$40p.$ Consider the network of $1^{st}$ order reactions given by

All four reactions are irreversible and have the same rate constant, $k.$ A cyclic system consisting of irreversible

reactions with identical rate constants would be extremely unlikely to occur, but it's a nice training problem$).$

$($a$)5$ p$.$ Write this set of reactions in matrix form $($i$.$e$.{}_A{?}_{?}=0_{?})$

$($b$)5$ p$.$ Write the rate equations in terms of the reaction rates in matrix form using the stoichiometric matrix

$($i$.$e$.,\frac{d{C}_{{?}_{?}}}{dt}={}_{{?}_{?}}^T{r}_{?}).$

$($c$)10$ p$.$ Write the rate equations in terms of a matrix of rate constants and a vector of concentrations $($i$.$e$.,$

$).$ How are the matrices ${}_{{?}_{?}}^T$ and $M_{?}$ related?

$($d$)15$ p$.$ Find an analytical solution to the rate equations for the initial condition

$C_{A^1}(t=0)=C_o$ and $C_{A^2}(t=0)=C_{A^2}(t=0)=C_{A^1}(t=0)=0$ using the matrix method discussed in class.

$($e$)5p.$ If there are $i{A}_1$ molecules, $j{A}_2$ molecules, and $k{A}_3$ molecules in the system at time $t,$ what is the

probability that at some very short time later $(t+t)$ there are $(j+1)A_2$ molecules? What is the probability

that there are $(j-1$ molecules? What is the probability that there are $j{A}_2$ molecules?