Problem 67 - Chemistry and Chemical Reactivity

Question

At temperatures below \(500 \mathrm{K},\) the reaction between carbon monoxide and nitrogen dioxide $$ \mathrm{NO}_{2}(\mathrm{g})+\mathrm{CO}(\mathrm{g}) \longrightarrow \mathrm{CO}_{2}(\mathrm{g})+\mathrm{NO}(\mathrm{g}) $$ has the following rate equation: Rate \(=k\left[\mathrm{NO}_{2}\right]^{2} .\) Which of the three mechanisms suggested here best agrees with the experimentally observed rate equation? Mechanism 1 \(\quad\) single, elementary step $$\mathrm{NO}_{2}+\mathrm{CO} \longrightarrow \mathrm{CO}_{2}+\mathrm{NO}$$ Mechanism \(2 \quad\) Two steps $$\begin{aligned}&\text { Slow } \quad \mathrm{NO}_{2}+\mathrm{NO}_{2} \longrightarrow \mathrm{NO}_{3}+\mathrm{NO}\\\&\text { Fast } \quad \mathrm{NO}_{3}+\mathrm{CO} \longrightarrow \mathrm{NO}_{2}+\mathrm{CO}_{2}\end{aligned}$$ Mechanism 3 \(\quad\) Two steps $$\begin{aligned}&\text { Slow } \quad \mathrm{NO}_{2} \longrightarrow \mathrm{NO}+\mathrm{O}\\\&\text { Fast } \quad \mathrm{CO}+\mathrm{O} \longrightarrow \mathrm{CO}_{2}\end{aligned}$$

Step-by-Step Solution

Answer

Mechanism 2 supports the observed rate equation.

Step 1: Identify the Experimental Rate Equation

The experimentally observed rate equation is given by: \(\text{Rate} = k[\mathrm{NO}_{2}]^{2}\). This indicates that the rate of reaction depends solely on the concentration of \(\mathrm{NO}_{2}\) squared, meaning two molecules of \(\mathrm{NO}_{2}\) are involved in the rate-determining step (slow step).

Step 2: Analyze Mechanism 1

Mechanism 1 is a single step reaction \(\mathrm{NO}_{2} + \mathrm{CO} \longrightarrow \mathrm{CO}_{2} + \mathrm{NO}\). If this were the correct mechanism, the rate equation would depend on both \(\mathrm{NO}_{2}\) and \(\mathrm{CO}\) concentrations, i.e., \(\text{Rate} = k[\mathrm{NO}_{2}][\mathrm{CO}]\). This does not match the experimentally observed rate equation.

Step 3: Analyze Mechanism 2

Mechanism 2 consists of two steps: a slow step \(\mathrm{NO}_{2} + \mathrm{NO}_{2} \longrightarrow \mathrm{NO}_{3} + \mathrm{NO}\), and a fast step \(\mathrm{NO}_{3} + \mathrm{CO} \longrightarrow \mathrm{NO}_{2} + \mathrm{CO}_{2}\). Here, the rate-determining step involves two \(\mathrm{NO}_{2}\) molecules, which matches the experimentally observed rate equation \(\text{Rate} = k[\mathrm{NO}_{2}]^{2}\).

Step 4: Analyze Mechanism 3

Mechanism 3 involves a slow step \(\mathrm{NO}_{2} \longrightarrow \mathrm{NO} + \mathrm{O}\), and a fast step \(\mathrm{CO} + \mathrm{O} \longrightarrow \mathrm{CO}_{2}\). In this mechanism, the rate-determining step only involves one molecule of \(\mathrm{NO}_{2}\), suggesting a rate equation as \(\text{Rate} = k[\mathrm{NO}_{2}]\). This contradicts the experimental rate equation.

Step 5: Determine the Best Mechanism

Based on the analysis, Mechanism 2 is the only mechanism that agrees with the experimentally observed rate equation \(\text{Rate} = k[\mathrm{NO}_{2}]^{2}\), as its rate-determining step involves two molecules of \(\mathrm{NO}_{2}\). Therefore, Mechanism 2 is the correct mechanism.

Key Concepts

Rate Equation

The rate equation of a chemical reaction reveals how the rate is influenced by the concentration of each reactant. It is expressed in the form \(\text{Rate} = k[A]^m[B]^n\), where \(k\) is the rate constant, \([A]\) and \([B]\) are the concentrations of reactants, and \(m\) and \(n\) are the orders of the reaction with respect to each reactant.
The order of a reaction can give insight into the mechanisms involved in the reaction.
For example, a second-order reaction with respect to a particular reactant indicates that two molecules of this reactant participate in the rate-determining step of the mechanism.
This is crucial because the experimentally determined rate equation: \(\text{Rate} = k[\mathrm{NO}_{2}]^2\), suggests that only \([\mathrm{NO}_2]^2\) impacts the rate, unlike a supposed equation \(\text{Rate} = k[\mathrm{NO}_2][\mathrm{CO}]\), which would imply involvement of both reactants in the rate-determining step.

Rate-determining Step

The rate-determining step is the slowest step in a reaction mechanism and acts like a bottleneck, controlling the overall reaction rate.
Identifying this step is critical for understanding reaction kinetics, as it defines which concentrations appear in the rate equation.

In the provided mechanisms, the experimentally observed rate equation, \(\text{Rate} = k[\mathrm{NO}_2]^2\), indicates that the rate-determining step involves two \(\mathrm{NO}_2\) molecules reacting. This aligns with Mechanism 2, where the slow step \(\mathrm{NO}_2 + \mathrm{NO}_2 \rightarrow \mathrm{NO}_3 + \mathrm{NO}\) clearly shows such a combination.
The conversion of two \(\mathrm{NO}_2\) molecules confirms why \(\mathrm{NO}_2\)'s concentration squared appears in the rate equation, reflecting its role and importance in the rate-determining step compared to the fast subsequent steps that don't affect the overall reaction speed significantly.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the speed of chemical reactions and the factors affecting them.
Understanding kinetics allows chemists to delve into reaction mechanisms, exploring how certain steps control reaction rates.

It considers variables like temperature, reactant concentration, and catalysts, which can significantly affect how quickly a reaction proceeds.
In our exercise, the focus is on how the concentration of \(\mathrm{NO}_2\) influences reaction speed, indicating its pivotal role in the initial phase of the reaction.

The stepwise sequences proposed in the mechanisms show real-world applications of kinetics.
By analyzing reaction orders and rate equations, chemists decipher complex mechanisms into understandable steps, ensuring that each step's speed is accurately represented by the experimentally observed kinetics.
This exploration acts as a blueprint to confirm theories regarding how specific molecular interactions lead to varying reaction speeds.