P$19\mathrm{.}32$ The chlorination of vinyl chloride, $C_2{H}_{3}Cl+C{l}_2{C}_2{H}_{3}C{l}_3,$ is believed to proceed by the following mechanism:

$C{l}_2{k}_{1}2Cl*$

$Cl*+C_2{H}_{3}Cl{k}_2{C}_2{H}_{3}C{l}_2*$

$C_2{H}_{3}C{l}_2*+C{l}_2{k}_3{C}_2{H}_{3}C{l}_3+Cl*$

$C_2{H}_{3}C{l}_2*+C_2{H}_{3}C{l}_2*k_4$ stable species

Derive the rate law expression for the chlorination of vinyl chloride based on this mechanism.