4 Gases: the density of states
In the last two chapters we have applied the statistical method as outlined in section 1. 5
to an assemblyofdistinguishable(localized) particles. We now embarkupon the
application of the same method to gases. This involves two new ideas.
The first concerns the one-particle states (step I of section 1.5). A gas particle is
confinedto alarge macroscopicbox ofvolumeV,whereas alocalizedparticleis
confined essentially to an atomic cell of volume(V/N). As a direct result, the energy
levelsofagas particlewill be extremelyclose together, withastronomicalnumbers
of them beingoccupied at anyreasonable temperature. This is in marked contrast
with the localized case, in which the energy levels are comparatively far apart and
the occupation ofonlyafew needtobe considered. Actually it turns out that one can
make a virtue of the necessityto consider verylarge numbers of states. This topic is
the subject of the present chapter.
The secondideaishow todealwiththeindistinguishabilityofgas particles. This
is vital to the countingof microstates (step III of section 1. 5 ). The ideas are essen-
tially quantum mechanical, explaining from our modern viewpoint why a correct
microscopic treatment ofgases was a matter ofsuchcontroversyand difficultyin
Boltzmann’s day. The countingof microstates and the derivation of the thermal
equilibrium distribution for a gas will be discussed in the following chapter.
4.1 Fittingwaves into boxes
We aregoingtobeabletodiscuss manytypes ofgaseous assembliesinthe nextfew
chapters, from hydrogengas to helium liquid, from conduction electrons to black-
body radiation. However, there is a welcome economy about the physics of such
diverse systems. In eachcase, the state ofagas particle canbediscussedin terms
of a wavefunction. And the basic properties of such wavefunctions are dominated by
pure geometry only, as we will now explain.
Consider a particlein one ofits possible statesdescribedbya wavefunction
ψ(x,y,z). The time-dependence of the wavefunction need not be explicitlyincluded
here, since in thermal physics we only need the stationary states (i.e. the normal
modes or theeigenstates) ofthe particle. Whatdoweknow about suchstates when
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