Question

Lead ( \(\mathrm{Pb})\) is one of the principal pollutants that the Clean Air Act (40 CFR part 50 ) requires the Environmental Protection Agency (EPA) to list in the National Ambient Air Quality Standards (NAAQS). The level of \(\mathrm{Pb}\) in ambient air set by the NAAQS is $0.15 \mu \mathrm{g} / \mathrm{m}^{3}$ for an average time of three months to protect public health and the environment. Beyond that level, Pb is considered harmful to public health and the environment. If the level of \(\mathrm{Pb}\) found in the ambient air of a residential place at \(20^{\circ} \mathrm{C}\) is $0.005 \mathrm{~mol} / \mathrm{kmol}$ for over a period of three months, determine whether the level of \(\mathrm{Pb}\) is harmful. The molar mass of \(\mathrm{Pb}\) is $207.2 \mathrm{~kg} / \mathrm{kmol} .$

Short answer

Answer: Yes, the calculated concentration of Pb in the ambient air (426.41 μg/m³) exceeds the NAAQS standard (0.15 μg/m³), which indicates that the level of Pb in this residential place is harmful to public health and the environment.

Step-by-step solution

Convert mol/kmol to kg/kmol

We are given the amount of lead (Pb) in mol/kmol, and we want to convert it to kg/kmol. The molar mass of lead is 207.2 kg/kmol. So, given the amount of lead in mol/kmol, its mass in kg/kmol can be calculated as follows: Mass of Pb = (Amount of Pb in mol/kmol) × (Molar mass of Pb) Mass of Pb = (0.005 mol/kmol) × (207.2 kg/kmol) = 1.036 kg/kmol

Convert kg/kmol to μg/m³ using the ideal gas law

Using the ideal gas law, we can convert the mass of Pb (in kg/kmol) into a concentration (in μg/m³). The ideal gas law states: \(PV = nRT\) We know the temperature of the ambient air is \(20^{\circ}\mathrm{C}\) or \(293.15 \mathrm{K}\). Also, the universal gas constant \(R\) can be given as \(8.314 \mathrm{J / (mol·K)}\). We want to find the value of the gas concentration, which is the lead's concentration in the ambient air. First, we'll convert the mass of Pb into moles. Let's denote the gas concentration in μg/m³ as \([\mathrm{Pb}]\): \(n = \frac{[\mathrm{Pb}] \times \text{Volume}}{\text{Molar mass of Pb}}\) Now, we'll substitute this into the ideal gas law: \(P \times V = \frac{[\mathrm{Pb}] \times V}{207.2 \times 10^3\,\mathrm{g}} \times RT\) Solving for \([\mathrm{Pb}]\): \([\mathrm{Pb}] = \frac{P \times 207.2 \times 10^3\,\mathrm{g} }{RT}\) We can convert pressure from atm to Pa by using the relationship 1 atm = 101325 Pa, and the given mass of Pb in kg/kmol (1.036 kg/kmol) to μg, we get: \([\mathrm{Pb}] = \frac{101325 \,\mathrm{Pa} \times 1.036 \times 10^6\, \mu\mathrm{g}}{8.314 \mathrm{J / (mol·K)} \times 293.15\,\mathrm{K}}\) \([\mathrm{Pb}] ≈ 426.41\, \mu\mathrm{g/m³}\)

Compare the calculated value to the NAAQS standard

Finally, we will compare the calculated concentration of Pb in the ambient air, which is 426.41 μg/m³, with the value set by the NAAQS, which is 0.15 μg/m³: 426.41 μg/m³ > 0.15 μg/m³ As the calculated value of Pb in the ambient air (426.41 μg/m³) exceeds the NAAQS standard (0.15 μg/m³), the level of Pb in this residential place is considered harmful to public health and the environment.