174 Canonical ensemble
5 10
r-50005001000u^0(r)+u( 1
r)u 0 (r)u 1 (r)0Fig. 4.6Plot of the potentialu 0 (r) +u 1 (r). The dashed line corresponds tor= 0.= 2πNρ∫∞
0r^2 u 1 (r)θ(r−σ) dr= 2πNρ∫∞
σr^2 u 1 (r) dr≡−aNρ,where
a=− 2 π∫∞
σr^2 u 1 (r) dr > 0. (4.7.30)Sinceu 1 (r)<0,amust be positive. Next, in order to determineA(0), it is necessary
to determineZ(0)(N,V,T). Note that ifσwere equal to 0, the potentialu 0 (r) would
vanish, andZ(0)(N,V,T) would just be the ideal gas configurational partition func-
tionZ(0)(N,V,T) =VN. Thus, in the low density limit, we might expect that the
unperturbed configurational partition function, to a good approximation, would be
given by
Z(0)(N,V,T)≈VavailableN , (4.7.31)
whereVavailableis the total available volume to the system. For a hard sphere gas,
Vavailable< Vsince there is a distance of closest approach between any pair of parti-
cles. Smaller interparticle separations are forbidden, as the potentialu 0 (r) suddenly
increases to∞. Thus, there is anexcludedvolumeVexcludedthat is not accessible to
the system, and the available volume can be reexpressed asVavailable=V−Vexcluded.
The excluded volume, itself, can be written asVexcluded=Nbwherebis the excluded
volume per particle. In order to see what this excluded volume is, consider Fig. 4.7,