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 ppb \((0.441\) 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 ppb if the atmospheric pressure is 0.67 atm. (b) How many ozone molecules are in \(1.0 \mathrm{~L}\) of air in Mexico City? Assume \(T=25^{\circ} \mathrm{C}\).

Short answer

(a) The partial pressure of ozone in Mexico City is \(2.95 \times 10^{-7}\) atm. (b) There are \(7.33 \times 10^{15}\) ozone molecules in 1.0 L of air in Mexico City.

Step-by-step solution

Calculate the mole fraction of ozone

To calculate the mole fraction of ozone, divide the ozone concentration (in ppb) by 1 billion. Mole fraction of ozone = \(\frac{441}{1,000,000,000} = 4.41 \times 10^{-7}\)

Calculate the partial pressure of ozone

To calculate the partial pressure of ozone, multiply the mole fraction of ozone by the atmospheric pressure. Partial pressure of ozone = Mole fraction of ozone × Atmospheric Pressure = \(4.41 \times 10^{-7} \times 0.67 \ \text{atm}\) Partial pressure of ozone = \(2.95 \times 10^{-7} \ \text{atm}\)

Convert temperature to Kelvin

Since the temperature is given in Celsius, we need to convert it to Kelvin. We convert the temperature from Celsius to Kelvin using the formula \(T_{K} = T_{C} + 273.15\). \(T_{K} = 25^{\circ} \mathrm{C} + 273.15 = 298.15 \ \mathrm{K}\)

Calculate the number of moles of ozone in 1.0 L of air

Now we can use the Ideal Gas Law equation to find the number of moles of ozone. \(PV = nRT\) Rearrange the equation to solve for the number of moles: \(n = \frac{PV}{RT}\) Plug in the values: \(n = \frac{(2.95 \times 10^{-7} \ \text{atm})(1.0 \ \text{L})}{(0.0821 \ \text{L} \cdot \text{atm/mol} \cdot \text{K})(298.15 \ \text{K})}\) \(n = 1.21 \times 10^{-8} \ \text{moles}\)

Calculate the number of ozone molecules in 1.0 L of air

Finally, multiply the number of moles by Avogadro's number to obtain the number of ozone molecules. Number of ozone molecules = Number of moles × Avogadro's number = \(1.21 \times 10^{-8} \ \text{moles} \times 6.022 \times 10^{23} \ \text{molecules/mol}\) Number of ozone molecules = \(7.33 \times 10^{15} \ \text{molecules}\) So, there are \(7.33 \times 10^{15}\) ozone molecules in 1.0 L of air in Mexico City.

Key concepts

Partial Pressure

Partial pressure is an essential concept in gas laws that refers to the pressure exerted by a single type of gas in a mixture of gases. When we talk about the total pressure in a container of gases, each gas contributes to that total pressure based on its mole fraction - this is known as its partial pressure. For example, in the exercise above, the partial pressure of ozone in the atmosphere of Mexico City can be derived using its mole fraction and the atmospheric pressure.

The formula to calculate the partial pressure of a gas is:
\[P_\text{gas} = \text{Mole Fraction} \times \text{Total Pressure}\]
This calculation is significant when analyzing air quality, where it helps us understand the concentration and pressure contribution of pollutants like ozone.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture. It's calculated as the ratio of the moles of one particular gas to the total number of moles of all gases present. Since mole fraction is a ratio, it does not have units.

The mole fraction can be represented by the formula:
\[\chi_i = \frac{n_i}{n_\text{total}}\]
In the solution above, you calculated the mole fraction of ozone by dividing its concentration in parts per billion (ppb) by one billion. This simple yet crucial step is essential for the accurate computation of the partial pressure of any gas in atmospheric studies.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry representing the number of constituent particles, typically atoms or molecules, that are contained in one mole of a substance. This number is approximately 6.022 x 10^23. It enables chemists to count entities at the atomic scale, making it possible to translate between atomic/molecular scale and macroscopic (measurable) quantities.

In the exercise, after determining the number of moles of ozone, Avogadro's number is used to convert that amount into molecules. This step is critical when quantifying the small-scale composition of gases in large volumes and is instrumental for practical applications such as calculating pollutants in the atmosphere.

Ideal Gas Law

The Ideal Gas Law is the cornerstone for much of gas chemistry, connecting the four variables of pressure (P), volume (V), number of moles (n), and temperature (T) in a simple equation:

\[PV = nRT\]
where \(R\) is the ideal gas constant. This equation assumes gases are "ideal," meaning they follow certain assumptions like negligible volume and no interaction between molecules. While no real gas perfectly aligns with these assumptions, the law provides an excellent approximation for many gases under a variety of conditions.

In the original solution, the Ideal Gas Law was used to derive the number of moles of ozone from the partial pressure calculated previously. This illustrates how the relationship helps in practical computations of gas-related problems, aiding in gauging behaviors of gases under specific environmental conditions.