Problem 40 - Chemistry The Central Science

Question

(a) If the pressure exerted by ozone, \(\mathrm{O}_{3}\), in the stratosphere is \(3.0 \times 10^{-3}\) atm 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 \((1.00\) atm \()\) 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) The number of ozone molecules in a 1 liter volume under the given conditions is \(8.79 \times 10^{19}\) O3 molecules. (b) The number of CO2 molecules in the 2.0 L sample collected from Earth's atmosphere is \(1.96 \times 10^{19}\) CO2 molecules.

Step 1: Use the Ideal Gas Law formula for part (a)

The Ideal Gas Law is given by the formula: \(PV=nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the amount of gas in moles, \(R\) is the universal gas constant, and \(T\) is temperature. Now, let's plug in the values given in the problem to find the number of moles of O3 in a 1-liter volume. Given, \(P = 3.0 \times 10^{-3} \mathrm{atm}\), \(V = 1.0 \mathrm{L}\), \(T = 250 \mathrm{K}\), and \(R = 0.0821 \frac{\mathrm{L.atm}}{\mathrm{mol.K}}\) Rearrange the formula to find "n": \( n = \frac{PV}{RT} \)

Step 2: Calculate the number of moles for part (a)

Plug in the values in the formula: \( n_\mathrm{O_3} = \frac{(3.0 \times 10^{-3}\mathrm{atm})(1.0\mathrm{L})}{(0.0821\frac{\mathrm{L.atm}}{\mathrm{mol.K}})(250\mathrm{K})} \) Calculate the number of moles: \(n_\mathrm{O_3} = 1.46 \times 10^{-4} \: \mathrm{mol}\)

Step 3: Calculate the number of ozone molecules for part (a)

We'll use Avogadro's number to convert moles to molecules. The Avogadro's number is approximately \(6.022 \times 10^{23}\) molecules per mole. Number of O3 molecules = \(n_\mathrm{O_3} \times N_\mathrm{A}\) where \(N_\mathrm{A}\) is Avogadro's number. Number of O3 molecules = \( (1.46 \times 10^{-4}\: \mathrm{mol}) \times (6.022 \times 10^{23} \: \mathrm{molecules/mol}) \) Calculate the number of O3 molecules: Number of O3 molecules = \(8.79 \times 10^{19}\) O3 molecules

Step 4: Use the Ideal Gas Law formula for part (b)

To find the number of CO2 molecules in the 2.0 L sample, first calculate the number of moles of CO2, and then convert it to molecules. Given, CO2 in Earth's atmosphere = \(0.04\%\) = \(0.0004\), collected sample = \(2.0\mathrm{L}\), pressure = \(1.00\mathrm{atm}\), and temperature = \(27^{\circ}\mathrm{C}\) = \(300\mathrm{K}\). Calculate the pressure exerted by CO2 only: \(P_\mathrm{CO2} = (0.0004)(1.00\mathrm{atm}) = 4.0 \times 10^{-4} \mathrm{atm}\) Now, plug in the values in the Ideal Gas Law formula and rearrange it to find the number of moles of CO2: \(n_\mathrm{CO2}=\frac{P_\mathrm{CO2}V}{RT}\)

Step 5: Calculate the number of moles for part (b)

Plug in the values in the formula: \( n_\mathrm{CO2} = \frac{(4.0\times10^{-4}\mathrm{atm})(2.0\mathrm{L})}{(0.0821\frac{\mathrm{L.atm}}{\mathrm{mol.K}})(300\mathrm{K})} \) Calculate the number of moles: \(n_\mathrm{CO2} = 3.25 \times 10^{-5} \mathrm{mol}\)

Step 6: Calculate the number of carbon dioxide molecules for part (b)

Use Avogadro's number to convert moles to molecules: Number of CO2 molecules = \(n_\mathrm{CO2} \times N_\mathrm{A}\) Number of CO2 molecules = \( (3.25 \times 10^{-5}\: \mathrm{mol}) \times (6.022 \times 10^{23} \: \mathrm{molecules/mol}) \) Calculate the number of CO2 molecules: Number of CO2 molecules = \(1.96 \times 10^{19}\) CO2 molecules

Key Concepts

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry and physics. It represents the number of atoms, ions, or molecules contained in one mole of a substance. The value of Avogadro's number is approximately \(6.022 \times 10^{23}\) molecules per mole.
This number is named after the Italian scientist Amedeo Avogadro, who provided insights into the molecular theory of gases.

This constant is crucial because it allows scientists and students to convert between moles and the actual number of atoms or molecules. For instance, if you know the number of moles of ozone \((\mathrm{O}_3)\) as calculated using the Ideal Gas Law, you can easily find out the number of individual ozone molecules by multiplying the moles by Avogadro's number.

• Easy conversion between moles and particles.
• It links macroscopic and microscopic quantities.
• Essential for chemical reactions and stoichiometry.

Moles Calculation

Calculating moles is an essential skill in chemistry, as it bridges the gap between the mass of a substance and the number of its constituent particles. The mole is a standard unit of measurement for amount of substance. It makes it feasible to count particles by weighing the substance.

In order to find the number of moles \(n\), we use the formula from the Ideal Gas Law \(PV = nRT\). Here, \(P\) is the pressure, \(V\) the volume, \(R\) the universal gas constant \((0.0821 \frac{\text{L atm}}{\text{mol K}})\), and \(T\) is the temperature.
To solve a problem using the Ideal Gas Law, rearrange it to find moles: \(n = \frac{PV}{RT}\). This step is essential in calculating moles in both high school and college-level chemistry classes.

• Simple method to calculate moles from gas variables.
• Relates pressure, volume, and temperature to \'moles\'.
• Helps understand gas behavior under different conditions.

Stratosphere

The stratosphere is the second major layer of Earth's atmosphere, sitting above the troposphere and below the mesosphere.
It extends roughly 10 to 50 kilometers above Earth's surface. This layer is known for containing the ozone layer, which absorbs and scatters ultraviolet solar radiation.

Understanding the stratosphere is crucial for environmental studies, especially when studying the effect of pollutants such as ozone or carbon dioxide. The pressure within the stratosphere is much lower than at Earth's surface, which impacts how gases behave at this altitude.
Conditions here are critical for aviation, climate science, and understanding Earth's radiation balance.

• Located 10-50 km above Earth.
• Home to the ozone layer.
• Vital in protecting Earth from UV radiation.

Atmospheric Pressure

Atmospheric pressure is the force exerted onto a surface by the weight of the air above it in the Earth's atmosphere.
It plays a fundamental role in weather patterns, flight dynamics, and biological processes. Atmospheric pressure changes with altitude, becoming lower as you move higher into the atmosphere.

At sea level, the standard atmospheric pressure is about 1.00 atm. This is the baseline for many calculations involving gases, including those using the Ideal Gas Law. In chemistry problems, adjusting for atmospheric pressure when dealing with gases can lead to accurate results in experiments and real-life applications.

• Measures how much air is pressing down on an area.
• Varies with altitude and weather conditions.
• Integral to predicting and understanding weather.