Consider again the chemical reaction mechanism from PA$9$:

$\frac{dA}{dt}=-k_{1}AB+k_{2}C$

$\frac{dB}{dt}=-k_{1}AB+k_{2}C-k_{3}BC$

$\frac{dC}{dt}=k_{1}AB-k_{2}C-k_{3}BC$

The goal of this problem is to analyze how the maximum value of $C$ depends on the reaction

rate $k_3.$ It's theorized that this dependence can be modeled with the following equation:

$C_{\text{m}\text{a}\text{x}}=\frac{Z}{k_3^m}$

Write a script $m-$file and any required function $m-$files to determine the model constants $Z$

and $m$ by doing the following:

Solve the system of ODEs for values of $k_3$ varying from $1$ to $20$ in increments of $2.$ Set $k1=$

$k2=0\mathrm{.}1$ for all cases. The only result you need to save from each case is the maximum value

$2,$ and $C(0)=0.$

Determine the model constants $Z$ and $m$ that result in a least$-$squares curve fit of the model

for $C_{\text{m}\text{a}\text{x}}$ to the numerical data generated for $C_{\text{m}\text{a}\text{x}}$ vs $k_3$ by the ODE solutions.

Generate a plot of the data $($markers only$)$ along with model equation $($line only$).$