than the recommended checking frequency for the battery, which is once a year. This means that the Am-241 material in the smoke detector will not decay substantially over a human's typical lifespan.
03

Discuss the implications of a large half-life and decay constant

The activity of Am-241 is governed by the decay constant, and since it has a small decay constant and a large half-life, that means it will remain effective for a long time. Since it doesn't decay substantially, there would be no need to check how much Am-241 remains undecayed in the smoke detector regularly.
04

Concluding remarks

In conclusion, the reason one is not encouraged to check the remaining amount of Am-241 is that it has a long half-life, meaning it doesn't lose its effectiveness significantly over time. Instead, the battery of the home smoke detector is more likely to run out before the Am-241 becomes ineffective, so checking the battery annually is more important to ensure the proper functionality of the smoke detector.

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