= Solution
At the <theta temperature>, use the long-chain ideal-coil relations
$$
\langle R^2\rangle\propto M,\qquad R_g^2=\frac{\langle R^2\rangle}{6}.
$$
The two stated molar masses have ratio $M_2/M_1=5$. Therefore
$$
R_{{\rm rms},2}=11.2\sqrt5\,\mathrm{nm},\qquad
\boxed{R_{g,2}=11.2\sqrt{\frac56}\,\mathrm{nm}\simeq10.2\,\mathrm{nm}.}
$$
The input $11.2\,\mathrm{nm}$ is an end-to-end root-mean-square distance, not a <radius of gyration>: omitting $\sqrt6$ would give the wrong result. The apparent missing superscript in the printed masses does not affect the factor-five ratio used here. The estimate assumes the same chain architecture and unperturbed segment statistics, and uses the long-chain rather than finite-link correction.
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