Thermal velocity

The thermal velocity or thermal speed is a typical velocity of the thermal motion of particles which make up a gas, liquid, etc. Thus, indirectly, thermal velocity is a measure of temperature. Technically speaking it is a measure of the width of the peak in the Maxwell–Boltzmann particle velocity distribution. Note that in the strictest sense thermal velocity is not a velocity, since velocity usually describes a vector rather than simply a scalar speed.

Since the thermal velocity is only a "typical" velocity, a number of different definitions can be and are used.

Taking $k_{B}$ to be the Boltzmann constant, $T$ is the temperature, and $m$ is the mass of a particle, then we can write the different thermal velocities:

In one dimension

If $v_{{th}}$ is defined as the root mean square of the velocity in any one dimension (i.e. any single direction), then

$v_{{th}}={\sqrt {{\frac {k_{B}T}{m}}}}$.

If $v_{{th}}$ is defined as the mean of the magnitude of the velocity in any one dimension (i.e. any single direction), then

$v_{{th}}={\sqrt {{\frac {2k_{B}T}{\pi m}}}}$.

In three dimensions

If $v_{{th}}$ is defined as the most probable speed, then

$v_{{th}}={\sqrt {{\frac {2k_{B}T}{m}}}}$.

If $v_{{th}}$ is defined as the root mean square of the total velocity (in three dimensions), then

$v_{{th}}={\sqrt {{\frac {3k_{B}T}{m}}}}$.

If $v_{{th}}$ is defined as the mean of the magnitude of the velocity of the atoms or molecules, then

$v_{{th}}={\sqrt {{\frac {8k_{B}T}{m\pi }}}}$.

By all of these definitions $v_{{th}}$ falls in the range of

$v_{th}=(1.6\pm 0.2)\sqrt{\frac{k_BT}{m}}$.

Thermal Velocity at Room Temperature

At 20 °C (293 Kelvin), the mean thermal velocity of common gasses is:^[1]

Hydrogen 1,754 m/s
Helium  1,245 m/s
Water vapor 585 m/s
Nitrogen 470 m/s
Air 464 m/s
Argon 394 m/s
Carbon dioxide 375 m/s

References

1. https://www.pfeiffer-vacuum.com/en/know-how/introduction-to-vacuum-technology/fundamentals/thermal-velocity/