68 Maxwell–Boltzmanngases
6 .3 The connection to thermodynamics
This is an important section. We can now calculate the properties of a gas of weakly
interacting spinless particlesinthelimitoflowdensity. These are preciselythe con-
ditionsfor anidealgas, one whichobeysthe equation ofstatePV =RTwithT
being the thermodynamic temperature. We shall find the comforting result that our
gas also obeys this equation ofstate, althoughourTwasdefinedstatisticallyfromβ.
Hence we shall fully justifythe statisticaldefinitions oftemperature andofentropy
(see section 2.3).
Thecalculation methodsfor Boltzmann statistics canbe usedagainhere, so refer-
ence should be made back to section 2.5. The onlycaution is that the expressions for
t∗are similar but not identical for the two cases. For localized particles (2.3) hasa
factor ofN!, whereas the corresponding(5.6) for agas of indistinguishable particles
does not.
6 .3.1 Internal energy andCCCV
Since wehavegone to the troubleofevaluatingZfor a monatomicspinlessgas (see
(6.6)), the quickest way toUandCVis to use Method 2 of section 2. 5. The derivation
of(2.27)dependedonlyonthedefinition ofZ,whichisthe samefor gases asfor
localizedparticles. Hence
U=Nd(lnZ)/dβ (2. 27 )=NkkkBT^2 d(lnZ)/dT (6.11)usingβ=− 1 /kkkBT.SinceZ∝T^3 /^2 (equation (6.6)), (6.11) at once gives
U=^32 NkkkBT (6.12)This is the result expected from the kinetic theorytreatment of ( 6 .9), sinceU=Nε ̄.
The internal energyUfor our perfect gas is a function ofTonly – it does not depend
onV;anditis provedtohave thevaluegivenbytheoldequipartitionlaw. The
expression forCVfollows at once
CV=dU/dT=^32 NkkkB (6.13)The heat capacityis a constant in the MB limit – a warningfor lovers of the third
law of thermodynamics that this limit cannot survive down to the absolute zero of
temperature.
- 2 Entropy
To find the entropy, we use Method 1 of section 2.5, namely a direct evaluation ofS
fromits statistical definitionS=kkkBln. For theMBgas, thecalculation proceeds