Question

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

Short answer

The rate law for the reaction mechanism is given by: Rate = \( \frac{k[NO_2]^3[CO_2]}{[CO]K_{eq}} \) The overall balanced equation for the reaction is: 2 NO₂ + CO → NO + CO₂

Step-by-step solution

Analyze the given reaction mechanism

The given reaction mechanism is composed of two elementary steps: 1. NO₂ + NO₂ → NO₃ + NO (Slow) 2. NO₃ + CO → NO₂ + CO₂ (Fast) We want to derive the rate law for this reaction mechanism and determine the overall balanced equation for the reaction.

Identify the slow step and its rate law

Since the slow step is the rate-determining step, we'll base the rate law on the reactants in this step. From the first step, we have: Rate = k[NO₂]² Here, k is the rate constant, and the rate law shows that the rate of reaction is proportional to the square of the concentration of nitrogen dioxide (NO₂).

Eliminate the intermediate term

We now need to eliminate the NO₃ intermediate that appears in both elementary steps. From the second step, we have the equation: NO₃ + CO → NO₂ + CO₂ (Fast) Since it's a fast step, we can assume that it's under equilibrium conditions: \(K_{eq} = \frac{[NO_2][CO_2]}{[NO_3][CO]}\) Isolating the [NO₃] term, we have: \( [NO_3] = \frac{[NO_2][CO_2]}{[CO]K_{eq}} \) Now, we can substitute the expression of [NO₃] into the rate law from Step 2: Rate = k[NO₂]² \(\left(\frac{[NO_2][CO_2]}{[CO]K_{eq}}\right)\)

Simplify the rate law

Now, we can simplify the rate law: Rate = \( \frac{k[NO_2]^3[CO_2]}{[CO]K_{eq}} \) This is the final rate law for the given reaction mechanism.

Determine the overall balanced equation

To determine the overall balanced equation, we can add the elementary reactions: (1) NO₂ + NO₂ → NO₃ + NO (Slow) (2) NO₃ + CO → NO₂ + CO₂ (Fast) -------------------------------------- Overall: NO₂ + NO₂ + CO → NO + CO₂ So, the overall balanced equation for the given reaction mechanism is: 2 NO₂ + CO → NO + CO₂

Key concepts

Rate Law Determination

Understanding the rate law determination is crucial for predicting how fast a chemical reaction proceeds under given conditions. It is a mathematical representation of the rate of reaction in terms of the concentration of reactants and the specific rate constant. To determine the rate law of a reaction, one must look at the reaction mechanism and identify the rate-determining or slowest step, as this dictates the overall rate.

In our exercise, the rate-determining step is the first step where two molecules of nitrogen dioxide (NO₂) react to form nitric oxide (NO) and a reaction intermediate NO₃. The rate of this slow step can be expressed as: \[ Rate = k[NO_{2}]^{2} \]Where \(k\) is the rate constant and \([NO_{2}]\) is the concentration of nitrogen dioxide. The rate law indicates that the reaction rate is directly proportional to the square of the concentration of NO₂. This illustrates the concept of reaction order, which in this case, is second order with respect to NO₂.

To arrive at a more comprehensive rate law, the reaction intermediate must be accounted for, which we achieve by placing the fast step (which involves the intermediate) under equilibrium conditions to solve for its concentration. The final rate law is found by substituting our expression for the concentration of the intermediate into the rate equation of the slow step, yielding a more complex expression that includes other reactants present in the reaction mechanism.

Reaction Intermediates

In chemical kinetics, a reaction intermediate is a transitory species formed during the conversion of reactants into products. It is neither a reactant nor a product of the overall chemical reaction, but it plays a critical role as it appears in the mechanism's elementary steps. Since intermediates are generally very reactive, they are consumed as quickly as they are formed and do not usually accumulate in the reaction mixture.

In our example, the NO₃ molecule formed in the first step is a reaction intermediate. Often, intermediates complicate the determination of the rate law because they do not appear in the overall balanced chemical equation. The step-by-step solution provides an elegant workaround. We used the fast step in which the intermediate reacts with CO to produce NO₂ and CO₂ to determine its concentration. Through an equilibrium assumption for the fast step: \[ K_{eq} = \frac{[NO_{2}][CO_{2}]}{[NO_{3}][CO]} \]We derive an expression for the intermediate concentration and then substitute it back into the slow step's rate law. This gives us a rate law that correlates the reaction rate with the concentration of reactants only, not intermediates, which is a more practical form for experimental purposes.

Balanced Chemical Equations

The balanced chemical equation is a representation of a chemical reaction using chemical formulas where the number of atoms for each element is equal on both sides of the equation. This conservation of mass principle is fundamental in chemistry and ensures that the reaction adheres to the atomic composition of reactants and products. To balance an equation, one must adjust the coefficients, which denote the number of molecules or moles, without altering the chemical formulas of the reactants or products.

In the mechanism provided, we have two elementary steps which lead to an overall reaction. By adding these steps, we cancel out the intermediates and align the coefficients to reflect the conservation of atoms. The slow and fast steps together yield the net equation:2 NO₂ + CO → NO + CO₂ This is the final overall balanced equation for the reaction. It shows that two molecules of NO₂ and one molecule of CO are required to produce one molecule of NO and one molecule of CO₂. Balancing chemical equations is a fundamental skill in chemistry that not only involves the arithmetic of coefficients but also an understanding of stoichiometry and the laws that conserve mass and charge in chemical reactions.