Statistical Physics, Second Revised and Enlarged Edition

66 Maxwell–Boltzmanngases

inwhichg(v)δvisdefinedas the number ofstatesinthe rangeofinterest,i.e. with
speeds betweenvandv+δv.
The number ofstatesisobtaineddirectlyfrom the‘fitting wavesintoboxes’result,
(6.3), with the simple linear transformation: momentum=k=Mv,i.e.k=(M/)v
(remembering that the rangeδkmust be transformed toδvas well as thek^2 termto
v^2 !). The energyε(v)equalsMv^2 / 2 ;andthe constantAisN/Z,withZgivenby
( 6. 6 ). Hence ( 6 .7) is reduced to be a function ofvonly

n(v)δv=Cv^2 exp(−Mv^2 / 2 kkkBT)δv (6.8)

with

C= 4 πN(M/ 2 πkkkBT)^3 /^2 (6.8a)

This is the required result, obtained originallybyMaxwell. It is an entirelyclassical
result, as can be seen by the fact thathappears nowhere in it. In our derivation, the
h−^3 factorinZ(adetailedquantumidea, asdiscussedabove) cancelledwithanh^3
fromthetransformationfromktov. Another way of obtaining the constantCmakes
this point clear. If (6.8) is to describe the properties ofNparticles (gas molecules),
thenit must satisfythe normalization requirement

N=

∑

i

ni=

∫∞

0

∫∫

n(v)dv

Integration of ( 6 .8), havingreplaced the rangeδvby dv,maybeachievedusingthe
integralIII 2 (Appendix C). It will be found that the value ofCwhich satisfies this
normalization condition is again given by (6.8a).
More comprehensivediscussions oftheproperties anduses ofthespeeddistribution
are found in many books on kinetic theory. Some of its properties are illustrated in
Fig. 6.1. The three different but representative speeds indicated on the graph are all
oforder (kkkBT/M)^1 /^2 =vT,say.Theyare asfollows:

1.vmax(=

√

2 vT),the most probable speedcorrespondingto the maximum ofthe
curve.
2.v ̄ (=

√

( 8 /π)vT), the mean speed ofthemolecules. Thisis againcalcu-

l∫atedusingtheintegralsofAppendixCwithb =M/ 2 kkkBT.Wewritev ̄ =

∞

0

∫∫

vn(v)dv/

∫∞

0

∫∫

n(v)dv=III 3 /III 2 , which gives the stated result.
3 .vrms(=

√

3 vT),the root mean square speed.Thisiscalculatedsimilarly(albeit

with even less difficultysince it involves the recurrence relation of Appendix C

only) since the mean square speedv^2 rms=III 4 /III 2.

The evaluation of the mean square speed is of particular significance to the thermal
properties, sinceit proves that the average (kinetic) energy per moleculeisgivenby

ε ̄=Mv^2 rms/ 2 = 3 kkkBT/2(6.9)