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

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

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Answer 1 · 2017-12-23T05:02:34

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

$\frac{d ¯ f (ν)}{d ν} = \sqrt{\frac{2}{π}} {(\frac{m}{k T})}^{\frac{3}{2}} [2 ν + (\frac{- m ν}{k T}) ν^{2}] e^{(- m ν^{2}) / (2 k T)} = 0$ $(dbar(f)(nu))/(dnu)=sqrt(2/pi)(m/(kT))^(3/2)[2nu+((-mnu)/(kT))nu^2]e^((-mnu^2)//(2kT))=0$

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

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

$ν_{m.p.} = \sqrt{\frac{2 k T}{m}}$ $color(blue)(nu_"m.p."=sqrt((2kT)/m))$

Mean speed

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

$< ν > = \int_{0}^{\infty} ν ¯ f (ν) d ν = \int_{0}^{\infty} e^{(- m ν^{2}) / (2 k T)} \sqrt{\frac{2}{π}} {(\frac{m}{k T})}^{\frac{3}{2}} ν^{3} d ν$ $< nu > = int_0^(oo)nubar(f)(nu)dnu=int_0^(oo)e^((-mnu^2)//(2kT))sqrt(2/pi)(m/(kT))^(3/2)nu^3dnu$

$\Rightarrow < ν > = \sqrt{\frac{8}{π}} \sqrt{\frac{k T}{m}}$ $=> color(blue)(< nu > = sqrt(8/pi)sqrt((kT)/m))$

Root mean square speed

A calculation of $< ν^{2} >$ $< nu^2 >$ proceeds as:

$< ν^{2} > = \int_{0}^{\infty} ν^{2} ¯ f (ν) d ν = 3 \frac{k T}{m}$ $< nu^2 > = int_0^(oo)nu^2bar(f)(nu)dnu=3(kT)/m$

$\Rightarrow \frac{1}{2} m < ν^{2} > = \frac{3}{2} k T$ $=>1/2m< nu^2 > = 3/2kT$

$\Rightarrow ν_{r.m.s} = \sqrt{\frac{3 k T}{m}}$ $=>color(blue)(nu_"r.m.s"=sqrt((3kT)/m))$

Note:

The mean speed $< ν >$ $< nu >$ is $13 %$ $13%$ larger than $ν_{m.p.}$ $nu_"m.p."$ and $ν_{r.m.s}$ $nu_"r.m.s"$ is $22 %$ $22%$ larger.
The common proportionality to $\sqrt{k T / m}$ $sqrt(kT//m)$ 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 $R$ $R$ are given in the figure above.

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

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