In free$-$radical polymerization,

$($a$)$ Calculate a number$-$average molar mass of polystyrene prepared by bulk radical polymerization. $(7pts$

$1279\frac{g}{m}ol$

$($b$)$ If the light lamp is suddenly broken in the middle of initiation stage, explain briefly possible variations in the rate of polymerization and degree of polymerization. No need to use equations. $(5$ pts$)$

Assumptions: $(1)$ The initial concentration of the benzophenone as an initiator is $3\mathrm{.}3{10}^{-3}mo\frac{l}{d}{m}^3$; $(2)$ Quantum yield of benzophenone is $0\mathrm{.}8$ ; $(3)$ Molar absorptivity of the initiator is $4\mathrm{.}34{10}^3\frac{L}{m}ol-cm$; $(4)$ Intensity of incident light is $1200mo\frac{l}{m^2}-s$; $(5)$ The rate coefficients for propagation and termination are $341$ and $4{10}^{7}d\frac{m^3}{m}ol-s,$ respectively; $(6)$ The density of styrene at the polymerization temperature is $0\mathrm{.}88\frac{g}{c}{m}^3$; $(7)$ Termination by disproportionation is negligible; $(8)$ Chain transfer is negligible; $(9)$ Neglect contributions from end groups; $(10)$ The ultraviolet light lamp $(365$ nm in wavelength$)$ is used in this experiment; $(11)$ Steady state condition.

Hint: $1L=1d{m}^3.$