$($a$)$ Calculate the concentration of benzoyl peroxide required to prepare polystyrene with a number$-$average molar mass of $750kgmo{l}^{-1}$ by bulk polymerization at $60C.$ Assume: $($i$)$ the initiator is $60\%$ efficient; $($ii$)$ the rate coefficients for initiator dissociation, propagation and termination are, respectively, $2{10}^{-6}{s}^{-1},341d{m}^{3}mo{l}^{-1}{s}^{-1}$ and $4{10}^{7}d{m}^{3}mo{l}^{-1}{s}^{-1}$; $($iii$)$ the density of styrene at the polymerization temperature is $0\mathrm{.}88gc{m}^{-3}$; and $($iv$)$ that termination by disproportionation, and chain transfer to initiator and monomer, are negligible.

$($b$)$ Calculate initial values of $($i$)$ the radical concentration and $($ii$)$ the rate of bulk polymerization expected for bulk polymerization of styrene at the concentration of benzoyl peroxide determined in $($a$).$