Problem 51 - Chemistry The Molecular Science

Question

At \(298 \mathrm{~K},\) a sealed \(750-\mathrm{mL}\) vessel contains equimolar amounts of \(\mathrm{O}_{2}, \mathrm{H}_{2}\), and He at a total pressure of 3.85 atm. Determine the partial pressure of the \(\mathrm{H}_{2}\) gas.

Step-by-Step Solution

Answer

The partial pressure of \( \mathrm{H}_2 \) is approximately 1.28 atm.

Step 1: Determine the Total Moles of Gas

Since the mixture contains equimolar amounts of three gases, we assume there are equal moles of \( \mathrm{O}_2, \mathrm{H}_2, \text{and} \ \mathrm{He} \) in the vessel. We can represent this total number of moles as \( n + n + n = 3n \), where \( n \) is the moles of each individual gas.

Step 2: Calculate the Mole Fraction

The mole fraction of \( \mathrm{H}_2 \), represented as \( X_{\mathrm{H}_2} \), is the ratio of moles of \( \mathrm{H}_2 \) to the total moles of all gases. So, \( X_{\mathrm{H}_2} = \frac{n}{3n} = \frac{1}{3} \).

Step 3: Use Dalton's Law of Partial Pressures

Dalton's law states that the partial pressure of a gas is equal to the mole fraction of that gas multiplied by the total pressure. Thus, for \( \mathrm{H}_2 \), its partial pressure can be calculated as:\[ P_{\mathrm{H}_2} = X_{\mathrm{H}_2} \times P_{\text{total}}\]Substitute \( X_{\mathrm{H}_2} = \frac{1}{3} \) and \( P_{\text{total}} = 3.85 \text{ atm} \) into the equation.

Step 4: Calculate the Partial Pressure of \( \mathrm{H}_2 \)

Calculate:\[ P_{\mathrm{H}_2} = \frac{1}{3} \times 3.85 \, \text{atm} = 1.2833 \, \text{atm} \]Rounding to significant figures based on the given data, \( P_{\mathrm{H}_2} \approx 1.28 \, \text{atm} \).

Key Concepts

Mole Fraction

In the context of gases, the mole fraction is a useful way to express the proportion of a particular gas within a mixture. It is defined as the number of moles of one component divided by the total number of moles in the mixture. For a gas mixture, calculating the mole fraction allows us to understand what fraction of the entire pressure is attributed to each gas.
This is essential because gas samples often contain multiple gases mixed together. To calculate the mole fraction of a gas in a mixture, use the formula:

• Mole Fraction (\( X_i \)) = \( \frac{\text{moles of gas } i}{\text{total moles of gas}} \)

For example, if you have a gas mixture with equimolar amounts of three gases such as oxygen, hydrogen, and helium, each gas will make up one third of the mixture. This means the mole fraction of each gas is \( \frac{1}{3} \).
This fraction is crucial in determining the partial pressures of each gas using Dalton's law of partial pressures.

Partial Pressure Calculation

Partial pressure is an important concept from Dalton's Law of Partial Pressures. It describes the pressure that each gas in a mixture would exert if it occupied the entire volume by itself, at the same temperature. Dalton's Law helps predict the behavior of gas mixtures more accurately. It states that the total pressure of a mixture of non-reacting gases is the sum of the partial pressures of each individual gas.To calculate the partial pressure of a specific gas in a mixture, you multiply its mole fraction by the total pressure of the gas mixture:

• \( P_i = X_i \times P_{\text{total}} \)

Where \( P_i \) is the partial pressure of the gas, \( X_i \) is the mole fraction, and \( P_{\text{total}} \) is the total pressure.
In our example, for hydrogen (\( \mathrm{H}_2 \)), the mole fraction is \( \frac{1}{3} \), and the total pressure is 3.85 atm. By applying Dalton’s Law, the partial pressure \( P_{\mathrm{H}_2} \) is calculated as:\[ P_{\mathrm{H}_2} = \frac{1}{3} \times 3.85 \, \text{atm} = 1.28 \, \text{atm} \]
This helps in finding out the contribution of hydrogen to the total pressure exerted by the mixture.

Equimolar Gas Mixture

An equimolar gas mixture is simply a mixture where all components (gases) have the same number of moles. This condition often simplifies calculations related to mixtures of gases, as each component will contribute equally to certain properties, like total moles or mole fractions.
When you have an equimolar mixture, as in our example with gases such as \( \mathrm{O}_2 \), \( \mathrm{H}_2 \), and \( \mathrm{He} \), each gas has the same number of moles. Therefore, the mole fraction of each gas would be the same, simplifying calculations involving total and partial pressures.Key points about equimolar mixtures include:

• All gases have equal moles: \( n_1 = n_2 = n_3 \) (if three gases).
• Mole fraction is simple: \( X_1 = X_2 = X_3 = \frac{1}{\text{number of gases}} \).
• Helps simplify calculations using Dalton's Law of Partial Pressures.

Understanding equimolar mixtures aids in simplifying analysis and understanding the physical behavior of gas mixtures, especially when computing partial pressures.