= Solution
\b[Meaning of the mixing <free energy>.] The supplied expression is the <Flory–Huggins solution theory> mixing <free energy> \i[per lattice site], or per solvent-sized reference cell. Although the question calls it “per particle,” it cannot mean per <polymer> molecule: one chain occupies $N$ sites. Let the site volume be $v_0$, the total number of sites be $M=V/v_0$, and the chain and solvent counts be $n_p,n_s$, so
$$
M=Nn_p+n_s,\qquad\phi=\frac{Nn_p}{M},\qquad F_{\rm mix}=Mk_BT\,g(\phi),
$$
where
$$
g(\phi)=\frac\phi N\ln\phi+(1-\phi)\ln(1-\phi)+\chi\phi(1-\phi).
$$
The first term is the <polymer>'s translational <entropy of mixing>. Its factor $1/N$ expresses that $N$ connected segments translate as one molecule. The second is the solvent's translational mixing <entropy>. Both are negative in the mixing <free energy> for $0<\phi<1$, favoring mixing.
The final term is a mean-field excess contact contribution. The <Flory interaction parameter> $\chi$ measures the relative preference for unlike polymer-solvent contacts compared with like contacts. In the simplest lattice model with coordination number $z$ and pair energies $\epsilon_{ps},\epsilon_{pp},\epsilon_{ss}$,
$$
\chi=\frac z{k_BT}\left[\epsilon_{ps}-\frac{\epsilon_{pp}+\epsilon_{ss}}2\right].
$$
Positive $\chi$ penalizes unlike contacts and promotes separation; negative $\chi$ favors them. Effective real-solution values can also include non-enthalpic effects, so their <temperature> dependence need not be exactly a constant divided by $T$. The subsequent derivatives assume composition-independent $\chi$ as in the supplied model.
\b[<Osmotic pressure> and the special value one half.] At fixed chain count, $\partial\phi/\partial V=-\phi/V$. Differentiating $F_{\rm mix}=(V/v_0)k_BTg(\phi)$ with respect to volume, relative to the pure-solvent reference, gives
$$
\pi=\frac{k_BT}{v_0}[\phi g'(\phi)-g(\phi)].
$$
The resulting <osmotic pressure in Flory-Huggins solution theory> is
$$
\boxed{\frac{\pi v_0}{k_BT}
=-\ln(1-\phi)-\left(1-\frac1N\right)\phi-\chi\phi^2
=\frac\phi N+\left(\frac12-\chi\right)\phi^2
+\frac{\phi^3}{3}+O(\phi^4).}
$$
The leading term is the ideal <osmotic pressure> of chains, since their <number density> is $\phi/(Nv_0)$. At $\chi=1/2$ the second-virial term vanishes: effective two-segment attraction and excluded-volume repulsion balance. This is the <theta solvent> condition for a sufficiently long dilute chain; the positive third-order contribution remains, so a finite-concentration solution is not a completely noninteracting gas. The site-volume factor is necessary to give physical <pressure> units.
For $\chi<1/2$, the effective pair <excluded volume> is positive. In a <good solvent>, the isolated chain expands relative to a Gaussian coil, with $R\sim bN^\nu$, $\nu\simeq0.588$ or $3/5$ in the <Flory approximation>. For $\chi>1/2$, attraction wins in a <poor solvent> and a sufficiently long chain collapses toward a dense <polymer globule>, with asymptotic size $R\propto N^{1/3}$. Close to the theta condition, finite-chain crossover effects matter. At $\chi=1/2$, $R\sim bN^{1/2}$ to leading scaling order. This theta criterion is not the exact finite-$N$ phase-separation critical point; the lattice model has $\phi_c=1/(1+\sqrt N)$ and $\chi_c=(1+1/\sqrt N)^2/2$.
\b[Overlap concentration.] The <polymer overlap concentration> is reached when there is roughly one isolated coil in its own pervaded volume. If the coil size is $R$ and a <repeat unit> occupies volume $v_0$,
$$
\phi^*\sim\frac{Nv_0}{R^3}.
$$
With $R\sim bN^\nu$, the general scaling law is
$$
\boxed{\phi^*\sim\frac{v_0}{b^3}N^{1-3\nu}.}
$$
The usual single-segment-volume estimate $v_0\sim b^3$ gives $\phi^*\sim N^{1-3\nu}$: $N^{-1/2}$ in a <theta solvent>, approximately $N^{-4/5}$ in a <good solvent>, and order one for compact globules. A coil has no sharp surface, so choosing a <radius of gyration> or a geometric coil volume changes numerical prefactors.
For the stated <polystyrene> at $N=1500$ and $\chi=1/2$, the conventional scaling estimate is
$$
\boxed{\phi^*\sim N^{-1/2}=1500^{-1/2}=0.0258\simeq2.6\%.}
$$
This is a volume-fraction estimate, not a molar concentration. It uses the implicit site-volume approximation; the question does not provide an independent repeat-unit volume to determine a more precise prefactor.
\b[Semidilute length scale.] Beyond overlap, a <semidilute polymer solution> is characterized locally by a <concentration blob> size $\xi$. Suppose one blob contains $g$ statistical units. At the theta condition the subchain is Gaussian, so $\xi\sim b\sqrt g$, while matching its internal concentration to the bulk gives $\phi\sim gv_0/\xi^3$. Eliminating $g$ gives the <concentration-blob size in a theta solvent>:
$$
\boxed{\xi\sim\frac{v_0}{b^2\phi}\sim\frac b\phi
\quad\text{when }v_0\sim b^3.}
$$
Using the supplied <statistical step length> $b=0.67\,\mathrm{nm}$ and $\phi=0.05$,
$$
\boxed{\xi\sim\frac{0.67}{0.05}\,\mathrm{nm}=13.4\,\mathrm{nm}\simeq13\,\mathrm{nm}.}
$$
The same convention gives $g\sim\phi^{-2}=400<N=1500$, and the whole <ideal chain> has $R\sim b\sqrt N\simeq26\,\mathrm{nm}$. Thus the blob is smaller than the chain. Because $\phi/\phi^*\simeq1.94$ is only modestly above overlap, the value is a scaling estimate in a crossover region, not a precision asymptotic measurement. A <statistical step length> is not automatically a physical monomer diameter: if $v_0$ differs from $b^3$, the more general estimates are $\phi^*\sim(v_0/b^3)N^{-1/2}$ and $\xi\sim v_0/(b^2\phi)$, with the missing prefactor left explicit.
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