title: Thermal velocity
text: # Thermal velocity
  
  Thermal velocity or thermal speed is a typical velocity of the thermal motion of particles that 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 $k_{\text{B}}$ to be the Boltzmann constant, $T$ the absolute temperature, and $m$ the mass of a particle, we can write the different thermal velocities:
  
  ## In one dimension
  
  If $v_{\text{th}}$ is defined as the root mean square of the velocity in any one dimension (i.e. any single direction), then
  
  $v_{\text{th}}={\sqrt {\frac {k_{\text{B}}T}{m}}}.$
  
  If $v_{\text{th}}$ is defined as the mean of the magnitude of the velocity in any one dimension (i.e. any single direction), then
  
  $v_{\text{th}}={\sqrt {\frac {2k_{\text{B}}T}{\pi m}}}.$
  
  ## In three dimensions
  
  If $v_{\text{th}}$ is defined as the most probable speed, then
  
  $v_{\text{th}}={\sqrt {\frac {2k_{\text{B}}T}{m}}}.$
  
  If $v_{\text{th}}$ is defined as the root mean square of the total velocity, then
  
  $v_{\text{th}}={\sqrt {\frac {3k_{\text{B}}T}{m}}}.$
  
  If $v_{\text{th}}$ is defined as the mean of the magnitude of the velocity of the atoms or molecules, then
  
  $v_{\text{th}}={\sqrt {\frac {8k_{\text{B}}T}{\pi m}}}.$
  
  All of these definitions are in the range
  
  $v_{\text{th}}=(1.6\pm 0.2){\sqrt {\frac {k_{\text{B}}T}{m}}}.$
  
  ## Thermal velocity at room temperature
  
  At 20 °C (293.15 kelvins), the mean thermal velocity of common gasses in three dimensions is:
  
  | Gas | Thermal velocity |
  | - | - |
  | Hydrogen | 1,754 m/s (5,750 ft/s) |
  | Helium | 1,245 m/s (4,080 ft/s) |
  | Water vapor | 585 m/s (1,920 ft/s) |
  | Nitrogen | 470 m/s (1,500 ft/s) |
  | Air | 464 m/s (1,520 ft/s) |
  | Argon | 394 m/s (1,290 ft/s) |
  | Carbon dioxide | 375 m/s (1,230 ft/s) |
  