Question

The plateau modulus of polystyrene has a value of 0.2 × 10⁶ Pa at 150 °C. Given, the density of polystyrene is 1.05 g/cm³, the universal gas constant, \( R = 8.3 \, \text{J K}^{-1} \text{mol}^{-1} \), and the monomer molecular weight is 104 g/mol. The molecular weight between entanglements (rounded off to the nearest integer) of polystyrene chains is \(\underline{\hspace{2cm}}\) g/mol.

Hint:

To calculate the molecular weight between entanglements using the plateau modulus, use the relationship \( G_{\text{plateau}} = \frac{\rho R T}{M_e} \).

Answer 1

Correct Answer: 18000

We know that the plateau modulus \( G_{\text{plateau}} \) is related to the molecular weight between entanglements \( M_e \) by the equation:
\[ G_{\text{plateau}} = \frac{\rho R T}{M_e} \] Where:
\[ G_{\text{plateau}} = 0.2 \times 10^6 \, \text{Pa}, \, \rho = 1.05 \, \text{g/cm}^3, \, R = 8.3 \, \text{J K}^{-1} \text{mol}^{-1}, \, T = 150 \, \text{°C} = 423 \, \text{K}. \] Rearranging the formula to solve for \( M_e \):
\[ M_e = \frac{\rho R T}{G_{\text{plateau}}} \] Substitute the known values:
\[ M_e = \frac{1.05 \times 8.3 \times 423}{0.2 \times 10^6} \] Solving this:
\[ M_e \approx 50.4 \, \text{g/mol}. \] Thus, the molecular weight between entanglements of polystyrene chains is approximately 50.4 g/mol.