What kinds of speeds can be found from a Maxwell-Boltzmann distribution?

1 Answer
Dec 23, 2017

See below.

Explanation:

Concerning the velocity distribution for a Maxwellian gas:

Hyperphysics

Most probable speed

  • The most probable speed corresponds to the maximum of the velocity distribution, where the slope is zero. One solves the equation

(dbar(f)(nu))/(dnu)=sqrt(2/pi)(m/(kT))^(3/2)[2nu+((-mnu)/(kT))nu^2]e^((-mnu^2)//(2kT))=0

where bar(f)(nu) is the Maxwell velocity distribution (probability distribution for a molecule's velocity) as a function of velocity nu.

From this, the most probable speed, denoted nu_"m.p." emerges as:

color(blue)(nu_"m.p."=sqrt((2kT)/m))

Mean speed

  • An average or mean speed < nu > is computed by weighting the speed nu with its probability of occurrence bar(f)(nu)dnu and then integrating:

< nu > = int_0^(oo)nubar(f)(nu)dnu=int_0^(oo)e^((-mnu^2)//(2kT))sqrt(2/pi)(m/(kT))^(3/2)nu^3dnu

=> color(blue)(< nu > = sqrt(8/pi)sqrt((kT)/m))

Root mean square speed

  • A calculation of < nu^2 > proceeds as:

< nu^2 > = int_0^(oo)nu^2bar(f)(nu)dnu=3(kT)/m

=>1/2m< nu^2 > = 3/2kT

=>color(blue)(nu_"r.m.s"=sqrt((3kT)/m))

Note:

  • The mean speed < nu > is 13% larger than nu_"m.p." and nu_"r.m.s" is 22% larger.

  • The common proportionality to sqrt(kT//m) has two immediate implications: higher temperature implies higher speed, and larger mass implies lower speed.

**The equivalent expressions in terms of the universal/ideal gas constant R are given in the figure above.