Polyethylene is used in many items such as 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 (the basic unit
is called a monomer) together (see p. 369 ). The initiation step is
$$
\mathrm{R}_{2} \stackrel{k_{1}}{\longrightarrow} 2 \mathrm{R} \cdot \quad
\text { initiation }
$$
The \(\mathrm{R}\) - species (called a radical) reacts with an ethylene molecule
\((\mathrm{M})\) to generate another radical
$$
\mathrm{R} \cdot+\mathrm{M} \longrightarrow \mathrm{M}_{1} \cdot
$$
Reaction of \(\mathrm{M}_{1}\). 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
$$
\mathrm{M}^{\prime} \cdot+\mathrm{M}^{\prime \prime} \cdot
\stackrel{k_{t}}{\longrightarrow} \mathrm{M}^{\prime}-\mathrm{M}^{\prime
\prime} \quad \text { termination }
$$
(a) The initiator used in the polymerization of ethylene is benzoyl peroxide
\(\left[\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COO}\right)_{2}\right]:\)
$$
\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COO}\right)_{2} \longrightarrow 2
\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COO} \cdot
$$
This is a first-order reaction. The half-life of benzoyl peroxide at
\(100^{\circ} \mathrm{C}\) is \(19.8 \mathrm{~min} .\) (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 above polymerization process and identify the
reactant, product, and intermediates. (d) What condition would favor the
growth of long high-molar-mass polyethylenes?
