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

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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

01

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
02

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.
03

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.
04

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.
05

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.
06

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.

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