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For a single-site solvent and polymer length $N$, expand $(1-\varphi )\ln (1-\varphi )=-\varphi +{\varphi }^2/2+{\varphi }^3/6+O({\varphi }^4)$ in Flory–Huggins solution theory. The displayed result follows. Its linear term changes reference chemical potentials; its quadratic term sets the effective two-segment excluded volume. The latter vanishes at the theta temperature, while the positive cubic term remains. Taking $\varphi f^{\prime }-f$ yields the osmotic pressure in Flory-Huggins solution theory, whose cubic coefficient is $1/3$ rather than the free energy's $1/6$.

Interpret the stated free energy as per lattice site of volume $v_0$. Flory–Huggins solution theory counts polymer chains, giving the entropy term $(\phi /N)\ln \phi$, and solvent molecules, giving $(1-\phi )\ln (1-\phi )$. The interaction term penalizes unlike contacts when $\chi >0$; a simple contact model has $\chi =z[{ϵ}_{ps}-({ϵ}_{pp}+{ϵ}_{ss})/2]/(k_{B}T)$. Connectivity reduces polymer translational entropy relative to independent monomers. This is an incompressible mean-field model with composition-independent $\chi$ for the calculations below.^[ref]

Expanding the dilute osmotic pressure identifies the effective second virial coefficient. A good solvent has $\chi <1/2$ and positive excluded volume, promoting swollen chains. A poor solvent has $\chi >1/2$, favouring contraction and potentially phase separation. The theta-solvent condition is $\chi =1/2$ in this long-chain mean-field description: the second virial term vanishes, but higher terms remain. For finite $N$, this condition is not the exact critical point, which instead has ${\chi }_c=(1+N^{-1/2}{)}^2/2$.

Let $g=f_{\mathrm{mix}}/(k_{B}T)$. The osmotic pressure in Flory-Huggins solution theory isThus $\Pi =(k_{B}T/v_0)[\phi /N+(1/2-\chi ){\phi }^2+{\phi }^3/3+\cdots ]$. For mass concentration $c={\rho }_p\phi$ and chain molar mass $M_n=N{\rho }_p{v}_0{N}_A$,osmometry uses a membrane permeable to solvent but retaining every polymer chain, at known uniform temperature. Extrapolate $\Pi /c$ to zero concentration to remove interactions; the intercept gives the number-average molar mass. Aggregation, leakage of small chains, ions with a Donnan contribution, or a poor-solvent two-phase sample invalidate the simple molecular-weight interpretation.

The measured ratios give an exactly linear reduced-pressure fit:The reduced osmotic-pressure intercept is $185\mathrm{Pa}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}$. The data determine $M_n/T=0.04494\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$; the measurement temperature is needed to convert this into an absolute molar mass. For an explicitly assumed room temperature $T=300\mathrm{K}$, $M_n\simeq 13.5\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}=1.35\times 10^4\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$. The boxed expression is the general answer for an arbitrary experimental temperature.

Meaning of the mixing free energy. The supplied expression is the Flory–Huggins solution theory mixing free energy 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 $n_p,n_s$, sowhereThe 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 $0<\varphi <1$, 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 ${ϵ}_{ps},{ϵ}_{pp},{ϵ}_{ss}$,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.

Osmotic pressure and the special value one half. At fixed chain count, $\partial \varphi /\partial V=-\varphi /V$. Differentiating $F_{\mathrm{mix}}=(V/v_0)k_{B}Tg(\varphi )$ with respect to volume, relative to the pure-solvent reference, givesThe resulting osmotic pressure in Flory-Huggins solution theory isThe leading term is the ideal osmotic pressure of chains, since their number density is $\varphi /(N{v}_0)$. 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 $R\sim b{N}^{\nu }$, $\nu \simeq 0.588$ 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 $\chi =1/2$, $R\sim b{N}^{1/2}$ to leading scaling order. This theta criterion is not the exact finite-$N$ phase-separation critical point; the lattice model has ${\varphi }_c=1/(1+\sqrt{N})$ and ${\chi }_c=(1+1/\sqrt{N}{)}^2/2$.

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 $v_0$,With $R\sim b{N}^{\nu }$, the general scaling law isThe usual single-segment-volume estimate $v_0\sim b^3$ gives ${\varphi }^{\ast }\sim N^{1-3\nu }$: $N^{-1/2}$ 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 isThis 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.

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 $\varphi \sim g{v}_0/{\xi }^3$. Eliminating $g$ gives the concentration-blob size in a theta solvent:Using the supplied statistical step length $b=0.67\mathrm{nm}$ and $\varphi =0.05$,The same convention gives $g\sim {\varphi }^{-2}=400<N=1500$, and the whole ideal chain has $R\sim b\sqrt{N}\simeq 26\mathrm{nm}$. Thus the blob is smaller than the chain. Because $\varphi /{\varphi }^{\ast }\simeq 1.94$ 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 ${\varphi }^{\ast }\sim (v_0/b^3)N^{-1/2}$ and $\xi \sim v_0/(b^2\varphi )$, with the missing prefactor left explicit.

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,where $z_c$ is the coordination number and the $ϵ$ 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.

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 giveFor a good solvent, the Flory approximation balances entropic stretching and two-segment interaction:Minimization gives $R^5\sim v_{\mathrm{eff}}{a}^2{N}^3$, henceThus the familiar good-solvent scaling is $R\sim a{N}^{3/5}$ 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 $\chi \uparrow 1/2$. The crossover occurs when $(1-2\chi )N^{1/2}$ is of order one. Nearer theta, the ideal-coil size supplies the appropriate limit.

Numerical sizes. The theta value isFor $\chi =0.3$, the plain good-solvent scaling givesIf the solvent-quality amplitude is retained with $v_0=a^3$, it instead givesThe 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 $(1-2\chi )\sqrt{N}\simeq 15.5$, so a swollen-coil estimate is reasonable. Neither result is an exact microscopic prediction from the supplied data alone.

Overlap concentration. The polymer overlap concentration corresponds to one chain per coil volume. Using the scaling convention $V_{\mathrm{coil}}\sim R_{\theta }^3$ givesThe first is chain number density; the second counts repeat units. Thus the repeat-unit molar concentration is approximately $0.143\mathrm{mol}{\mathrm{L}}^{-1}$. If the lattice-site volume is taken as $v_0=a^3$, this is equivalentlyFor a mass-concentration estimate, a polystyrene repeat unit ${\mathrm{C}}_8{\mathrm{H}}_8$ has molar mass about $104.15\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$, so this scaling convention givesThere 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 $R_g=R_{\theta }/\sqrt{6}=10.6\mathrm{nm}$, gives $c^{\ast }\simeq 52\mathrm{g}{\mathrm{L}}^{-1}$. The invariant content is one chain per coil volume and the theta scaling ${\varphi }^{\ast }\propto N^{-1/2}$; the radius and coil-volume convention must accompany a numerical value.

Osmotic pressure. In the given lattice expression, $\varphi$ 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 $M=N_s+N{N}_p$ is the number of sites and $\varphi =N{N}_p/M$, differentiation of $F_{\mathrm{mix}}=Mf(\varphi )$ with respect to the solvent count at fixed chain count givesBalancing this change in solvent chemical potential against the pressure contribution $v_0\Pi$ gives $\Pi =(\varphi f^{\prime }-f)/v_0$. For composition-independent $\chi$,and therefore the osmotic pressure in Flory-Huggins solution theory isExpanding the logarithm as $\varphi \to 0$ givesThe first term is $n_p{k}_{B}T$ with $n_p=\varphi /(N{v}_0)$ 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.