What kinds of speeds can be found from a Maxwell-Boltzmann distribution?
1 Answer
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
d¯f(ν)dν=√2π(mkT)32[2ν+(−mνkT)ν2]e(−mν2)/(2kT)=0
where
From this, the most probable speed, denoted
νm.p.=√2kTm
Mean speed
- An average or mean speed
<ν> is computed by weighting the speedν with its probability of occurrence¯f(ν)dν and then integrating:
<ν>=∫∞0ν¯f(ν)dν=∫∞0e(−mν2)/(2kT)√2π(mkT)32ν3dν
⇒<ν>=√8π√kTm
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> is13% larger thanνm.p. andνr.m.s is22% 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