Hard-disc excluded-area approximation ($b=2\pi r^2$)

At low density, sequential insertion gives a configurational integral approximated by ${\prod }_{j=0}^{N-1}(a-4j\pi r^2)$. Expanding its logarithm to first order in occupied area gives $N\ln a-2\pi r^{2}N(N-1)/a$. Replacing $N-1$ by $N$ and matching a free-volume expression gives $(a-Nb{)}^N$, with $b=2\pi r^2$. This matching is controlled only at low density; its pole at packing fraction $1/2$ is not a physical close-packing limit.