168 Dealingwith interactions
The extendedproductis not so easyto evaluate after theleadingterm, whichgives
simply unity and hence a contribution toQofVNas for the perfect gas considered
above. Theinteractions are all inthefffijs.The termsinvolvingf factors are often
groupedusingtheidea of‘clusters’, whichspecifythe number ofinteractingparticles
which are involved together. The order of a cluster is chosen to be equal to the number
ofinteracting particles minus 1. Forinstance,inthe expansion of(14.9) there are terms
withjust oneffactor (((ffrom onebracket, withfactors of 1 from everyotherbracket,
e.g.fff 12 ). These are single first-order cluster terms, since only one pair of interacting
particlesisinvolved. We can (given theform ofφij)inprinciple evaluate suchterms.
Sincefffij =0 outside a typical atomic diameter, it is not difficult to see that each
such term contributes toQan addition of orderVN−^1 bwherebis the molecular
volume. Next consider termswithtwoffactors. There are two types ofthese,for
examplefff 1222 fff 45 andfff 1222 fff 23 .The first of these is aproduct of two separate first-order
clusters, whereas the second corresponds to a single second-order cluster. These terms
similarlyeachgenerate additions toQoforderVN−^2 b^2.
Thus one can see that apower series of the formQ=VN[ 1 +a 1 (b/V)+a 2 (b/V)^2 +
···]emerges. This expansionis oneinwhichbisindependent ofV,TandN,and
inwhichwe can expect the coefficienta 1 tobeproportionaltoN(the number of
singlefterms in the expandedproduct) and to depend onTthrough the exponent
exp(−φ/kkkBT).Therefore when we come to evaluate the pressure, usingP/kkkBT=
(∂lnZZA/∂V)T,N(compare (12.7) andrememberP=−(∂F/∂V)T,N),we recover
an expansion of the form of the virial expansion, (14.4).
Actually, thewhole evaluation ofthe configurationalintegralgets very toughand
technical,andwe shallnot pursueitfurtherhere. However stretchingon theimagi-
nation is the detail,this cluster classification nevertheless does enable calculations to
bemadewhichgive usefulresultsforimperfect gases andwhichgive reliablevalues
fromfirstprinciplesfor thevirialBcoefficients of(14.4).