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: $r=\frac{1}{2}\frac{d[HBr]}{dt}$.
\begin{equation} \frac{d[HBr]}{dt}=k_{p1}[Br][H_2]+k_{p2}[H][Br_2]-k_{r}[H][HBr]\label{ec115} \end{equation}
We apply the steady state approximation for H.
\begin{equation} \frac{d[H]}{dt}=0=k_{p1}[Br][H_2]-k_{p2}[H ][Br_2]-k_r [H][HBr]\label{ec116} \end{equation}
Solving equation $(\ref{ec116})$:
\begin{equation} k_{p2}[H][Br_2]=k_{p1}[Br][H_2 ]-k_{r}[H][HBr] \end{equation}
and substituting into $(\ref{ec115})$:
\begin{equation} \frac{d[HBr]}{dt}=2k_{p2}[H][ Br_2] \end{equation}
Applying the steady-state approximation to Br:
\begin{equation} \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] \end{equation}
Solving for $[Br]$:
\begin{equation} [Br]={\left(\frac{k_i}{k_t}\right)}^{1/2}[Br]^{1/ 2} \end{equation}
Solving for $[H]$ from equation $(\ref{ec116})$:
\begin{equation} [H]=\frac{k_{p1}[Br][H_2]}{k_{p2}[ Br_2]+k_r[HBr]} \end{equation}
Substituting $[Br]$ into this last equation, we get:
\begin{equation} [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]} \end{equation}
Therefore, the variation of [HBr] in time will be given by:
\begin{equation} \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] \end{equation}
As $r =\frac{1}{2}\frac{d[HBr]}{dt}$
\begin{equation} 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] \end{equation}
Dividing the latter equation for $k_{p2}$ and $[Br_2]$ is obtained:
\begin{equation} 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]}} \end{equation}