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

$v_{th}=\sqrt{\frac{k_BT}{m}}$

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

$v_{th}=\sqrt{\frac{2 k_BT}{\pi m}}$

If $v_{th}$ is defined as the $1/e$ half-width of the thermal distribution or

if $v_{th}$ is defined such that a particle with this speed has an energy of $k_B T$ , then

$v_{th}=\sqrt{\frac{2k_BT}{m}}$ .

In three dimensions

If $v_{th}$ is defined as the root mean square of the total velocity (in three dimensions), then

$v_{th}=\sqrt{\frac{3k_BT}{m}}$ .

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

$v_{th}=\sqrt{\frac{8k_B T}{m\pi}}$