Question

When air pollution is high, ozone \(\left(\mathrm{O}_{3}\right)\) contents can reach 0.60 ppm (i.e., 0.60 mol ozone per million mol air). How many molecules of ozone are present per liter of polluted air if the barometric pressure is \(755 \mathrm{~mm} \mathrm{Hg}\) and the temperature is \(79^{\circ} \mathrm{F}\) ?

Short answer

Answer: There are \(1.46 \times 10^{16}\) ozone molecules in 1 liter of polluted air.

Step-by-step solution

Convert Temperature to Kelvin

To convert the temperature from Fahrenheit to Kelvin, we first need to convert it to Celsius: \(C = \frac{5}{9}(F - 32)\). Then, we can use the formula \(K = C + 273.15\) to convert Celsius to Kelvin.

Convert Pressure to Atm

Next, we need to convert the pressure from mmHg to atm. The formula for this is \(P_\text{atm} = \frac{P_\text{mmHg}}{760}\).

Find Moles of Ozone

To find the number of moles of ozone in 1 liter of air, we first need to find the moles of air using the ideal gas law formula: \(PV = nRT\). Then, we use the given ozone concentration (in ppm) to calculate the moles of ozone: \(n_\text{ozone} = n_\text{air} \times \frac{0.60}{10^6}\).

Find the Number of Ozone Molecules

Finally, we use Avogadro's number (\(N_A\)) to find the number of ozone molecules: \(N_\text{ozone} = n_\text{ozone} \times N_A\). Now, let's apply these steps to the given values:

Convert Temperature to Kelvin

\(C = \frac{5}{9}(79 - 32) = 26^{\circ} \mathrm{C}\). Therefore, \(K = 26 + 273.15 = 299.15 \, \mathrm{K}\).

Convert Pressure to Atm

\(P_\text{atm} = \frac{755}{760} = 0.993 \, \mathrm{atm}\).

Find Moles of Ozone

First, we find the moles of air using the ideal gas law formula: \(PV = nRT \Longrightarrow \frac{0.993\, \text{atm} \times 1\, \text{L}}{0.0821\, \frac{\text{L} \cdot \text{atm}}{\text{mol} \cdot \text{K}} \times 299.15\, \text{K}} = n_\text{air} \Longrightarrow n_\text{air}=0.0404\, \text{mol}\). Now, we can find the moles of ozone: \(n_\text{ozone} = n_\text{air} \times \frac{0.60}{10^6} = 0.0404 \, \text{moles} \times \frac{0.60}{10^6} = 2.424 \times 10^{-8}\, \text{moles}\).

Find the Number of Ozone Molecules

Now, we use Avogadro's number to find the number of ozone molecules: \(N_\text{ozone} = n_\text{ozone} \times N_A = 2.424 \times 10^{-8}\, \text{moles} \times 6.022 \times 10^{23}\, \frac{\text{molecules}}{\text{mol}} = 1.46 \times 10^{16}\, \text{molecules}\). Thus, in 1 liter of polluted air, there are \(1.46 \times 10^{16}\) ozone molecules.