Two chemical reactions can be written as

$A+CB,CD$

The first Yeaction is reversible. It has a forward rate constant $k_{1f}=10\frac{L}{m}$mol$-s$ and a backwards rate constant $k_{1b}=1s^{-1}.$ The second reaction is irreversible, i$.$e$.$ it can only proceed from reactant $C$ toward product $D.$ It has a forward rate constant $k_{2f}=2s^{-1}.$

The evolution of the concentrations of species A$,$ B$,$ and C can be written by these differential equations,

$\frac{del{c}_A}{\text{d}\text{e}\text{l}\text{t}}=-{10}_{f1}{c}_A{c}_C+1c_B$

$\frac{del{c}_B}{\text{d}\text{e}\text{l}\text{t}}={10}_{f1}{c}_A{c}_C-1c_B$

$\frac{del{c}_C}{\text{d}\text{e}\text{l}\text{t}}=-{10}_{f1}{c}_A{c}_C+1c_B-2c_C$

which incorporate the rate constants directly.

Write a MATLAB function that computes the right$-$hand side of these differential equations for a given set of concentrations $c=[c_A{c}_B{c}_C].$ Then use a MATLAB function $($ode $45$ or ode$15$s$)$ to solve for the concentrations of $A,B,$ and $C$ over times of ${10}^{-4}$ to $3$ seconds. $($See class notes for MATLAB reminders.$)$ Use initial conditions $c_A=50$mmo$\frac{l}{L},c_B=0$mmo$\frac{l}{L},$ and $c_C=40$mmo$\frac{l}{L}$ at $t={10}^{-4}s.$

Request a narrow spacing in time near $t={10}^{-4}s.$ To decide how small, compute the initial slope $(del\frac{c_A}{d}$elt$)$ at $t=0$ and choose a step $t$ such that $c_A$ would not change very much over one step. Requesting results for times that have a logarithmic spacing rather than a linear spacing could be useful.