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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 \( {\displaystyle k_{B}} \) to be the Boltzmann constant, \( {\displaystyle T} \) is the temperature, and \( {\displaystyle m} \) is the mass of a particle, then we can write the different thermal velocities:
In one dimension
If \( {\displaystyle v_{th}} \) is defined as the root mean square of the velocity in any one dimension (i.e. any single direction), then
\( {\displaystyle v_{th}={\sqrt {\frac {k_{B}T}{m}}}} \) .
If \( {\displaystyle v_{th}} \) is defined as the mean of the magnitude of the velocity in any one dimension (i.e. any single direction), then
\( {\displaystyle v_{th}={\sqrt {\frac {2k_{B}T}{\pi m}}}} \).
If
In three dimensions
If \( {\displaystyle v_{th}} \) is defined as the most probable speed, then
\( {\displaystyle v_{th}={\sqrt {\frac {2k_{B}T}{m}}}} \) .
If \( {\displaystyle v_{th}} \) is defined as the root mean square of the total velocity (in three dimensions), then
\( {\displaystyle v_{th}={\sqrt {\frac {3k_{B}T}{m}}}} . \)
If \( {\displaystyle v_{th}} \) is defined as the mean of the magnitude of the velocity of the atoms or molecules, then
\( {\displaystyle v_{th}={\sqrt {\frac {8k_{B}T}{m\pi }}}} \) .
By all of these definitions v t h {\displaystyle v_{th}} falls in the range of
\( {\displaystyle v_{th}=(1.6\pm 0.2){\sqrt {\frac {k_{B}T}{m}}}} \) .
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
https://www.pfeiffer-vacuum.com/en/know-how/introduction-to-vacuum-technology/fundamentals/thermal-velocity/
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