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{{Unreferenced|date=December 2009}}
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 distribution#Distribution of the velocity vector|Maxwell-Boltzmann particle velocity distribution]]. Note that in the strictest sense ''thermal velocity'' is not a [[velocity]], since ''velocity'' usually describes a [[vector (geometric)|vector]] rather than simply a [[scalar (physics)|scalar]] [[speed]].

Since the thermal velocity is only a "typical" velocity, a number of different definitions can be and are used.

Taking <math>k_B</math> to be the [[Boltzmann constant]], <math>T</math> is the temperature, and <math>m</math> is the mass of a particle, then we can write the different thermal velocities:

== In one dimension ==

If <math>v_{th}</math> is defined as the [[root mean square]] of the velocity in any one dimension (i.e. any single direction), then

:<math>v_{th}=\sqrt{\frac{k_BT}{m}}</math>.

If <math>v_{th}</math> is defined as the [[mean]] of the magnitude of the velocity in any one dimension (i.e. any single direction), then

:<math>v_{th}=\sqrt{\frac{2 k_BT}{\pi m}}</math>.

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

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

:<math>v_{th}=\sqrt{\frac{2k_BT}{m}}</math>.

== In three dimensions ==

If  <math>v_{th}</math> is defined as the most probable speed, then

:<math>v_{th}=\sqrt{\frac{2k_BT}{m}}</math>.

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

:<math>v_{th}=\sqrt{\frac{3k_BT}{m}}</math>.

If <math>v_{th}</math> is defined as the mean of the magnitude of the velocity of the atoms or molecules, then

:<math>v_{th}=\sqrt{\frac{8k_B T}{m\pi}}</math>.



{{DEFAULTSORT:Thermal Velocity}}
[[Category:Thermodynamics]]
[[Category:Statistical mechanics]]


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