In kinetic theory of gases, atoms are modeled as point masses m with mean speed v related to temperature T by m v^2 = 3 k T, where k is Boltzmann’s constant. Assuming gas atoms travel several thousand Angstroms between collisions with each other, how cool would hydrogen gas need to be before quantum mechanics would have to be taken into consideration?

In kinetic theory of gases, atoms are modeled as point masses m with mean speed v related to temperature T by m v^2 = 3 k T, where k is Boltzmann’s constant. Assuming gas atoms travel several thousand Angstroms between collisions with each other, how cool would hydrogen gas need to be before quantum mechanics would have to be taken into consideration?