Viscosity-average molecular weight from dilute-solution viscometry: 23-Chem-B8 Polymer Engineering, December 2014, Q1

Question 1: Viscosity-average molecular weight from dilute-solution viscometry (Part A — 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.

Given. Efflux (flow) times in an Ostwald/Ubbelohde capillary viscometer for polystyrene (PS) in toluene at 30 °C; the pure-solvent time is $t_0 = 67.04\ \text{s}$. In a capillary viscometer the relative viscosity is the ratio of efflux times (equal-density dilute-solution approximation), $\eta_r = \eta/\eta_0 = t/t_0$. Mark–Houwink constants $K = 1.2\times10^{-4}$, $\alpha = 0.71$.

Find. The viscosity-average molecular weight $M_v$ from the intrinsic viscosity $[\eta]$ via $[\eta] = K M_v^{\alpha}$.

Double extrapolation to infinite dilution: the Huggins line ($\eta_{sp}/c$, rising) and the Kraemer line ($\ln\eta_r/c$, falling) share a common intercept $[\eta]\approx1.28\ \text{dL/g}$.

Approach. Convert each efflux time to relative, specific, reduced and inherent viscosities; extrapolate the reduced viscosity $\eta_{sp}/c$ (Huggins) and inherent viscosity $\ln\eta_r/c$ (Kraemer) to $c\to0$; their common intercept is the intrinsic viscosity $[\eta]$; invert Mark–Houwink for $M_v$.

Relative and specific viscosities. With $\eta_r=t/t_0$ and $\eta_{sp}=\eta_r-1$, e.g. at $c=0.402$: $\eta_r=107.70/67.04=1.6065$, $\eta_{sp}=0.6065$. The full set of reduced viscosities $\eta_{sp}/c$ is $1.509,\,1.595,\,1.648,\,1.750,\,1.987\ \text{dL/g}$ and inherent viscosities $\ln\eta_r/c$ are $1.179,\,1.170,\,1.148,\,1.093,\,1.014\ \text{dL/g}$ (for $c=0.402\ldots1.207\ \text{g/dL}$).
Huggins extrapolation. A least-squares fit of $\eta_{sp}/c = [\eta] + k_H[\eta]^2\,c$ gives intercept $[\eta]_H = 1.293\ \text{dL/g}$ and slope $0.576$, i.e. a Huggins constant $k_H = 0.576/[\eta]^2 = 0.34$ — the value expected for a flexible coil in a good solvent (toluene is a good solvent for PS).
Kraemer extrapolation. Fitting $\ln\eta_r/c = [\eta] - k_K[\eta]^2\,c$ gives intercept $[\eta]_K = 1.272\ \text{dL/g}$. The two intercepts agree to within 2 %, confirming the common value $$[\eta] = \tfrac12\big([\eta]_H+[\eta]_K\big) = \boxed{1.28\ \text{dL/g}}$$
Invert Mark–Houwink. With $[\eta]=K M_v^{\alpha}$, $$M_v = \left(\frac{[\eta]}{K}\right)^{1/\alpha} = \left(\frac{1.28}{1.2\times10^{-4}}\right)^{1/0.71} = \boxed{4.7\times10^{5}\ \text{g/mol}}$$ where $[\eta]$ and $K$ are taken in the same units of dL/g (see the Verify note).

Quantity	Value
Intrinsic viscosity $[\eta]$ (Huggins / Kraemer)	1.293 / 1.272 → 1.28 dL/g
Huggins constant $k_H$	0.34 (good-solvent range)
Viscosity-average molecular weight $M_v$	≈ 4.7 × 10⁵ g/mol

Check

The tabulated constant $K=1.2\times10^{-4},\ \alpha=0.71$ is the standard literature Mark–Houwink pair for PS/toluene when $[\eta]$ is expressed in dL/g; the problem's “cm³/g” label is a units slip. Computing $[\eta]$ in dL/g (as done above) gives the physically reasonable $M_v\approx4.7\times10^5$. Taking the constant literally in cm³/g ($[\eta]=128\ \text{cm}^3/\text{g}$) would give $M_v\approx3\times10^{8}$ — far above any real polystyrene — so the dL/g convention is intended.

Topic: Dilute-solution viscometry and the Mark–Houwink–Sakurada relation for molecular-weight determination.

Key relations: $\eta_r=t/t_0$; $\eta_{sp}=\eta_r-1$; Huggins $\eta_{sp}/c=[\eta]+k_H[\eta]^2c$; Kraemer $\ln\eta_r/c=[\eta]-k_K[\eta]^2c$ (with $k_H+k_K=\tfrac12$); Mark–Houwink $[\eta]=KM_v^{\alpha}$.

Why this works: Intrinsic viscosity measures the hydrodynamic volume a single isolated coil sweeps out, so it must be evaluated at infinite dilution where chains do not interact. Both the Huggins (specific) and Kraemer (logarithmic) functions are linear in $c$ and by construction extrapolate to the same intercept; using both is a self-consistency check on the data. The exponent $\alpha$ (0.5 at theta conditions, up to ~0.8 in good solvents) encodes coil expansion, and $M_v$ lies between $M_n$ and $M_w$, closer to $M_w$.

Common pitfall: Mixing concentration units. If $c$ is in g/dL then $[\eta]$ is in dL/g and $K$ must be the dL/g value; a stray factor of 100 (dL vs. cm³) throws $M_v$ off by $100^{1/\alpha}\approx10^{2.8}$. Also, never extrapolate a single concentration — the intercept, not any single reduced-viscosity point, is $[\eta]$.

Source: Rudin & Choi, Elements of Polymer Science & Engineering 3rd ed., Ch. 3 (solution viscosity); Sperling, Physical Polymer Science 4th ed., §3.3.

Quantity	Value
Intrinsic viscosity \([\eta]\) (Huggins / Kraemer)	1.293 / 1.272 → 1.28 dL/g
Huggins constant \(k_H\)	0.34 (good-solvent range)
Viscosity-average molecular weight \(M_v\)	≈ 4.7 × 10⁵ g/mol

Question 1 of 7: Viscosity-average molecular weight from dilute-solution viscometry

Question 1: Viscosity-average molecular weight from dilute-solution viscometry (Part A — 20 marks)