= Solution
Interpret the stated <free energy> as per lattice site of volume $v_0$. <Flory–Huggins solution theory> counts <polymer> chains, giving the <entropy> term $(\varphi/N)\ln\varphi$, and <solvent> molecules, giving $(1-\varphi)\ln(1-\varphi)$. The interaction term penalizes unlike contacts when $\chi>0$; a simple contact model has $\chi=z[\epsilon_{ps}-(\epsilon_{pp}+\epsilon_{ss})/2]/(k_BT)$. Connectivity reduces <polymer> <translational entropy> relative to independent <monomers>. This is an incompressible mean-field model with composition-independent $\chi$ for the calculations below.<Thermodynamics of High Polymer Solutions by Paul J. Flory>[1942, connected-chain combinatorial entropy and mixing free energy]{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 $g=f_{\rm mix}/(k_BT)$. The <osmotic pressure in Flory-Huggins solution theory> is
$$
\boxed{\Pi=\frac{k_BT}{v_0}(\varphi g'-g)
=-\frac{k_BT}{v_0}\left[\ln(1-\varphi)+(1-1/N)\varphi+\chi\varphi^2\right].}
$$
Thus $\Pi=(k_BT/v_0)[\varphi/N+(1/2-\chi)\varphi^2+\varphi^3/3+\cdots]$. For <mass> <concentration> $c=\rho_p\varphi$ and chain molar <mass> $M_n=N\rho_pv_0N_A$,
$$
\frac\Pi c=\frac{RT}{M_n}+B c+O(c^2).
$$
<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:
$$
\boxed{\Pi/c=185+750c\quad\text{in the stated SI concentration units},\qquad
M_n=\frac{RT}{185}.}
$$
The <reduced osmotic-pressure intercept> is $185\,\mathrm{Pa\,m^3\,kg^{-1}}$. The data determine \b[$M_n/T=0.04494\,\mathrm{kg\,mol^{-1}\,K^{-1}}$]; the measurement <temperature> is needed to convert this into an absolute <molar mass>. For an explicitly assumed room <temperature> $T=300\,\mathrm K$, \b[$M_n\simeq13.5\,\mathrm{kg\,mol^{-1}}=1.35\times10^4\,\mathrm{g\,mol^{-1}}$]. The boxed expression is the general answer for an arbitrary experimental <temperature>.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2014/paper-1-osmometry.png]
{title=Original reduced-osmotic-pressure regression and its zero-concentration intercept}
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