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

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

In three dimensions
If $$v_\text{th}$$ is defined as the most probable speed, then $$v_\text{th} = \sqrt{\frac{2k_\text{B} T}{m}}.$$

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: