The most significant source of natural radiation is radon-222. \(^{222} \mathrm{Rn},\) a decay product of \(^{238} \mathrm{U},\) is continuously generated in the earth's crust, allowing gaseous Rn to seep into the basements of buildings. Because \(^{222} \mathrm{Rn}\) is an \(\alpha\) -particle producer with a relatively short half-life of 3.82 days, it can cause biological damage when inhaled. a. How many \(\alpha\) particles and \(\beta\) particles are produced when \(^{238} \mathrm{U}\) decays to \(^{222} \mathrm{Rn} ?\) What nuclei are produced when \(^{222} \mathrm{Rn}\) decays? b. Radon is a noble gas so one would expect it to pass through the body quickly. Why is there a concern over inhaling \(^{222} \mathrm{Rn} ?\) c. Another problem associated with \(^{222} \mathrm{Rn}\) is that the decay of \(^{222} \mathrm{Rn}\) produces a more potent \(\alpha\) -particle producer \(\left(t_{1 / 2}=\right.\) 3.11 min) that is a solid. What is the identity of the solid? Give the balanced equation of this species decaying by \(\alpha\) particle production. Why is the solid a more potent \(\alpha\) -particle producer? d. The U.S. Environmental Protection Agency (EPA) recommends that \(^{222}\) Rn levels not exceed 4 pCi per liter of air (1 \(\mathrm{Ci}=1\) curie \(=3.7 \times 10^{10}\) decay events per second; \(1 \mathrm{pCi}=1 \times 10^{-12} \mathrm{Ci}\). Convert \(4.0 \mathrm{pCi}\) per liter of air into concentrations units of \(^{222} \mathrm{Rn}\) atoms per liter of air and moles of \(^{222}\) Rn per liter of air.

Step by step solution

01

Finding α and β particles for U to Rn decay

In order to determine the number of α and β particles produced when U-238 decays to Rn-222, we will use the difference of both mass numbers and atomic numbers of Uranium and Radon. U-238: - Mass number: A = 238 - Atomic number: Z = 92 Rn-222: - Mass number: A = 222 - Atomic number: Z = 86 Since α particles have a mass number of 4 (two protons and two neutrons) and a charge of 2 (two protons), we can find the number of α particles: Number of α particles = (238 - 222) / 4 = 16 / 4 = 4 Now we will find the difference in atomic numbers: Number of β particles = 92 - 86 - (2 * Number of α particles) = 92 - 86 - 8 = 6 There are 4 α particles and 6 β particles produced when U-238 decays to Rn-222.

02

Decay of Rn to other nuclei

When Rn-222 decays, it undergoes alpha decay since it is an alpha-particle producer: \( ^{222}Rn -> ^{218}Po + ^4He \) Polonium-218 is the other nucleus produced in this decay process.

03

Reason for radon-222 inhalation concern

Even though radon-222 is a noble gas that can pass through the body quickly, it can cause biological damage because it is an alpha-particle producer with a short half-life of 3.82 days. Inhalation exposes lung tissue to ionizing radiation contained in Radon's alpha particles, increasing the risk of lung cancer.

04

Identifying the solid and its decay equation

The solid produced when Rn-222 decays is Polonium-218 (^{218}Po). Po-218 has a half-life of 3.11 minutes and undergoes alpha decay: \( ^{218}Po -> ^{214}Pb + ^4He \) Lead-214 gets produced after the decay of Po-218. The reason why Polonium-218 is a more potent α-particle producer is its short half-life, causing it to decay at a higher rate and produce more alpha particles in a given time period.

05

Converting pCi to concentration units

We need to convert the given concentration of 4.0 pCi per liter of air to number of Rn-222 atoms per liter and moles of Rn-222 per liter. First, we need to convert pCi to decays per second: \( 4.0 \times 10^{-12} Ci \times \frac{3.7 \times 10^{10} decays}{1 Ci} = 148 \, decays/s \) Now, we can use the Avogadro's number to convert decays per second to moles per liter and atoms per liter: Atoms per liter = \( \frac{148 \, decays/s \times 3600 \, s}{1 \, hr} \times 6.022 \times 10^{23} atom/mol \) Atoms per liter = \( 3.22 \times 10^6 \, atoms/L \) Moles per liter = \( \frac{3.22 \times 10^6 atoms/L}{6.022 \times 10^{23}atoms/mol} \) Moles per liter = \( 5.35 \times 10^{-18} \, mol/L \)

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