Question

You have two containers each with 1 mol 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 \(\mathrm{Xe}\) 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

a. The average kinetic energy of the Xe atoms is the same in both containers since they have the same temperature. b. The force with which Xe atoms collide with the container walls is higher in container B due to its higher pressure. c. The root mean square velocity of the Xe atoms is the same in both containers since they have the same temperature and molar mass. d. The collision frequency of the Xe atoms (with other atoms) is higher in container B due to its higher concentration of atoms, resulting from its smaller volume. e. The pressure of the Xe sample is higher in container B (24.51 atm) than in container A (8.17 atm) as calculated using the Ideal Gas Law.

Step-by-step solution

Calculate pressure in 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 universal gas constant (0.0821 L atm / K mol), and T is the temperature in Kelvin. We can rearrange the formula to solve for pressure in each container: P_A = nRT / V_A P_B = nRT / V_B By substituting the given values, we can calculate the pressure in each container: P_A = (1 mol)(0.0821 L atm / K mol)(288.15 K) / 3.0 L ≈ 8.17 atm P_B = (1 mol)(0.0821 L atm / K mol)(288.15 K) / 1.0 L ≈ 24.51 atm

Comparing the average kinetic energy of the Xe atoms

According to the Kinetic Molecular Theory, the average kinetic energy of the atoms in an ideal gas is directly proportional to the temperature. Since the temperature in both containers is the same, the average kinetic energy of the Xe atoms is also the same.

Comparing the force with which the Xe atoms collide with the container walls

Force exerted by the Xe atoms (per unit area) on the container wall is related to pressure. Since the pressure in container B is higher than in container A (24.51 atm vs. 8.17 atm), the force with which the Xe atoms collide with the container walls is greater in container B.

Comparing the root mean square velocity of the Xe atoms

The root mean square (rms) velocity is given by the formula: v_rms = \sqrt{3RT / M} Where R is the universal gas constant, T is the temperature, and M is the molar mass. Since the temperature and molar mass of Xe are the same in both containers, the root mean square velocity of the Xe atoms is also the same in both containers.

Comparing the collision frequency of the Xe atoms (with other atoms)

Collision frequency is affected by the concentration of atoms and their velocities. The concentration of Xe atoms is higher in container B because of its smaller volume. Since temperature (and thus velocity of atoms) is the same in both containers, the collision frequency will be higher in container B due to its higher concentration of atoms.

Comparing the pressure of the Xe sample

Based on the calculations in Step 1, the pressure of the Xe sample is higher in container B (24.51 atm) than in container A (8.17 atm). In summary: a. The average kinetic energy of the Xe atoms is the same in both containers. b. The force with which Xe atoms collide with the container walls is higher in container B. c. The root mean square velocity of the Xe atoms is the same in both containers. d. The collision frequency of the Xe atoms (with other atoms) is higher in container B. e. The pressure of the Xe sample is higher in container B.

Key concepts

Kinetic Molecular Theory

The Kinetic Molecular Theory offers a vital explanation for gas behaviors. According to this theory, gases consist of numerous tiny particles in constant, random motion. These particles' motions are directly related to temperature. At higher temperatures, particles move more vigorously. There are a few key assumptions regarding ideal gases:

• Gas particles are in constant random motion and collisions between particles are perfectly elastic.
• The volume of the gas particles themselves is negligible compared to the volume of their container.
• There are no forces of attraction or repulsion between the particles.
• The average kinetic energy of gas particles is proportional to the absolute temperature of the gas.

Given these assumptions, it’s clear how the conditions in different containers, even when filled with the same gas at the same temperature, can produce varied behaviors. For example, in our exercise, even though xenon in both containers has the same average kinetic energy due to equal temperature, pressure and collision frequency differ due to volume changes. Each assumption of the Kinetic Molecular Theory plays a distinct role in shaping the behaviors of gases under varying conditions.

root mean square velocity

Root mean square velocity is a concept connected closely with the gas molecules' speed and motion. It is calculated from the kinetic molecular theory and is given by the expression:\[ v_\text{rms} = \sqrt{\frac{3RT}{M}} \]Where:

• \(R\) is the universal gas constant
• \(T\) is the absolute temperature in Kelvin
• \(M\) is the molar mass of the gaseous substance

This equation shows that the rms velocity is dependent on both the temperature and the mass of the gas molecules. Although both temperature and molar mass are constant for the xenon gas in our example, the root mean square velocity remains constant across different containers. The temperature dictates average movement energy, while molar mass affects the perceived velocity. Nonetheless, even if the velocities remain unchanged, different volumes, as seen in our example, alter attributes like pressure and collision frequency. Understanding the root mean square velocity is thus key to determining how gas particle motion interrelates with thermodynamic temperature changes.

collision frequency

Collision frequency refers to how often gas molecules collide with one another. This frequency is more complex than it might first appear and is tied to a few crucial factors:

• Molecular velocity, which depends on temperature and molar mass
• Concentration, which is the number of molecules per unit volume

In our scenario, the collision frequency is higher in container B than in container A. The reason behind this is the lower volume in container B, which leads to a greater molecule concentration. Even with a constant molecule velocity due to equal temperature, a higher concentration implies more frequent collisions. It means molecules are closely packed, increasing the chances of them hitting each other. Such behavior of gas particles helps in understanding practical scenarios, such as how containment affects gas pressure and why reactions speed up or slow down in response to varied environmental fabrics.

xenon gas

Xenon is a noble gas that is colorless, odorless, and tasteless, known for its lack of reactivity under normal conditions. In the context of our exercise, xenon is used to illustrate principles of gas laws and kinetic theory. Some properties of xenon include:

• It has a relatively high molar mass (approximately 131.29 g/mol), affecting its velocity in gaseous states according to kinetic theory.
• It is monoatomic, meaning each molecule consists of a single atom, simplifying calculations and behaviors in its gaseous state.

Xenon’s characteristics make it apt for studying ideal gas behavior since it typically exhibits negligible interaction between its atoms. This means it’s a nearly ideal gas, perfectly fitting in theories like Kinetic Molecular Theory. Understanding xenon’s behavior gives insight into more complex or reactive gases. Moreover, by observing how xenon's properties affect parameters such as pressure or collision frequency related to volume changes, students gain a clearer understanding of gas law applications in practical, controlled environments.