Problem 113 - Chemistry and Chemical Reactivity

Question

You have two gas-filled balloons, one containing He and the other containing \(\mathrm{H}_{2} .\) The \(\mathrm{H}_{2}\) balloon is twice the volume of the He balloon. The pressure of gas in the \(\mathrm{H}_{2}\) balloon is 1 atm, and that in the He balloon is 2 atm. The \(H_{2}\) balloon is outside in the snow \(\left(-5^{\circ} \mathrm{C}\right),\) and the He balloon is inside a warm building \(\left(23^{\circ} \mathrm{C}\right) .\) (a) Which balloon contains the greater number of molecules? (b) Which balloon contains the greater mass of gas?

Step-by-Step Solution

Answer

(a) The \(H_2\) balloon has more molecules. (b) The He balloon has greater mass.

Step 1: Use Ideal Gas Law

First, use the ideal gas law equation for both balloons: \( PV = nRT \). To compare the number of molecules, calculate the number of moles \( n \) using this equation for both gases. Remember, \( R \) is the gas constant \( 0.0821 \frac{L \, atm}{mol \, K} \).

Step 2: Calculate Moles in He Balloon

For helium (He) balloon, substitute \( P = 2 \, atm \), \( V = V_{He} \), and \( T = 23 + 273 = 296 \, K \). Rearrange the equation to solve for \( n_{He} \): \[ n_{He} = \frac{PV}{RT} = \frac{2 \times V_{He}}{0.0821 \times 296} \]

Step 3: Calculate Moles in H2 Balloon

For hydrogen (\(H_2\)) balloon, remember \( V = 2V_{He} \). Use \( P = 1 \, atm \) and \( T = -5 + 273 = 268 \, K \): \[ n_{H_2} = \frac{PV}{RT} = \frac{1 \times 2V_{He}}{0.0821 \times 268} \]

Step 4: Compare Number of Moles

Solve the expressions for \( n_{He} \) and \( n_{H_2} \). Since \( V_{He} \) is common in both, compare their coefficients: \[ n_{He} = \frac{2}{0.0821 \times 296}V_{He} \approx 0.082V_{He} \]\[ n_{H_2} = \frac{2}{0.0821 \times 268}V_{He} \approx 0.091V_{He} \]Thus, \( n_{H_2} > n_{He} \), so the \( H_2 \) balloon has more molecules.

Step 5: Calculate Mass for He Balloon

Calculate the mass of helium using moles obtained earlier. He has a molar mass of 4 g/mol:\[ \text{mass}_{He} = n_{He} \times 4 \approx 0.082V_{He} \times 4 \]

Step 6: Calculate Mass for H2 Balloon

Calculate the mass of hydrogen using moles obtained earlier. \(H_2\) has a molar mass of 2 g/mol:\[ \text{mass}_{H_2} = n_{H_2} \times 2 \approx 0.091V_{He} \times 2 \]

Step 7: Compare Masses

Compare the calculated masses:\[ \text{mass}_{He} = 0.328V_{He} \]\[ \text{mass}_{H_2} = 0.182V_{He} \]Helium has the greater mass because \( 0.328V_{He} > 0.182V_{He} \).

Key Concepts

Moles Calculation

The process of calculating the number of moles (")) in a gas sample is central to understanding gas behavior through the Ideal Gas Law, which is expressed as \( PV = nRT \). In this equation, \( P \) stands for pressure, \( V \) for volume, \( n \) for number of moles, \( R \) for the ideal gas constant (0.0821 L atm/mol K), and \( T \) for temperature in Kelvin. To determine the number of moles in each balloon, we can rearrange the equation to solve for \( n \):

• For the helium balloon, substitute its known values: pressure \( P \), volume \( V \), and temperature \( T \), and compute \( n \) using \( n = \frac{PV}{RT} \).
• For the hydrogen balloon, remember the volume is twice that of the helium balloon. Using the rearranged equation can help to compute \( n \).
• Compare the expressions for \( n \) between the two balloons.

The ideal gas law easily lets us understand which balloon contains more moles by just comparing the total amounts calculated. Since moles are directly related to the number of molecules, finding \( n \) helps us derive that the hydrogen balloon holds more molecules than the helium balloon.

Gas Properties

Gas properties such as pressure, volume, and temperature crucially influence the physical state and behavior of gases. When analyzing the behavior of gases in balloons, these characteristics are vital:

• **Pressure (P):** Measured in atmospheres (atm), it indicates the gas force exerted inside the balloon walls. The helium balloon in this problem has twice the pressure of the hydrogen balloon.
• **Volume (V):** Represents the space a gas occupies. In our case, the hydrogen balloon's volume is noted to be twice that of the helium, implying more room for gas expansion.
• **Temperature (T):** Measured in Kelvin and impacting gas energy. Even though expressed in Celsius initially, always convert Celsius to Kelvin by adding 273.

The combination of these properties, as illustrated by the ideal gas law, directly correlates with how many moles or molecules are inside each balloon. Understanding these interrelations helps us predict and explain why the hydrogen balloon contains a greater number of molecules, thanks to its larger volume and lower relative pressure.

Molar Mass

Molar mass links the number of moles of a substance to its mass and is imperative when comparing substances like helium and hydrogen gases in balloons. The concept can be described as:

• **Definition of Molar Mass:** The mass of one mole of a given substance expressed in grams/mole (g/mol). Each element has a unique molar mass.
• **Helium vs. Hydrogen: He weighs 4 g/mol, whereas \(H_2\) weighs 2 g/mol.** This distinction plays a crucial role when measuring the mass of gases contained within each balloon.

To compute the mass of each gas in their respective balloons, multiply the number of moles by its molar mass. This results in:

• For helium: \( \text{mass}_{He} = n_{He} \times 4 \)
• For hydrogen: \( \text{mass}_{H_2} = n_{H_2} \times 2 \)
• Despite the number of molecules being greater in the hydrogen balloon, helium's greater molar mass results in a higher overall mass in its balloon.

Understanding molar mass gives insights into why even with fewer molecules, helium weighs more than hydrogen. This concept is pivotal when dealing with gas weight comparisons in chemical equations and reactions.