Consider the following set of elementary reactions between hydrogen and chlorine:

$C{l}_2+M{k}_{1}Cl+Cl+M$

$Cl+H_2{k}_{2}HCl+H$

$Cl+Cl+M{?}_{k_3}^{kCl}+Cl$

$C{l}_2+M$

$($a$)$ Show that this scheme leads to the rate expression

$\frac{d{C}_{HC}}{dt}=2k_2\sqrt[\phantom{2}]{\frac{k_1}{k_4}}(C_{C{l}_2}{)}^{\frac{1}{2}}{C}_{H_2}$

$($b$)NC{l}_3$ is a potential inhibitor for the reaction, according to

$NC{l}_3+Cl{k}_{5}NC{l}_2+C{l}_2$

Derive the kinetic rate expression for the scheme including reaction $($R$5).$ What is the form of the rate expression when $k_5{k}_1?$

$($c$)O_2$ also inhibits the reaction through

$H+O_2+Mlongrightarro{w}^{k_6}H{O}_2+M$

Why should $NC{l}_3$ be a much stronger inhibitor than $O_2?$