Statistical Physics, Second Revised and Enlarged Edition

6

Maxwell–Boltzmann gases

As a first application of the groundwork of the two previous chapters, we consider
thesimplest situation. Thisis a gasfor whichthe Maxwell–Boltzmann (dilute)limit
isvalid. Furthermore we shallconsider onlymonatomicgasesinthe present chapter,
leaving the complications (and the interest!) of diatomic gases until Chapter 7. First
we needtodecidethe practicalrange ofvalidity oftheMBlimit. Next we canderive
theMBdistribution ofspeedsinthegas. Finallywe mayworkout thethermodynamic
properties of the gas and compare these to results from the ideal gas laws.

6 .1 The validityof the Maxwell–Boltzmann limit

The MB distribution ((5.12) and (5.13)) may be written as

fffi=Aexp(−εi/kkkBT) (6.1)

with the constantA = exp(α)= 1 /B. Our statistical method so far(and until

Chapter 14) applies onlytoperfect gases,inthe sense ofgases whose particles are

weakly interacting. Butinaddition theMBdistribution applies onlyto agas suffi-

ciently dilute that all the occupation numbersniaremuchless thanthenumberof

statesgi,i.e. that allfffi 1 .Taking the groundstate energy as the energy zero (or

close toit), thedilute condition therefore means that the constantAin ( 6 .1) should

be 1.

Clearlyinorder to explorefurther we needto calculateA.Thisisdone usingits

associated condition (5.7):

N=

∑

i

ni

=

∑

i

giiifffi

=A

∑

i

giexp(−εi/kkkBT)

=AZ (6.2)

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