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 (293.15 Kelvin), the mean thermal velocity of common gasses is:^[1]

Hydrogen 1,754 m/s (5,750 ft/s)
Helium   1,245 m/s (4,080 ft/s)
Water vapor  585 m/s (1,920 ft/s)
Nitrogen  470 m/s (1,500 ft/s)
Air  464 m/s (1,520 ft/s)
Argon  394 m/s (1,290 ft/s)
Carbon dioxide  375 m/s (1,230 ft/s)

References

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

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