Domain: Polymer physical chemistry

Consider a single macromolecular chain of polystyrene with molar mass \(\text{(M)} = 10^6\) dissolved under theta conditions at \(25^\circ C\), where the polymer behaves as an ideal random coil. Assuming that the root-mean-square \((\text{RMS})\) end-to-end distance effectively represents the diameter of the equivalent spherical coil, determine the molecular-scale volume occupied by a single chain within the dilute regime. The system parameters are as follows : \(\text{C-C }\) bond length \(({l})\) \( = 1.54 \times 10^{-8} \ cm\), bond angle \((\theta)\) \( = 109.5^\circ\), and the steric hindrance parameter for polystyrene at \(25^\circ C\), steric parameter (\(\sigma)\) \(= 2.3\) and molar mass of styrene is \(104 \ g/mol\) . Assume ideal chain statistics and neglect solvent–polymer volume interactions.

Grading rubric:
criterion: Calculates the fully extended contour length (\(L_{max}\)) of the polymer chain.
weight: 20
criterion: Explicitly classifies the polystyrene stereochemistry as atactic in the structural analysis.
weight: 20
criterion: Correctly calculates the number of backbone bonds (n) as exactly 19230. 
weight: 20
criterion: Correctly calculates the bond angle geometric expansion factor \((\frac{1 - \cos\theta}{1 + \cos\theta})\) as exactly 2. 
weight: 10
criterion: Correctly calculates the unperturbed mean-square end-to-end distance (\(\langle r^2 \rangle_0\)) as \(4.82 \times 10^{-11} \text{cm}^2\).
weight: 20
criterion: Reports the final molecular volume as approximately \(1.75 \times 10^{-16} \text{cm}^3\).
weight: 10

Answer:
1.75 × 10^-16 cm^3