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 ${\displaystyle k_{\text{B}}}$ to be the Boltzmann constant, ${\displaystyle T}$ the absolute temperature, and ${\displaystyle m}$ the mass of a particle, we can write the different thermal velocities:

If ${\displaystyle v_{\text{th}}}$ is defined as the root mean square of the velocity in any one dimension (i.e. any single direction), then^[1][2] $${\displaystyle v_{\text{th}}={\sqrt {\frac {k_{\text{B}}T}{m}}}.}$$

If ${\displaystyle v_{\text{th}}}$ is defined as the mean of the magnitude of the velocity in any one dimension (i.e. any single direction), then $${\displaystyle v_{\text{th}}={\sqrt {\frac {2k_{\text{B}}T}{\pi m}}}.}$$

If ${\displaystyle v_{\text{th}}}$ is defined as the most probable speed, then^[2] $${\displaystyle v_{\text{th}}={\sqrt {\frac {2k_{\text{B}}T}{m}}}.}$$

If ${\displaystyle v_{\text{th}}}$ is defined as the root mean square of the total velocity, then $${\displaystyle v_{\text{th}}={\sqrt {\frac {3k_{\text{B}}T}{m}}}.}$$

If ${\displaystyle v_{\text{th}}}$ is defined as the mean of the magnitude of the velocity of the atoms or molecules, then $${\displaystyle v_{\text{th}}={\sqrt {\frac {8k_{\text{B}}T}{\pi m}}}.}$$

All of these definitions are in the range $${\displaystyle v_{\text{th}}=(1.6\pm 0.2){\sqrt {\frac {k_{\text{B}}T}{m}}}.}$$

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

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