Question

What is the rate law for the overall process?

(1) NO(g) + `O_3`(g) `->` `NO_2`(g) + `O_2`(g) (slow)
(2) O(g) + `NO_2`(g) `->` NO(g) + `O_2` (fast)

Answer 1

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

The best answer is:

`r(t) = k_1["NO"]["O"_3]`

but it shows no explicit dependency on `["O"]` in this form. In another form,

`r(t) = k_2["NO"_2]["O"]`

which shows no explicit dependency on `["O"_3]`. However, this form has `["NO"_2]`, an intermediate, in it, so arguably the first form is more experimentally convenient.

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First off, the overall reaction is:

`"NO"(g) + "O"_3(g) stackrel(k_1" ")(->) "NO"_2(g) + "O"_2(g)` (slow)
`ul("O"(g) + "NO"_2(g) stackrel(k_2" ")(->) "NO"(g) + "O"_2(g))` (fast)
`"O"(g) + "O"_3(g) stackrel(k_"obs")(->) 2"O"_2(g)`

So, the rate law ought to contain `["O"]` and `["O"_3]` 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:

`color(blue)(r(t) = k_1["NO"]["O"_3])`

Now, it can be seen that `"NO"` is a catalyst, and `"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 `["NO"_2]` is approximately constant, i.e. assume the steady-state approximation for the intermediate (`"NO"_2`):

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

Solving for `["NO"]`:

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

So far we would have:

`r(t) = cancel(k_1)k_2/cancel(k_1) (["NO"_2]["O"])/cancel(["O"_3])cancel(["O"_3])`

`= k_2["NO"_2]["O"]`

But we see that although the approximation is valid, we get no new information from this... if we found an expression for `["NO"_2]`, it would end up being in terms of `["NO"]` again, and we find ourselves in an endless cycle, solving over and over again to get rid of `["NO"]` only to find `["NO"_2]` again, and vice versa.

The mechanism proposed is therefore not good, because it does not show the dependency of the rate law on `[O_3]`, a key reactant, and does not allow for representation of the rate based on ONLY REACTANTS.