Question

The rate law for the following reaction: $$\mathrm{CO}(g)+{\mathrm{NO}}_2(g)⟶{\mathrm{CO}}_2(g)+\mathrm{NO}(g)$$ is rate $=k{\left[{\mathrm{NO}}_2\right]}^2$. Suggest a plausible mechanism for the reaction, given that the unstable species ${\mathrm{NO}}_3$ is an intermediate.

Short answer

Possible mechanism: 1) $2{\text{NO}}_2\to {\text{NO}}_3+\text{NO}$; 2) ${\text{NO}}_3+\text{CO}\to {\text{CO}}_2+{\text{NO}}_2$.

Step-by-step solution

Understand the Rate Law

The given rate law is $\text{rate}=k[{\text{NO}}_2{]}^2$. This indicates that the reaction is second order with respect to ${\text{NO}}_2$ and suggests that two ${\text{NO}}_2$ molecules are involved in the rate-determining step.

Identify Possible Intermediates and Products

We know that ${\text{NO}}_3$ is an intermediate. This means that ${\text{NO}}_3$ is not in the initial or final products but forms temporarily. The final products are ${\text{CO}}_2$ and $\text{NO}$.

Propose the First Step of the Mechanism

In the first step, assume that two ${\text{NO}}_2$ molecules react to form ${\text{NO}}_3$ and $\text{NO}$. This can be written as: ${\text{NO}}_2+{\text{NO}}_2\to {\text{NO}}_3+\text{NO}$. This step involves two ${\text{NO}}_2$ molecules, fitting the rate law.

Propose the Second Step of the Mechanism

In the second step, ${\text{NO}}_3$ reacts with $\text{CO}$ to form ${\text{CO}}_2$ and ${\text{NO}}_2$. This can be written as: ${\text{NO}}_3+\text{CO}\to {\text{CO}}_2+{\text{NO}}_2$. ${\text{NO}}_3$ is used up and ${\text{NO}}_2$ is reformed, consistent with ${\text{NO}}_3$ as an intermediate.

Verify Consistency with Rate Law

The first step is the rate-determining step (slow step), involving two ${\text{NO}}_2$ molecules: $\text{rate}=k[{\text{NO}}_2{]}^2$. This is directly consistent with the provided rate law. The second step is fast and regenerates ${\text{NO}}_2$, maintaining overall stoichiometry.

Key concepts

Rate Law

When a chemical reaction takes place, its rate can often be described by a rate law. The rate law is an equation that links the reaction rate with the concentrations of reactants. For the reaction involving carbon monoxide ($\text{CO}$) and nitrogen dioxide (${\text{NO}}_2$), the rate law is given by:
$$\text{rate}=k[{\text{NO}}_2{]}^2$$
This means the rate of the reaction depends on the concentration of ${\text{NO}}_2$ squared. The coefficient 2 indicates the reaction is second-order with respect to ${\text{NO}}_2$.
There are key aspects to consider about this rate law:

• The constant $k$ is the rate constant, which varies with temperature and provides insight into the speed of the reaction.
• The second-order dependency implies that for the formation of products, two ${\text{NO}}_2$ molecules are necessary in the rate-determining step.

Understanding this helps in proposing the sequence of steps involved in a reaction mechanism.

Intermediate Species

An intermediate species in a chemical reaction is a molecule that is formed in one of the steps and consumed in another, never appearing in the overall reaction equation. In the given problem, ${\text{NO}}_3$ is identified as an intermediate.Why is ${\text{NO}}_3$ crucial in this context?

• It is formed from the reaction of ${\text{NO}}_2$ molecules, which supports the second-order kinetic expression.
• It does not appear in the final products, indicating it is consumed during the process.
• Its temporary formation allows for the overall mechanism to work, facilitating the transformation of $\text{CO}$ into ${\text{CO}}_2$.

Understanding intermediates help us piece together how individual steps lead to the overall reaction. They provide a clearer picture of the transition and transformation states involved in the mechanism.

Rate-Determining Step

In reaction mechanisms, the rate-determining step is the slowest step which controls the speed of the entire reaction. It is like a bottleneck that decides how fast the overall process can proceed.For the reaction of $\text{CO}$ and ${\text{NO}}_2$, the rate-determining step involves two molecules of ${\text{NO}}_2$ forming ${\text{NO}}_3$ and $\text{NO}$. This is significant because:

• It correlates directly with the rate law $\text{rate}=k[{\text{NO}}_2{]}^2$, showing that the concentration of ${\text{NO}}_2$ directly impacts the rate.
• The reaction cannot proceed faster than this step, making it critical to understand and analyze this step when modifying or optimizing conditions to speed up the reaction.

The concept of a rate-determining step helps chemists identify which steps in a reaction mechanism need more focus to either accelerate or control the reaction efficiently.