= Solution
\b[Solvent quality.] The <Flory interaction parameter> $\chi$ measures the free-energy cost of polymer–solvent contacts relative to like contacts. In the simplest lattice model,
$$
\chi=\frac{z_c}{k_BT}\left[\epsilon_{ps}-\frac{\epsilon_{pp}+\epsilon_{ss}}2\right],
$$
where $z_c$ is the coordination number and the $\epsilon$ values are contact energies. For lattice-site volume $v_0$, the effective two-segment <excluded volume> is $v_{\mathrm{eff}}=v_0(1-2\chi)$. Its sign determines the leading tendency of a long dilute flexible chain.
For $\chi<1/2$, $v_{\mathrm{eff}}>0$: a <good solvent> favors a swollen coil because segment encounters are effectively repulsive. For $\chi=1/2$, the leading attraction and repulsion cancel, giving approximately <ideal chain> statistics. For $\chi>1/2$, effective attraction favors contraction in a <poor solvent>, eventually producing a compact <polymer globule> under sufficiently poor conditions. This does not mean that every finite-chain solution with $\chi>1/2$ is automatically phase-separated; that also depends on composition and chain length.
The condition $\chi=1/2$ is the <theta temperature> condition for that polymer–solvent pair. For an energetic model $\chi=A/T$, it occurs at $T=\Theta=2A$. More generally, if $\chi=A/T+B$, it occurs when $A/\Theta+B=1/2$. It is not the absence of all intermolecular interactions and does not require the polymer and solvent to be chemically identical.
\b[Chain dimensions.] Interpret $R$ as the root-mean-square <end-to-end distance of a polymer chain>. Let $a$ denote the supplied statistical step length, calibrated to the same count $N$ as the <degree of polymerization>. At the theta condition, random-walk statistics give
$$
\boxed{R_\theta=aN^{1/2}}.
$$
For a <good solvent>, the <Flory approximation> balances entropic stretching and two-segment interaction:
$$
\frac{F(R)}{k_BT}\sim\frac{R^2}{Na^2}+\frac{v_{\mathrm{eff}}N^2}{R^3}.
$$
Minimization gives $R^5\sim v_{\mathrm{eff}}a^2N^3$, hence
$$
\boxed{R_{\mathrm{good}}\sim a\left(\frac{v_{\mathrm{eff}}}{a^3}\right)^{1/5}N^{3/5}}.
$$
Thus the familiar good-solvent scaling is $R\sim aN^{3/5}$ when its nonuniversal solvent-quality amplitude is suppressed. With the additional lattice estimate $v_0\simeq a^3$, the more explicit form is $R\sim a(1-2\chi)^{1/5}N^{3/5}$. This swollen-coil estimate is used away from the narrow theta crossover; it must not be extrapolated to predict $R=0$ as $\chi\uparrow1/2$. The crossover occurs when $(1-2\chi)N^{1/2}$ is of order one. Nearer theta, the ideal-coil size supplies the appropriate limit.
\b[Numerical sizes.] The theta value is
$$
\boxed{R_\theta=(0.67\,\mathrm{nm})\sqrt{1500}=25.9\,\mathrm{nm}}.
$$
For $\chi=0.3$, the plain good-solvent scaling gives
$$
\boxed{R_{\mathrm{good}}\sim(0.67\,\mathrm{nm})(1500)^{3/5}=53.9\,\mathrm{nm}}.
$$
If the solvent-quality amplitude is retained with $v_0=a^3$, it instead gives
$$
\boxed{R_{\chi=0.3}\sim(0.67\,\mathrm{nm})(0.4)^{1/5}(1500)^{3/5}=44.9\,\mathrm{nm}}.
$$
The distinction is an explicitly stated modeling convention: the question does not give a microscopic site volume or an absolute swollen-coil prefactor. Both estimates describe a coil larger than the theta coil. Here $(1-2\chi)\sqrt N\simeq15.5$, so a swollen-coil estimate is reasonable. Neither result is an exact microscopic prediction from the supplied data alone.
\b[Overlap concentration.] The <polymer overlap concentration> corresponds to one chain per coil volume. Using the scaling convention $V_{\mathrm{coil}}\sim R_\theta^3$ gives
$$
n_p^*\sim R_\theta^{-3}=5.72\times10^{22}\,\mathrm{m^{-3}},\qquad
n_m^*\sim\frac{N}{R_\theta^3}=8.58\times10^{25}\,\mathrm{m^{-3}}.
$$
The first is chain <number density>; the second counts <repeat units>. Thus the repeat-unit <molar concentration> is approximately $0.143\,\mathrm{mol\,L^{-1}}$. If the lattice-site volume is taken as $v_0=a^3$, this is equivalently
$$
\boxed{\phi^*\sim\frac{Nv_0}{R_\theta^3}\simeq N^{-1/2}=2.58\times10^{-2}}.
$$
For a mass-concentration estimate, a <polystyrene> repeat unit $\mathrm{C_8H_8}$ has <molar mass> about $104.15\,\mathrm{g\,mol^{-1}}$, so this scaling convention gives
$$
\boxed{c^*\sim14.8\,\mathrm{g\,L^{-1}}}.
$$
There is no sharp coil surface fixing an exact numerical prefactor. For example, using the often adopted geometric definition $V_{\mathrm{coil}}=4\pi R_g^3/3$, with the <ideal chain> <radius of gyration> $R_g=R_\theta/\sqrt6=10.6\,\mathrm{nm}$, gives $c^*\simeq52\,\mathrm{g\,L^{-1}}$. The invariant content is one chain per coil volume and the theta scaling $\phi^*\propto N^{-1/2}$; the radius and coil-volume convention must accompany a numerical value.
\b[Osmotic pressure.] In the given lattice expression, $\phi$ is a dimensionless polymer <volume fraction>, not a dimensional <number density>, and $f$ is the mixing <free energy> per lattice site. Introduce $v_0$ to obtain a physical <osmotic pressure>. If $M=N_s+NN_p$ is the number of sites and $\phi=NN_p/M$, differentiation of $F_{\mathrm{mix}}=Mf(\phi)$ with respect to the solvent count at fixed chain count gives
$$
\Delta\mu_s=f-\phi f'.
$$
Balancing this change in solvent <chemical potential> against the pressure contribution $v_0\Pi$ gives $\Pi=(\phi f'-f)/v_0$. For composition-independent $\chi$,
$$
\frac{f'}{k_BT}=\frac{\ln\phi+1}{N}-\ln(1-\phi)-1+\chi(1-2\phi),
$$
and therefore the <osmotic pressure in Flory-Huggins solution theory> is
$$
\boxed{\Pi=\frac{k_BT}{v_0}\left[\frac\phi N-\ln(1-\phi)-\phi-\chi\phi^2\right]}.
$$
Expanding the <logarithm> as $\phi\to0$ gives
$$
\boxed{\Pi=\frac{k_BT}{v_0}\left[\frac\phi N+\left(\frac12-\chi\right)\phi^2+\frac{\phi^3}{3}+O(\phi^4)\right]}.
$$
The first term is $n_pk_BT$ with $n_p=\phi/(Nv_0)$ now denoting chain <number density>: a whole chain supplies one translational contribution. The sign of the quadratic correction reproduces the solvent-quality classification. It is positive for a <good solvent>, zero at the <theta temperature>, and negative for a <poor solvent>. This is the same two-segment interaction that swells or contracts a single coil. At theta, a positive third-order contribution remains; the chain is approximately ideal in its large-scale conformation, not a gas of segments with no interactions at all.
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