Question

Nitramide, \(\mathrm{NO}_{2} \mathrm{NH}_{2},\) decomposes slowly in aqueous solution according to the following reaction: $$\mathrm{NO}_{2} \mathrm{NH}_{2}(\mathrm{aq}) \rightarrow \mathrm{N}_{2} \mathrm{O}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\ell)$$ The reaction follows the experimental rate law $$\text { Rate }=\frac{k\left[\mathrm{NO}_{2} \mathrm{NH}_{2}\right]}{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}$$ (a) What is the apparent order of the reaction in a buffered solution? (b) Which of the following mechanisms is the most appropriate for the interpretation of this rate law? Explain. Mechanism 1 \(\mathrm{NO}_{2} \mathrm{NH}_{2} \stackrel{k_{1}}{\longrightarrow} \mathrm{N}_{2} \mathrm{O}+\mathrm{H}_{2} \mathrm{O}\) Mechanism 2 \(\begin{array}{l}\mathrm{NO}_{2} \mathrm{NH}_{2}+\mathrm{H}_{3} \mathrm{O}^{+} \frac{k_{2}}{k_{2}^{\prime}} \mathrm{NO}_{2} \mathrm{NH}_{3}^{+}+\mathrm{H}_{2} \mathrm{O} \\\\\mathrm{NO}_{2} \mathrm{NH}_{3}+\stackrel{k_{3}}{\longrightarrow} \mathrm{N}_{2}\mathrm{O}+\mathrm{H}_{3} \mathrm{O}^{+}(\text {rapid equilibrium })\end{array}\) Mechanism 3 \(\mathrm{NO}_{2} \mathrm{NH}_{2}+\mathrm{H}_{2} \mathrm{O} \frac{k_{4}}{k_{4}^{\prime}} \mathrm{NO}_{2} \mathrm{NH}^{-}+\mathrm{H}_{3} \mathrm{O}^{+}\) \(\mathrm{NO}_{2} \mathrm{NH}^{-} \longrightarrow \mathrm{N}_{2} \mathrm{O}+\mathrm{OH}^{-} \quad\) (rapid equilibrium) \(\mathrm{H}_{3} \mathrm{O}^{+}+\mathrm{OH}^{-} \stackrel{k_{6}}{\longrightarrow} 2 \mathrm{H}_{2} \mathrm{O} \quad\) (very fast reaction) (c) Show the relationship between the experimentally observed rate constant, \(k\), and the rate constants in the selected mechanism. (Note that when writing the expression for \(K\), the equilibrium constant, \(\left[\mathrm{H}_{2} \mathrm{O}\right]\) is not involved. \(\rightarrow\) Chapter \(16 .\) ) (d) Show that hydroxide ions catalyze the decomposition of nitramide.

Short answer

(a) First-order in buffered solution. (b) Mechanism 2 matches rate law. (c) \(k = \frac{k_2 k_3}{k_2'}\). (d) \(\mathrm{OH}^-\) enhances decomposition.

Step-by-step solution

Determine the Apparent Order in Buffered Solution

In a buffered solution, the concentration of \([\mathrm{H}_3\mathrm{O}^+]\) is constant. Therefore, the rate law becomes \(\text{Rate} = k' [\mathrm{NO}_2 \mathrm{NH}_2]\), where \(k' = \frac{k}{[\mathrm{H}_3\mathrm{O}^+]}\). This indicates that the reaction is first-order with respect to \(\mathrm{NO}_2 \mathrm{NH}_2\) in a buffered solution.

Analyze Mechanism 1

Mechanism 1 suggests a simple decomposition without involvement of \([\mathrm{H}_3\mathrm{O}^+]\), contradicting the dependence of the rate law on \([\mathrm{H}_3\mathrm{O}^+]\). Thus, Mechanism 1 is inconsistent with the observed rate law.

Analyze Mechanism 2

In Mechanism 2, the first step involves \(\mathrm{H}_3\mathrm{O}^+\) to form \(\mathrm{NO}_2\mathrm{NH}_3^+\), and the second step regenerates \(\mathrm{H}_3\mathrm{O}^+\). Applying the steady-state approximation for \(\mathrm{NO}_2\mathrm{NH}_3^+\), we can derive that the rate law matches the observed form. Mechanism 2 is consistent with the observed rate law.

