If I’m given a multi-step reaction, say:
$$\ce{2 NO + O2 -> 2 NO2} \label{rxn:R1}\tag{R1}$$
with the following mechanism:
$$ \begin{align} \ce{NO + O2 &<=>[$K$] NO3} &\quad &\text{(fast)} \label{rxn:R1.1}\tag{R1.1}\\ \ce{NO3 + NO &->[$K_1$] NO2 + NO2} &\quad &\text{(slow)} \, , \label{rxn:R1.2}\tag{R1.2} \end{align} $$
then will the rate constant of the whole reaction be equal to the product of the equilibrium constant of the fast reaction and the rate constant of the slower reaction?
Because this is how they made us solve it at my institute:
$$K_\mathrm{eq} = \frac{[\ce{NO3}]}{[\ce{NO}][\ce{O2}]}$$
$$ \begin{align} \text{Rate of slow step \eqref{rxn:R1.2}} &= K[\ce{NO3}][\ce{NO}] \\ &= KK_\mathrm{eq}[\ce{NO}][\ce{O2}][\ce{NO}] \\ \Rightarrow\text{Rate of reaction} &= K'[\ce{NO}]^2[\ce{O2}] \end{align} $$
where $K' = KK_\mathrm{eq}$, and the order is $2 + 1 = 3$.
So, is $K'$ the rate constant for the overall reaction? And is the reaction a 3rd order reaction?