Real imperfect gases 165theory,sothat we aregettingbackto Boltzmann. Step twoisthen to start to apply
the classical treatment to an imperfect gas, although we shall give up when the going
gets too technical.
1 4.4.1 Classical statistical mechanics
The classical approach to statistical physics was to use the concept ofphase space.
Considerfirst the state ofone particle, a gas moleculeinabox. Its classicalstateis
specified bythe position and momentum of the particle, somethingwhich Heisen-
berg’sUncertaintyPrinciplenowadaystellsuscanonlybedonewithlimitedprecision.
Phase spaceisdefinedas a combinedposition andmomentum space. For thephase
space of oneparticle, there are six dimensions to the space, and the state of theparticle
is then specified by a single point in the space. For a quantum mechanic, there would
be worriesifthepoint were not cloudy(uncertainin positioninphase space) overa
volume of abouth^3. This is because we expectpxx∼hand similarly for they
andzpairs ofdimensions, givingasix-dimensionalvolume uncertainty ofh^3.
Itisinstructive to seehow thisidea tiesinwithourdiscussion ofthequantum
states for a particle in an ideal gas in Chapter 4. There we saw that the states could
be representedbypointsink-space witha separation of 2 π/a(aisacubicalbox
dimension andperiodicboundaryconditions are used). Thisgave an equalvolume
occupied ink-space for every state of( 2 π)^3 /V,whereV=a^3 is the volume of the
cube. Hence the resultofChapter 4: there areV/( 2 π)^3 states per unitvolumein
k-space, whatever thesize andshapeofthe realspace volumeV.
How does this translate into phase space? Consider a small volume in phase space
δv=δpxδpyδpzδxδyδz.Thisisthe product ofa smallvolumein realspace (δxδyδz)
and an element of momentum space(δpxδpyδpz). But, usingde Broglie’s relationp=
k, the momentum space volume is simply^3 times thek-space volume. Therefore,
usingthek-space resultabove, we can at once workout that thesix-dimensional
volumeδvcontains a number of states equal toδv/[^3 ( 2 π)^3 ]=δv/h^3.
Thus, as suggested from hand-waving earlier, each and everyh^3 volume of
phase spacehas one quantum state associatedwithit. Thisis a remarkablysimple
geometrical result.
The classical treatment for the statistical physics of an ideal gas then proceeds as
follows. Themicrostate (state ofthe assemblyofNparticles) canbe specifiedby
a single point in a 6N-dimensionalphase space; thispoint defines the momentum
andposition ofeachandevery particle. The energy constraint ofthemicrocanonical
ensemble means that the representativepoint must lie on a ( 6 N–1)-dimensionalcon-
stant energy ‘surface’ in this space. The oldaveragingpostulatewas to giveequal
weight to equal‘areas’ofthis surface. From a suitable generalization ofdiscussionin
the previous paragraph,itis now easyto see that this use ofvolumesinphase space as
the statistical weight for classical microstates is identical to the quantum assignment
ofequalweighttoeachindividualmicrostate. The connectionisthat equalvolumes
h^3 Nofphase space eachcontain onequantum state.