Question

Someone claims that the mass and the mole fractions for a mixture of \(\mathrm{CO}_{2}\) and \(\mathrm{N}_{2} \mathrm{O}\) gases are identical. Do you agree? Explain.

Short answer

Answer: Yes, mass and mole fractions for a mixture of CO2 and N2O gases are identical. However, this applies specifically to these gases because their molar masses are the same.

Step-by-step solution

Recall the concept of mass fraction and mole fraction

Mass fraction is the ratio of the mass of a component to the total mass of the mixture. Mole fraction is the ratio of the moles of a component to the total moles of the mixture.

Identify the molar masses of \(\mathrm{CO}_2\) and \(\mathrm{N}_2\mathrm{O}\)

The molar mass of carbon dioxide (\(\mathrm{CO}_2\)) is obtained as follows: C: 12 g/mol O: 2 * 16 g/mol = 32 g/mol Molar mass \(\mathrm{CO}_2\) = 12 + 32 = 44 g/mol The molar mass of dinitrogen monoxide (\(\mathrm{N}_2\mathrm{O}\)) is obtained like this: N: 2 * 14 g/mol = 28 g/mol O: 16 g/mol Molar mass \(\mathrm{N}_2\mathrm{O}\) = 28 + 16 = 44 g/mol

Consider a hypothetical mixture of \(\mathrm{CO}_2\) and \(\mathrm{N}_2\mathrm{O}\) gases

Let's imagine we have a mixture with 1 mol of \(\mathrm{CO}_2\) and 1 mol of \(\mathrm{N}_2\mathrm{O}\). The total mass of the mixture is: Total mass = mass of \(\mathrm{CO}_2\) + mass of \(\mathrm{N}_2\mathrm{O}\) = (1 mol * 44 g/mol) + (1 mol * 44 g/mol) = 88 g

Calculate the mass and mole fractions of \(\mathrm{CO}_2\) and \(\mathrm{N}_2\mathrm{O}\) in the mixture

For \(\mathrm{CO}_2\): Mass fraction of \(\mathrm{CO}_2\) = Mass of \(\mathrm{CO}_2\) / Total mass = (1 mol * 44 g/mol) / 88 g = 0.5 Mole fraction of \(\mathrm{CO}_2\) = moles of \(\mathrm{CO}_2\) / total moles = 1 mol / (1 mol + 1 mol) = 0.5 For \(\mathrm{N}_2\mathrm{O}\): Mass fraction of \(\mathrm{N}_2\mathrm{O}\) = Mass of \(\mathrm{N}_2\mathrm{O}\) / Total mass = (1 mol * 44 g/mol) / 88 g = 0.5 Mole fraction of \(\mathrm{N}_2\mathrm{O}\) = moles of \(\mathrm{N}_2\mathrm{O}\) / total moles = 1 mol / (1 mol + 1 mol) = 0.5

Compare the mass fractions and mole fractions

Since the molar masses are equal for both \(\mathrm{CO}_2\) and \(\mathrm{N}_2\mathrm{O}\), in this specific case, the mass fractions and mole fractions are identical (both equal to 0.5) for both components in the mixture. Therefore, we agree with the claim that mass and mole fractions for a mixture of \(\mathrm{CO}_2\) and \(\mathrm{N}_2\mathrm{O}\) gases are identical, but it must be noted that this is true because the molar masses of both gases are the same. If the molar masses were different, this claim would not be valid.