Thermal velocity

Thermal velocity or thermal speed is a typical velocity of the thermal motion of particles that 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 $k_{\text{B}}$ to be the Boltzmann constant, $T$ the absolute temperature, and $m$ the mass of a particle, we can write the different thermal velocities:

In one dimension

If $v_{\text{th}}$ is defined as the root mean square of the velocity in any one dimension (i.e. any single direction), then[1][2]

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

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

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

In three dimensions

If $v_{\text{th}}$ is defined as the most probable speed, then[2]

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

If $v_{\text{th}}$ is defined as the root mean square of the total velocity, then

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

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

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

All of these definitions are in the range

v_{\text{th}}=(1.6\pm 0.2){\sqrt {\frac {k_{\text{B}}T}{m}}}.

Thermal velocity at room temperature

At 20 °C (293.15 kelvins), the mean thermal velocity of common gasses in three dimensions is:[3]

Gas	Thermal velocity
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

Baumjohann, Wolfgang; Treumann, Rudolf A. (2006). Basic Space Plasma Physics (Reprinted ed.). London: Imperial College Press. ISBN 978-1-86094-079-8.
Gurnett, Donald A.; Bhattacharjee, Amitava (2017). Introduction to Plasma Physics: With Space, Laboratory and Astrophysical Applications (2nd ed.). Cambridge: Cambridge University Press. ISBN 978-1-107-02737-4.
"Thermal velocity". www.pfeiffer-vacuum.com. Retrieved 2023-05-28.

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[Baumjohann-1] Baumjohann, Wolfgang; Treumann, Rudolf A. (2006). Basic Space Plasma Physics (Reprinted ed.). London: Imperial College Press. ISBN 978-1-86094-079-8.

[Gurnett-2] Gurnett, Donald A.; Bhattacharjee, Amitava (2017). Introduction to Plasma Physics: With Space, Laboratory and Astrophysical Applications (2nd ed.). Cambridge: Cambridge University Press. ISBN 978-1-107-02737-4.

[3] "Thermal velocity". www.pfeiffer-vacuum.com. Retrieved 2023-05-28.