What is the rate law for the overall process?

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.

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.