The estimated average concentration of \(\mathrm{NO}_{2}\) in air in the United States in 2015 was 0.010 ppm. (a) Calculate the partial pressure of the \(\mathrm{NO}_{2}\) in a sample of this air when the atmospheric pressure is \(101 \mathrm{kPa} .(\mathbf{b})\) How many molecules of \(\mathrm{NO}_{2}\) are present under these conditions at \(25^{\circ} \mathrm{C}\) in a room that measures \(10 \mathrm{~m} \times 8 \mathrm{~m} \times 2.50 \mathrm{~m} ?\)

Since we are given the concentration of \(\mathrm{NO}_2\) in the air (0.010 ppm) and the atmospheric pressure (101 kPa), we can use the following formula to find the partial pressure of \(\mathrm{NO}_2\): Partial pressure of \(\mathrm{NO_2}\) = (Concentration of \(\mathrm{NO_2}\)) × (Total atmospheric pressure) First, we need to convert the concentration of \(\mathrm{NO}_2\) from ppm (parts per million) to a ratio: \(0.010 \ ppm = 0.010 \times 10^{-6}\) Now, we can find the partial pressure of \(\mathrm{NO}_2\): Partial pressure of \(\mathrm{NO_2}\) = \( 0.010 \times 10^{-6} \times 101\ \mathrm{kPa}\) Partial pressure of \(\mathrm{NO_2}\) = \(\mathrm{1.01 \times 10^{-3} \ kPa}\)

To calculate the number of molecules of \(\mathrm{NO}_2\) in the room, we need to find the number of moles of \(\mathrm{NO}_2\) present and then convert that to the number of molecules using Avogadro's number. We can start by finding the volume of the room: Volume = length × width × height = \(10\ \mathrm{m} \times 8\ \mathrm{m} \times 2.50\ \mathrm{m} = 200\ \mathrm{m^3}\) Next, we need to convert the volume to liters: \(200\ \mathrm{m^3} \times \frac{1000\ \mathrm{L}}{1\ \mathrm{m^3}} = 200,000\ \mathrm{L}\) Now, we can use the ideal gas law formula to find the number of moles of \(\mathrm{NO_2}\) present in the room: \(PV = nRT\) We will first isolate n (moles) in the equation: \(n = \frac{PV}{RT}\) Where: P = Partial pressure of \(\mathrm{NO_2}\) = \(\mathrm{1.01 \times 10^{-3} \ kPa}\) V = Volume of the room = \(200,000\ \mathrm{L}\) R = Ideal gas constant = \(\mathrm{8.314\ \frac{J}{mol \cdot K}}\) (we will need to convert kPa to J/L) T = Temperature = \(25^{\circ}\mathrm{C} = 298.15\ \mathrm{K}\) First, let's convert the pressure in kPa to J/L: \(\mathrm{1.01 \times 10^{-3} \frac{kPa}{L}} \times \frac{1000\ \mathrm{J}}{1\ \mathrm{kPa \cdot L}} = 1.01 \times 10^{-3} \ \frac{J}{L}\) Now we can substitute the known values into the formula: \(n = \frac{(1.01 \times 10^{-3}\ \mathrm{J/L})(200,000\ \mathrm{L})}{(8.314\ \mathrm{J/mol \cdot K})(298.15\ \mathrm{K})}\) Solving for n: \(n \approx 0.0812\ \mathrm{moles}\) Now, we can use Avogadro's number to convert moles to molecules: Number of molecules = (moles) × (Avogadro's number) = \(0.0812\ \mathrm{moles} \times 6.022 \times 10^{23}\ \mathrm{molecules/mol}\) Number of molecules of \(\mathrm{NO_2}\) = \(\mathrm{\approx 4.88 \times 10^{22} \ molecules}\)