The Maxwell–Boltzmann distribution ofspeeds 65Itis worthputtinginthe numbersfor the worst possible case! Consider^4 Heat1
atmosphere pressure and at a temperature of 5 K, at which it is still a gas (the boiling
pointis 4.2 K). Substituting the numbers (checkitusing the necessary constantsfrom
Appendix D?)gives
A=(N/V)·(h^2 / 2 πMkkkBT)^3 /^2 (6. 6 a)
=0.0 9
for this case.
Thisis a usefulcalculation. Whatitshowsisthat evenfor^4 He (andsimilarly for(^3) He which boils at 3.2 K)the value ofAis sufficientlysmall tojustifythe use of MB
statistics as a first approximation (and as we see later in section 6.3, MB statistics lead
to the perfectgaslaws).Aisoften calledthedegeneracy parameter,A 1 beingthe
classical or ‘non-degenerate’ limit.
For helium gas near to its boiling point, the valueA= 0 .09 suggests that degeneracy
will be a small but significant cause ofdeviationfrom the perfectgaslaws. Itis not
too straightforward to identify experimentally, since just above the boiling point
one unsurprisinglyfindsthat correctionsdue to non-ideality (interactionsbetween
atoms,finite size ofatoms) also cause significantdeviations. However, thedegeneracy
corrections are particularly interesting; since^4 Heis a bosonand^3 He is a fermion,
thedeviations maybe expectedtobeofopposite signs. Andsoitisfound.
For allother realchemicalgases, andforhelium at more reasonable temperatures,
the value ofAis even smaller,since the massMand temperatureTbothenteras
inverse powers. For example, air at room temperature andpressurehasA≈ 10 −^5.
The nearest competitor toheliumishydrogengas,but thisboils around20 K. On the
other hand, in the free electron gas model of a metal, one uses an electron gas at the
metallicdensity. HereA1since the massissosmall,sothat thegasisdegenerate
and FD statistics must be used (see Chapter 8).
To conclude the section,let us note thatAandZare truly quantum quantities. They
dependon Planck’s constanth,andon thespinfactorG(=1for^4 He). But when the
numbers are substituted, we find that the dilute MB limit is entirelyjustified for real
gases. Hence, on that basis, the rest of the chapter is worth the effort!
6 .2 The Maxwell–Boltzmann distribution of speeds
Without really trying, wehaveinfactderivedthedistribution ofspeedsofgas
moleculesinanidealgas, thedistribution whichplaysanimportant partinthekinetic
theory of gases.
Inkinetictheory, one often requires the number ofmolecules whichhave (scalar)
velocitiesbetweenvandv+δv.Thisisgivendirectly bytheMBdistribution –it
is as easy asni=gi×fffi. The number we require is conveniently written asn(v)δv,
definingthefunctionn(v)asadensityofparticlesin speedv. Hencewehave
n(v)δv=g(v)δv·Aexp(−ε(v)/kkkBT) (6.7)