= Solution
For the stipulated <effective-sphere polymer drag>, the radius is the end-to-end RMS distance, $R=2a\sqrt N$, rather than the single-monomer radius. Its <Stokes law> and translational <diffusion coefficient> are
$$
\zeta_2=6\pi\eta R=12\pi\eta a\sqrt N,\qquad
D_2=\frac{k_BT}{12\pi\eta a\sqrt N}.
$$
Under the same three-dimensional RMS convention as used above,
$$
\boxed{d_2=\sqrt{6D_2t}=\sqrt{\frac{k_BTt}{2\pi\eta a\sqrt N}},\qquad
\frac{d_2}{d_1}=\frac{N^{1/4}}{\sqrt2}.}
$$
Thus for sufficiently long chains ($N>4$), this model gives a larger displacement than independent <monomer> drag: <solvent> motion couples the parts of a coil, reducing its effective friction relative to $N$ isolated moving beads. The $D\propto N^{-1/2}$ theta-solvent scaling is consistent with the dilute-coil hydrodynamic scaling underlying the <Zimm model>, but treating a permeable fluctuating coil as a solid sphere with radius exactly $R_{\rm rms}$ gives only an order-one hydrodynamic approximation. The <hydrodynamic radius> and end-to-end radius are not generally equal. Neither model by itself restores the long-time constraints of a truly entangled solution.
If the one-dimensional contour statistic in part (iii) is used literally, the comparison of this three-dimensional $d_2$ with that contour RMS is $\sqrt{3/2}\,N^{1/4}$; it compares different coordinates. Using mean radial displacements in both three-dimensional models instead changes both prefactors equally and leaves their ratio unchanged.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2021/paper-2-polymer-scaling.png]
{title=Original idealized polymer size and translational-diffusion scalings under explicitly different drag models}
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