1. Use the relation for rms speed

For any gas molecule, the rms speed is determined by translational kinetic energy only:

$v_{\mathrm{rms}} = \sqrt{\frac{3k_B T}{m}}$

Given that the rms velocity is $\overline v$, we have

$\overline v^2 = \frac{3k_B T}{m}$

2. Solve for temperature

Rearranging,

$T = \frac{m\overline v^2}{3k_B}$

3. Why rotational/vibrational motions do not change this relation

Although HCl has rotational, translational, and vibrational degrees of freedom, the molecular speed $\overline v$ is connected only with the translational motion of the center of mass. Hence,

$\frac{1}{2}m\overline v^2 = \frac{3}{2}k_B T$

not $\frac{f}{2}k_B T$ with $f=5,6,7$.

4. Check options

• A: $\dfrac{m\overline v^2}{5k_B}$ — incorrect
• B: $\dfrac{m\overline v^2}{6k_B}$ — incorrect
• C: $\dfrac{m\overline v^2}{7k_B}$ — incorrect
• D: $\dfrac{m\overline v^2}{3k_B}$ — correct

Therefore, the correct answer is:

$\boxed{\frac{m\overline v^2}{3k_B}}$