Question

Energy of 10 non rigid diatomic molecules at temperature T is :

• A. $\frac{7}{2} \, RT$
• B. $70 \, K_B T$
• C. $35 \, RT$
• D. $35 \, K_B T$

Answer 1

The Correct Option is D

For a non-rigid diatomic molecule, the degrees of freedom \( f \) are given by:
\[f = 5 + 2(3N - 5)\]
Since \( N = 2 \) (for diatomic molecules):
\[f = 5 + 2(3 \times 2 - 5) = 7\]
The energy of one molecule is:
\[\text{Energy} = \frac{f}{2} k_B T = \frac{7}{2} k_B T\]
For 10 molecules, the total energy is:
\[10 \times \frac{7}{2} k_B T = 35 k_B T\]

Answer 2

To determine the energy of 10 non-rigid diatomic molecules at temperature \( T \), we need to consider the degrees of freedom available to a non-rigid diatomic molecule and apply the principles of statistical mechanics.

Diatomic molecules, unlike monatomic molecules, have additional degrees of freedom due to their rotational and vibrational motions. In the case of non-rigid diatomic molecules:

• They have 3 translational degrees of freedom.
• They have 2 rotational degrees of freedom (since rotation axis along the bond is not considered significant in the classical limit).
• They also have 2 vibrational degrees of freedom (both kinetic and potential energy contribute).

Total degrees of freedom for a non-rigid diatomic molecule = 3 (translational) + 2 (rotational) + 2 (vibrational) = 7.

According to the equipartition theorem, each degree of freedom contributes \(\frac{1}{2} k_B T\) to the energy at thermal equilibrium, where \( k_B \) is the Boltzmann constant.

Hence, the energy per molecule is:

\(E = \frac{7}{2} k_B T\)

For 10 molecules, the total energy will be:

\(E_{\text{total}} = 10 \times \frac{7}{2} k_B T = 35 k_B T\)

Thus, the total energy for 10 non-rigid diatomic molecules at temperature \( T \) is:

\(35 \, K_B T\)

Therefore, the correct answer is:

$35 \, K_B T$