Analyze Mechanism 3

Mechanism 3 introduces \(\mathrm{OH}^-\) in the equilibrium and involves \(\mathrm{H}_3\mathrm{O}^+\) in the rapid step; however, it doesn't directly explain the inverse dependence on \([\mathrm{H}_3\mathrm{O}^+]\) in the rate law. Mechanism 3 is less likely to match the observed rate law as well as Mechanism 2.

Show Relationship of Observed k with Mechanism 2 Constants

For Mechanism 2, the steady-state approximation gives us \(k = \frac{k_2 k_3}{k_2'}\). This results from assuming the intermediate \(\mathrm{NO}_2\mathrm{NH}_3^+\) is in rapid equilibrium compared to the slow step that controls the rate.

Demonstrate Hydroxide Ion Catalysis

Hydroxide ions can increase the rate of decomposition by neutralizing \(\mathrm{H}_3\mathrm{O}^+\), thus shifting the equilibrium in Mechanism 3 and promoting the breakdown of \(\mathrm{NO}_2\mathrm{NH}_2\) via increasing the proportion of deprotonated intermediates. This enhances the decomposition of nitramide.

Key concepts

Rate Laws

Rate laws describe how the concentration of reactants affects the rate of a chemical reaction. In this context, the rate law for nitramide decomposition is given by:
\[\text{Rate} = \frac{k[\mathrm{NO}_2\mathrm{NH}_2]}{[\mathrm{H}_3\mathrm{O}^+]}\]
This formula indicates that the rate of reaction is directly proportional to the concentration of nitramide \([\mathrm{NO}_2\mathrm{NH}_2]\) and inversely proportional to the concentration of hydronium ions \([\mathrm{H}_3\mathrm{O}^+]\). Thus, even though hydronium ions are not reactants in the overall reaction, they significantly influence the reaction rate.

• Order of Reaction: The reaction is first-order concerning nitramide, meaning if the concentration of nitramide doubles, the reaction rate also doubles.
• Equilibrium Impact: Keep in mind that in a buffered solution, the \([\mathrm{H}_3\mathrm{O}^+]\) remains constant, reinforcing the observation that the reaction appears first-order relative to nitramide.

Understanding these interactions is crucial for predicting how changes in concentration affect reaction speeds.

Equilibrium in Reactions

In many chemical reactions, temporary states of balance, known as equilibria, are established. These include rapid formations and breakdowns of intermediates before reaching the final products. This concept is expertly illustrated in the proposed mechanism 2 for nitramide decomposition.
This mechanism describes rapid equilibrium:

• Initially, nitramide and hydronium ions establish an equilibrium to form an intermediate \(\mathrm{NO}_2\mathrm{NH}_3^+\).
• This intermediate then quickly decomposes back to the hydronium ion and the final product, nitrous oxide \(\mathrm{N}_2\mathrm{O}\).

The equilibrium concept helps to understand why hydronium ions are inversely affecting the rate of reaction in the rate law. By maintaining constant hydronium ion concentration, any absorbed hydronium ions are rapidly released back into the solution following this equilibrium. Recognizing these rapid equilibria is central to understanding more complex kinetics.

Chemical Kinetics

Chemical kinetics is a domain of chemistry focused on understanding the speeds of reactions and the conditions affecting those speeds. For nitramide, the decomposition reaction delineates several key principles:
- **Rate-Determining Steps**: The slowest step in a reaction mechanism, often determines the overall reaction rate. In mechanism 2, the formation and decomposition of the intermediate \(\mathrm{NO}_2\mathrm{NH}_3^+\) highlight this principle.
- **Steady-State Approximation**: This useful assumption states that the concentration of an unstable intermediate remains relatively constant throughout the reaction. This assumption helps break down complex mechanisms into simpler rate laws.
- **Catalysis**: Introduction of a catalyst, such as hydroxide ions in mechanism 3, can modify the reaction pathway, effectively increasing the rate by altering rate-determining steps or shifting equilibria.In chemical kinetics, understanding the interplay of these factors allows chemists to predict and control reaction rates effectively, an essential for optimizing reactions in both industrial and laboratory settings.