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Chapter 10: Problem 65

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

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

Step by step solution

01

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: \[V_{CO2} = 407 L \]
02

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): \[n_{CO2} = \frac{V_{CO2}}{Vm} = \frac{407 L}{22.41 L/mol} = 18.16\,mol\]
03

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: \[V_{rest} = 10^6 L - 407 L = 999,593 L\]
04

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: \[n_{rest} = \frac{V_{rest}}{Vm} = \frac{999,593 L}{22.41 L/mol} = 44,632.64\,mol\]
05

Calculate the total number of moles in the atmosphere

Add the number of moles of CO₂ and the rest of the atmosphere components together: \[n_{total} = n_{CO2} + n_{rest} = 18.16\,mol + 44,632.64\,mol = 44,650.80\,mol\]
06

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 \(4.07 \times 10^{-4}\).

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Most popular questions from this chapter

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