The rate equation for a reaction mechanism is determined by the slowest step (the rate-determining step).

The rate equation for a reaction mechanism is determined by the slowest step (the rate-determining step).

The given mechanism is:

1. $\text{N}_2\text{O}_5\text{(g)} \to \text{N}_2\text{O}_3\text{(g)} + \text{O}_2\text{(g)}$ (Slow step)
2. $\text{N}_2\text{O}_3\text{(g)} \to \text{NO(g)} + \text{NO}_2\text{(g)}$ (Fast step)
3. $\text{NO(g)} + \text{N}_2\text{O}_5\text{(g)} \to 3\text{NO}_2\text{(g)}$ (Fast step)

The slow step is the first step:

$$\text{N}_2\text{O}_5\text{(g)} \to \text{N}_2\text{O}_3\text{(g)} + \text{O}_2\text{(g)}$$

For an elementary step, the rate law can be written directly from the stoichiometry of the reactants in that step. In this slow step, one molecule of $\text{N}_2\text{O}_5$ reacts.

Therefore, the rate equation for the reaction is:

$$\text{Rate} = k[\text{N}_2\text{O}_5]$$