Question

The atmospheric concentration of \(\mathrm{CO}_{2}\) gas is presently 407 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 can be found directly from the given ppm value. Simply convert the ppm value to a fraction and use the same fraction for the mole fraction. Therefore, the mole fraction of CO₂, \(X_{CO₂} = \frac{407}{10^6}\), which simplifies to approximately \(4.07 \times 10^{-4}\).

Step-by-step solution

Convert ppm value to a fraction

In the problem, we are given that 407 L of CO₂ is present in every 10⁶ L of the atmosphere. We can write this concentration as a fraction by placing the volume of CO₂ over the total volume of the atmosphere, which is: \( \frac{407}{10^6} \)

Find the number of moles of CO₂ and air

Knowing that the volume of an ideal gas is proportional to the amount of moles, we can convert the volume fraction to moles fraction. Since the ratio of both moles will be the same as the ratio of volumes, then the mole fraction of CO₂ can be found directly from the volume fraction: \( \frac{n_{CO₂}}{n_{Total}} = \frac{407}{10^6} \)

Find the mole fraction of CO₂

To find the mole fraction of CO₂, we simply use the same fraction as calculated in step 2: Mole fraction of CO₂, \(X_{CO₂} = \frac{n_{CO₂}}{n_{Total}} = \frac{407}{10^6}\)

Simplify the mole fraction

We have found the mole fraction of CO₂. Now let's simplify the fraction: \(X_{CO₂} = \frac{407}{10^6}\) The mole fraction of CO₂ in the atmosphere is approximately \(4.07 \times 10^{-4}\).

Key concepts

Atmospheric Chemistry

Atmospheric chemistry explores the chemical composition of the Earth's atmosphere and the processes occurring within it. It studies how different gases interact, both with each other and with external factors, such as sunlight.
One important aspect of atmospheric chemistry involves studying gases like carbon dioxide (\(\mathrm{CO}_2\)).
These gases play significant roles in climate change, as they affect the Earth's energy balance.In our example, we are examining the concentration of \(\mathrm{CO}_2\), a greenhouse gas, in the atmosphere.
Understanding its concentration is fundamental in predicting climate implications and crafting environmental policies.
Atmospheric chemists use various concentration measurements to monitor gases like \(\mathrm{CO}_2\).
Among these, the mole fraction provides a useful sense of the relative amount of a gas compared to the entire mixture.

Concentration Units

Concentration units help us quantify how much of a particular substance is present in a mixture.
When discussing gases in the atmosphere, expressing concentrations in parts per million (ppm) is common.
For example, a \(\mathrm{CO}_2\) concentration of 407 ppm means that for every million units of air, 407 units are \(\mathrm{CO}_2\).Understanding ppm can be enriched by recognizing its relation to the mole fraction, another concentration measure.
To convert ppm to a mole fraction, one can interpret it as a ratio of volumes or correspondingly moles for ideal gases.
The simple relationship between these concentration units allows scientists to compare and analyze gas concentrations efficiently, using whichever is most convenient for the task at hand.
This conversion becomes easy because, under ideal conditions, the volume ratios can directly translate into mole ratios due to the principles laid down by the Ideal Gas Law.

Ideal Gas Law

The Ideal Gas Law is a key principle used to describe the behavior of gases under different conditions.
It is expressed by the equation \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is temperature.
This equation provides a relationship between these variables, showing how changes in one can affect the others.In atmospheric problems, particularly conversion of concentration units like ppm to mole fractions, the Ideal Gas Law simplifications come in handy.
Under conditions where gases behave ideally (i.e., low pressure and high temperature), the volume of gas is directly proportional to its number of moles.
This is why in the problem at hand, the volume fraction could be used directly as the mole fraction for \(\mathrm{CO}_2\).By understanding the principles behind the Ideal Gas Law, students can grasp how chemists use this foundational relationship to understand the behavior of gases in different settings, including our atmosphere.
It showcases how theoretical models guide practical measurements and conversions in atmospheric chemistry.