Obtain the kinetic equation for the radical linear chain reaction, $H_2 + Br_2 \rightarrow 2HBr$, which occurs through the following mechanism: \begin{eqnarray} Br_2 + M & \stackrel{k_i}{\rightarrow} & 2Br + M\label{ec110}\\ Br + H_2 & \stackrel{k_{p1}}{\rightarrow} & HBr + H\label{ec111}\\ H + Br_2 & \stackrel{k_{p2}}{\rightarrow} & HBr + Br\label{ec112}\\ H + HBr & \stackrel{k_{r}}{\rightarrow} & H_2 + Br\label{ec113}\\ 2Br + M & \stackrel{k_{t}}{\rightarrow} & Br_2 + M\label{ec114} \end{eqnarray}

$(\ref{ec110})$ Initiation stage

$(\ref{ec111})$ and $(\ref{ec112})$ Stages of propagation

$(\ref{ec113})$ Delay stage

$(\ref{ec114})$ Termination stage

We define the rate of the reaction for HBr: $r=\frac{1}{2}\frac{d[HBr]}{dt}$.

$$\frac{d[HBr]}{dt}=k_{p1}[Br][H_2]+k_{p2}[H][Br_2]-k_{r}[H][HBr]\label{ec115}$$

We apply the steady state approximation for H.

$$\frac{d[H]}{dt}=0=k_{p1}[Br][H_2]-k_{p2}[H ][Br_2]-k_r [H][HBr]\label{ec116}$$

Solving equation $(\ref{ec116})$:

$$k_{p2}[H][Br_2]=k_{p1}[Br][H_2 ]-k_{r}[H][HBr]$$

and substituting into $(\ref{ec115})$:

$$\frac{d[HBr]}{dt}=2k_{p2}[H][ Br_2]$$

Applying the steady-state approximation to Br:

$$\frac{d[Br]}{dt}=0=2k_{i}[Br_2][M]-\cancel{k_ {p1}[Br][H_2]}+\cancel{k_{p2}[H][Br_2]}+\cancel{k_r[H][HBr]}-2k_{t}[Br]^2[M]$$

Solving for $[Br]$:

$$[Br]={\left(\frac{k_i}{k_t}\right)}^{1/2}[Br]^{1/ 2}$$

Solving for $[H]$ from equation $(\ref{ec116})$:

$$[H]=\frac{k_{p1}[Br][H_2]}{k_{p2}[ Br_2]+k_r[HBr]}$$

Substituting $[Br]$ into this last equation, we get:

$$[H]=\frac{k_{p1}\left(\frac{ k_i}{k_ {t}}\right)^{1/2}[Br_2]^{1/2}[H_2]}{k_{p2}[Br_2]+k_r[HBr]}$$

Therefore, the variation of [HBr] in time will be given by:

$$\frac{d[HBr]}{dt}=2k_{p2}\frac{k_{p1}\left(\frac{k_i}{k_{ t}}\right)^{1/2}[Br_2]^{1/2}[H_2]}{k_{p2}[Br_2]+k_r[HBr]}[Br_2]$$

As $r =\frac{1}{2}\frac{d[HBr]}{dt}$

$$r=k_{p2}\frac{k_{p1}\left(\frac{k_i}{k_{ t}}\right)^{1/2}[Br_2]^{1/2}[H_2]}{k_{p2}[Br_2]+k_r[HBr]}[Br_2]$$

Dividing the latter equation for $k_{p2}$ and $[Br_2]$ is obtained:

$$r=\frac{1}{2}\frac{d[HBr]}{dt}=\frac{k_{p1 }\left(\frac{k_i}{k_t}\right)^{1/2}[Br_2]^{1/2}[H_2]}{1+\frac{k_r}{k_{p2}}\frac {[HBr]}{[Br_2]}}$$