Osmotic Pressure of a Polymer Fraction: Number-Average Molar Mass and Solvent Quality
A polymer sample is dissolved in toluene at 298 K, and osmotic pressure is measured with a membrane osmometer. The following data are obtained (c in g L^-1, π in Pa): c = 2.00, 4.00, 6.00, 8.00; π = 0.252, 0.512, 0.780, 1.056. Each pressure is the average of three readings with ±0.005 Pa reproducibility. The membrane is calibrated, and the solvent vapor pressure is negligible. A student assumes ideal behavior and calculates M = RT/(π/c) at c = 2.00 g/L, obtaining 19,700 g/mol. A colleague notes that the result changes with concentration and suspects polymer-solvent interactions. The second virial coefficient is expected to be small but non-zero. The student has a linear regression routine but is unsure whether to extrapolate π/c to c = 0 and whether osmotic pressure gives M_n or M_w. No aggregation or dissociation is expected. R = 8.314 J mol^-1 K^-1.
1. Using the data, determine the number-average molar mass and the second virial coefficient A2. State any assumptions and explain why extrapolation to c = 0 is necessary.
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For osmotic pressure, π/c = RT/M_n + RT A2 c. Compute π/c: 0.126, 0.128, 0.130, 0.132 Pa L g^-1. A linear least-squares fit gives intercept 0.124 Pa L g^-1 and slope 0.00100 Pa L^2 g^-2. Thus M_n = RT/intercept = (8.314 × 298)/0.124 = 19,981 g mol^-1 ≈ 2.00 × 10^4 g mol^-1. A2 = slope/(RT) = 0.00100/2477.6 = 4.04 × 10^-7 L mol g^-2. Assumptions: ideal dilute solution, no aggregation, membrane impermeable to polymer, and π small enough that the gas law applies. Extrapolation to c = 0 is necessary because at finite concentration excluded-volume and polymer-solvent interactions add a concentration-dependent term; the intercept isolates the ideal colligative contribution. A single-point measurement at c = 2 g/L gives an apparent M_n about 1.6% low, which is the size of the A2 term at that concentration.
2. The student's single-point calculation gave 19,700 g/mol. Is this error consistent with the measured A2? Quantify the discrepancy and explain whether a single-point measurement at c = 2 is acceptable for a polymer of this molar mass.
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At c = 2.00 g/L, π/c = 0.126 Pa L g^-1. If A2 is ignored, M_app = RT/(π/c) = 2477.6/0.126 = 19,663 g mol^-1, vs 19,981 from extrapolation, a difference of 318 g mol^-1 or 1.6%. This is exactly the direction expected for positive A2: π/c is above the intercept, so the apparent molar mass is lower. The discrepancy is consistent with A2 = 4.04 × 10^-7 L mol g^-2 because RT A2 c = 2477.6 × 4.04×10^-7 × 2.00 = 0.0020 Pa L g^-1, which is 1.6% of 0.124. A single-point measurement may be acceptable for a rough order-of-magnitude check, but not for a reliable M_n; extrapolation or at least two concentrations is required. For a polymer of 20,000 g/mol, the effect is small but measurable.
3. If the polymer were partially dimerized through hydrogen bonding in toluene, how would the plot of π/c vs c change? Would the apparent M_n increase or decrease? Justify.
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Dimerization reduces the number of solute particles for a given total mass. Osmotic pressure counts particles, so π would be lower than for the same mass of monomer. Hence π/c would fall below the ideal line, giving downward curvature at higher c. The apparent molar mass M_app = RT/(π/c) would increase because fewer particles carry the same mass. At c → 0, the monomer-dimer equilibrium shifts toward monomer, so the intercept would still approach the true monomer M_n, but the slope would become negative or the plot nonlinear. If association is strong even at low c, the intercept itself could be elevated, and extrapolation would overestimate the true M_n. Thus the observed positive A2 (good solvent) would be masked by association; the plot would not be linear over the full range.