Membrane Osmometry — M̄ n and Second Virial Coefficient: 23-Chem-B8 Polymer Engineering, May 2016, Q3

Question 3: Membrane Osmometry — M̄_n and Second Virial Coefficient (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Find. M̄_n (from the c → 0 intercept) and the second virial coefficient A₂ (from the slope) of the osmotic virial expansion.

Approach. The van 't Hoff–virial form $\pi/c = RT/\bar M_n + RT A_2\,c + \dots$ is linear in c, so a least-squares fit of π/c against c yields M̄_n from the intercept and A₂ from the slope. Work in CGS units to match the data (π in dyne/cm², c in g/cm³, R = 8.314×10⁷ erg·mol^-1K^-1).

Form π/c. Dividing each pressure by its concentration:

c (g/cm³) 0.002 0.004 0.006 0.008 0.010

π/c (erg/g) 254 000 260 000 263 333 268 750 274 000
Least-squares line $\pi/c = \text{intercept}+\text{slope}\cdot c$: $$\text{slope}=2.44\times10^{6}\ \text{erg}\cdot\text{cm}^3\text{g}^{-2},\qquad \text{intercept}=2.494\times10^{5}\ \text{erg}\cdot\text{g}^{-1}$$
Number-average molecular weight from the intercept, with $RT=(8.314\times10^{7})(296.15)=2.462\times10^{10}$ erg/mol: $$\bar M_n=\frac{RT}{\text{intercept}}=\frac{2.462\times10^{10}}{2.494\times10^{5}}=\boxed{9.87\times10^{4}\ \text{g}\cdot\text{mol}^{-1}}$$
Second virial coefficient from the slope: $$A_2=\frac{\text{slope}}{RT}=\frac{2.44\times10^{6}}{2.462\times10^{10}}=\boxed{9.9\times10^{-5}\ \text{cm}^3\cdot\text{mol}\cdot\text{g}^{-2}}$$

c (g/cm³)	0.002	0.004	0.006	0.008	0.010
π/c (erg/g)	254 000	260 000	263 333	268 750	274 000

Fig. 3 — osmometry plot of π/c versus c. The intercept (c → 0) gives RT/M̄_n; the positive slope RT·A₂ shows the solvent is thermodynamically good (A₂ > 0).

Quantity	Value
Intercept, RT/M̄_n	2.494×10⁵ erg/g
Slope, RT·A₂	2.44×10⁶ erg·cm³/g²
Number-average MW, M̄_n	≈ 9.87×10⁴ g/mol
Second virial coefficient, A₂	≈ 9.9×10^-5 cm³·mol/g²

Topic: Membrane osmometry for number-average molecular weight and solution thermodynamics.

Key relations: Osmotic virial expansion $\dfrac{\pi}{c}=\dfrac{RT}{\bar M_n}+RT A_2 c+RT A_3 c^2+\cdots$; A₂ measures polymer–solvent interaction (A₂ > 0 good solvent, = 0 at the theta point).

Why this works: Osmotic pressure is a colligative property counting the number of dissolved chains, so extrapolating to infinite dilution — where interchain interactions vanish — recovers RT/M̄_n exactly, i.e. a number-average. The leading correction at finite c is the pairwise excluded-volume interaction, captured by A₂.

Common pitfall: Mixing unit systems (using R = 8.314 J/mol·K with CGS pressures), or reading M̄_n off a single data point instead of the true c → 0 intercept — the individual π/c values already sit 3–10% above the intercept.

Source: Rudin & Choi, The Elements of Polymer Science and Engineering, 3rd ed., Ch. 3 (osmometry); Sperling, §3.

Question 3 of 6: Membrane Osmometry — M̄ n and Second Virial Coefficient

Question 3: Membrane Osmometry — M̄n and Second Virial Coefficient (20 marks)

Question 3: Membrane Osmometry — M̄_n and Second Virial Coefficient (20 marks)