$7.($a$)$ The reaction $2$NO $+$ O$2->2$NO$2$ is third order. Assuming that a small amount of NO$3$ exists in rapid

reversible equilibrium with NO and O$2$ and that the rate$-$determining step is the slow bimolecular

reaction NO$3+$ NO $->2$NO$2,$ derive the rate expression for this mechanism.

$($b$)$ Another possible mechanism for the reaction $2$NO $+$ O$2->2$NO$2$ is

$(1)$ NO $+$ NO $->$ N$2$O$2$ k$1$

$(2)$ N$2$O$2->2$NO k$2$

$(3)$ N$2$O$2+$ O$2->2$NO$2$ k$3$

Apply the steady$-$state approximation to $[$N$2$O$2]$ to obtain the rate law for d$[$NO$2]/$dt$.$

$($c$)$ How would you distinguish experimentally between the mechanisms suggested in parts $($a$)$ and

$($b$)?$