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 $$ \begin{aligned} &\mathrm{NO}_{2}+\mathrm{NO}_{2} \longrightarrow \mathrm{NO}_{3}+\mathrm{NO} \\\ &\mathrm{NO}_{3}+\mathrm{CO} \longrightarrow \mathrm{NO}_{2}+\mathrm{CO}_{2} \end{aligned} $$ 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, assuming the first step is the rate-determining step, is \(Rate = k_1 [NO_2]^2\). The overall balanced equation for the reaction is \(NO_2 + CO \longrightarrow NO + CO_2\).

Step-by-step solution

We are given a two-step reaction mechanism for the gas-phase reaction of nitrogen dioxide (NO₂) with carbon monoxide (CO) to form nitric oxide (NO) and carbon dioxide (CO₂): \[ NO_2 + NO_2 \longrightarrow NO_3 + NO \] \[ NO_3 + CO \longrightarrow NO_2 + CO_2 \] #Step 2: Write the rate law for each elementary step#

For each elementary step, the rate law is given by the product of the rate constant (k) and the concentrations of the reactants, raised to the power of their stoichiometric coefficients. The rate laws for the two steps are therefore: \[Rate_1 = k_1[NO_2]^2 \] \[Rate_2 = k_2[NO_3][CO] \] #Step 3: Find the overall rate law for the reaction#

The overall rate law depends on the slowest (rate-determining) step in the reaction mechanism. Let's assume the first step is the slowest step, so the overall rate law is: \[Rate = k_1 [NO_2]^2\] #Step 4: Find the overall balanced equation#

The overall balanced equation for the reaction can be found by summing the elementary steps and canceling out any species that appear on both sides: \[NO_2 + NO_2 + NO_3 + CO \longrightarrow NO_3 + NO + NO_2 + CO_2\] After canceling out the species, we are left with: \[ NO_2 + CO \longrightarrow NO + CO_2 \] Thus, the overall balanced equation of the reaction is: \[ NO_2 + CO \longrightarrow NO + CO_2 \]

Key concepts

Rate Law

Understanding the rate law is crucial in the study of chemical kinetics, the branch of chemistry that deals with the speed or rate of a reaction. For a simple reaction where the coefficients of the reactants in the balanced equation are indicative of the reaction order, the rate law can often be directly deduced. In more complex mechanisms involving multiple steps, however, the rate law must be inferred from the experimental data and the mechanism.

The rate law expresses the rate of a reaction in terms of the concentration of reactants. Each reactant's concentration is raised to an exponent that corresponds to its reaction order. For example, in the mechanism provided, if the first step is the slowest or rate-determining step, the rate law would be \( Rate = k_1[NO_2]^2 \). This shows that the reaction is second-order with respect to NO₂, and the reaction rate depends only on the concentration of NO₂, and not on NO₃ or CO, despite them being part of the mechanism.

Elementary Steps in a Reaction

Complex reactions often proceed through a series of simpler, elementary steps. Each of these steps involves a direct rearrangement and collision of reactant molecules to form products. They are the basic building blocks of a reaction mechanism.

In our given mechanism, there are two elementary steps. The first step involves two molecules of NO₂ reacting to form NO₃ and NO, while the second step involves the reaction of NO₃ with CO to produce NO₂ and CO₂. It is essential to recognize that the actual observable rate law for a reaction could be influenced by any one of these steps, usually the slowest one, which is often termed the rate-determining step.

When analyzing reaction mechanisms, chemists focus on understanding each elementary step: the order of the reaction with respect to each reactant, the stoichiometry, and the resulting products, to determine the overall progression of the chemical reaction.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It is based on the conservation of mass where the total mass of reactants equals the total mass of products. Stoichiometry calculations allow chemists to predict the amounts of substances consumed and produced in a reaction.

In practice, stoichiometry requires a balanced chemical equation, which provides the mole ratio of the reactants and products. An understanding of stoichiometry is essential when deriving the rate law for a chemical reaction, as it dictates the coefficients used in the rate equation. In the given reaction, stoichiometry helps us understand that for every mole of NO₂ reacting with CO, one mole each of NO and CO₂ will be produced, assuming the reaction goes to completion.

Balanced Chemical Equation

A balanced chemical equation provides a clear picture of a chemical reaction, showcasing the reactants, products, and their respective proportions. For a chemical equation to be balanced, it must obey the law of conservation of mass, meaning the number of atoms of each element must be the same on both sides of the equation.

In our example, the equation \( NO_2 + CO \longrightarrow NO + CO_2 \) is the result of balancing the complex series of reactions detailed in the mechanism. After adding up the elementary steps and canceling out intermediate species like NO₃ (which appears on both sides), we find the simplest whole-number coefficients that balance the equation. The balanced equation is critical for calculating the amounts of reactants needed and products formed and is central to both theoretical and practical aspects of chemistry.