Question 16.2: Calculation of Initiator Required Calculate the amount of be......

One benzoyl peroxide molecule produces two free radicals that initiate two chains that then combine to form one polyethylene chain. Thus, there is a 1:1 ratio between benzoyl peroxide molecules and polyethylene chains. If the initiator were 100% effective, one molecule of benzoyl peroxide would be required per polyethylene chain.
To determine the amount of benzoyl peroxide required, the number of chains with an average molecular weight of 200,000 g/mol in 1 kg of polyethylene must be calculated.
The molecular weight of ethylene = (2 C atoms)(12 g/mol) + (4 H atoms) (1 g/mol) = 28 g/mol. The number of molecules per chain (also known as the degree of polymerization) is given by
\frac{200,000 \ g/mol}{28 \ g/mol} = 7143 ethylene molecules/chain
The total number of monomers required to form 1 kg of polyethylene is
\frac{(1000 \ g)(6.022 × 10^{23} \ \text{monomers}/\text{mol})}{28 \ g/\text{mol}} = 2.15 × 10^{25} monomers
Thus, the number of polyethylene chains in 1 kg is
\frac{2.15 × 10^{25} \ \text{ethylene molecules}}{7143 \ \text{ethylene molecules/chain}} = 3.0 × 10^{21} chains
Since the benzoyl peroxide initiator is only 20% effective, 5(3.0 × 10^{21})=1.5 × 10^{22} benzoyl peroxide molecules are required to initiate 3.0 × 10^{21} chains. The molecular weight of benzoyl peroxide is (14 C atoms)(12 g/mol) + (10 H atoms) (1 g/mol) + (4 O atoms)(16 g/mol) = 242 g/mol. Therefore, the amount of initiator required is
\frac{1.5 × 10^{22} \ \text{molecules}\ (242 g/\text{mol})}{6.022 × 10^{23} \ \text{molecules/mol}} = 6.0 \ g