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.

Hi mason and alex and chase and logan! my name i sbob!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

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

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

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

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

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

In three dimensions

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

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

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

$V_{t}={\sqrt {\frac {8k_{B}T}{m\pi }}}$