Question

A collapsed balloon is filled with He to a volume of 12.5 L. at a pressure of 1.00 atm. Oxygen, \(\mathrm{O}_{2}\), is then added so that the final volume of the balloon is \(26 \mathrm{L}\) with a total pressure of 1.00 atm. The temperature, which remains constant throughout, is \(21.5^{\circ} \mathrm{C}\) (a) What mass of He does the balloon contain? (b) What is the final partial pressure of He in the balloon? (c) What is the partial pressure of \(\mathrm{O}_{2}\) in the balloon? (d) What is the mole fraction of each gas?

Short answer

(a) 2.068 g of He; (b) 0.481 atm He; (c) 0.519 atm O2; (d) Mole fractions: 0.481 He, 0.519 O2.

Step-by-step solution

Find Mass of He

Use the ideal gas law \( PV = nRT \) to calculate the number of moles of Helium initially present.Given: - \( P = 1.00 \text{ atm} \)- \( V = 12.5 \text{ L} \)- \( R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \)- \( T = 21.5^{\circ}C \) or \( 294.65 \text{ K} \)\[ n_{He} = \frac{PV}{RT} = \frac{1.00 \times 12.5}{0.0821 \times 294.65} \approx 0.517 \text{ mol} \]The molar mass of He is approximately \(4.00 \text{ g/mol} \), so:\[ \text{Mass of He} = n_{He} \times 4.00 \approx 2.068 \text{ g} \]

Calculate Final Partial Pressure of He

Since the temperature and amount of Helium remain constant, use the partial volume relationship:\[ P_{He} = \left( \frac{V_{He}}{V_{\text{final}}} \right) P_{\text{total}} \]Here, \( V_{He} = 12.5 \text{ L} \) and \( V_{\text{final}} = 26.0 \text{ L} \), thus:\[ P_{He} = \left( \frac{12.5}{26.0} \right) \times 1.00 \approx 0.481 \text{ atm} \]

Find Partial Pressure of O2

Using Dalton's Law, the total pressure of the mixture is the sum of the partial pressures of the gases.\[ P_{\text{total}} = P_{He} + P_{O_2} \rightarrow 1.00 = 0.481 + P_{O_2} \]Solving for \( P_{O_2} \):\[ P_{O_2} = 1.00 - 0.481 \approx 0.519 \text{ atm} \]

Calculate Mole Fraction of Each Gas

Calculate the mole fraction of Helium and Oxygen using their partial pressures.Mole fraction of He:\[ \chi_{He} = \frac{P_{He}}{P_{\text{total}}} = \frac{0.481}{1.00} \approx 0.481 \]Mole fraction of O2:\[ \chi_{O_2} = \frac{P_{O_2}}{P_{\text{total}}} = \frac{0.519}{1.00} \approx 0.519 \]

Key concepts

Partial Pressure

Partial pressure essentially refers to the pressure exerted by a single type of gas in a mixture of gases. When you have a balloon filled with more than one gas, like helium and oxygen, each type of gas contributes to the total pressure inside the balloon. The individual pressures that each gas contributes are known as their partial pressures.

The concept of partial pressure can be easily understood by considering that each gas in a mixture behaves independently of the others. So, the partial pressure of a gas is simply the pressure it would exert if it were the only gas in the container. This means that it is dependent on both the volume the gas occupies and the temperature, as per the Ideal Gas Law.

In the case of helium in our balloon problem, we calculated the partial pressure using its own initial volume compared to the total volume of the gas mixture in the balloon. The equation we used: \( P_{He} = \left( \frac{V_{He}}{V_{\text{final}}} \right) P_{\text{total}} \) made it possible to find the contribution of helium to the total pressure, reflecting its partial pressure.

Mole Fraction

The mole fraction is a way of expressing the concentration of a particular gas in a mixture. It is defined as the ratio of the moles of one specific gas to the total moles of all gases present. This provides insight into the proportion of each constituent gas in the mixture.

Calculating the mole fraction is straightforward. It involves the formula: \( \chi = \frac{n_{\text{gas}}}{n_{\text{total}}} \) where \( \chi \) is the mole fraction, and \( n \) represents the number of moles. In our example, we used the partial pressures of the gases instead, as they are directly proportional to mole fractions when pressure and temperature conditions are constant.

Each mole fraction tells us the percentage presence of that specific gas. For helium and oxygen, it allowed us to understand how much each gas contributes relative to the entire mixture's gas molecules, representing it with fractions like 0.481 for helium and 0.519 for oxygen, indicating that each gas occupies a reasonably equal share in the balloon.

Dalton's Law

Dalton's Law of Partial Pressures is a fundamental principle in understanding gas mixtures. It states that the total pressure of a gaseous mixture is equal to the sum of the partial pressures of each individual gas in the mixture. This law can be quite helpful, especially in cases like our balloon problem, to calculate unknown pressures.

All gases in a mixture behave independently of each other and each exerts its own pressure as if the others were not present. This total pressure can be expressed as: \( P_{\text{total}} = P_{1} + P_{2} + P_{3} + \ldots \) where each \( P \) represents the partial pressure of the gases in the mixture.

In our example, Dalton's Law was used to determine the partial pressure of oxygen by subtracting the partial pressure of helium from the total pressure, providing insights into the blending of gases. This calculation: \( P_{\text{O}_2} = 1.00 - 0.481 = 0.519 \text{ atm} \) highlighted how much of the total balloon pressure was due to the oxygen present, ensuring a complete understanding of the gas interactions within the balloon.