Consider a CSTR with the following reaction sequence: $2AB,BA,BC.$ The reaction rates per unit volume are $r_1=k_1{C}_A^2,r_2=k_2{C}_B$ and $r_3=k_3{C}_B,$ where $C_A$ and $C_B$ are the reactor concentrations. The feed concentrations are $C_{Af}$ and $C_{Bf},$ and the volumetric flow rate in and out of the reactor is $q.$ The reactor has constant volume $V.$

$($a$)$ Show that the steady$-$state mass balance can be written as:

$\left[\begin{array}{ccc}2C_A^2& -C_B& 0\\ -C_A^2& C_B& C_B\end{array}\right]k=\left[\begin{array}{c}\frac{q}{V}(C_{Af}-C_A)\\ \frac{q}{V}(C_{Bf}-C_B)\end{array}\right]A(C_A,C_B)k=b(C_A,C_B,C_{Af},C_{Bf})$

where: $k={\left[\begin{array}{ccc}{k}_1& k_2& k_3\end{array}\right]}^T.$

$($b$)$ Assume that $4$ steady$-$state experiments have been performed with constant $\frac{q}{V}$ and different combinations of the feed concentrations and that the corresponding reactor concentrations have been measured to yield data sets $\{C_{Af,i},C_{Bf,i},C_{A,i},C_{B,i}\}$ for $i[1,4].$ Formulate the least$-$squares solution to estimate the parameters $k.$