Check statement A

A says: The motion of gas molecules freezes at $0^\circ\text{C}$.

This is false. In kinetic theory, gas molecules possess random thermal motion at all temperatures above absolute zero.

$0^\circ \text{C} = 273\,\text{K}$

Molecular motion would only cease ideally at:

$T = 0\,\text{K}$

not at $0^\circ \text{C}$.

Check statement B

Mean free path of gas molecules is given by:

$\lambda = \frac{1}{\sqrt{2}\,\pi d^2 n}$

where $d$ is molecular diameter and $n$ is number density.

If density (hence number density $n$) increases, then $\lambda$ decreases.

So B is true.

Check statement C

Using ideal gas law at constant pressure:

$P = nk_B T$

so at constant $P$,

$n \propto \frac{1}{T}$

Since

$\lambda \propto \frac{1}{n}$

therefore,

$\lambda \propto T$

Hence, if temperature increases at constant pressure, mean free path increases.

So C is true.

Check statement D

By equipartition theorem, average kinetic energy per degree of freedom is:

$\frac{1}{2}k_B T$

For a monoatomic gas, total translational degrees of freedom are 3, so total average kinetic energy per molecule is:

$\frac{3}{2}k_B T$

Thus statement D is false, because it incorrectly says per degree of freedom is $\frac{3}{2}k_B T$.

Conclusion

• A: False
• B: True
• C: True
• D: False

Therefore the correct combination is:

$\boxed{\text{B and C only}}$

So the correct option is:

$\boxed{\text{Option B}}$