Statistical Physics, Second Revised and Enlarged Edition

64 Maxwell–Boltzmanngases

where thepartitionfunctionZisdefinedas the sum over allone-particle states of
the Boltzmann factors exp(−εi/kkkBT), just as in the Boltzmann statistics of localized
particles (section 2.4). Hence the problems ofcalculatingAandZareidentical.To
make progress we now needthedetailsofgi,andthat was thetopicofChapter 4.
To be specific, consider a monatomic, zero-spin gas such as^4 He. The states of a
gas particle are then precisely those of ‘fitting waves into boxes’, (4.4) or (4.5) with
G=1. We have

g(k)δk=V/( 2 π)^3 · 4 πk^2 δk (6.3)

as the number of states betweenk andk+δk, the wavevectork goingfrom
0to∞.
Withinthedensityofstates approximation, the partitionfunctionisthen calculated
as an integral

Z=

∫∞

0

∫∫

V/( 2 π)^3 · 4 πk^2 exp(−ε(k)/kkkBT)dk (6.4)

Theintegralmaybe evaluatedin (atleast) two ways. Oneis to transform thedensity
ofstatesfromkto energyε,i.e. to evaluate

Z=

∫∞

0

∫∫

g(ε)exp(−ε/kkkBT)dε

withg(ε)precisely that worked out in section 4.3 (see (4.9)). The other entirely
equivalent routeis to use thedispersion relationε(k)=^2 k^2 / 2 M, (4.7), to transform
the energyin ( 6 .4), leavinga tractable integral overk. Usingthe latter method, we
obtain

Z=V/( 2 π)^3 · 4 π

∫∞

0

∫∫

k^2 exp(−bk^2 )dk (6.5)

with b = ^2 /( 2 MkkkBT) = h^2 /( 8 π^2 MkkkBT). The integral in (6. 5 )ispre-
cisely equalto the standardintegralIII 2 discussedin AppendixC,sothat the
expressionbecomes

Z=V/( 2 π)^3 · 4 π·(III 2 /III 0 )·III 0

=V/( 2 π)^3 · 4 π·( 1 / 2 b)·(π/ 4 b)^1 /^2

=V( 2 πMkkkBT/h^2 )^3 /^2 (6.6)

Equation (6.6) is a central result for the MB gas, and we shall use it later in the
chapter to calculate thethermodynamicfunctions ofthegas. Meanwhile we return to
the question of the validity of MB statistics. Having calculatedZ, we have effectively
calculatedthe constantA=N/Z(equation (6.2)). For MB statistics to be valid we
requireA1.