Question

An activity of 20 picocuries \(\left(20 \times 10^{-12} \mathrm{Ci}\right)\) of radon- 222 per liter of air in a house constitutes a health hazard to anyone living there. The half-life of radon-222 is 3.82 days. Calculate the concentration of radon in air (moles per liter) that corresponds to a 20 -picocurie activity level.

Short answer

Answer: To calculate the concentration of radon-222 in air, you will need to first convert its activity from picocuries to decays per second, then find its decay constant using the given half-life, find the number of atoms, and finally convert atoms to moles to arrive at the concentration. Following these steps, you can determine the concentration of radon-222 in moles per liter.

Step-by-step solution

Convert activity from picocuries to decays per second

We are given the activity of radon-222 as 20 picocuries. We will first convert this value from picocuries to decays per second. Since 1 curie is equal to 3.7 x 10^10 decays per second, we multiply the given activity by the conversion factor to obtain the decays per second: Activity (decays/s) = \((20 \times 10^{-12}\,\mathrm{Ci}) \times (3.7 \times 10^{10}\,\mathrm{decays/s/Ci})\)

Calculate the decay constant

To find the decay constant, we'll use the half-life formula: \(T_{1/2} = \frac{ln(2)}{\lambda}\) Rearranging the formula to solve for the decay constant: \(\lambda = \frac{ln(2)}{T_{1/2}}\) Replacing the given value of half-life: \(\lambda = \frac{ln(2)}{3.82\,\mathrm{days}}\) We'll need to convert the decay constant to seconds: \(\lambda = \frac{ln(2)}{3.82 \times 24 \times 3600\,\mathrm{s}}\)

Find the number of moles of radon-222

Now we can use the decay constant to find the number of moles of radon-222. The activity (A) is given by: \(A = \lambda N\) Where A is the activity in decays per second, λ is the decay constant, and N is the number of atoms of radon-222. We can rearrange the formula to solve for N: \(N = \frac{A}{\lambda}\) Substituting the given values: \(N = \frac{(20 \times 10^{-12}\,\mathrm{Ci}) \times (3.7 \times 10^{10}\,\mathrm{decays/s/Ci})}{\frac{ln(2)}{3.82 \times 24 \times 3600\,\mathrm{s}}}\)

Calculate the concentration of radon-222 in air

Now that we have the number of atoms (N), we can convert it to moles to find the concentration. We do this by dividing N by Avogadro's number (NA), which is 6.022 x 10^23 atoms per mole: \(n = \frac{N}{N_A}\) Replacing the given values: \(n = \frac{(20 \times 10^{-12}\,\mathrm{Ci}) \times (3.7 \times 10^{10}\,\mathrm{decays/s/Ci})}{(6.022\times10^{23}\,\frac{\mathrm{atoms}}{\mathrm{mol}})(\frac{ln(2)}{3.82 \times 24 \times 3600\,\mathrm{s}})}\) Finally, since the concentration is in moles per liter, we can assume that the activity is for 1 liter of air. So, the concentration of radon-222 in moles per liter is equal to the number of moles (n): Concentration (moles/L) = \(n\) Following these steps, you will be able to calculate the concentration of radon-222 in air that corresponds to a 20-picocurie activity level.