Thermal velocity

{{Short description|Typical velocity of the thermal motion of particles}}

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.

Definitions

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{{cite book | last = Baumjohann | first = Wolfgang | title = Basic Space Plasma Physics | last2 = Treumann | first2 = Rudolf A. | date = 2006 | publisher = Imperial College Press | isbn = 978-1-86094-079-8 | edition = Reprinted | location = London}}{{cite book | last = Gurnett | first = Donald A. | title = Introduction to Plasma Physics: With Space, Laboratory and Astrophysical Applications | last2 = Bhattacharjee | first2 = Amitava | date = 2017 | publisher = Cambridge University Press | isbn = 978-1-107-02737-4 | edition = 2nd | location = Cambridge}}

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

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

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

= 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:{{Cite web |title=Thermal velocity |url=https://www.pfeiffer-vacuum.com/en/know-how/introduction-to-vacuum-technology/fundamentals/thermal-velocity/ |access-date=2023-05-28 |website=www.pfeiffer-vacuum.com |archive-url= https://web.archive.org/web/20230131163522/https://www.pfeiffer-vacuum.com/en/know-how/introduction-to-vacuum-technology/fundamentals/thermal-velocity/ |archive-date= 2023-01-31}}

class="wikitable"
Gas ! Thermal velocity
Hydrogen \| {{convert\|1,754\|m/s\|ft/s\|abbr=on}}
Helium \| {{convert\|1,245\|m/s\|ft/s\|abbr=on}}
Water vapor \| {{convert\|585\|m/s\|ft/s\|abbr=on}}
Nitrogen \| {{convert\|470\|m/s\|ft/s\|abbr=on}}
Air \| {{convert\|464\|m/s\|ft/s\|abbr=on}}
Argon \| {{convert\|394\|m/s\|ft/s\|abbr=on}}
Carbon dioxide \| {{convert\|375\|m/s\|ft/s\|abbr=on}}

class="wikitable"

Gas

! Thermal velocity

Hydrogen

| {{convert|1,754|m/s|ft/s|abbr=on}}

Helium

| {{convert|1,245|m/s|ft/s|abbr=on}}

Water vapor

| {{convert|585|m/s|ft/s|abbr=on}}

Nitrogen

| {{convert|470|m/s|ft/s|abbr=on}}

Air