Question

The atmospheric concentration of \(\mathrm{CO}_{2}\) gas is presently 407 \(\mathrm{ppm}(\) parts per million, by volume; that is, 407 \(\mathrm{L}\) of every \(10^{6} \mathrm{L}\) of the atmosphere are \(\mathrm{CO}_{2}\) . What is the mole fraction of \(\mathrm{CO}_{2}\) in the atmosphere?

Short answer

The mole fraction of CO₂ in the atmosphere is approximately \(4.07 \times 10^{-4}\).

Step-by-step solution

Find the volume of CO₂

Given that there are 407 L of CO₂ per every 1,000,000 L of the atmosphere, let's calculate the volume of CO₂ present in 1 million liters of the atmosphere: \[V_{CO2} = 407 L \]

Calculate the number of moles of CO₂

Using the ideal gas law, \(PV = nRT\), where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is the temperature. Since we are given the volume fraction, we don't have to worry about the values of P, R, and T. They will cancel out in the mole fraction calculation. So we can find the number of moles of CO₂ by dividing the volume of CO₂ by the molar volume at given conditions (assuming standard conditions with the molar volume of an ideal gas being 22.41 L/mol): \[n_{CO2} = \frac{V_{CO2}}{Vm} = \frac{407 L}{22.41 L/mol} = 18.16\,mol\]

Calculate the volume of the rest of the atmosphere components

Subtract the volume of CO₂ from 1 million liters to find the volume of the rest of the atmosphere components: \[V_{rest} = 10^6 L - 407 L = 999,593 L\]

Calculate the number of moles of the rest of the atmosphere components

As we did for CO₂, find the number of moles of the rest of the atmosphere components: \[n_{rest} = \frac{V_{rest}}{Vm} = \frac{999,593 L}{22.41 L/mol} = 44,632.64\,mol\]

Calculate the total number of moles in the atmosphere

Add the number of moles of CO₂ and the rest of the atmosphere components together: \[n_{total} = n_{CO2} + n_{rest} = 18.16\,mol + 44,632.64\,mol = 44,650.80\,mol\]

Calculate the mole fraction of CO₂ in the atmosphere

Finally, calculate the mole fraction of CO₂ by dividing the number of moles of CO₂ by the total number of moles: \[\chi_{CO2} = \frac{n_{CO2}}{n_{total}} = \frac{18.16\,mol}{44,650.80\,mol} = 4.07 \times 10^{-4}\] The mole fraction of CO₂ in the atmosphere is approximately \(4.07 \times 10^{-4}\).

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental principle in chemistry that helps to describe the behavior of gases under different conditions. It is expressed as \(PV = nRT\), where:

• \(P\) is the pressure of the gas
• \(V\) is the volume of the gas
• \(n\) is the number of moles of the gas
• \(R\) is the ideal gas constant (approximately 0.0821 L·atm/mol·K)
• \(T\) is the temperature in Kelvin

This formula allows us to relate pressure, volume, and temperature of a gas, assuming that the gas behaves ideally: meaning there are no interactions between the gas molecules and they occupy no space. In real-world scenarios, this assumption is an approximation. However, it works well under conditions of moderate temperature and pressure, especially when calculating the properties of gases in atmospheric chemistry.
For example, in this exercise, we used the ideal gas law implicitly to approach the calculation of moles of carbon dioxide, since the pressure, temperature, and the gas constant would cancel out in a mole fraction scenario. This example simplifies complex calculations and provides valuable insights into the composition of gases, such as carbon dioxide in the atmosphere.

Carbon Dioxide Concentration

Carbon dioxide concentration in the atmosphere is often quantified in parts per million (ppm). This measure indicates how prevalent CO₂ is in a million parts of the air by volume. Presently, the concentration is around 407 ppm, meaning for every million liters of atmospheric air, 407 liters are carbon dioxide. Understanding CO₂ concentration is essential because it plays a crucial role in climate change and environmental sciences.
By calculating the mole fraction of CO₂, we gain an insight into its proportion relative to other gases in the atmosphere. While ppm is a volumetric measure, the mole fraction gives us a dimensionless number that depicts this fraction. Why is this important? Because carbon dioxide is a significant greenhouse gas. Even small changes in its atmospheric concentration can have large impacts on global temperatures and ecosystems.
In this exercise, finding the mole fraction involved converting volumes to moles, using principles such as the molar volume of an ideal gas at standard conditions. This conversion helps in representing the concentration in terms based on constant discipline standards like molarity.

Atmospheric Chemistry

Atmospheric chemistry focuses on the chemical processes occurring in our atmosphere. It encompasses a variety of topics, from the composition of the atmosphere to the reactions and interactions that take place among atmospheric constituents.
Carbon dioxide is one such component; it is not just a stable gas but actively participates in processes like photosynthesis, respiration, and the earth's carbon cycle. Moreover, its role as a greenhouse gas makes it an essential study focus in atmospheric science. This field examines how CO₂, among other gases, contributes to phenomena such as global warming and climate change.
Understanding atmospheric chemistry can help us predict environmental changes and develop strategies to mitigate harmful impacts. The mole fraction of CO₂ is a small, yet significant, puzzle piece in this complex picture. Chemistry helps us understand the balance and transitions happening in the atmospheric layers, underscoring the importance of these calculations.