A gas$-$phase reaction involving all ideal gases synthesizes a valuable product P according to the

general reaction:

A $+$ B $->$ P $+$ R

This reaction is conducted in an isothermal reactor. And the proposed mechanism for the reaction

is via the elementary steps below:

A $+$ B $$ I$1$

I$1$I$2$

I$2$ P $+$R

PART $1$

$(1\mathrm{.}1)$ Derive the rate expression based on this mechanism for rate of production P as a function of the concentrations of the reactants$/$products$.$

A$)$ Assume all $3$ steps are reversible using PSSH alone

B$)$ Assume all $3$ steps are irreversible. What is the rigorous justification for this?

C$)$ Assume step $1$ is quasi$-$equilibrated and steps $2$ and $3$ are irreversible. What is the rigorous justification for this?

$(1\mathrm{.}2)$ Show your work on reaction rate arrow diagram for all three cases above.