Question

A sample of air contains only \(\mathrm{N}_{2}, \mathrm{O}_{2}\) and \(\mathrm{H}_{2} \mathrm{O} .\) It is saturated with water vapour and pressure is 640 torr. The vapour pressure of water is 40 torr and the molar ratio of \(\mathrm{N}_{2}: \mathrm{O}_{2}\) is \(3: 1\). The partial pressure of \(\mathrm{N}_{2}\) in the sample is (a) 540 torr (b) 900 torr (c) 1080 torr (d) 450 torr

Short answer

The partial pressure of \(\mathrm{N}_{2}\) in the sample is 450 torr.

Step-by-step solution

Identify Total Pressure

The total pressure given for the sample of air is 640 torr.

Account for Water Vapor Pressure

Subtract the vapor pressure of water (40 torr) from the total pressure to find the combined pressure of \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\). This yields 640 torr - 40 torr = 600 torr.

Calculate Molar Fraction of \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\)

Using the molar ratio \(\mathrm{N}_{2}:\mathrm{O}_{2} = 3:1\), find the molar fraction of each gas. The sum of mole fractions is always equal to 1, so there are 3 parts \(\mathrm{N}_{2}\) and 1 part \(\mathrm{O}_{2}\), totaling 4 parts. Thus, the mole fraction of \(\mathrm{N}_{2}\) is 3/4, and for \(\mathrm{O}_{2}\), it's 1/4.

Calculate Partial Pressure of \(\mathrm{N}_{2}\)

Multiply the mole fraction of \(\mathrm{N}_{2}\) (3/4) by the combined pressure of \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\): \(\frac{3}{4} \times 600 \(\text{torr}\) = 450\) torr. This is the partial pressure of \(\mathrm{N}_{2}\) in the sample.

Key concepts

Vapor Pressure of Water

Vapor pressure is a property of water—and all liquids—indicating how much the liquid tends to evaporate at a given temperature. It's the pressure exerted by water vapor when it is in equilibrium with its liquid form. Think of it as the force that water molecules exert when they're working to escape into the air as gas. At different temperatures, the vapor pressure of water changes; as temperature goes up, so does the vapor pressure.

When we deal with a sample of air saturated with water vapor, like in the exercise, the vapor pressure of water is crucial for determining the moisture content. If we know the total pressure of the system, subtracting the vapor pressure of water gives us the pressure due to other gases in the mixture. That's because the total pressure is the sum of the partial pressures of all gases present, according to Dalton's Law.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture of substances. It's defined as the number of moles of a particular component divided by the total number of moles of all components in the mixture. To illustrate, if you have a mix of gases, the mole fraction tells you what portion of the whole mixture is made up of a single type of gas molecule.

Mathematically, the mole fraction, represented by the Greek letter chi \(\chi\), is calculated as \(\chi = \frac{n_i}{n_{total}}\), where \(|n_i|\) is the number of moles of a component and \(|n_{total}|\) is the total number of moles in the mixture. One key point is that mole fractions are dimensionless and always sum to one, which reflects the idea that you're essentially dividing a part by the whole.

Gas Law Applications

Gas laws help us understand and predict how gases behave under various conditions of pressure, volume, temperature, and amount. Applying these laws enables us to determine properties like partial pressures in a mixture of gases. For example, in the exercise, we relied on Dalton's Law of Partial Pressures. This law states that the total pressure of a mixture of non-reacting gases is equal to the sum of their individual pressures—which would be exerted if each gas occupied the entire volume on its own.

Another common gas law is the Ideal Gas Law, represented by \(PV = nRT\), relating pressure \(P\), volume \(V\), number of moles \(n\), the universal gas constant \(R\), and the temperature \(T\). In our case, temperature and volume are not directly involved, but an understanding of the basic properties of gases is critical for interpreting and solving the given problem.

Chemical Equilibrium

Chemical equilibrium refers to the state of a reaction when the rates of the forward and reverse reactions are equal, resulting in no overall change in the concentration of reactants and products. It's essential to recognize that chemical equilibrium can apply to the phase changes, like evaporation and condensation, occurring within the system—the water vapor is in equilibrium with liquid water when its vapor pressure is defined.

Our specific problem doesn't involve a chemical reaction, but understanding equilibrium concepts helps us grasp how gases establish partial pressures when mixed. Equilibrium can help explain how, given enough time, the gases in a container will spread out evenly, and each gas's partial pressure will become stable, based on its mole fraction in the mixture.