142 Chemical thermodynamics
this treatmentitis not necessary(or even convenient) togroup the states together (the
inotation of section 5 .1). Rather we discuss the occupation ofindividualone-particle
states (hence thejnotation as Chapter 1). For compactness, we choose todefinefor
eachsuchstate a quantityγγγjdefined as
γγγj=exp[(μ−εεj)/kkkBT] (13.7)Wemaynote that, withthe expectedphysicalidentifications withour earlier approach,
thisisthesameasγγγj=exp(α+βεεj), a frequently occurring quantity.
Consider one particular microstate ofthe assembly. Suppose that,inthismicrostate,
state∑ jis occupied bynnj(identical,gaseous) particles. The microstate thus has a total
nnj=NNNlparticles and a total energy
∑
nnjεεj =EEk, where the sums go over all
the one-particle statesj.The contribution ofthis one microstate toZZZG(see (13.4))is
exp(μNNNl/kkkBT)exp(−EEk/kkkBT),i.e. it is simplyequal to
∏jγγjnj (13.8)verifiedbysubstitution. Thereis one suchtermfor everymicrostate ofthegrand
assembly; and there are no restrictions whatever on thennjsince allNNNlandEEkareto
beincludedinthe sum. To obtainZZZGwe needto sum over allpossiblemicrostates,a
taskwhichsoundsdauntingbut turns out tobe amazinglyeasy.
The easy answer to the sum is different, depending on whether the identical gas
particles arefermions orbosons, andwe must treat the two cases separately.
Fermi–Dirac. Here, the Pauliexclusion principle (antisymmetric wavefunction)
tells us that only two occupation numbers are possible. We must havennj= 0 (empty
states) ornnj=1(fullstates). Hence alltheγfactorsfor eachin (13.8) must either
beγ^0 =1for emptystates orγ^1 =γforfullstates.There are no totalnumber
restrictions,so that the sum over all microstates can therefore be written as
ZZZG(FD)=∏
j( 1 +γγγj) (13.9)The appearance of1andγin everybracketinthis product ensures that everyallowable
microstate of the form prescribed by(13.8) is included in (13.9), and that it is included
once only.
Bose–Einstein. The same technique also worksfor the BE case. Theonly difference
is that now all occupation numbers arepossible, i.e.nnjcan take anyinteger value,
and there are many more microstates which must be counted. The answer, however,
is easy
ZZZG(BE)=∏
j( 1 +γγγj+γγγj^2 +γγj^3 ...) (13. 10 a)