= Solution
For a square lattice, $A=N\ell^2$. Substituting $\ell=\sqrt{A/N}$ into the preceding result gives
$$
F=-2Nk_BT\ln\left[\frac2\Lambda\frac{\sqrt A-\sigma\sqrt N}{\sqrt N}\right].
$$
Thus
$$
\boxed{B_{\square}=N\sigma^2,\qquad C=2/\Lambda.}
$$
Although the requested parameterization emphasizes particle and thermal parameters, $B$ necessarily includes $N$ because it is a total close-packed area. At \b[$A=B_{\square}$], neighbours touch, the cell free area vanishes and this classical cell model gives $F\to+\infty$. The two logarithms in the printed form are dimensionful separately but their combined argument is dimensionless. For $A<B_{\square}$ the presumed square lattice is geometrically impossible.
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