The reversible reaction $A+BC$ is carried out in an adiabatic batch reactor. The rate function is given by:

$r_A=-k_1{c}_A^{0\mathrm{.}5}{c}_B^{0\mathrm{.}5}+k_2{c}_C.$

The following information is provided:

The initial temperature is $100C.$

At time $0,$ the concentrations in the reactor are: $,C_{Ao}=0\mathrm{.}1$mol$/$L and $,C_{Bo}=0\mathrm{.}125$mol$/$L

The rate constants are: $,k_1(373K)=2{10}^{-3}{s}^{-1},$ and $,k_2(373K)=3{10}^{-5}{s}^{-1}$

The activation energies are: $,E_{a1}=100k\frac{j}{m}ol$ and $E_{a2}=150k\frac{J}{m}ol$

The heat of reaction is: $,H_R(298K)=-40,000\frac{J}{m}$olA

The heat capacities of the components present in the system can be considered constant over the temperature

range of the reaction. The values are:

$C_{pA}=25\frac{J}{m}ol*K,C_{pB}=25\frac{J}{m}ol*K,C_{pC}=40\frac{J}{m}ol*K$

a$)$ Plot the conversion and concentrations of $A,B$ and $C$ as a function of time.

b$)$ What is the maximum conversion achievable in the reactor?

c$)$ How does this maximum conversion compare to the maximum conversion attainable in an isothermal case?