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

d¯f(ν)dν=2π(mkT)32[2ν+(mνkT)ν2]e(mν2)/(2kT)=0

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

From this, the most probable speed, denoted νm.p. emerges as:

νm.p.=2kTm

Mean speed

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

<ν>=0ν¯f(ν)dν=0e(mν2)/(2kT)2π(mkT)32ν3dν

<ν>=8πkTm

Root mean square speed

  • A calculation of <ν2> proceeds as:

<ν2>=0ν2¯f(ν)dν=3kTm

12m<ν2>=32kT

νr.m.s=3kTm

Note:

  • The mean speed <ν> is 13% larger than νm.p. and νr.m.s is 22% larger.

  • The common proportionality to 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.