Question 42.33: The air in some caves includes a significant amount of radon......
The air in some caves includes a significant amount of radon gas, which can lead to lung cancer if breathed over a prolonged time. In British caves, the air in the cave with the greatest amount of the gas has an activity per volume of 1.55 × 10^5 Bq/m³. Suppose that you spend two full days exploring (and sleeping in) that cave. Approximately how many ^{222}Rn atoms would you take in and out of your lungs during your two-day stay? The radionuclide ^{222}Rn in radon gas has a half-life of 3.82 days. You need to estimate your lung capacity and average breathing rate.
We note that 3.82 days is 330048 s, and that a becquerel is a disintegration per second (see Section 42-3). From Eq. 34-19, we have
\theta_1=\alpha+\beta \quad \text { and } \quad \beta=\theta_2+\gamma (34-19)
\frac{N}{\mathcal{V}}=\frac{R}{\mathcal{V}} \frac{T_{1 / 2}}{\ln 2}=\left(1.55 \times 10^5 \frac{\mathrm{Bq}}{\mathrm{m}^3}\right) \frac{330048 \mathrm{~s}}{\ln 2}=7.4 \times 10^{10} \frac{\text { atoms }}{\mathrm{m}^3}
where we have divided by volume v. We estimate v (the volume breathed in 48 h = 2880 min) as follows:
\left(2 \frac{\text { liters }}{\text { breath }}\right)\left(\frac{1 \,m ^3}{1000 L }\right)\left(40 \frac{\text { breaths }}{\min }\right)(2880 \,min )
which yields v \approx 200 m ^3 . Thus, the order of magnitude of N is
\left(\frac{N}{ V }\right)( V ) \approx\left(7 \times 10^{10} \frac{\text { atoms }}{ m ^3}\right)\left(200 \,m ^3\right) \approx 1 \times 10^{13} \text { atoms. }