Explain the following for the chain$-$growth polymerization:

a$)$ It is found that polymerization progresses even without an initiator when the vinyl monomers are

exposed to high temperatures.

i$)$ How does the initiation occur in this case? Please explain briefly.

ii$)$ Model the rate of polymerization $($rp$).$ Find the average molecular weight of the polymer.

b$)$ Assume that the temperature of polymerization is increased from $50\backslash$deg C to $100\backslash$deg C for pure styrene, and

the polymerization is carried out to completion.

i$)$ Do you expect the equilibrium monomer concentration to reduce or increase?

ii$)$ Calculate the equilibrium monomer concentration as a function of temperature.