Discussion
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 ________.
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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