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A three-dimensional ideal chain of $N$ independent segments of length $b$ has $⟨R^2⟩=N{b}^2=Lb$. The isotropic small-force susceptibility gives $⟨z⟩=F⟨R^2⟩/(3k_{B}T)$, so $k_s=3k_{B}T/(Lb)$. For a long worm-like chain, the statistical segment is $b=2l_p$. This stiffness describes entropic end-to-end extension, not axial stretching of the material backbone, whose spring constant is $EA/L$.

For a planar polymer with $N$ segments of length $\ell$, approximate its coil free energy by $F/(k_{B}T)\sim R^2/(N{\ell }^2)+v_2{N}^2/R^2$. The first term is the entropic stretching cost of an ideal chain, while $v_2$ is an excluded-area coefficient. Minimizing gives $R^4\sim v_2{\ell }^2{N}^3$ and size exponent $\nu =3/4$. For $v_2\sim {\ell }^2$, this becomes $R\sim \ell N^{3/4}$. A parameter multiplying $N^2/R^2$ must carry area units; using an unsquared segment length there would be dimensionally inconsistent.