Question

Apply your knowledge of the kinetic theory of gases to the following situations. (a) Two flasks of volumes \(V_{1}\) and \(V_{2}\left(V_{2}>V_{1}\right)\) contain the same number of helium atoms at the same temperature. (i) Compare the rootmean-square (rms) speeds and average kinetic energies of the helium (He) atoms in the flasks. (ii) Compare the frequency and the force with which the He atoms collide with the walls of their containers. (b) Equal numbers of He atoms are placed in two flasks of the same volume at temperatures \(T_{1}\) and \(T_{2}\left(T_{2}>T_{1}\right) .\) (i) Compare the rms speeds of the atoms in the two flasks. (ii) Compare the frequency and the force with which the He atoms collide with the walls of their containers. (c) Equal numbers of He and neon (Ne) atoms are placed in two flasks of the same volume, and the temperature of both gases is \(74^{\circ} \mathrm{C}\). Comment on the validity of the following statements: (i) The rms speed of He is equal to that of Ne. (ii) The average kinetic energies of the two gases are equal. (iii) The rms speed of each He atom is \(1.47 \times 10^{3} \mathrm{~m} / \mathrm{s}\)

Short answer

(a) RMS speeds and kinetic energies are the same; smaller container has higher collision frequency and force. (b) Higher temp increases speeds, collisions, and force. (c) He and Ne diff. speeds, same kinetic energy; He rms is valid.

Step-by-step solution

Understand Root-Mean-Square Speed Formula

The root-mean-square speed for a gas is calculated using the formula \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( k \) is the Boltzmann constant, \( T \) is the temperature, and \( m \) is the mass of a gas molecule.

Comparing RMS Speeds in Situation (a)

Since both flasks are at the same temperature and contain helium, which has a constant mass, the rms speeds of He atoms in both flasks will be equal, irrespective of the volume.

Comparing Average Kinetic Energies in Situation (a)

The average kinetic energy of a gas is given by \( KE = \frac{3}{2}kT \). Since the temperature is identical in both flasks, the average kinetic energies of the helium atoms are the same.

Comparing Collision Frequency and Force in Situation (a)

Collision frequency depends on both speed and the volume of the flask. Even though the rms speed is the same, in the larger flask \( V_2 \), the atoms have more space, leading to a lower collision frequency. The force of collision depends on average kinetic energy and collision frequency, and since they're moving more freely in \( V_2 \), the frequency of collisions is less, thus exerting less force per unit area as compared to \( V_1 \).

Comparing RMS Speeds in Situation (b)

Using the rms speed equation \( v_{rms} = \sqrt{\frac{3kT}{m}} \), since \( T_2 > T_1 \), the rms speed of atoms in the flask at \( T_2 \) is greater than at \( T_1 \).

Comparing Collision Frequency and Force in Situation (b)

A higher temperature \( T_2 \) increases both the atoms' speeds and the frequency of wall collisions, thus increasing the force exerted by the He atoms on the container walls compared to \( T_1 \).

Validity of Given Statements in Situation (c)

i) The rms speed of He and Ne is not equal since \( v_{rms} = \sqrt{\frac{3kT}{m}} \) and Ne has a greater molar mass than He, resulting in lower speeds. ii) The average kinetic energy depends solely on temperature (\( KE = \frac{3}{2}kT \)), so it's equal for both gases at the given temperature. iii) To calculate the rms speed of helium: \( v_{rms} = \sqrt{\frac{3 \times k \times 347}{4 \times 10^{-26}}} \). This calculation should result in 1470 m/s, confirming the statement.

Key concepts

Root-Mean-Square Speed

The root-mean-square (rms) speed is a crucial concept in the kinetic theory of gases. It gives an idea of the average speed of gas molecules in a container. The formula used to calculate rms speed is \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where:

• \( k \) represents the Boltzmann constant, a key factor in statistical mechanics.
• \( T \) is the absolute temperature of the gas, measured in Kelvin. Temperature plays a significant role in determining molecular speed.
• \( m \) denotes the mass of a single gas molecule.

According to the formula, the rms speed is directly proportional to the square root of the temperature and inversely proportional to the square root of molecular mass. This implies that at the same temperature, lighter molecules will move faster than heavier ones.

Average Kinetic Energy

Average kinetic energy is another fundamental concept linked to the kinetic theory of gases. It represents the energy possessed by each gas molecule due to its motion. The equation for average kinetic energy is given by \( KE = \frac{3}{2}kT \), where:

• \( k \) is the Boltzmann constant, once again emphasizing its importance in molecular physics.
• \( T \) is the absolute temperature of the gas.

The temperature here is paramount, as average kinetic energy is directly proportional to it. Regardless of the gas type, at the same temperature, every gas will have the same average kinetic energy. This is highlighted in situations where different gases at the same temperature have different masses, yet their average kinetic energies remain identical due to the constant temperature factor.

Collision Frequency

Collision frequency refers to how often gas molecules collide with each other or the walls of the container. It is influenced by several factors including:

• The speed of the molecules - higher speeds generally result in more frequent collisions.
• The volume of the container - a larger volume allows molecules more space, reducing collision frequency.

While the rms speed remains the same for gases in containers of different sizes at the same temperature, the collision frequency can differ. For instance, in a larger container, the molecules have more room to move, so they collide with the walls less frequently compared to a smaller container. This variation plays a role in how often and how forcefully gas particles strike the container walls.

Temperature Effect on Gases

Temperature is a key element that significantly impacts the behavior of gases. It affects both the speed and kinetic energy of gas molecules:

• A rise in temperature leads to an increase in the rms speed \( (v_{rms} = \sqrt{\frac{3kT}{m}}) \), meaning molecules move faster.
• It also escalates the average kinetic energy \( (KE = \frac{3}{2}kT) \), resulting in more energetic particles.

This increase in both speed and energy causes molecules to impact the container walls more often and with greater force. Consequently, the pressure inside the container rises as temperature increases. Essentially, temperature determines how energetically and rapidly gas molecules operate, directly influencing overall gas behavior.