5 Gases: the distributions
In this chapter the statistical method outlined in section 1. 5 is used to derive the
thermalequilibriumdistributionfor a gas. The results will be appliedtoawide
varietyof physical situations in the next four chapters.
The method follows the usual four steps. Step I concerns the one-particle states.
These werediscussedinthe previous chapter, andinfact nofurther specificdiscussion
iscalledforuntilwecometoparticularapplications. Theonevitalfeaturetorecallhere
is that the states are very numerous. Hence we shall discuss gases in terms of a grouped
distribution as explained in section 5.1. This discussion of possible distributions is
step II of the argument. Step III involves countingmicrostates, i.e. quantum states for
theN-particle assembly. In section 5 .2 we briefly review the quantum mechanics of
systems containingmore than oneidenticalparticles. The countingofmicrostatesis
then outlined in section 5 .3 and finally(step IV) we derive the thermal (most probable)
distribution in section 5 .4.5 .1 Distribution in groups
Asdiscussedinthelast chapter, the number ofrelevant one-particle statesinagas
is enormous, often very much greater than the numberNof gas particles. Under
suchconditions, thedefinition ofadistributionin states containsfar too muchdetail
for comfort. Insteadwe use effectivelythedistributioninlevels, {ni}asdefinedin
section 1.4.
Thepointisthatitdoes no violence to thephysics togroupthe states. In theform
ofthedistribution usedfor computation theithgroupistaken to containgistates of
average energyεi. The only difference from the true specification of the states is that
inthegroupeddistribution allgistates are consideredtohavethesameenergyεi,
rather than their correct individual values. Thisgrouped distribution is illustrated in
Fig. 5 .1. Let us note several points before proceeding.
1 .Thegroupingis one ofconvenience, not ofnecessity. We are choosingto use
large numbers of statesgitaken together and also large numbers of particlesni
occupying these states. Thiswillenableusbelow to use the mathematics oflarge
numbers (e.g. Stirling’s approximation) in discussingniandgiin order to work
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