Copolymerisation Questions and Answers

Q: The monomers, A and B with reactivity ratios rA and rB, form alternate copolymers when,

Alternating copolymers form when \(r_{A} = r_{B} = 0\).The reactivity ratio \(r_{A} = k_{AA}/k_{AB}\) compares how fast a chain ending in A adds another A with how fast it adds B. If both ratios are zero, a chain end never adds its own monomer, so each A must be followed by a B and each B by an A: a perfectly alternating chain.\(r_{A} = r_{B} = 1\) is the ideal case, where each chain end adds either monomer equally and the copolymer is random. If both ratios exceed 1 each end prefers its own monomer, giving long runs (block-like) or separate homopolymers. Both below 1 gives a tendency to alternate, but strict alternation needs the limit of zero.

Q: The propagation step of a free radical copolymerization is represented by the following possible reaction steps:\(P_1\): \(M_1^{*} + M_1 \to M_1^{*}\), rate constant = \(k_{11}\)\(P_2\): \(M_1^{*} + M_2 \to M_2^{*}\), rate constant = \(k_{12}\)\(P_3\): \(M_2^{*} + M_1 \to M_1^{*}\), rate constant = \(k_{21}\)\(P_4\): \(M_2^{*} + M_2 \to M_2^{*}\), rate constant = \(k_{22}\)where, \(M_1\) and \(M_2\) are two monomers and \(M_1^{*}\) and \(M_2^{*}\) are the active radicals of \(M_1\) and \(M_2\), respectively. \(k_{ij}\) (\(i,j\) = 1,2) represents the rate constant of each step as shown above.If the reactivity ratios of \(M_1\) and \(M_2\) are expressed as: \(r_1 = k_{11}/k_{12}\) and \(r_2 = k_{22}/k_{21}\), respectively, and feed mole ratio, \(F\) is expressed as \(F = [M_1]/[M_2]\) (where, \([M_1]\) and \([M_2]\) are the concentrations of \(M_1\) and \(M_2\), respectively), the probability of the reaction \(P_2\) is ________.

The probability of reaction \(P_2\) is \(\frac{1}{r_1F + 1}\).A growing \(M_1^{*}\) radical can add either \(M_1\) (step \(P_1\)) or \(M_2\) (step \(P_2\)). The probability of \(P_2\) is its rate divided by the total rate of the two competing steps:\[ \begin{aligned} p_{12} &= \frac{k_{12}[M_2]}{k_{11}[M_1] + k_{12}[M_2]} \\ &= \frac{1}{r_1F + 1} \end{aligned} \]after dividing through by \(k_{12}[M_2]\). \(\frac{r_1}{r_1F + 1}\) has the wrong numerator. \(\frac{1}{r_2 + F}\) and \(\frac{r_2}{F + r_2}\) describe the steps of an \(M_2^{*}\) radical, \(P_3\) and \(P_4\), not \(P_2\).

Q: Determine the mole fraction of vinyl chloride in a copolymer of vinyl chloride (CH2CHCl) and vinyl acetate (CH2-CH-OCOCH3) having molecular weight of 10520 g/mol and degree of polymerization of 160.

The mole fraction of vinyl chloride is about 0.86.The average mass per repeat unit is 10520/160 = 65.75 g/mol. The repeat unit masses are about 62.5 g/mol for vinyl chloride (C2H3Cl) and 86.1 g/mol for vinyl acetate (C4H6O2). With f the mole fraction of vinyl chloride:\[ \begin{aligned} 62.5f + 86.1(1 - f) &= 65.75 \\ 23.6f &= 20.35 \\ f &\approx 0.86 \end{aligned} \]0.14 is the mole fraction of vinyl acetate, the complement. 0.30 and 0.70 do not satisfy the average mass; a 70 percent vinyl chloride copolymer would have a mean repeat mass of about 69.6 g/mol.

The monomers, A and B with reactivity ratios r_A and r_B, form alternate copolymers when,

A.\(r_{A} = r_{B} = 0\)
B.\(r_{A} = r_{B} = 1\)
C.\(r_{A} \gt 1,\ r_{B} \gt 1\)
D.\(r_{A} \lt 1,\ r_{B} \lt 1\)

Exam Year(s)

GATE XE - 2018,

Answer: A

Alternating copolymers form when \(r_{A} = r_{B} = 0\).

The reactivity ratio \(r_{A} = k_{AA}/k_{AB}\) compares how fast a chain ending in A adds another A with how fast it adds B. If both ratios are zero, a chain end never adds its own monomer, so each A must be followed by a B and each B by an A: a perfectly alternating chain.

