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 is given by

:$V\_t=sqrt\{frac\{8k\_B\; T\}\{mpi$