= Hard-disc excluded-area approximation
{title2=$b=2\pi r^2$}
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 $N\ln a-2\pi r^2N(N-1)/a$. 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.
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