Problem 38 - Chemistry The Central Science In Si Units

Question

(a) If the pressure exerted by ozone, \(\mathrm{O}_{3}\), in the stratosphere is \(304 \mathrm{~Pa}\) and the temperature is \(250 \mathrm{~K}\), how many ozone molecules are in a liter? (b) Carbon dioxide makes up approximately \(0.04 \%\) of Earth's atmosphere. If you collect a \(2.0-\mathrm{L}\) sample from the atmosphere at sea level ( \(101.33 \mathrm{kPa}\) ) on a warm day \(\left(27^{\circ} \mathrm{C}\right),\) how many \(\mathrm{CO}_{2}\) molecules are in your sample?

Step-by-Step Solution

Answer

(a) Approximately \(8.79 \times 10^{19}\) ozone molecules, (b) \(1.95 \times 10^{19}\) \(CO_2\) molecules.

Step 1: Convert Pressure to Atmospheres (Part A)

Given the pressure of ozone is 304 Pa, convert this to atmospheres using the conversion factor: 1 atm = 101325 Pa. \[P = \frac{304 \text{ Pa}}{101325 \text{ Pa/atm}} = 0.003 \text{ atm}\]

Step 2: Use Ideal Gas Law (Part A)

Using the Ideal Gas Law \(PV = nRT\), solve for \(n\), the number of moles. Given: \(P = 0.003 \text{ atm}, V = 1 \text{ L}, R = 0.0821 \text{ L}\cdot\text{atm/mol}\cdot\text{K}, T = 250 \text{ K}\).\[n = \frac{PV}{RT} = \frac{0.003 \times 1}{0.0821 \times 250} \approx 1.46 \times 10^{-4} \text{ moles}\]

Step 3: Calculate Number of Molecules (Part A)

Convert moles to molecules using Avogadro's number \(6.022 \times 10^{23} \text{ molecules/mol}\).\[\text{Number of molecules} = 1.46 \times 10^{-4} \times 6.022 \times 10^{23} \approx 8.79 \times 10^{19} \text{ molecules}\]

Step 4: Convert Temperature to Kelvin (Part B)

Convert the temperature from Celsius to Kelvin. Given \(T = 27^{\circ} \text{C}\), the conversion is \(T = 27 + 273 = 300 \text{ K}\).

Step 5: Calculate Moles of Air Sample (Part B)

Using the Ideal Gas Law \(PV = nRT\), solve for \(n\), the number of moles for the entire air sample: \(P = 101.33 \text{ kPa (or 0.999 atm)}\), \(V = 2.0 \text{ L}\), \(T = 300 \text{ K}\), \(R = 0.0821 \text{ L atm/mol K}\).\[n = \frac{0.999 \times 2.0}{0.0821 \times 300} \approx 0.081 \text{ moles}\]

Step 6: Calculate Moles of \(CO_2\) (Part B)

Since \(CO_2\) makes up 0.04% of the atmosphere, calculate the moles of \(CO_2\): \[n_{CO_2} = 0.0004 \times 0.081 \approx 3.24 \times 10^{-5} \text{ moles}\]

Step 7: Calculate Number of \(CO_2\) Molecules (Part B)

Convert moles of \(CO_2\) to molecules using Avogadro's number:\[\text{Number of molecules} = 3.24 \times 10^{-5} \times 6.022 \times 10^{23} \approx 1.95 \times 10^{19} \text{ molecules}\]

Key Concepts

Ozone

Ozone, scientifically represented as \(\text{O}_3\), is a molecule composed of three oxygen atoms. It plays a crucial role in Earth's stratosphere by absorbing most of the Sun's harmful ultraviolet radiation. In the context of the Ideal Gas Law, which is crucial for understanding the behavior of gases, ozone can be described by parameters such as pressure, volume, and temperature. In our example, we calculated the number of ozone molecules in a liter given a specific pressure and temperature using the Ideal Gas Law equation: \(PV = nRT\).
This allowed us to first determine the amount in moles before using Avogadro's number to find the total number of molecules. Understanding ozone is not only important for chemical calculations but also for environmental science and protecting life on Earth.

Carbon Dioxide

Carbon dioxide, or \(\text{CO}_2\), is a trace gas in Earth's atmosphere that is vital for life and climate. It's produced by both natural and human activities, such as respiration and burning fossil fuels. In the provided exercise, \(\text{CO}_2\) makes up approximately 0.04% of Earth's atmosphere, which impacts the calculation of how many molecules are in a given volume of air.

• First, we calculated the total number of moles for the 2.0-liter air sample using the Ideal Gas Law.
• Next, we discerned the tiny fraction of \(\text{CO}_2\) within this sample since it is just a small component of the air.
• This fractional concentration was then multiplied by the total number of moles to get \(\text{CO}_2\) moles.

Finally, Avogadro's number helped convert these moles to molecules, providing a clear picture of its representation in volume. Understanding \(\text{CO}_2\) interactions and quantities is crucial for studying atmospheric sciences and global environmental changes.

Avogadro's Number

Avogadro's number, \(6.022 \times 10^{23}\), is the key to bridging the microscopic world of molecules and atoms with the macroscopic world we experience every day. It tells us how many molecules are in one mole of a substance, acting like a count in much the same way a dozen refers to 12 items. In chemistry, it's crucial for converting between the amount of substance and its molecular content.In our exercises, once we calculated the moles of ozone and carbon dioxide, Avogadro's number allowed us to find the precise number of molecules within a given volume. It provided a tangible understanding of gas quantities based on moles. It not only supports the calculations we performed but also enhances the comprehension of how vast and numerous molecules are in real-world samples.Key points about Avogadro's number:

• Essential for chemical quantification.
• Connects the microscopic atomic scale to more observable scales.
• Allows calculations from moles to discrete molecules.