Kinetic Theory

To compare the kinetic energies and of carbon monoxide (CO) and nitrogen (N ) at the same temperature, we must understand the relation between temperature and kinetic energy for gases.

According to the kinetic theory of gases, the average kinetic energy of an ideal gas's molecules is proportional to the absolute temperature of the gas. Mathematically, it is expressed as:

where is the average kinetic energy per molecule, is the Boltzmann constant, and is the absolute temperature.

Since the problem states that both gases are at the same temperature, we have:

Here, , leading to the conclusion that:

This indicates that the average kinetic energy per molecule is the same for both gases at the same temperature.

Hence, the correct answer is:

To determine the mean energy per molecule for a diatomic gas, we can use the concept of degrees of freedom. A diatomic molecule in thermal equilibrium has translational, rotational, and vibrational degrees of freedom. However, at typical room temperatures, vibrations can often be ignored, leaving us with:

• Translational degrees of freedom: 3 (one for each spatial dimension)
• Rotational degrees of freedom: 2 (diatomic molecules can rotate about two axes perpendicular to the bond axis)

Total degrees of freedom = 3 (translational) + 2 (rotational) = 5

According to the equipartition theorem, each degree of freedom contributes to the energy per molecule, where is the Boltzmann constant and is the temperature. Therefore, the mean energy per molecule is:

Thus, the mean energy per molecule for a diatomic gas is .

To solve this problem, we use the kinetic theory of gases. The number of hits made by one atom in 1 second is related to the pressure and the volume of the container.
Step 1: Calculate the number of atoms per unit volume.
The pressure of a gas is related to the number of collisions by the equation: Where: - is the pressure, - is the number density of the gas (number of atoms per unit volume), - is the mass of one atom of Boron, - is the root mean square speed of the gas molecules. For Boron, the atomic mass is approximately 10.81 u, which is .
Step 2: Calculate the mass of the gas.
We know that the number of collisions made by a single atom with a wall in one second is 500. This gives us the number of atoms per unit volume because we can use the equation for collisions between gas molecules and a wall. Now, using the relationships from the kinetic theory of gases, we calculate the total mass in the box: Where: - is the volume of the box, - is the Boltzmann constant, - is the temperature.
After solving the equations, the total mass of the Boron gas is found to be approximately .