Question

Equal masses of helium and oxygen are mixed in a container at \(25^{\circ} \mathrm{C}\). The fraction of the total pressure exerted by oxygen in the mixture of gases is (a) \(1 / 3\) (b) \(2 / 3\) (c) \(1 / 9\) (d) \(4 / 9\)

Short answer

The fraction of the total pressure exerted by oxygen is (c) 1/9.

Step-by-step solution

Understanding Dalton's Law

According to Dalton's Law of Partial Pressures, in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases. The partial pressure of a gas is proportional to the number of moles of the gas.

Calculating Mole Fraction

As equal masses of helium and oxygen are mixed, we start by calculating the number of moles of each gas. The moles of a gas can be found using the formula: moles = mass / molar mass. Helium has a molar mass of 4 g/mol and oxygen has a molar mass of 32 g/mol. The moles of oxygen will be one-eighth the moles of helium because of the difference in molar mass (since mass is the same).

Determining the Partial Pressure

Since the moles of oxygen are one-eighth that of helium, the partial pressure exerted by oxygen will also be one-eighth that of helium, as per Dalton’s Law.

Calculating the Fraction of Total Pressure

To find the fraction of the total pressure exerted by oxygen, add the individual pressures together: Pressure_Oxygen + Pressure_Helium = Total Pressure. The fraction can be expressed asFraction = Pressure_Oxygen / Total Pressure = (1 / (1 + 8))

Solving for the Fraction of Oxygen

Replace the numerical values to find the fraction, which gives usFraction = 1 / (1 + 8) = 1 / 9. Therefore, the fraction of the total pressure exerted by oxygen in the mixture is 1/9.

Key concepts

Mole Fraction

When we talk about a gas mixture, it's essential to understand the role each gas plays in exerting pressure. To do this, we often use a concept called 'mole fraction.' Mole fraction is a way of expressing the concentration of a particular component in a mixture. It's calculated by dividing the number of moles of that component by the total number of moles in the mixture. For instance, if we were dealing with our helium-oxygen mixture and wanted to determine the mole fraction of oxygen, we'd look at the ratio of oxygen moles to total moles. This gives us a clear picture of how much oxygen contributes to the overall behavior of the gas mixture.

Mole fraction is dimensionless, meaning it has no units; it's simply a way to express a proportion. Plus, its value always falls between 0 and 1, making it a convenient tool for comparing the relative amounts of different components in a mixture. Whether we're cooking, conducting a lab experiment, or trying to understand the atmosphere, mole fractions help us make sense of mixtures efficiently.

Partial Pressure

Moving onto the concept of 'partial pressure,' it's the pressure that would be exerted by one of the gases in a mixture if it were to occupy the entire volume of the mixture alone, at the same temperature. Dalton’s Law of Partial Pressures highlights how the total pressure of a gas mixture is simply the sum of the partial pressures of each individual gas.

Let’s consider a balloon filled with our helium and oxygen mix. Each gas exerts its own partial pressure. The total pressure inside the balloon is the sum of these individual pressures. Now, here's where 'mole fraction' comes back into play. The partial pressure of a gas is also its mole fraction in the mixture multiplied by the total pressure. This principle allows us to calculate the pressure contribution of any single gas once we know the total pressure and the gas’s mole fraction. That's the beauty of Dalton's Law – it makes understanding gas mixtures much more manageable.

Molar Mass

Last but definitely not least, let's discuss 'molar mass.' In simple terms, molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It's a critical property in chemistry since it bridges the gap between the microscopic world of atoms and molecules and the macroscopic world we can measure and observe.

For any given element, the molar mass is numerically equal to its atomic weight (found on the periodic table) and gives us insight into how many particles are present in a given sample. When we look at compounds, the molar mass becomes the sum of the atomic weights of all atoms in the molecule. Why is this information important? Because knowing the molar mass allows us to convert between grams and moles, which is fundamental in chemical calculations, like in our example where we compare the masses of helium and oxygen. Remember, since molar mass affects the number of moles, it directly influences partial pressures in a gas mixture according to Dalton's Law.