= 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> is
$$
\frac F{k_BT}\sim\frac{R^2}{N\ell^2}+\frac{v_2N^2}{R^2}.
$$
The first term is the entropic stretching cost and the second counts repulsive monomer contacts. Minimization gives $2R/(N\ell^2)-2v_2N^2/R^3=0$, hence
$$
R\sim(v_2\ell^2)^{1/4}N^{3/4},\qquad\boxed{\nu_{2d}=3/4.}
$$
For $v_2$ 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.
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