# What is the rate law for the overall process?

## (1) NO(g) + ${O}_{3}$(g) $\to$ $N {O}_{2}$(g) + ${O}_{2}$(g) (slow) (2) O(g) + $N {O}_{2}$(g) $\to$ NO(g) + ${O}_{2}$ (fast)

Jul 26, 2018

The mechanism is flawed, because it does not allow one to write a rate law in terms of ONLY reactants.

$r \left(t\right) = {k}_{1} \left[{\text{NO"]["O}}_{3}\right]$

but it shows no explicit dependency on $\left[\text{O}\right]$ in this form. In another form,

$r \left(t\right) = {k}_{2} \left[\text{NO"_2]["O}\right]$

which shows no explicit dependency on $\left[{\text{O}}_{3}\right]$. However, this form has $\left[{\text{NO}}_{2}\right]$, an intermediate, in it, so arguably the first form is more experimentally convenient.

First off, the overall reaction is:

${\text{NO"(g) + "O"_3(g) stackrel(k_1" ")(->) "NO"_2(g) + "O}}_{2} \left(g\right)$ (slow)
$\underline{{\text{O"(g) + "NO"_2(g) stackrel(k_2" ")(->) "NO"(g) + "O}}_{2} \left(g\right)}$ (fast)
${\text{O"(g) + "O"_3(g) stackrel(k_"obs")(->) 2"O}}_{2} \left(g\right)$

So, the rate law ought to contain $\left[\text{O}\right]$ and $\left[{\text{O}}_{3}\right]$ in some form.

Using the slow step, we should be able to write a preliminary rate law based on the coefficients, only because it is the rate-determining step:

$\textcolor{b l u e}{r \left(t\right) = {k}_{1} \left[{\text{NO"]["O}}_{3}\right]}$

Now, it can be seen that $\text{NO}$ is a catalyst, and ${\text{NO}}_{2}$ is an intermediate. We shouldn't leave this in terms of a catalyst...

Since the first step is slow and the second is fast, we could try to assume that $\left[{\text{NO}}_{2}\right]$ is approximately constant, i.e. assume the steady-state approximation for the intermediate (${\text{NO}}_{2}$):

(d["NO"_2])/(dt) ~~ 0 = k_1["NO"]["O"_3] - k_2["O"]["NO"_2]

Solving for $\left[\text{NO}\right]$:

["NO"] = k_2/k_1 (["NO"_2]["O"])/(["O"_3])

So far we would have:

$r \left(t\right) = \cancel{{k}_{1}} {k}_{2} / \cancel{{k}_{1}} \left(\left[{\text{NO"_2]["O"])/cancel(["O"_3])cancel(["O}}_{3}\right]\right)$

$= {k}_{2} \left[\text{NO"_2]["O}\right]$

But we see that although the approximation is valid, we get no new information from this... if we found an expression for $\left[{\text{NO}}_{2}\right]$, it would end up being in terms of $\left[\text{NO}\right]$ again, and we find ourselves in an endless cycle, solving over and over again to get rid of $\left[\text{NO}\right]$ only to find $\left[{\text{NO}}_{2}\right]$ again, and vice versa.

The mechanism proposed is therefore not good, because it does not show the dependency of the rate law on $\left[{O}_{3}\right]$, a key reactant, and does not allow for representation of the rate based on ONLY REACTANTS.