Problem 40 - Chemistry and Chemical Reactivity

Question

A collapsed balloon is filled with He to a volume of \(12.5 \mathrm{L}\) at a pressure of 1.00 atm. Oxygen, \(\mathrm{O}_{2}\) is then added so that the final volume of the balloon is 26 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?

Step-by-Step Solution

Answer

(a) 2.064 g He; (b) 0.481 atm He; (c) 0.519 atm O₂; (d) Mole fractions: 0.478 He, 0.522 O₂.

Step 1: Calculate moles of He using PV=nRT

First, we determine the number of moles of helium. Using the ideal gas law \( PV = nRT \), where \( P = 1.00 \text{ atm} \), \( V = 12.5 \text{ L} \), \( R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \), and \( T = 21.5^{\circ} C = 294.65 \text{ K}\):\[ n = \frac{PV}{RT} = \frac{1.00 \times 12.5}{0.0821 \times 294.65} \approx 0.516 \text{ moles} \]

Step 2: Calculate mass of He

Using the moles of He calculated, we find the mass. The molar mass of He is approximately \(4.00 \text{ g/mol}\). \[ \text{Mass of He} = 0.516 \text{ moles} \times 4.00 \text{ g/mol} = 2.064 \text{ g} \]

Step 3: Determine final partial pressure of He

The final volume of the balloon is 26 L with a total pressure of 1.00 atm. Since helium initially occupied a volume of 12.5 L at 1.00 atm, the partial pressure in the larger balloon volume is calculated using Boyle's Law:\[ P_{\text{He}} = \frac{V_{\text{initial}}}{V_{\text{final}}} \times P_{\text{initial}} = \frac{12.5}{26} \times 1.00 = 0.481 \text{ atm} \]

Step 4: Calculate partial pressure of O2

The total pressure is 1.00 atm, so the partial pressure of \(\text{O}_2\) is obtained by subtracting the partial pressure of He from the total pressure:\[ P_{\text{O}_2} = 1.00 - 0.481 = 0.519 \text{ atm} \]

Step 5: Calculate mole fraction of each gas

The mole fraction is the ratio of the number of moles of each gas to the total moles of all gases. The total moles (from PV = nRT) is given by:\[ n_{\text{total}} = \frac{P_{\text{total}} \times V}{R \times T} = \frac{1.00 \times 26}{0.0821 \times 294.65} = 1.08 \text{ moles} \]Mole fraction of He:\[ \chi_{\text{He}} = \frac{0.516}{1.08} \approx 0.478 \]Mole fraction of \(\text{O}_2\):\[ \chi_{\text{O}_2} = \frac{1.08 - 0.516}{1.08} \approx 0.522 \]

Key Concepts

Partial Pressure

In any gas mixture, each individual gas contributes to the total pressure based on its presence in the container. This is known as its partial pressure. When you have gases mixed together, like helium and oxygen in a balloon, the total pressure can be seen as the sum of each gas's partial pressures. This concept uses Dalton's Law of Partial Pressures, which states that: - The total pressure of a gas mixture equals the sum of the partial pressures of each component gas. - For helium in particularly, if it initially has a pressure of 1.00 atm in a volume of 12.5 L, then later, when the volume increases to 26 L but the total pressure remains constant, the partial pressure of helium changes. Using Boyle's Law, you can find the new partial pressure of helium once the balloon expands. It originally had 1.00 atm pressure, but with its volume doubled, its partial pressure was reduced proportionately, becoming approximately 0.481 atm in the final balloon setup.

Mole Fraction

The mole fraction empowers us to express the concentration of a specific gas in a gas mixture. It is calculated as the ratio of the number of moles of a component gas to the total number of moles of all gases in the mixture. This is a dimensionless quantity that provides a sense of how much of each gas makes up the entire mixture. - In the helium and oxygen system of the balloon, the mole fraction of helium, for instance, is determined by the number of moles of helium in comparison to the total moles of gas present in the balloon. - Once the final total number of moles is determined using the ideal gas law, you can compare the helium's moles to get its mole fraction. This process reflects a clear picture of each gas's presence and allows us to compute the proportions simply and precisely. - For helium, given the provided computations, the mole fraction is around 0.478, indicating it makes up about 47.8% of the gas mix.

Boyle's Law

Boyle's Law is a fundamental concept that describes how the pressure of a gas tends to decrease when its volume increases, provided the temperature remains the same. It is represented mathematically as:\[ P_1V_1 = P_2V_2 \]Where \( P \) stands for pressure and \( V \) for volume. Essentially, this law tells us that if a gas is allowed to expand, its pressure drops provided no other variables like temperature change.- In the scenario of the balloon, helium was initially at a volume of 12.5 liters with 1.00 atm pressure. When the volume expanded to 26 liters, Boyle's law allowed us to calculate the new partial pressure of helium in the larger volume. This essential principle helps us understand how gases behave under changing volumes, offering a straightforward tool to predict pressure changes.

Molar Mass

Molar mass is a property of a substance, expressed in units of grams per mole, that reflects the mass per amount of substance. It's essential in converting between grams of a substance and moles, allowing for calculations concerning amounts of reactants or products. - For example, in helium's case, it has a molar mass of approximately 4.00 g/mol. This small molar mass means helium atoms weigh very little per mole, which aligns with helium being a light and less dense gas. - To find the actual mass of helium in the balloon given the number of moles, we simply multiply the moles by the molar mass. In scenarios such as this exercise, understanding the molar mass allows us to transition smoothly from calculating moles to determining the gas's physical mass within the balloon. - Ensuring calculations using the molar mass are accurate, helps maintain consistency across various gas laws and equations, giving precise results in chemical equations and gas scenarios.