\(r_{A} = r_{B} = 1\) is the ideal case, where each chain end adds either monomer equally and the copolymer is random. If both ratios exceed 1 each end prefers its own monomer, giving long runs (block-like) or separate homopolymers. Both below 1 gives a tendency to alternate, but strict alternation needs the limit of zero.

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The propagation step of a free radical copolymerization is represented by the following possible reaction steps:

\(P_1\): \(M_1^{*} + M_1 \to M_1^{*}\), rate constant = \(k_{11}\)
\(P_2\): \(M_1^{*} + M_2 \to M_2^{*}\), rate constant = \(k_{12}\)
\(P_3\): \(M_2^{*} + M_1 \to M_1^{*}\), rate constant = \(k_{21}\)
\(P_4\): \(M_2^{*} + M_2 \to M_2^{*}\), rate constant = \(k_{22}\)

where, \(M_1\) and \(M_2\) are two monomers and \(M_1^{*}\) and \(M_2^{*}\) are the active radicals of \(M_1\) and \(M_2\), respectively. \(k_{ij}\) (\(i,j\) = 1,2) represents the rate constant of each step as shown above.

If the reactivity ratios of \(M_1\) and \(M_2\) are expressed as: \(r_1 = k_{11}/k_{12}\) and \(r_2 = k_{22}/k_{21}\), respectively, and feed mole ratio, \(F\) is expressed as \(F = [M_1]/[M_2]\) (where, \([M_1]\) and \([M_2]\) are the concentrations of \(M_1\) and \(M_2\), respectively), the probability of the reaction \(P_2\) is ________.

A.\(\frac{r_1}{r_1F + 1}\)
B.\(\frac{1}{r_1F + 1}\)
C.\(\frac{1}{r_2 + F}\)
D.\(\frac{r_2}{F + r_2}\)

Exam Year(s)

GATE XE - 2024,

Answer: B

The probability of reaction \(P_2\) is \(\frac{1}{r_1F + 1}\).

A growing \(M_1^{*}\) radical can add either \(M_1\) (step \(P_1\)) or \(M_2\) (step \(P_2\)). The probability of \(P_2\) is its rate divided by the total rate of the two competing steps:

\[ \begin{aligned} p_{12} &= \frac{k_{12}[M_2]}{k_{11}[M_1] + k_{12}[M_2]} \\ &= \frac{1}{r_1F + 1} \end{aligned} \]

after dividing through by \(k_{12}[M_2]\). \(\frac{r_1}{r_1F + 1}\) has the wrong numerator. \(\frac{1}{r_2 + F}\) and \(\frac{r_2}{F + r_2}\) describe the steps of an \(M_2^{*}\) radical, \(P_3\) and \(P_4\), not \(P_2\).

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Useful data

Boltzmann's constant	\(1.38 \times 10^{-23}\) J K^-1
Charge on an electron	\(1.602 \times 10^{-19}\) C
Gas Constant	8.314 J mol^-1 K^-1
Electron rest mass	\(9.1 \times 10^{-31}\) kg
Permittivity of vacuum (ε_o)	\(8.854 \times 10^{-12}\) F m^-1
Bohr Magneton	\(9.274 \times 10^{-24}\) A/m²

Determine the mole fraction of vinyl chloride in a copolymer of vinyl chloride (CH₂CHCl) and vinyl acetate (CH₂-CH-OCOCH₃) having molecular weight of 10520 g/mol and degree of polymerization of 160.

A.0.14
B.0.30
C.0.70
D.0.86

Exam Year(s)

GATE XE - 2012,

Answer: D

The mole fraction of vinyl chloride is about 0.86.

The average mass per repeat unit is 10520/160 = 65.75 g/mol. The repeat unit masses are about 62.5 g/mol for vinyl chloride (C₂H₃Cl) and 86.1 g/mol for vinyl acetate (C₄H₆O₂). With f the mole fraction of vinyl chloride:

\[ \begin{aligned} 62.5f + 86.1(1 - f) &= 65.75 \\ 23.6f &= 20.35 \\ f &\approx 0.86 \end{aligned} \]

0.14 is the mole fraction of vinyl acetate, the complement. 0.30 and 0.70 do not satisfy the average mass; a 70 percent vinyl chloride copolymer would have a mean repeat mass of about 69.6 g/mol.

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Copolymerisation Questions and Answers

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