Problem $1$: $100(50+50)$

Write a Python script to solve a system of ODE where up to $5$ ODEs can be solved simultaneously using $4^{th}$ order Runge$-$Kutta Method. Use your code and please solve the following set of ODEs $($any two problem$).$

Chemical Engineering Problem:

An adiabatically operated reactor contains a mixture of chemicals. A chemical $\text{'}N\text{'}$ transfers to $\text{'}M\text{'}$ over a duration of time. $\text{'}C(t)\text{'}$ is the concentration of the chemical $N$ at any point of time. Rate of reduction of the chemical $N$ can be written as $r=kC,$ where $k=k_0$exp$(-\frac{A}{RT}),T$ is the temperature of the mixture, $R$ is Universal gas constant, $A$ is activation energy. Applying the mass balance criterion we get the following ODEs

$\frac{dC}{dt}=-k_0$Cexp$(-\frac{A}{RT}),$ with I.C: $C(0)=10$gmo$\frac{l}{l}it,k_0=0\mathrm{.}1$ per sec$.$

$\frac{dT}{dt}=Cexp(-\frac{A}{RT}),$ with I.C: $T(0)=200K,=1\frac{K}{g}$molsec,$\frac{A}{R}=200K$

Solve the ODES and plot the result in graphical form $($both $C$ and $T)$ for first $200$ secs$.$ Use different values of $t$ and show which value of $t$ will converge the results. Also plot $C-$T graph for all values of $t.$ Can you see any pattern in it or it's just a chaos?

Environmental Engineering Problem:

A simple Lorenz equation for atmospheric fluid dynamics can be written as

$\frac{dT}{dt}=-10T+10y$

$\frac{dy}{dt}=28T-y-Tz$

$\frac{dz}{dt}=-2\mathrm{.}6667z+Ty$

I.C: $T=y=z=5$ at $t=0.t$ is in hours. $T$ is temperature variation in atmosphere, $y$ and $z$ are horizontal and vertical directions.

Solve the ODES and plot the result in a graphical form $(T,y$&$z)$ for first $20$ hours. Use different values of $t$ and show which values of $t$ will converge the results. Also plot $T-y$ graph and $T-z$ graph for all ts$.$ Can you see any pattern in it or it's just a chaos?