Statistical Physics, Second Revised and Enlarged Edition

Summary 71

2.Extensivity.Next,afew wordsabout that extraN!in ( 6 .1 6 b). For our treatments
of gases or of localized particles to make sense, we require that the extensivity is
correct. Ifwedoublethe amount ofthesubstance, we expectfunctionslikeU,S
andFalso todouble, solongas thedoublingisdone at constantTandatconstant
densityN/V.
Nowforasolid,the equationF=−NkkkBTlnZ, (2.23), satisfies this require-
ment. The energy levelsεεjdependonlyon the specificvolume,V/N.Hence
Z=

∑

exp(−εεj/kkkBT)is unchanged at givenTandN/V. ThereforeF∝Nas
required.
However, for agas the extraN!term is needed. Since all thegas particles are
competing for the same space, we haveZ∝V,but independent ofN,(6.6).It
isnowZ/Nwhichis unchangedwhenNisalteredat constantdensity.Therefore
(6.16):F=−NkkkBT(lnZ/N+ 1 )has the correctproperties. The bracket remains
unchanged and againF∝N.
3 .TheGibbs paradox.Thiswasaclassicalproblem ofindistinguishability, perhaps a
paradox no longer. Consider a box of fixed volumeVand temperatureTcontaining
amixture of two ideal gases A and B. We adopt a notation in which the subscript A
refertothepropertieswhichgasAwouldhaveintheabsenceofgasB;andsimilarly
for subscript B. If the two gases are different then they behave independently, they
occupydifferent states. Hence=A×B,andS=SA+SB,F=FFA+FFFB,
P=PA+PB,etc. The twogasesbehave asiftheother were not present. Even
an isotopic mixture, say of^3 Heand^4 He, behaves in this way.
However, thesituationisdifferentifthe two gasesAandB areidentical.Itis true
thatP= 2 PA,butS= 2 SA,andF= 2 FFA.Themolecules are now competingfor
states, so the statistical properties of the second gas are modified by the existence
of the first. In fact we can see from ( 6 .14) thatS= 2 SA−NkkkBln 2 (arisingfrom
theNlnNterm,i.e.from theN!)whereNisthe totalnumber ofAmolecules.
The degree of disorder is lessened by the competition.

6 .4 Summary

Thischapterderives the properties ofanidealmonatomicgasinthedilutelimit.

1. Thedilutelimitisfoundtobeavalidapproximationfor allrealchemicalgases.


2. Thepartition functionZ,summingBoltzmann factors (exp(−εi/kkkBT))over all
   states, againplays a usefulrole.


3. MB statisticsleadsdirectlyto the speeddistribution ofgas molecules,firstderived
   by Maxwell.


4. TheMBgasisshown tohave the equation ofstatePV=RT.


5. Our statistical temperature (based onβ =− 1 /kkkBT)is thus identical to the
   thermodynamic Kelvin temperature.


6. The kinetic energy of the gas molecules in thermal equilibrium gives an illustration
   oftheclassicalprincipleofequipartition ofenergy.