PYTHON QUESITON $3.$ Solving a system of ODEs for a series of reactions in a batch reactor. $(50$

pts$)$

The following series of reactions takes place in a batch reactor where our goal is to make product

$D,$ but a side reaction in which one of the reactants $(A)$ creates an undesired product $(C)$ that

reacts with $D$ to make another undesired product $(E).$

$A+B{k}_{1}2D$

$2A{k}_{2}C$

$C+D{k}_{3}E$

The governing equations are:

$\frac{d{C}_A}{dt}=-k_1{C}_A{C}_B-2k_2{C}_A^2$

$\frac{d{C}_B}{dt}=-k_1{C}_A{C}_B$

$\frac{d{C}_C}{dt}=k_2{C}_A^2-k_3{C}_C{C}_D$

$\frac{d{C}_D}{dt}=2k_1{C}_A{C}_B-k_3{C}_C{C}_D$

$\frac{d{C}_E}{dt}=k_3{C}_C{C}_D$

The initial concentrations of species is $\{50$;$15$;$0$;$0$;$0\}($all in mol$/$L$).$ The reaction time is $10hrs.$

The rate constants are $k_1=5{10}^{-5}\frac{L}{m}o\frac{l}{h}r,k_2=0\mathrm{.}009\frac{L}{m}o\frac{l}{h}r,k_3=0\mathrm{.}007\frac{L}{m}o\frac{l}{h}r.$

Assume constant volume and isothermal conditions.

a$)$ Solve the system in Python and provide a plot of concentration versus time. The plot can

use symbols or lines but should be labeled, with a legend, and be neat.

b$)$ What is the maximum concentration of $D$ in your reactor and at what time does it reach

this value? Your answer should be correct within $2$ decimal places to receive full credit

$($i$.$e$.$ do not guess your number off your plot$).$ Show the Python code you used to determine this.

Please use

import numpy as np

import matplotlib.pyplot as plt

from scipy.integrate import solve$\_$ivp