Question

The mechanism for the gas-phase reaction of nitrogen dioxide with carbon monoxide to form nitric oxide and carbon dioxide is thought to be $$ \begin{array}{c}{\mathrm{NO}_{2}+\mathrm{NO}_{2} \longrightarrow \mathrm{NO}_{3}+\mathrm{NO}} \\ {\mathrm{NO}_{3}+\mathrm{CO} \longrightarrow \mathrm{NO}_{2}+\mathrm{CO}_{2}}\end{array} $$ Write the rate law expected for this mechanism. What is the overall balanced equation for the reaction?

Short answer

The rate law for the given reaction mechanism is: Overall Rate = (k₂/k₁)(Rate₁)[CO] Where Rate₁ = k₁[NO₂]² and k₁ and k₂ are the rate constants for the elementary steps. The overall balanced equation for the reaction is: 2NO₂ + CO → 2NO + CO₂

Step-by-step solution

Write the rate law for each elementary step

To find the rate law for each elementary step, we need to look at the coefficients of the reactants in the step. For an elementary reaction, the rate law can be directly obtained from the reaction equation. 1. For the first elementary step, NO₂ + NO₂ → NO₃ + NO, we have two molecules of NO₂ involved. Rate₁ = k₁[NO₂]² 2. For the second elementary step, NO₃ + CO → NO₂ + CO₂, we have one molecule of NO₃ and one molecule of CO involved. Rate₂ = k₂[NO₃][CO]

Identify the reaction intermediates

Reaction intermediates are species that are produced during one elementary step and consumed during another, so they don't appear in the overall balanced equation. In this case, NO₃ is the reaction intermediate, as it is produced in the first step and consumed in the second step.

Write the overall rate law

Since NO₃ is a reaction intermediate, we need to express [NO₃] in terms of other concentrations. From the first step, we have: [NO₃] = (Rate₁)/(k₁[NO₂]) Now we can substitute this into the Rate₂ expression: Overall Rate = Rate₂ = k₂[(Rate₁)/(k₁[NO₂])][CO] This simplifies to: Overall Rate = (k₂/k₁)(Rate₁)[CO]

Write the overall balanced equation

Now we can find the overall balanced equation for the reaction by adding the two elementary steps: (NO₂ + NO₂ → NO₃ + NO) + (NO₃ + CO → NO₂ + CO₂) The NO₃ intermediate is produced in the first step and consumed in the second step. Therefore, it does not appear in the overall balanced equation. The equation simplifies to: NO₂ + NO₂ + CO → NO + NO₂ + CO₂ Further simplifying it, we have: 2NO₂ + CO → 2NO + CO₂ So, the overall balanced equation is: 2NO₂ + CO → 2NO + CO₂

Key concepts

Rate Law

The rate law helps us understand how the concentration of reactants influences the rate of a chemical reaction.
In the reaction mechanism, each elementary step has its own rate law. An elementary reaction is a single reaction step with its own participating molecules.
In such a mechanism, the rate law can be directly written from the stoichiometry of the reactants involved. For example:

• The first step in the given reaction mechanism: \(\text{NO}_2 + \text{NO}_2 \rightarrow \text{NO}_3 + \text{NO}\) has two \(\text{NO}_2\) molecules. Thus, its rate law is \(\text{Rate}_1 = k_1 [\text{NO}_2]^2\).
• For the second step: \(\text{NO}_3 + \text{CO} \rightarrow \text{NO}_2 + \text{CO}_2\), the molecules \(\text{NO}_3\) and \(\text{CO}\) participate, leading to the rate law: \(\text{Rate}_2 = k_2 [\text{NO}_3][\text{CO}]\).

It's important to note that reaction intermediates, such as \(\text{NO}_3\) in this mechanism, must be expressed in terms of the concentrations of stable species when determining the overall rate law.

Elementary Reaction

An elementary reaction is a fundamental concept in the study of reaction mechanisms. These are single events where reactants convert to products in a single step with no intermediates. The rate law for such a reaction can be directly extracted from its molecularity, which is the number of molecules colliding in that step.
In the mechanism you analyzed:

• The step \( \text{NO}_2 + \text{NO}_2 \rightarrow \text{NO}_3 + \text{NO} \) is an example of a bimolecular reaction as it involves two molecules of \(\text{NO}_2\).
• The step \( \text{NO}_3 + \text{CO} \rightarrow \text{NO}_2 + \text{CO}_2 \) is also a bimolecular reaction, involving one \(\text{NO}_3\) and one \(\text{CO}\) molecule.

The stoichiometry of these elementary reactions is essential in deducing the rate laws and understanding the overall mechanism of a complex reaction.

Reaction Intermediates

Reaction intermediates are species formed in one step of a mechanism and consumed in another step, never appearing in the overall balanced equation of the reaction. In your mechanism, \(\text{NO}_3\) is a key intermediate.
Intermediates like \(\text{NO}_3\) are crucial for constructing a correct overall rate law:

• \(\text{NO}_3\) is produced during the first step \(\text{NO}_2 + \text{NO}_2 \rightarrow \text{NO}_3 + \text{NO}\) and consumed in the second \(\text{NO}_3 + \text{CO} \rightarrow \text{NO}_2 + \text{CO}_2\).
• In the overall rate law, you must eliminate intermediates like \(\text{NO}_3\) by substituting from other steps, as seen with \(\text{NO}_3 = \frac{\text{Rate}_1}{k_1[\text{NO}_2]}\).

Understanding reaction intermediates allows chemists to piece together how and why reactions proceed the way they do.

Overall Balanced Equation

The overall balanced equation provides the simplified stoichiometric representation of all reactants and products in a reaction, without intermediates. To derive this, add each elementary step and cancel out intermediates.
For the reaction mechanism you are exploring:

• The elementary steps are combined: \(\text{NO}_2 + \text{NO}_2 \rightarrow \text{NO}_3 + \text{NO}\) and \(\text{NO}_3 + \text{CO} \rightarrow \text{NO}_2 + \text{CO}_2\).
• By adding these, the intermediates \(\text{NO}_3\) cancel, as it is formed and then consumed across steps.
• The resulting overall balanced equation is: \(2\text{NO}_2 + \text{CO} \rightarrow 2\text{NO} + \text{CO}_2\).

This balanced equation reflects the total change from initial reactants to final products, omitting the transient intermediates.