Statistical Physics, Second Revised and Enlarged Edition

Real imperfect gases 167

is evaluatedas theintegral

ZZA=

1

h^3 N

1

N!

∫

exp(−E/kkkBT)dp 1 ...dpNdr 1 ...drN (14.5)

We use the contracted notationdrifordxidyidzi,the volume element for theithpar-
ticle, and similarly for its momentum elementdpi.The factorh^3 Nis the phase space
to statesfactor mentionedabove. TheN!factor comesfrom theindistinguishability
of the identical gas molecules, since without it the same state would be countedN!
timesintheintegral.
The assemblyenergy is workedout as the sum ofkineticandpotentialenergy,i.e.

E=

∑

i

p^2 i

2 M

+(r 1 ,r 2 ...rN) (14.6)

When (14.6) is substituted back into (14. 5 ), the integral splits into two parts, a space
part anda momentum part. The momentum partis preciselythe same as thatfor an
idealgas, andgives the product of 3Nidentical one-dimensional integrals of the form
∫∞

−∞

∫∫

exp(−p^2 / 2 MkkkBT)ddp

wherepis a momentum component of one particle. This integral is readily evaluated
(III 0 in AppendixC)as( 2 πMkkkBT)^1 /^2 .Hence the partition function, (14.5), can be
written as

ZZA=

1

h^3 N

1

N!

·( 2 πMkkkBT)^3 N/^2 ·Q (14.7)

whereQis called theconfiguration integral, defined as

Q=

∫

exp(−/kkkBT)dr 1 ,r 2 ...drN (14.8)

Itisanintegralover thewholeofa3N-dimensionalrealspacegivingthe positions
ofalltheNparticles (so-called configuration space). We maynote at once that for
a perfect gas, i.e. one for whichis zero for all particle positions, the integrand in
(14.8)isunityandhenceQ=VN.We thus recover the usualresult (see section 12.2):
ZZA=ZN/N!,with Z=V( 2 πMkkkBT/h^2 )^3 /^2.
Intheinteracting situation, the configurationalintegraldependsontherelative posi-
tions ofthe particles. Thewaythe storynow unfoldsis to concentrate on aninteraction
functionfffijbetween apair ofparticlesiandj, defined byfffij=[exp(−φ(rij)/kkkBT)− 1 ].
Thisfunctionis arrangedso thatit equals zero atlarge separations,itis positive at
moderateseparations(whenthemoleculesattract)anditequals–1whenthemolecules
are very close (when the particles’ hard cores exclude each other). Equation (14.8)
thenbecomes

Q=

∫ ∏

all pairs

( 1 +fffij)dr 1 ...drN (14.9)