The data in the table represent osmotic pressure measurements on a polystyrene sample in the indicated solvents, obtained with the intention of extracting the third virial coefficient, $B_3$. (Data based in part on work of J. Li, Y. Wan, Z. Xu, and J.W. Mays, Macromolecules, $28,5347,1995$ ). One theory holds that in a theta solvent, $B_3$ should be zero.
(Table cant copy)
Analyze the data by two means. First, fit directly to the appropriate polynomial. Then, use the so-called "Bawn plot," first proposed over 50 years ago (C.E.H. Bawn, R.F.J. Freeman, and A.R. Kamaliddin, Trans. Faraday Soc., 46, 862, 1950):
$$
\left(\frac{\Pi}{c_i R T}-\frac{\Pi}{c_j R T}\right) /\left(c_i-c_j\right) \text { vs. }\left(c_i+c_j\right)
$$
where $c_{\mathrm{i}}$ and $c_{\mathrm{j}}$ are any two of the measured concentrations. Derive the expression that should result, and explain why it might have been a useful approach 50 years ago. Then answer the following questions:
- Provide your best estimate of $M_{\mathrm{n}}$, and the associated uncertainty, for this sample.
- Provide your best estimate of the second virial coefficients in the two solvents.
- Provide your best estimate of the third virial coefficients. Is the one in cyclohexane zero within error?
- Using appropriate data from Figure 7.19, estimate the range of reduced concentration, $c / c^*$, over which these data were obtained. Comment on the suitability of this range; what do you have to be concerned about if $c / c^*$ is too large?