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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 to be the Boltzmann constant, is the temperature, and is the mass of a particle, then we can write the different thermal velocities:
In one dimension
If is defined as the root mean square of the velocity in any one dimension (i.e. any single direction), then
If is defined as the mean of the magnitude of the velocity in any one dimension (i.e. any single direction), then
If is defined as the half-width of the thermal distribution or
if is defined such that a particle with this speed has an energy of , then
In three dimensions
If is defined as the most probable speed, then
If is defined as the root mean square of the total velocity (in three dimensions), then
If is defined as the mean of the magnitude of the velocity of the atoms or molecules, then
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