 Solution

For a square lattice, $A=N{\ell }^2$. Substituting $\ell =\sqrt{A/N}$ into the preceding result givesThusAlthough the requested parameterization emphasizes particle and thermal parameters, $B$ necessarily includes $N$ because it is a total close-packed area. At $A=B_{\square }$, 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.