A gas mixture used for anesthesia contains 2.83 moles oxygen, O₂, and 8.41 moles nitrous oxide, N₂O. The total pressure of the mixture is 192 kilopascals. What are the mole fractions of O₂ and N₂O?
So in this question, we have a mixture of two different gases in the same container, oxygen and nitrous oxide. The total pressure of the mixture is 192 kilopascals, which is about 1.9 times atmospheric pressure. Our job is to take the amounts of oxygen and nitrous oxide and work out the mole fractions. A mole fraction of a chemical X is equal to the amount of X in moles divided by the total amount in moles. In this question, the total amount refers to the amount of oxygen plus the amount of nitrous oxide.
So we can start off working out the total amount of substance in the container. Here, we’re counting discrete gas particles, O₂ or N₂O. This gives us 11.24 moles of gas molecules. Mole fraction is sometimes given the symbol X. So that’s what I’ll use going forward. The mole fraction of oxygen, O₂, is equal to the amount of O₂ in moles divided by the total amount, which equals 2.83 moles divided by 11.24 moles. This gives us a mole fraction of oxygen of 0.25. And we can do exactly the same with nitrous oxide. Where the mole fraction of nitrous oxide is equal to the amount of nitrous oxide in the mixture divided by the total amount in the mixture in moles. This gives us 8.41 moles divided by 11.24 moles and a mole fraction for N₂O of 0.75.
All the values in our calculation are given to three significant figures. So we should give our answers to the same precision. So the final mole fractions for O₂ and N₂O in this mixture is oxygen 0.252 and nitrous oxide 0.748.
I’m going to store away the prerounded numbers for the next part of the question. You could use the rounded figures for future calculations. But since we’ve got a higher precision value, we should use this to reduce the error in the final calculation.
What are the partial pressures of O₂ and N₂O?
The term partial pressure can be confusing. It’s not the contribution of the component to the total pressure. It’s actually the pressure of that component if it was the only thing in the same container in the same amount. Fortunately, from the ideal gas law, we know that all else being equal, the pressure of a gas is proportional to the amount of gas. So for mixtures of ideal gases, the partial pressure of anyone component is equal to its mole fraction multiplied by the total pressure. This can be written in symbols as 𝑝 of X is equal to 𝑥 of X times the total pressure, 𝑝.
From part a), we already have the accurate mole fractions for the two gases. The partial pressure of O₂ is equal to its mole fraction times the total pressure, which is equal to 0.251779 times 192 kilopascals. Meaning that the partial pressure of oxygen in this system is about 48 kilopascals. So we can move on to do the same for nitrous oxide. The partial pressure of nitrous oxide is equal to the mole fraction of nitrous oxide multiplied by the total pressure. Which is equal to 0.7482 to one times 192 kilopascals. Which is equal to about 144 kilopascals.
The original data for this question was still given to three significant figures. So we should give our final answer to three significant figures also. So our final answer for the partial pressures of O₂ and N₂O in this mixture is O₂ 48.3 kilopascals and N₂O 144 kilopascals.
As a result of the way gases behave, we have 25 mole percent oxygen in our mixture. And about 25 percent of the total pressure is because of oxygen. If the ratios of mole fractions don’t translate into the partial pressures for a gas mixture, there’s been a mistake.