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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 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鈥檚 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 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.

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.