Question

The mean energy per molecule for a diatomic gas is

• A. \( \frac{5k_B T}{N} \)
• B. \( \frac{5k_B T}{2N} \)
• C. \( \frac{5k_B T}{2} \)
• D. \( \frac{3k_B T}{2} \)

Hint:

For diatomic gases, the total energy per molecule includes contributions from translational and rotational degrees of freedom. For each degree of freedom, the energy is \( \frac{k_B T}{2} \).

Answer 1

The Correct Option is C

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 \(\frac{1}{2}k_BT\) to the energy per molecule, where \(k_B\) is the Boltzmann constant and \(T\) is the temperature. Therefore, the mean energy per molecule is:

\[\frac{1}{2} k_B T \times 5 = \frac{5k_B T}{2}\]

Thus, the mean energy per molecule for a diatomic gas is \(\frac{5k_B T}{2}\).

Answer 2

In a diatomic gas, the energy contributions come from both translational and rotational motions. According to the equipartition theorem, each degree of freedom contributes \( \frac{1}{2}k_B T \) to the energy per molecule, where \( k_B \) is the Boltzmann constant and \( T \) is the temperature in Kelvin.

A diatomic gas has:

• 3 translational degrees of freedom
• 2 rotational degrees of freedom

This totals to 5 degrees of freedom.

The energy per molecule can be calculated as:

\( E = \frac{5}{2} k_B T \)

This means the mean energy per molecule for a diatomic gas is \( \frac{5k_B T}{2} \), which corresponds to option:

\( \frac{5k_B T}{2} \)

Thus, the correct answer is: \( \frac{5k_B T}{2} \).