Problem 41 - Chemistry and Chemical Reactivity

Question

You have two flasks of equal volume. Flask A contains \(\mathrm{H}_{2}\) at \(0^{\circ} \mathrm{C}\) and 1 atm pressure. Flask \(\mathrm{B}\) contains \(\mathrm{CO}_{2}\) gas at \(25^{\circ} \mathrm{C}\) and 2 atm pressure. Compare these two gases with respect to each of the following: (a) average kinetic energy per molecule (b) average molecular velocity (c) number of molecules (d) mass of gas

Step-by-Step Solution

Answer

(a) Flask B; (b) Flask A; (c) Flask A; (d) Flask B.

Step 1: Determine Average Kinetic Energy Per Molecule

The average kinetic energy per molecule of any gas can be determined using the formula:\[ KE = \frac{3}{2}kT \]where \( k \) is the Boltzmann constant and \( T \) is the absolute temperature in Kelvin. Note that kinetic energy depends only on the temperature and not on the type or pressure of the gas.- Convert \(0^{\circ}C\) for \( \mathrm{H}_{2} \) to Kelvin: \(T_A = 273 \, \text{K}\).- Convert \(25^{\circ}C\) for \( \mathrm{CO}_{2} \) to Kelvin: \(T_B = 298 \, \text{K}\).Since \( T_B > T_A \), the average kinetic energy per molecule is greater in flask B.

Step 2: Calculate Average Molecular Velocity

The average molecular velocity can be determined using the root mean square speed formula:\[ v_{rms} = \sqrt{\frac{3kT}{m}} \]where \( m \) is the mass of a single molecule.- Since \( T_B > T_A \), \( \mathrm{CO}_{2} \) molecules have higher kinetic energy due to higher temperature.- However, \( \mathrm{H}_{2} \) has a much lower molar mass compared to \( \mathrm{CO}_{2} \).- Consequently, \( \mathrm{H}_2 \) molecules move faster due to their lighter mass despite the lower temperature.

Step 3: Determine Number of Molecules using Ideal Gas Law

Using the ideal gas equation: \( PV = nRT \), where \( n \) is the number of moles:- For the same volume, \( V \), and assuming \( R \) (universal gas constant) is constant, the number of moles \( n \) depends on \( P/T \).- For \( \mathrm{H}_2 \), \( n_A = \frac{1}{273}\).- For \( \mathrm{CO}_2 \), \( n_B = \frac{2}{298}\).- Since \( \frac{1}{273} > \frac{2}{298} \), flask A (\( \mathrm{H}_2 \)) contains more moles, and hence more molecules.

Step 4: Compare Mass of Gas

Again, applying the ideal gas law, we find the moles of each gas.- Multiply the moles by the molar mass to find the total mass: - Mass of \( \mathrm{H}_2 \) in flask A: \( \mathrm{m_A} = \left( \frac{1}{273} \right) \times 2 = \frac{2}{273} \). - Mass of \( \mathrm{CO}_2 \) in flask B: \( \mathrm{m_B} = \left( \frac{2}{298} \right) \times 44 = \frac{88}{298} \). - Comparing, \( \mathrm{m_B} > \mathrm{m_A} \). Thus, flask B has a larger mass.

Key Concepts

Average Kinetic Energy

The concept of average kinetic energy revolves around the energy that each molecule in a gas possesses due to its motion. According to the kinetic molecular theory, the average kinetic energy of gas molecules is directly proportional to the absolute temperature of the gas. This relation is expressed with the formula: \[ KE = \frac{3}{2}kT \] where:

• \( KE \) stands for average kinetic energy per molecule,
• \( k \) is the Boltzmann constant,
• and \( T \) is the temperature in Kelvin.

Thus, the temperature solely dictates the average kinetic energy, irrespective of the gas's type or pressure.
In the given scenario, comparing two gases—\( \mathrm{H}_2 \) and \( \mathrm{CO}_2 \)—we notice that \( \mathrm{CO}_2 \) is at a higher temperature. Hence, it inherently possesses greater average kinetic energy per molecule than \( \mathrm{H}_2 \). This emphasizes that temperature is a crucial factor determining the energy within molecular movement.

Molecular Velocity

Understanding molecular velocity aids in comprehending how fast the molecules within a gas move. The formula often used to calculate this is the root mean square speed: \[ v_{rms} = \sqrt{\frac{3kT}{m}} \] where:

• \( v_{rms} \) symbolizes the root mean square velocity,
• \( k \) again denotes the Boltzmann constant,
• \( T \) is the temperature in Kelvin,
• and \( m \) indicates the mass of a gas molecule.

It's pivotal to acknowledge that for lighter molecules, speeds are generally higher due to a lesser mass contributing to faster velocities.
Despite \( \mathrm{CO}_2 \) having more kinetic energy per molecule because of a higher temperature, the lighter \( \mathrm{H}_2 \) molecules exhibit a higher velocity. This is because their lighter mass counterbalances their lower temperature, facilitating more rapid motion.

Ideal Gas Law

The ideal gas law is a fundamental equation that helps determine the behavior of gases under varying conditions. This law is expressed as: \[ PV = nRT \] where:

• \( P \) stands for pressure,
• \( V \) symbolizes volume,
• \( n \) is the number of moles,
• \( R \) is the universal gas constant,
• and \( T \) denotes temperature in Kelvin.

This equation helps evaluate how the number of molecules in a gas correlates with its pressure, volume, and temperature.
For equivalent volumes, the number of moles, and thus the number of molecules, relates directly to \( \frac{P}{T} \). In our comparative scenario:

• For \( \mathrm{H}_2 \), fewer moles means fewer molecules than in \( \mathrm{CO}_2 \) when computed over equal volumes.
• The number of molecules depends significantly on both temperature and pressure within a fixed space.

Molecular Mass Comparison

Comparing molecular masses plays a vital role in understanding how gas properties affect overall mass in contained conditions. When we appeal again to the ideal gas law, the masses of gases can be inferred by multiplying the number of moles by the molar mass of the respective gas.
In this equation:

• Knowing the number of moles of each gas helps us calculate the total mass by using their specific molar mass—\( \mathrm{H}_2 \) has a molar mass of 2 g/mol and \( \mathrm{CO}_2 \) is 44 g/mol.
• \( \mathrm{CO}_2 \) has a much higher molar mass compared to \( \mathrm{H}_2 \), leading to a higher overall mass in flask B than in flask A when temperature and pressure are accounted for.

Thus, even with fewer molecules, a greater molar mass results in a larger mass of gas, as is the case with \( \mathrm{CO}_2 \) compared to \( \mathrm{H}_2 \).