= Solution
At the theta condition, effective <excluded volume> approximately cancel and a freely jointed <ideal chain> has $\langle R^2\rangle=Nb^2$. With the stipulated segment length $b=2a$,
$$
\boxed{R_{\rm rms,\theta}=2a\sqrt N.}
$$
In a <good solvent> repulsive <excluded volume> swells the coil, giving $R\sim bN^\nu$ with $\nu\simeq0.588$ in three dimensions, or the <Flory approximation> $\nu=3/5$. In a <poor solvent> effective segment attraction favours a compact globule, whose linear size scales as $bN^{1/3}$ at roughly fixed internal density. The latter are scaling laws with model-dependent prefactors. A real chemical <monomer> need not be a statistical <Kuhn length>; using $2a$ as the independent segment length is part of this question's idealized model.
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