Question

The combination of \(\text{NO}_{2}(g)\) and \(\text{CO}(g)\) occurs in two steps as follows: What is the rate law for the reaction?
\(2\text{NO}_{2}\rightarrow \text{NO}+\text{NO}_{3}\) \((k_{1})\) - slow
\(\text{NO}_{3}+\text{CO}\rightarrow \text{CO}_{2}+\text{NO}_{2}\) \((k_{2})\) - fast

• A. Rate \(=k_{2}[\text{NO}_{3}][\text{CO}]\)
• B. Rate \(=k_{1}[\text{NO}_{2}]\)
• C. Rate \(=k_{1}[\text{NO}_{2}]^{3}\)
• D. Rate \(=k_{1}[\text{NO}_{2}]^{2}\)

Hint:

The overall rate law is written from the slow (rate-determining) step, using its molecularity for the powers.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
The overall rate of a multi-step reaction is controlled by the slowest step, called the rate-determining step.
For an elementary step, the power of each reactant in the rate law equals its coefficient in that step.

Step 2: Identify the slow step
The slow step is \(2\text{NO}_{2}\rightarrow \text{NO}+\text{NO}_{3}\) with rate constant \(k_{1}\).
CO takes part only in the fast step, so it does not appear in the rate law.

Step 3: Calculation
Two \(\text{NO}_{2}\) molecules react in the slow step, so \(\text{Rate}=k_{1}[\text{NO}_{2}]^{2}\).

Step 4: Compare the options
Option (A) is built from the fast step and contains the intermediate \(\text{NO}_{3}\).
Options (B) and (C) give the wrong power of \(\text{NO}_{2}\).
Only option (D) matches the slow step.

Final Answer:
The rate law is k1 times the square of the nitrogen dioxide concentration, which is option (D). \[ \boxed{\text{(D) }\text{Rate }=k_{1}[\text{NO}_{2}]^{2}} \]