Statistical Physics, Second Revised and Enlarged Edition

Idealgases in thegrand ensemble 145

1 3.3.3 Thermodynamics of an ideal MB gas

The gas laws for the dilute (MB) ideal gas can now be rederived in short order from
the grandcanonicalensemble approach. In (13.11) wehavefoundthe appropriate
expressionforZZZG,thegrandpartitionfunction. We can now substituteZZZGinto the
basic equation, (13. 5 ), to findPVand hence other thermodynamic quantities. For
FD andBE gases, the same methodworkswell,but the mathematics getinvolved.
Howeverfor anidealgasintheMBlimit, the answers are elementaryandimmediate
(so we may as well do it!).
Equation (13.5) tells us thatPV=kkkBT ln ZZZG.The great simplificationfor the
MB limitisthatlnZZZGworks out so easily. Takingthe logarithm of (13.11) we obtain

lnZZZG=

∑

j

γγγj=exp(μ/kkkBT)

∑

j

exp(−εεj/kkkBT)

=exp(μ/kkkBT)Z (13.12)

The resultisalarminglysimple. Thelogarithmofthe grandpartitionfunctionisafac-
tor exp(μ/kkkBT)timesZ(V,T),theordinary‘sum-over-states’partitionfunction,first
introduced in Chapter 2 (equation (2.24)) and which played a central role in our earlier
discussion of the MB gas in Chapter 6 (e.g. section 6 .1).Zisdeterminedonlybythe
temperature (energyscalekkkBT)and bythe quantum states of one particle, involving
theboxsizeVthrough the allowed quantum energies of the particle. We recall from
Chapters 6 and 7 that for a spinless monatomicgasZ=V( 2 πMkkkBT/h^2 )^3 /^2 .For
a more complicated perfectgas, this expression, since it derives from translational
kinetic energy only, must be multiplied by another factorZZZintto allow for the other
internaldegrees offreedom (spin, rotation, vibration etc). For agivengas,ZZZintfor
rotation and vibration varies with temperature only(see Chapter 7), whilst for spin it
will give a constant factorG. Thus for any perfect gas we have

Z(V,T)=constVTn

where 2nis the number of degrees of freedom excited.
Let us nowlookat thethermodynamicfunctions. In thegrandcanonicalensemble
approach, we are startingoff byfixingμ,VandT. That enables us to evaluateZZZG
andhencelnZZZGas in(13.12)above, and thus

PV=kkkBTexp(μ/kkkBT)Z(V,T) (13.13)

Equation (13.6) now gives the prescription for calculation of other thermodynamic
quantities. ForinstanceS,PandNare obtainablefromitbysimple partial differen-
tiation. We shallnotlabour overShere,but this approachwould directlyrederive
the Sackur–Tetrode equation (6.1 5 ) for an ideal monatomic gas, and it would lead
usinto theheat capacitydiscussions ofChapter 7, withtheindexnappearingfrom
differentiation withrespect toT.Anyfurtherderivation ofPis unnecessaryhere,