Hard disc

The two-dimensional counterpart of a hard sphere: two equal discs of radius $r$ cannot have centre separation less than $2r$. Their pair-excluded area is $4\pi r^2$, but the second-virial excluded-area parameter per particle is half that value because a pair would otherwise be counted twice.

A circular free-area model for a disc surrounded by fixed neighbours a centre-to-centre distance $R$ away. The inscribed circle available to the moving centre has radius $R-2r$. It is an approximation to the actual curved cage boundary, not the area of a disc whose radius is $R-r$.

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.