 Solution

Write $\ell$ for a segment length and $v_2$ for the excluded-area coefficient, so that the interaction term has the correct dimensions. For a coil of planar size $R$ and area of order $R^2$, the two-dimensional Flory approximation isThe first term is the entropic stretching cost and the second counts repulsive monomer contacts. Minimization gives $2R/(N{\ell }^2)-2v_2{N}^2/R^3=0$, henceFor $v_2$ of order ${\ell }^2$ this becomes $R\sim \ell N^{3/4}$. The parameter multiplying $N^2/A$ in a two-dimensional interaction energy must have dimensions of area; it cannot simultaneously denote an unsquared bond length.