Problem 13 - Chemistry: The Central Science in SI Units, Global Edition

Question

The Environmental Protection Agency (EPA) has established air quality standards. For ozone \(\left(\mathrm{O}_{3}\right),\) the 8 -hour average concentration permitted under the standards is 0.085 parts per million (ppm). (a) Calculate the partial pressure of ozone at 0.085 ppm if the atmospheric pressure is \(100 \mathrm{kPa}\). (b) How many ozone molecules are in \(1.0 \mathrm{~L}\) of air? Assume \(T=25^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

(a) The partial pressure of ozone at 0.085 ppm is \(8.5 \times 10^{-6}\) kPa. (b) There are approximately \(2.05 \times 10^{15}\) ozone molecules in 1.0 L of air at 25°C.

Step 1: (a) Calculate the partial pressure of ozone at 0.085 ppm

To calculate the partial pressure of ozone at 0.085 ppm, we need to find the concentration of ozone in parts per million and convert it to partial pressure by the assumption that ozone in air behaves as an ideal gas. 1. Convert the concentration of ozone in parts per million to a fraction: \[0.085 \, \text{ppm} = \frac{0.085}{10^6}\] 2. Multiply the atmospheric pressure by the fraction of ozone concentration to find the partial pressure of ozone: \[P_\text{O3} = (100 \, \text{kPa}) \times \frac{0.085}{10^6}\]

Step 2: Calculation of the partial pressure

Calculate the partial pressure of ozone (O3): \[P_\text{O3} =(100 \, \text{kPa}) \times \frac{0.085}{10^6} = 8.5 \times 10^{-6} \, \text{kPa}\]

Step 3: (b) Calculate the number of ozone molecules in 1.0 L of air

To find the number of ozone molecules in 1.0 L of air, we will first find the number of moles of O3 using the ideal gas law, then convert moles of O3 to molecules using Avogadro's number. 1. Convert the partial pressure of ozone to pascals: \[P_\text{O3} = (8.5 \times 10^{-6} \, \text{kPa})\times (10^3 \, \text{Pa/kPa}) = 8.5 \times 10^{-3} \, \text{Pa}\] 2. Convert the volume of air to cubic meters: \[V = 1.0\, \text{L} \times \left( \frac{1 \, \text{m}^3}{10^3\, \text{L}}\right) = 1.0 \times 10^{-3} \, \text{m}^3\] 3. Convert the temperature to Kelvin: \[T = 25 \, \text{°C} + 273.15 = 298.15 \, \text{K}\] 4. Use the ideal gas law to find the number of moles of ozone: \[PV=nRT \Rightarrow n=\frac{PV}{RT}\] 5. Substitute the known values to find the number of moles of ozone: \[n=\frac{(8.5\times10^{-3}\, \text{Pa})(1.0\times10^{-3}\, \text{m}^3)}{(8.314\, \text{J/mol K})(298.15\, \text{K})}\] 6. Convert the number of moles of ozone to molecules using Avogadro's number: \[N_\text{ozone} = n\left(6.022\times10^{23}\, \text{molecules/mol}\right)\]

Step 4: Calculation of the number of ozone molecules

Calculate the number of ozone molecules in 1.0 L of air: \[N_\text{ozone} = \frac{(8.5\times10^{-3}\, \text{Pa})(1.0\times10^{-3}\, \text{m}^3)}{(8.314\, \text{J/mol K})(298.15\, \text{K})}(6.022\times10^{23}\, \text{molecules/mol}) \approx 2.05 \times 10^{15} \, \text{ozone molecules}\]

Key Concepts

Ideal Gas Law

The ideal gas law is a fundamental equation used to describe the behavior of gases. It is often represented as \( PV = nRT \), where:

• \( P \) is the pressure of the gas in units like Pascals or kiloPascals.
• \( V \) is the volume of the gas in cubic meters.
• \( n \) is the number of moles of the gas.
• \( R \) is the universal gas constant, approximately \( 8.314 \text{ J/mol K} \).
• \( T \) is the temperature in Kelvin.

The ideal gas law assumes that gases composed of randomly moving particles are perfect, meaning they adhere to this law without deviation.
Even though real gases can show slight deviations from ideal behavior, this law is an excellent approximation for many applications, particularly when gases are at high temperatures and low pressures. In our exercise, this law helps determine the number of moles of ozone in a given volume of air.
By rearranging the formula \( PV = nRT \) to \( n = \frac{PV}{RT} \), we can solve for \( n \), the number of moles. Once the number of moles is determined, other factors like particle number and partial pressure are easily found.

Ozone Concentration

Ozone concentration is crucial for understanding air quality. The Environmental Protection Agency (EPA) monitors ozone levels due to their impact on health and the environment. Ozone concentration is often measured in parts per million (ppm), which tells us how much ozone is present in a million parts of air. For instance, 0.085 ppm of ozone means there are 0.085 parts of ozone per one million parts of air.
To find the partial pressure of ozone in the atmosphere, it is crucial to convert ozone levels from ppm to a fraction. This fraction can then be used to calculate the partial pressure by multiplying it by the total atmospheric pressure.
In the given exercise, the concentration of 0.085 ppm is converted into a partial pressure by using the fraction \( \frac{0.085}{10^6} \) and multiplying it by the atmospheric pressure (e.g., 100 kPa). This conversion allows for understanding ozone's behavior in the air when treating it as an ideal gas.

Avogadro's Number

Avogadro's number is a constant that defines the number of particles, like atoms or molecules, found in one mole of a substance. It is approximately \( 6.022 \times 10^{23} \text{ molecules/mol} \). This number is vital for converting between the macroscopic scale of moles and the microscopic scale of individual molecules or atoms.
In our specific problem, Avogadro's number helps convert the number of moles of ozone calculated using the ideal gas law, into the actual number of ozone molecules in a given volume of air. Essentially, once the number of moles \( n \) is known, multiplying it by Avogadro's number \( N_A \) provides the total number of molecules:
\[ N = n \times N_A \]
This conversion is important for understanding the real presence of ozone molecules in varying atmospheric conditions and is a key step in assessing air quality and adhering to environmental standards.