Question

The estimated average concentration of \(\mathrm{NO}_{2}\) in air in the United States in 2006 was \(0.016\) ppm. (a) Calculate the partial pressure of the \(\mathrm{NO}_{2}\) in a sample of this air when the atmospheric pressure is 755 torr \((99.1 \mathrm{kPa}) .\) (b) How many molecules of \(\mathrm{NO}_{2}\) are present under these conditions at \(20^{\circ} \mathrm{C}\) in a room that measures \(15 \times 14 \times 8 \mathrm{ft}\) ?

Short answer

The partial pressure of NO2 in the air is \( 1.208\times10^{-2} \) torr. There are \( 1.904\times 10^{20} \) NO2 molecules in the room measuring \( 15 \times 14 \times 8 \mathrm{ft} \) at \( 20^{\circ} \mathrm{C} \) and 755 torr atmospheric pressure.

Step-by-step solution

(a) Calculate the partial pressure of NO2

To calculate the partial pressure of NO2, we can use the given concentration in ppm and the atmospheric pressure. The concentration in ppm can be understood as a ratio of the number of NO2 molecules to the total number of air molecules. Therefore, we can find the partial pressure by multiplying the atmospheric pressure by the concentration (as a fraction): Partial Pressure of NO2 = Atmospheric Pressure × (Concentration of NO2 / 10^6) Given concentration of NO2 = 0.016 ppm Atmospheric Pressure = 755 torr (99.1 kPa) Partial Pressure of NO2 = 755 torr × (0.016 / 10^6) = \( 1.208\times10^{-2} \) torr

(b) Find the number of NO2 molecules in the room

To find the number of NO2 molecules in the room, first, we'll use the Ideal Gas Law to determine the number of moles of NO2 present. Then, we will multiply the number of moles by Avogadro's number to obtain the number of molecules. Ideal Gas Law: PV = nRT Where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature in Kelvin. Given room dimensions, Volume (V) = 15ft x 14ft x 8ft = 1680 ft³ = 47.60 m³ (converting ft³ to m³ by multiplying by 0.0283168) Technically room temperature is 20°C, so, T = 293.15 K (converting to Kelvin by adding 273.15) We already have the partial pressure of NO2 (P) in torr, so we first convert it to kPa by multiplying with the conversion factor (1 torr = 0.133322 kPa): Partial Pressure of NO2 = \( 1.208\times10^{-2} \) torr × 0.133322 = \( 1.613\times10^{-3} \) kPa Now, we can solve for the number of moles (n) using the Ideal Gas Law: n = PV / RT n = \((1.613\times10^{-3} \mathrm{kPa})(47.60 \mathrm{m^3}) / ((8.314 \mathrm{J/(mol.K)})(293.15 \mathrm{K})\) = 3.16 × 10^-4 mol Lastly, we'll multiply the moles by Avogadro's number to determine the number of NO2 molecules: Number of NO2 molecules = \(3.16 \times 10^{-4} \mathrm{mol} \) × \( 6.022 \times 10^{23} \mathrm{molecules/mol} \) = \( 1.904\times 10^{20} \) NO2 molecules