Question

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

Answer 1

See below.

Explanation:

Concerning the velocity distribution for a Maxwellian gas:

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.