Question

You have two containers each with 1 mole of xenon gas at \(15^{\circ} \mathrm{C}\) . Container A has a volume of \(3.0 \mathrm{L},\) and container \(\mathrm{B}\) has a volume of \(1.0 \mathrm{L}\) . Explain how the following quantities compare between the two containers. a. the average kinetic energy of the Xe atoms b. the force with which the Xe atoms collide with the container walls c. the root mean square velocity of the Xe atoms d. the collision frequency of the Xe atoms (with other atoms) e. the pressure of the Xe sample

Short answer

In summary: a. The average kinetic energy of Xe atoms is the same in both containers, as they have the same temperature. b. The force with which Xe atoms collide with the container walls is greater in container B, as it has a higher pressure. c. The root mean square velocity of Xe atoms is the same in both containers, as they have the same temperature and gas type. d. The collision frequency of Xe atoms (with other atoms) is greater in container B, as it has a smaller volume. e. The pressure of the Xe sample is greater in container B, due to its smaller volume compared to container A.

Step-by-step solution

Find the temperature in Kelvin.

Convert the given temperature from Celsius to Kelvin: T(K) = T(°C) + 273.15 T(K) = 15°C + 273.15 = 288.15 K

Calculate the pressure of each container using the Ideal Gas Law.

The ideal gas law is given by: PV = nRT Where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant (8.314 J/(mol K)), and T is the temperature in Kelvin. For container A: P_A = (1 mol)(8.314 J/(mol K))(288.15 K) / (3.0 L * 0.001 m^3/L) ≈ 7835 Pa For container B: P_B = (1 mol)(8.314 J/(mol K))(288.15 K) / (1.0 L * 0.001 m^3/L) ≈ 23505 Pa

Compare the average kinetic energy of Xe atoms in both containers.

The average kinetic energy of gas particles is given by: KE = (3/2)kT, where k is the Boltzmann constant (1.38 × 10⁻²³ J/K). Since both containers have the same temperature, the average kinetic energy will be the same for both containers.

Compare the force with which Xe atoms collide with container walls.

The force of particles colliding with the container walls is directly proportional to the gas pressure. Since P_B > P_A, the force exerted by the Xe atoms on the wall of container B is greater than that on the wall of container A.

Compare the root mean square velocity of Xe atoms in both containers.

Root mean square velocity is given by: v_rms = √(3kT/m), where m is the molar mass of the gas. Since both containers have the same temperature and gas, the root mean square velocity of Xe atoms will be the same in both containers.

Compare the collision frequency of Xe atoms (with other atoms) in both containers.

Collision frequency is inversely proportional to the square root of the volume. As the volume of container A is three times greater than the volume of container B, the collision frequency will be higher in container B.

Compare the pressure of the Xe sample in both containers.

We have already calculated the pressures in containers A and B in Step 2. The pressure of the Xe sample in container B is greater than that in container A. To summarize the results: a. The average kinetic energy of Xe atoms is the same in both containers. b. The force with which Xe atoms collide with the container walls is greater in container B. c. The root mean square velocity of Xe atoms is the same in both containers. d. The collision frequency of Xe atoms (with other atoms) is greater in container B. e. The pressure of the Xe sample is greater in container B.

Key concepts

Kinetic Energy

Kinetic energy is a fundamental concept in the study of gases. At the molecular level, it refers to the energy that an atom or molecule possesses due to its motion. For an ideal gas, the average kinetic energy is directly related to the temperature of the gas. The relationship is given by the formula:

• \(KE = \frac{3}{2} k T\)

where:

• \(KE\) = average kinetic energy
• \(k\) = Boltzmann constant (\(1.38 \times 10^{-23} \text{J/K}\))
• \(T\) = temperature in Kelvin

Since the temperature of an ideal gas determines its kinetic energy, and both containers of xenon gas in the exercise are at the same temperature, the kinetic energy of atoms in both containers is identical. This means, regardless of the volume or pressure differences, as long as the temperature is constant, the kinetic energy remains constant.

Pressure

Pressure in gases is the result of collisions of gas particles with the walls of their container. Each collision exerts a force, and the total of these forces over the surface area of the container leads to pressure. The Ideal Gas Law, given by \(PV = nRT\), helps us understand how pressure relates to volume for a given temperature and quantity of gas.
In the exercise, the pressures calculated for the two containers were different due to their different volumes:

• Container A (3 L) has a pressure of approximately 7835 Pa.
• Container B (1 L) has a pressure of approximately 23505 Pa.

With both containers having the same amount of gas and at the same temperature, the pressure in the smaller container (B) is higher. This demonstrates how reducing the volume of the gas increases its pressure, a principle known as Boyle’s Law, for a given temperature. This increased pressure also means a greater force exerted on the walls of container B than on A.

Collision Frequency

Collision frequency in gases is a measure of how often gas molecules collide with each other. It is influenced by factors such as the density of the gas and the volume of the container. A simple way to understand this is to think of collision frequency as being higher where there's less space for those atoms to move around freely.
Mathematically, collision frequency is inversely related to the square root of the volume of the container. Therefore:

• Smaller volumes (like Container B) have higher collision frequencies. This is because particles have less space, increasing the likelihood of collisions.
• Larger volumes (like Container A) have lower collision frequencies as the atoms have more room to move.

In the exercise, with container B having a smaller volume, it naturally has a higher collision frequency compared to container A. This increased frequency also plays a role in the higher pressure observed in container B.

Root Mean Square Velocity

The root mean square (RMS) velocity is a way to quantify the speed of particles in a gas. It's essentially the measure of the average speed of particles, accounting for the fact that particles move in random directions and have varying speeds.
The RMS velocity is given by the formula:

• \(v_{rms} = \sqrt{\frac{3kT}{m}}\)

where:

• \(v_{rms}\) = root mean square velocity
• \(k\) = Boltzmann constant
• \(T\) = temperature in Kelvin
• \(m\) = molar mass of the gas

Because the RMS velocity depends only on the temperature and molar mass, and both containers in the exercise maintain the same temperature and gas type, the RMS velocity of xenon atoms in both containers is identical. Despite differing volumes or pressures, the speed at which the Xe atoms move remains constant.