Nitrogen dioxide reacts with carbon monoxide to produce nitrogen monoxide and carbon dioxide.

NO2(g) + CO(g) ==>NO(g) + CO2(g)

A proposed mechanism for this reaction is

2No2(g) = NO3(g) + NO(g) (fast,equilibrium)
NO2(g) + CO(g) ==> NO2(g) + CO2(g) (slow)

The rate law consitent with the propose mechanism is

rate = k[NO2]^2[CO]/[NO]

How?

1.The two equations you have written does not add to the final equation. Something is amiss.

2. I haven't seen any rate equations that look like equilibrium constants.
Check your post please.

The rate law for a proposed mechanism is determined by looking at the slowest step in the reaction mechanism. In this case, the slow step is the second step:

NO2(g) + CO(g) ==> NO(g) + CO2(g) (slow step)

The coefficients in this balanced equation give us the stoichiometry of the reaction. From the equation, we can see that one molecule of NO2 and one molecule of CO react to produce one molecule of NO and one molecule of CO2.

Based on the stoichiometry, we can write the rate expression for the slow step as:

rate = k[NO2][CO]

However, the proposed mechanism also includes a fast equilibrium step:

2NO2(g) ⇌ NO3(g) + NO(g) (fast, equilibrium step)

Since this step is fast, it quickly reaches equilibrium, which means the concentrations of the reactants and products in this step will remain relatively constant over time. Therefore, the concentration of NO3 can be considered constant and included in the rate constant, k.

After simplifying and rewriting the rate expression, we get:

rate = k'[NO2]^2[CO]

Here, k' represents the modified rate constant that incorporates the concentration of NO3.

To summarize, the rate law consistent with the proposed mechanism is:

rate = k[NO2]^2[CO]/[NO]

This rate law takes into account both the slow step and the equilibrium step in the reaction mechanism.

To determine the rate law for a reaction, you need to look at the proposed mechanism and identify the slowest step, also known as the rate-determining step. In this case, the slow step is:

NO2(g) + CO(g) → NO(g) + CO2(g) (slow)

The rate-determining step provides the coefficients that are used to determine the order of reaction for each reactant.

From the balanced chemical equation of the slow step, you can see that the reaction involves one molecule of NO2 and one molecule of CO. Therefore, the order of reaction for NO2 and CO is 1.

However, the rate law also includes a term with NO in the denominator:

rate = k[NO2]^2[CO]/[NO]

This suggests that NO is involved in a previous step of the reaction mechanism. Looking at the proposed mechanism, you can see that the formation of NO2 from NO3 and NO is a fast equilibrium step:

2NO2(g) ⇌ NO3(g) + NO(g) (fast, equilibrium)

Since NO appears in the denominator of the rate law, it means that NO is a product of a fast step and is consumed in the rate-determining step. Therefore, the order of reaction for NO is either zero or a negative value.

Overall, the rate law for this proposed mechanism is:

rate = k[NO2]^2[CO]/[NO]

This rate law accounts for the slow step as well as the fast equilibrium step involving NO2 and NO3.