Question

You have two containers each with 1 mole of xenon gas at \(15^{\circ} \mathrm{C} .\) Container \(\mathrm{A}\) has a volume of \(3.0 \mathrm{L},\) and container \(\mathrm{B}\) has a volume of 1.0 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

The average kinetic energy of the Xe atoms in both containers is the same. The force with which the Xe atoms collide with the container walls is three times greater in container A compared to container B. The root mean square velocities of the Xe atoms in both containers are the same. The collision frequency of the Xe atoms is three times greater in container B than in container A. The pressure in container A is three times greater than container B.

Step-by-step solution

a. The average kinetic energy of the Xe atoms:

The average kinetic energy (KE) of a gas is directly related to its temperature. Since both containers have the same temperature, the average kinetic energy of the xenon atoms in both the containers will be the same. Mathematically, KE = \( \frac{3}{2} kT \) where k is the Boltzmann constant and T is the temperature. Since the temperature of both containers is the same, their average kinetic energy will also be the same.

b. The force with which the Xe atoms collide with the container walls:

For a gas, the force exerted by its particles on the container wall is directly proportional to the pressure and the volume of the container. The ideal gas law states that: PV = nRT Where P is the pressure, V is the volume, n is moles of gas, R is the gas constant, and T is temperature. Since the temperature and the number of moles of gas are the same in both containers, we can compare their pressures: \( \frac{P_A}{P_B} = \frac{V_B}{V_A} = \frac{1.0}{3.0} \) So, the pressure in container A is three times greater than container B. Therefore, the force with which xenon atoms collide with the container wall is three times greater in container A compared to container B.

c. The root mean square velocity of the Xe atoms:

The root mean square velocity (v_rms) of a gas is given by: \( v_{rms} = \sqrt{\frac{3RT}{M}} \) Where R is the gas constant, T is temperature, and M is the molar mass of the gas. Since both containers have the same temperature and gas, their root mean square velocities will be the same.

d. The collision frequency of the Xe atoms (with other atoms):

The collision frequency of a gas depends on its concentration. A higher concentration of gas particles (in terms of particles per unit volume) results in more frequent collisions between the gas molecules. For an ideal gas, the concentration (C) is: C = n/V The concentration of xenon atoms in container A is: \( C_A = \frac{1}{3} \) In container B, the concentration of xenon atoms is: \( C_B = \frac{1}{1} \) So, the collision frequency of Xe atoms in container B is three times greater than in container A.

e. The pressure of the Xe sample:

We can use the ideal gas law to find the pressure of the xenon gas in both containers: PV = nRT For container A, \(P_A = \frac{nRT_A}{V_A} = \frac{1\cdot R\cdot (15+273)(K)}{3.0L}\) For container B, \(P_B = \frac{nRT_B}{V_B} = \frac{1\cdot R\cdot (15+273)(K)}{1.0L}\) As we found earlier, the pressure in container A is three times greater than container B.

Key concepts

Kinetic Theory of Gases

The kinetic theory of gases provides a comprehensive framework to help us understand the behavior of gases. At its core, this theory presents gases as a collection of tiny particles, or molecules, that are in constant random motion. This movement of molecules is the primary source of force when they collide with the walls of their container.

These collisions are what we measure as gas pressure. A pivotal detail of the kinetic theory is that the temperature of a gas is actually a measure of the average kinetic energy of the gas molecules. Therefore, when we look at two containers of xenon gas at the same temperature, like in our exercise, the kinetic theory tells us that the average kinetic energy of the gas particles in both containers will be the same, even if the volumes are different.

How Volume and Pressure Interact

The kinetic theory also implies that gas molecules will have more room to move in a larger volume, leading to fewer collisions with the container walls per unit time compared to a gas in a smaller volume at the same temperature. However, the average kinetic energy—and thus the temperature—remains consistent across different volumes when no energy is added or removed.

This understanding is crucial in explaining why, in our exercise, container A, with a larger volume, exerts the same average kinetic energy as container B but with different pressures and collision forces against the container walls.

Ideal Gas Law

The ideal gas law, represented by the equation PV = nRT, provides the relationship between pressure (P), volume (V), gas moles (n), the gas constant (R), and temperature (T). This equation can be a powerful tool in predicting the behavior of an ideal gas under various conditions.

The term 'ideal gas' refers to a hypothetical gas whose molecules exhibit no intermolecular attraction and occupy no volume. While no gas is truly ideal, many real gases behave in a manner closely approximating this at high temperatures and low pressures. The ideal gas law is derived from combining several empirical gas laws, such as Boyle's Law, Charles’s Law, and Avogadro's Law. Each of these describes gas behavior in response to changes in variable pairs (such as pressure and volume, volume and temperature, or number of moles and volume).

Applying the Ideal Gas Law

In our exercise, we use the ideal gas law to determine the pressure and force exerted by the gas molecules within the two containers. Despite container A having a larger volume, the pressure in container B is found to be three times greater due to its smaller volume. This is because, according to the ideal gas law, pressure is inversely proportional to volume when temperature and moles of gas remain constant.

Root Mean Square Velocity

Root mean square velocity (RMS velocity) is a way to quantify the speed of gas particles. Statistically, it reflects the average speed of a collection of gas particles at a given temperature, regardless of their individual directions of travel.

The RMS velocity is calculated using the formula: \( v_{rms} = \sqrt{\frac{3RT}{M}} \),where R is the universal gas constant, T is the absolute temperature in Kelvin, and M is the molar mass of the gas. This equation arises from the kinetic theory of gases and plays an indispensable role in understanding gas dynamics.

Significance in Different Conditions

Given our textbook exercise, this calculation shows that the root mean square velocity of the xenon atoms in both containers remains consistent because the temperature and the molar mass of xenon do not change. It's essential to note that volume and the number of moles don't affect the RMS velocity directly; it's solely dependent on the type of gas (its molar mass) and the temperature of the system.

So, even though container A and container B have different volumes and pressures, the RMS velocity of the xenon gas in both scenarios stays the same, illustrating the uniformity of particle speed within the ideal gas model at a constant temperature.