Polyethylene is used in many items, including water pipes, bottles, electrical insulation, toys, and mailer envelopes. It is a polymer, a molecule with a very high molar mass made by joining many ethylene molecules together. (Ethylene is the basic unit, or monomer for polyethylene.) The initiation step is
$$\mathrm{R}_{2} \stackrel{k_{1}}{\longrightarrow} 2 \mathrm{R} \cdot \quad \text { (initiation) }$$
The $\mathrm{R} \cdot$ species (called a radical) reacts with an ethylene molecule (M) to generate another radical
$$\mathrm{R} \cdot+\mathrm{M} \longrightarrow \mathrm{M}_{1}$$
The reaction of $M_{1}$. with another monomer leads to the growth or propagation of the polymer chain:
$$\mathrm{M}_{1} \cdot+\mathrm{M} \stackrel{k_{\mathrm{p}}}{\longrightarrow} \mathrm{M}_{2} \cdot \quad \text { (propagation) }$$
This step can be repeated with hundreds of monomer units. The propagation terminates when two radicals combine
$$\mathrm{M}^{\prime} \cdot+\mathrm{M}^{\prime \prime} \cdot \stackrel{k_{1}}{\longrightarrow} \mathrm{M}^{\prime}-\mathrm{M}^{\prime \prime} \quad \text { (termination) }$$
The initiator frequently used in the polymerization of ethylene is benzoyl peroxide $\left[\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COO}\right)_{2}\right]$
$$\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COO}\right)_{2} \longrightarrow 2 \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COO}$$
$$\mathrm{M}_{1} \cdot+\mathrm{M} \stackrel{k_{\mathrm{p}}}{\longrightarrow} \mathrm{M}_{2} \cdot \quad \text { (propagation) }$$
This step can be repeated with hundreds of monomer units. The propagation terminates when two radicals combine
$$\mathrm{M}^{\prime} \cdot+\mathrm{M}^{\prime \prime} \cdot \stackrel{k_{1}}{\longrightarrow} \mathrm{M}^{\prime}-\mathrm{M}^{\prime \prime} \quad \text { (termination) }$$
The initiator frequently used in the polymerization of ethylene is benzoyl peroxide $\left[\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COO}\right)_{2}\right]$
$$
\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COO}\right)_{2} \longrightarrow 2 \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COO}
$$
This is a first-order reaction. The half-life of benzoyl peroxide at $100^{\circ} \mathrm{C}$ is 19.8 min. (a) Calculate the rate constant (in $\min ^{-1}$ ) of the reaction. (b) If the half-life of benzoyl peroxide is $7.30 \mathrm{h}$ or $438 \mathrm{min},$ at $70^{\circ} \mathrm{C},$ what is the activation energy (in $\mathrm{kJ} / \mathrm{mol}$ ) for the decomposition of benzoyl peroxide? (c) Write the rate laws for the elementary steps in the preceding polymerization process, and identify the reactant, product, and intermediates. (d) What condition would favor the growth of long, high-molar-mass polyethylenes?