Problem 13 - Chemistry: The Central Science

Question

Air pollution in the Mexico City metropolitan area is among the worst in the world. The concentration of ozone in Mexico City has been measured at \(441 \mathrm{ppb}(0.441 \mathrm{ppm})\). Mexico City sits at an altitude of 7400 feet, which means its atmospheric pressure is only \(0.67\) atm. (a) Calculate the partial pressure of ozone at \(441 \mathrm{Ppb}\) if the atmospheric pressure is \(0.67 \mathrm{~atm}\). (b) How many ozone molecules are in \(1.0 \mathrm{~L}\) of air in Mexico City? Assume \(T=25^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

The partial pressure of ozone is found to be approximately \(2.95 * 10^{-7}\) atm, using the formula for partial pressure and the given data. By utilizing the ideal gas law and converting the given temperature to Kelvin, we can determine that there are approximately 1.20 x \(10^{-8}\) moles of ozone in 1.0 L of air. Using Avogadro's number, this translates to approximately 7.22 x \(10^{15}\) molecules of ozone in 1.0 L of air in Mexico City.

Step 1: Calculate the partial pressure of ozone

To calculate the partial pressure of ozone, we need to use the given concentration of ozone (in ppm) and the total atmospheric pressure. The formula to find the partial pressure (in atm) is: Partial pressure of ozone = (Concentration of ozone / total concentration) * Total atmospheric pressure where the total concentration of gases is assumed to be in parts per million (ppm) and is equal to \(10^6\).

Step 2: Insert the given values

Insert the given values from the exercise into the formula: Partial pressure of ozone = (0.441 ppm / \(10^6\) ppm) * 0.67 atm Now we can calculate the partial pressure of ozone: Partial pressure of ozone = \(|\frac{0.441}{10^6}|\) * 0.67 atm ≈ \(2.95 * 10^{-7}\) atm

Step 3: Calculate the number of ozone molecules per liter using the ideal gas law

To calculate the number of ozone molecules per liter, we will use the ideal gas law given below: PV = nRT where P is the partial pressure of ozone, V is the volume (1.0 L in this case), n is the number of moles of ozone, R is the ideal gas constant (0.08206 L atm/mol K for this problem), and T is the temperature in Kelvin. We'll first convert the Celsius temperature to Kelvin: T = 25°C + 273.15 = 298.15 K Now we will rearrange the ideal gas law to solve for the number of moles of ozone, n: n = \(|\frac{PV}{RT}|\)

Step 4: Insert values into the rearranged ideal gas law formula

Insert the values into the formula: n = \(|\frac{(2.95 * 10^{-7} \, \text{atm})(1.0 \, \text{L})}{(0.08206 \, \text{L}\, \text{atm/mol K})(298.15 \, \text{K})}|\) Now we can calculate the number of moles of ozone in 1.0 L of air: n ≈ 1.20 x 10^{-8} moles

Step 5: Convert moles to molecules

To find the number of ozone molecules, we will use Avogadro's number (6.022 x \(10^{23}\) molecules/mol): Number of ozone molecules = Number of moles * Avogadro's number Number of ozone molecules = (1.20 x \(10^{-8}\) moles) * (6.022 x \(10^{23}\) molecules/mol) ≈ 7.22 x \(10^{15}\) molecules The number of ozone molecules in 1.0 L of air in Mexico City is approximately 7.22 x \(10^{15}\) molecules.

Key Concepts

Ideal Gas Law

When discussing gases, the ideal gas law is a super useful equation. It helps us understand the relationship between pressure, volume, temperature, and number of moles for a given amount of gas.
In mathematical terms, the ideal gas law is expressed as:

• \[ PV = nRT \]
• where:
• \( P \) is the pressure of the gas in atmospheres (atm),
• \( V \) is the volume in liters (L),
• \( n \) is the number of moles of the gas,
• \( R \) is the ideal gas constant. For our purpose, it's usually \( 0.08206 \, \text{L atm/mol K} \),
• \( T \) is the temperature in Kelvin (K).

We learned how to rearrange this formula in our original problem to find the number of moles. By rearranging, we isolated \( n \):

• \[ n = \frac{PV}{RT} \]
• This rearrangement helps us find out how many moles of ozone were present under the given conditions of pressure and temperature.

Remember, always convert any temperature in Celsius to Kelvin by adding \( 273.15 \) to your Celsius temperature. This ensures our values work correctly in the equation.

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of the atmosphere on Earth. When we calculate pressures, especially in the context of gases like ozone, understanding how atmospheric pressure plays a role is crucial.

• Atmospheric pressure varies with altitude. In Mexico City, it is only 0.67 atm due to its high altitude at 7400 feet above sea level.
• Compared to the standard atmospheric pressure (1 atm), it's lower because the air is thinner at higher elevations owing to decreased gravitational pull.
• This difference directly impacts the concentration and behavior of gases within a given volume of air.

When calculating the partial pressure of ozone in the original exercise, we used the atmospheric pressure specific to Mexico City. This was necessary to accurately compute how much of the total atmospheric pressure was due to ozone.

Avogadro's Number

Avogadro's number is a big deal when you want to know the count of molecules. It tells us how many molecules are in one mole of a substance. To be precise:

• Avogadro's number is approximately \( 6.022 \times 10^{23} \) molecules per mole.

In our original exercise, after calculating the number of moles of ozone using the ideal gas law, we needed to find out how many individual molecules that corresponded to.

• We multiplied the number of moles by Avogadro's number to find the actual molecular count.
• This process helped us determine there were about \( 7.22 \times 10^{15} \) ozone molecules in a liter of air in Mexico City.

Using Avogadro’s number is essential for converting between the moles of a substance and the actual count of particles within a sample.