Rayleigh ratios \((R_\theta)\) were obtained at \(25^\circ\text{C}\) for a series of solutions of a polystyrene sample in benzene, with the detector situated at various angles \((\theta)\) to the incident beam of unpolarized monochromatic light of wavelength \(546.1\text{ nm}\). The results are tabulated below:

\[
\begin{array}{|c|c|c|c|c|}
\hline
\text{Polystyrene Concentration (g dm}^{-3}\text{)} & 30^\circ & 60^\circ & 90^\circ & 120^\circ \\
\hline
0.5 & 72.3 & 69.4 & 66.2 & 64.3 \\
1.0 & 89.8 & 85.7 & 81.3 & 78.2 \\
1.5 & 100.8 & 97.1 & 92.0 & 88.1 \\
2.0 & 108.7 & 103.8 & 99.7 & 95.9 \\
\hline
\end{array}
\]

Under the conditions of these measurements, the Rayleigh ratio and refractive index of benzene are \(46.5 \times 10^{-4} \text{ m}^{-1}\) and \(1.502\) respectively, and the refractive index increment for the polystyrene solutions is \(1.08 \times 10^{-4} \text{ dm}^3 \text{ g}^{-1}\).

Using a Zimm plot, determine:
(a) the weight average molar mass of the polystyrene sample,
(b) the radius of gyration \(((s^2)^{1/2})\) of polystyrene molecules in benzene at \(25^\circ\),
(c) the second virial coefficient.

The question revolves around the use of a Zimm plot to determine the weight-average molar mass, radius of gyration, and the second virial coefficient of a polystyrene sample in benzene. It involves using light scattering data and the Zimm plot extrapolation method, along with equations from the light scattering theory.

The student's question pertains to using a Zimm plot to determine the properties of a polystyrene sample in benzene, namely, the weight-average molar mass (Mw), the radius of gyration (Rg), and the second virial coefficient (A2).

To calculate these parameters, the Rayleigh ratio of the sample at various concentrations and angles is used, in conjunction with the refractive index of the solvent (benzene), and the refractive index increment of the polystyrene solutions. The Zimm plot analysis utilizes light scattering data to extrapolate these values. It involves plotting the reciprocal of the Rayleigh ratio (1/Rθ) against the sin2(θ/2) and concentration (c). The intercept and slope of these plots at θ = 0 and c = 0 provide the necessary information to calculate Mw, Rg, and A2.

To perform this analysis, the following equations from the light scattering theory are used:

KC/R(0) = 1/Mw(P(0) + 2A2C + 3A3C2 + ...)
K = 4π2n2 (dn/dC)2/NAλ2
The particle scattering function P(0) is related to the radius of gyration Rg.

The specifics of creating and analyzing a Zimm plot typically require advanced understanding of physical chemistry and polymer science and are often completed using specialized software or computational tools.