Write the free radical mechanism for the polymerization of ethene.

(N/A) The polymerization of ethene to polythene involves a free radical mechanism initiated by benzoyl peroxide. The process occurs in three steps:
$1$. Chain Initiation Step:
Benzoyl peroxide decomposes to form phenyl radicals,which then react with ethene to form a new radical.
$(C_6H_5COO)_2 \to 2C_6H_5COO\cdot \to 2C_6H_5\cdot + 2CO_2$
$C_6H_5\cdot + CH_2=CH_2 \to C_6H_5-CH_2-\dot{C}H_2$
$2$. Chain Propagating Step:
The radical reacts with more ethene molecules to grow the chain.
$C_6H_5-CH_2-\dot{C}H_2 + n(CH_2=CH_2) \to C_6H_5-(CH_2-CH_2)_n-CH_2-\dot{C}H_2$
$3$. Chain Terminating Step:
Two long-chain radicals combine to form the final polymer.
$C_6H_5-(CH_2-CH_2)_n-CH_2-\dot{C}H_2 + \dot{C}H_2-CH_2-(CH_2-CH_2)_n-C_6H_5 \to C_6H_5-(CH_2-CH_2)_n-CH_2-CH_2-CH_2-CH_2-(CH_2-CH_2)_n-C_6H_5$