Let's break down and answer each part step by step: --- ## **3(a) HDPE Catalysts and Isomerism** ### **Preferred Catalysts for HDPE** - **Type:** **Ziegler-Natta catalysts** or **metallocene catalysts** are preferred for producing high-density polyethylene (HDPE). - **Example:** A common Ziegler-Natta catalyst is **TiCl₄ with Al(C₂H₅)₃ (triethylaluminum)**. ### **Why No Isomerism in Polyethylene?** - **No positional, geometrical, or configurational isomerism** in polyethylene because: - **Polyethylene structure:** –[CH₂–CH₂]ₙ– - All carbons in the backbone are identical, each bonded only to hydrogens and other carbons in a linear chain. - No double bonds (so no cis-trans/geometric isomerism). - No side groups (so no positional or configurational isomerism). --- ## **3(b) Copolymerization of Styrene and p-Methylstyrene** ### **Given Data** - **Monomer 1 (Styrene):** \( r_1 = .98 \) - **Monomer 2 (p-Methylstyrene):** \( r_2 = .95 \) --- ### **(i) Sketch the Copolymer Composition Diagram** **Reactivity Ratios:** Both \( r_1 \) and \( r_2 \) are close to 1, meaning both monomers have nearly equal tendencies to react with themselves and each other. **Expected Copolymer Composition Plot:** - Plot **F₁** (mole fraction of styrene in copolymer) vs. **f₁** (mole fraction of styrene in feed). - For \( r_1 \approx r_2 \approx 1 \), the plot is **almost a straight line**, indicating near-ideal (random) copolymerization. **Sketch:** ```plaintext | \ | \ | \ | \ |_______\ 1 ``` - **x-axis:** \( f_1 \) (mole fraction of styrene in feed) - **y-axis:** \( F_1 \) (mole fraction of styrene in copolymer) - The line is slightly curved but almost diagonal, indicating random incorporation. --- ### **(ii) Type of Copolymers Expected** - Since \( r_1 \) and \( r_2 \) are both close to 1, **random (statistical) copolymers** are expected. - **Explanation:** Both monomers react with themselves and each other with nearly equal probability, so the resulting copolymer has a random sequence of styrene and p-methylstyrene units. --- ### **(iii) Feed Composition Calculation** Given: - \( F_2 = .20 \) (fraction of p-methylstyrene in copolymer) - Find **feed composition**: \( f_1 \) and \( f_2 \) (mole fractions of styrene and p-methylstyrene in feed) #### **Using the Copolymer Composition Equation:** \[ F_2 = \frac{r_2 f_2^2 + f_1 f_2}{r_1 f_1^2 + 2 f_1 f_2 + r_2 f_2^2} \] But it's easier to use the **simplified Mayo–Lewis equation** for two monomers: \[ \frac{F_2}{F_1} = \frac{f_2}{f_1} \cdot \frac{r_2 f_2 + f_1}{r_1 f_1 + f_2} \] Given \( F_2 = .20 \), so \( F_1 = .80 \): \[ \frac{.20}{.80} = \frac{f_2}{f_1} \cdot \frac{.95 f_2 + f_1}{.98 f_1 + f_2} \] \[ .25 = \frac{f_2}{f_1} \cdot \frac{.95 f_2 + f_1}{.98 f_1 + f_2} \] Let \( f_2 = x \), \( f_1 = 1 - x \): \[ .25 = \frac{x}{1 - x} \cdot \frac{.95x + (1-x)}{.98(1-x) + x} \] \[ .25 = \frac{x}{1-x} \cdot \frac{.95x + 1 - x}{.98 - .98x + x} \] \[ .25 = \frac{x}{1-x} \cdot \frac{1 - .05x}{.98 + .02x} \] Now solve for \( x \): Let’s denote \( x = f_2 \): \[ .25 = \frac{x(1 - .05x)}{(1-x)(.98 + .02x)} \] Cross-multiply: \[ .25 (1-x)(.98 + .02x) = x(1 - .05x) \] Expand: \[ .25(.98 - .98x + .02x - .02x^2) = x - .05x^2 \] \[ .245 - .245x + .005x - .005x^2 = x - .05x^2 \] \[ .245 - .24x - .005x^2 = x - .05x^2 \] Bring all terms to one side: \[ .245 - .24x - .005x^2 - x + .05x^2 = \] \[ .245 - 1.24x + .045x^2 = \] Now solve the quadratic: \[ .045x^2 - 1.24x + .245 = \] Use quadratic formula: \[ x = \frac{1.24 \pm \sqrt{(-1.24)^2 - 4 \cdot .045 \cdot .245}}{2 \cdot .045} \] \[ x = \frac{1.24 \pm \sqrt{1.5376 - .0441}}{.09} \] \[ x = \frac{1.24 \pm \sqrt{1.4935}}{.09} \] \[ \sqrt{1.4935} \approx 1.222 \] \[ x_1 = \frac{1.24 + 1.222}{.09} = \frac{2.462}{.09} \approx 27.36 \] \[ x_2 = \frac{1.24 - 1.222}{.09} = \frac{.018}{.09} \approx .20 \] Only \( x_2 = .20 \) is a valid mole fraction. So, - \( f_2 = .20 \) - \( f_1 = .80 \) --- ### **Final Answers** #### **3(a)** - **Catalysts for HDPE:** Ziegler-Natta or metallocene (e.g., TiCl₄/Al(C₂H₅)₃) - **No isomerism:** All backbone carbons and hydrogens are identical; no double bonds or side groups. #### **3(b)** - **(i) Copolymer composition diagram:** Nearly straight line (random copolymerization). - **(ii) Type:** Random (statistical) copolymer. - **(iii) Feed required for \( F_2 = .20 \):** - \( f_2 = .20 \) (p-methylstyrene) - \( f_1 = .80 \) (styrene) --- Let me know if you need the plot explicitly drawn!