Question

\(\mathrm{A}\) and \(\mathrm{B}\) are ideal gases. The molecular weights of \(\mathrm{A}\) and \(\mathrm{B}\) are in the ratio of \(1: 4\). The pressure of a gas mixture containing equal weights of \(\mathrm{A}\) and \(\mathrm{B}\) is \(\mathrm{P}\) atm. What is the partial pressure (in atm) of B in the mixture? (a) \(\mathrm{P} / 5\) (b) \(\mathrm{P} / 2\) (c) \(\mathrm{P} / 2.5\) (d) \(3 \mathrm{P} / 4\)

Short answer

The partial pressure of B is \( \frac{P}{5} \) atm.

Step-by-step solution

Understand the Relationship of Molecular Weights

Let the molecular weight of gas A be M. According to the problem, the molecular weight of gas B is then 4M, as they are in the ratio of 1:4.

Determine Moles of Each Gas

Suppose we have x grams of A and x grams of B (equal weights). The number of moles of A is given by \( \frac{x}{M} \), and the number of moles of B is given by \( \frac{x}{4M} \).

Calculate Total Moles in the Mixture

The total number of moles in the mixture, n, is the sum of moles of A and B:\[n = \frac{x}{M} + \frac{x}{4M} = \frac{4x}{4M} + \frac{x}{4M} = \frac{5x}{4M}.\]

Calculate the Mole Fraction of B

The mole fraction of B, \(X_B\), is given by the ratio of the moles of B to the total moles:\[X_B = \frac{\text{moles of B}}{\text{total moles}} = \frac{\frac{x}{4M}}{\frac{5x}{4M}} = \frac{1}{5}.\]

Find the Partial Pressure of B

The partial pressure of B, \(P_B\), is given by the mole fraction of B times the total pressure P:\[P_B = X_B \times P = \frac{1}{5} \times P = \frac{P}{5}.\]

Key concepts

Molecular Weight Ratio

To understand how gases behave in a mixture, we must first examine their molecular weights. In our exercise, we see that the molecular weights of gases A and B are in the ratio 1:4. This means if the molecular weight of A is assumed to be a certain mass (let's say \( M \)), then the molecular weight of B would simply be four times that size (\( 4M \)).
This ratio is fundamental as it lays the groundwork for understanding how many moles of each gas would be present in a given weight. A higher molecular weight means fewer moles per unit weight because it takes more mass to make up a mole of the substance. It’s like comparing the weight of a golf ball to an exercise ball; the ratio indicates how much more massive one is relative to the other, which directly impacts how they behave in a mixture.

Mole Fraction

Mole fraction is a way of expressing the concentration of a gas in a mixture. It is calculated as the ratio of the number of moles of a particular gas to the total number of moles of all gases present.In our example, gases A and B are added in equal weights. For gas B, with a molecular weight four times that of A, fewer moles are present for the same weight. Specifically, the number of moles of B is \( \frac{x}{4M} \), where \(x\) is the weight in grams. The total number of moles in the mixture is the sum of moles of A and B, calculated as \( \frac{5x}{4M} \). Thus, the mole fraction of B, \( X_B \), becomes \( \frac{1}{5} \). Understanding mole fractions is crucial because they influence how each component of a gas mixture behaves under various conditions.

Partial Pressure

Partial pressure refers to the pressure exerted by a single type of gas in a mixture of gases. It is determined by the mole fraction of the gas multiplied by the total pressure of the mixture.In our scenario, we want to find the partial pressure of gas B. Given that the mole fraction \( X_B \) is \( \frac{1}{5} \) and the total pressure is \( P \), the partial pressure of B, \( P_B \), is calculated using the formula: \[ P_B = X_B \times P = \frac{1}{5} \times P = \frac{P}{5}. \]This calculation tells us how much of the total pressure is contributed by gas B. Partial pressures are vital in understanding phenomena such as diffusion and reaction rates in gaseous mixtures.

Gas Mixture Calculation

Calculating properties of gas mixtures involves assembling various aspects that include molecular weight ratios, moles, and partial pressures. In the exercise, we start with the molecular weights and move through a calculation process that considers equal weights to derive moles. We assessed the total moles present in terms of the known ratio, leading to a straightforward calculation for the mole fraction of each gas. From there, using the Ideal Gas Law and our understanding of partial pressures, we were able to determine the partial pressure contributions of each gas.

• Molecular weights provide insight into how much of a gas is present per given weight.
• Mole fractions help understand the composition of the gas mixture.
• Partial pressures determine how each component affects total system pressure.

This holistic approach enables us to predict and manipulate conditions in gas mixtures accurately, catering to various scientific and industrial applications.