Domain: Polymer kinetics

In a free-radical polymerization of vinyl acetate at \(45^\circ \mathrm{C}\) in the presence of \(0.2~\mathrm{M}\) benzoyl peroxide and \(9.3 \times 10^{-4}~\mathrm{M}\) 2,3,5,6-tetramethylbenzoquinone (duroquinone), used as an inhibitor, the following data of monomer conversion versus time was obtained (Flory, 1953): At \(0~\mathrm{min}\), the conversion was \(0\%\); at \(100~\mathrm{min}\), it was \(0.10\%\); at \(200~\mathrm{min}\), it was \(0.30\%\); at \(300~\mathrm{min}\), it was \(0.66\%\); at \(400~\mathrm{min}\), it was \(1.18\%\); and at \(500~\mathrm{min}\), it was \(2.45\%\). Determine the inhibitor constant \(C_Z\) at \(45^\circ \mathrm{C}\). [For benzoyl peroxide at \(45^\circ \mathrm{C}\) take \(k_d = 2.8 \times 10^{-7}~\mathrm{s^{-1}}\), \(f = 0.25\).]

Grading rubric:
criterion: The solution explicitly assumes standard free-radical polymerization kinetics with inhibition.
weight: 10
criterion: The solution assumes that duroquinone suppresses radical concentration and reduces the polymerization rate during the inhibited period.
weight: 10
criterion: The solution correctly identifies the rate of initiation using the formula $$R_i = 2 f (k_d \times 60)[I].$$
weight: 10
criterion: The solution correctly identifies the slope of the inhibited region as $$\text{Slope} = -\frac{C_Z}{x}. $$
weight: 10
criterion: The solution accurately determines the slope from the initial linear region of the conversion versus time data.
weight: 10
criterion: The solution correctly writes the intercept expression relating inhibitor constant and rate of initiation as $$\text{Intercept} = \frac{C_Z [Z]_0}{R_i}. $$
weight: 10
criterion: The solution correctly identifies the numerical value of the intercept from the plot as $$5.1 \times 10^4\,\mathrm{min}. $$
weight: 10
criterion: The solution correctly substitutes numerical values to calculate the rate of initiation as $$R_i = 1.68 \times 10^{-6}\,\mathrm{mol\,L^{-1}\,min^{-1}}. $$
weight: 10
criterion: The solution correctly substitutes numerical values to calculate the inhibitor constant as $$C_Z = \frac{(5.1 \times 10^4\,\mathrm{min})(1.68 \times 10^{-6}\,\mathrm{mol\,L^{-1}\,min^{-1}})}{9.3 \times 10^{-4}\,\mathrm{mol\,L^{-1}}}. $$
weight: 10
criterion: The solution correctly reports the value of the inhibitor constant at \(45^\circ \mathrm{C}\) as $$C_Z = 92.1.$$

weight: 10

Answer:
C_Z = 92.1