For living anionic polymerization of styrene at $25C$ using butyl lithium as an initiator,

$($a$)$ Calculate the initial rate of polymerization and number$-$average molar mass of the polystyrene formed when $90\%$ of the styrene monomers are converted. $(6$ pts$)R_p=2\mathrm{.}4mo\frac{l}{d}{m}^3*s\frac{,}{b}ar(M{)}_n=78000\frac{g}{m}ol$

$($b$)$ Assuming that initial molar concentration of butyl lithium is identical to that of sodium naphthalide and the other conditions are constant, explain briefly possible variations in initial rate of polymerization and numberaverage molar mass when sodium naphthalide is used as initiator instead of butyl lithium and provide reason$($s$)$ for your answer. $(5$ pts$)$

Assumptions: $(1)$ Neglect contributions from end groups; $(2)$ The molar mass of butyl lithium is $64\frac{g}{m}ol$; $(3)$ The rate coefficient for propagation is $320d\frac{m^3}{m}ol-s$ and is regardless of the type of initiator; $(4)$ The initial mass concentration of styrene is $260\frac{g}{d}{m}^3$ and that of butyl lithium is $0\mathrm{.}192\frac{g}{d}{m}^3$; $(5)$ At initial stage of polymerization, $[M]=[M{]}_0$; $(6)$ Tetrahydrofuran is used as solvent for polymerization using sodium naphthalide initiator.