= Solution
For a long <ideal chain> of $N$ statistical segments of <length> $b$, $R_G^2=Nb^2/6$. At fixed repeat chemistry and statistical segment <length>, <molar mass> is proportional to $N$. The <molecular-weight scaling of an ideal-chain gyration radius> thus gives
$$
\boxed{R_G(10^6)=120\,\mathrm{nm}\sqrt{\frac{10^6}{10^5}}=379.5\,\mathrm{nm}\simeq3.8\times10^2\,\mathrm{nm}.}
$$
The supplied size and ideal statistics are used as stipulated; an independently tabulated coil size would not replace the given baseline.
<solvent quality> changes the balance of segment–segment and segment–<solvent> contacts. In a <good solvent>, excluded-volume repulsion swells a sufficiently long isolated chain, with $R_G\propto N^\nu$, $\nu\simeq0.588$ in three dimensions; the <Flory approximation> gives $\nu=3/5$. At theta conditions the effective two-segment <excluded volume> vanishes and the ideal exponent $1/2$ is recovered for long chains. A <poor solvent> favors a compact <polymer globule>, whose large-chain size scales approximately as $N^{1/3}$ at fixed internal <density>. <solvent> and <temperature> can also change local <persistence length> and hence the prefactor. These different conformations need not have the same radius at the same <molar mass>.
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