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The reaction is:

$$\ce{NO(g) + NO2(g) + O2(g) -> N2O5(g)}$$

From the experimental data, I have found the rate law to be: $$\ce{Rate~=~k[NO][NO2]}$$

This means the overall reaction does not depend on $\ce{O2(g)}$.

Can someone suggest how I go about building the elementary steps to form the reaction mechanism?

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First step - the rate determining step or the slow step (activation energy is high). $$\ce{NO(g) + NO2(g) -> N2O3(g)}$$

2nd step - the fast step. $$\ce{N2O3(g) + O2(g) -> N2O5(g)}$$

Since the rate determining step does not include $\ce{O2(g)}$, the rate is independent of the initial $\ce{O2(g)}$ concentration.

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