Problem 110 - Chemistry: An Atoms-Focused Approach

Question

The rate laws for the thermal and photochemical decomposition of \(\mathrm{NO}_{2}\) are different. Which of the following mechanisms are possible for the thermal decomposition of \(\mathrm{NO}_{2},\) and which are possible for the photochemical decomposition of \(\mathrm{NO}_{2}\) ? For the thermal decomposition, Rate \(=k\left[\mathrm{NO}_{2}\right]^{2},\) and for the photochemical decomposition, Rate \(=k\left[\mathrm{NO}_{2}\right]\). a. \(\mathrm{NO}_{2}(g)+\mathrm{NO}_{2}(g) \stackrel{\text { slow }}{\longrightarrow} \mathrm{N}_{2} \mathrm{O}_{4}(g)\) \(\mathrm{N}_{2} \mathrm{O}_{4}(g) \stackrel{\text { fast }}{\longrightarrow} \mathrm{N}_{2} \mathrm{O}_{3}(g)+\mathrm{O}(g)\) \(\mathrm{N}_{2} \mathrm{O}_{3}(g)+\mathrm{O}(g) \stackrel{\text { fast }}{\mathrm{N}_{2} \mathrm{O}_{2}(g)} \stackrel{\mathrm{fast}}{\longrightarrow} \mathrm{N}_{2} \mathrm{O}_{2}(g)+\mathrm{O}_{2}(g)\) \(\quad \quad \mathrm{NO}(g)\) b. \(\mathrm{NO}_{2}(g)+\mathrm{NO}_{2}(g) \stackrel{\text { slow }}{\longrightarrow} \mathrm{NO}(g)+\mathrm{NO}_{3}(g)\) \(\mathrm{NO}_{3}(g) \stackrel{\mathrm{fast}}{\longrightarrow} \mathrm{NO}(g)+\mathrm{O}_{2}(g)\) c. \(\quad \mathrm{NO}_{2}(g) \stackrel{\text { slow }}{\longrightarrow} \mathrm{N}(g)+\mathrm{O}_{2}(g)\) \(\begin{aligned} \mathrm{N}(g)+& \mathrm{NO}_{2}(g) \frac{\mathrm{fast}}{\mathrm{N}_{2} \mathrm{O}_{2}(g)} \mathrm{N}_{2} \mathrm{O}_{2}(g) \\ & \stackrel{\text { fast }}{\longrightarrow} \mathrm{NO}(g) \end{aligned}\)

Step-by-Step Solution

Answer

Answer: Mechanism A and mechanism B correspond to the thermal decomposition of NO2, while mechanism C corresponds to the photochemical decomposition of NO2.

Step 1: Mechanism A: Analyzing for Thermal or Photochemical Decomposition

In mechanism A, the slow step is: \(\mathrm{NO}_{2}(g)+\mathrm{NO}_{2}(g) \stackrel{\text { slow }}{\longrightarrow} \mathrm{N}_{2} \mathrm{O}_{4}(g)\) The rate of this step depends on the concentration of \(\mathrm{NO}_{2}\) squared, so the rate law for this slow step is: Rate \(=k\left[\mathrm{NO}_{2}\right]^{2}\). This corresponds to the rate law for the thermal decomposition. Therefore, mechanism A is possible for the thermal decomposition of \(\mathrm{NO}_{2}\).

Step 2: Mechanism B: Analyzing for Thermal or Photochemical Decomposition

In mechanism B, the slow step is: \(\mathrm{NO}_{2}(g)+\mathrm{NO}_{2}(g) \stackrel{\text { slow }}{\longrightarrow} \mathrm{NO}(g)+\mathrm{NO}_{3}(g)\) The rate of this step also depends on the concentration of \(\mathrm{NO}_{2}\) squared, so the rate law for this slow step is: Rate \(=k\left[\mathrm{NO}_{2}\right]^{2}\). This corresponds to the rate law for the thermal decomposition. Therefore, mechanism B is possible for the thermal decomposition of \(\mathrm{NO}_{2}\).

Step 3: Mechanism C: Analyzing for Thermal or Photochemical Decomposition

In mechanism C, the slow step is: \(\mathrm{NO}_{2}(g) \stackrel{\text { slow }}{\longrightarrow} \mathrm{N}(g)+\mathrm{O}_{2}(g)\) The rate of this step depends on the concentration of \(\mathrm{NO}_{2}\), so the rate law for this slow step is: Rate \(=k\left[\mathrm{NO}_{2}\right]\). This corresponds to the rate law for the photochemical decomposition. Therefore, mechanism C is possible for the photochemical decomposition of \(\mathrm{NO}_{2}\).

Step 4: Summary

Mechanism A and mechanism B are possible for the thermal decomposition of \(\mathrm{NO}_{2}\), while mechanism C is possible for the photochemical decomposition of \(\mathrm{NO}_{2}\).

Key Concepts

Thermal Decomposition

Thermal decomposition involves breaking down a chemical compound with the application of heat. This process is crucial in many industrial and chemical applications.
For the thermal decomposition of \(\text{NO}_2\), the rate law is represented as \(\text{Rate} = k[\text{NO}_2]^2\). This implies that the reaction rate is dependent on the concentration of \(\text{NO}_2\) squared.
This demonstrates a second-order reaction, where two molecules of \(\text{NO}_2\) interact simultaneously in the slow step.

• Mechanism A: Begins with two \(\text{NO}_2\) molecules forming \(\text{N}_2\text{O}_4\), which correlates to the thermal decomposition rate law.
• Mechanism B: Also fits thermal decomposition, involving two \(\text{NO}_2\) molecules resulting in \(\text{NO}\) and \(\text{NO}_3\).

Understanding these mechanisms helps in predicting how the reaction proceeds under thermal conditions.

Photochemical Decomposition

Photochemical decomposition occurs when a chemical compound is broken down due to the absorption of light energy. This process is key in natural phenomena such as photosynthesis.
For photochemical decomposition of \(\text{NO}_2\), the rate law is \(\text{Rate} = k[\text{NO}_2]\). This suggests a first-order reaction, with the rate depending linearly on the concentration of \(\text{NO}_2\).

• Mechanism C: Represents photochemical decomposition with a slow step of single \(\text{NO}_2\) breaking into \(\text{N}\) and \(\text{O}_2\). This aligns well with the observed rate law.

Photochemical processes often require specific wavelengths of light, which can vary depending on the compound involved.

Chemical Mechanisms

Chemical mechanisms explain the step-by-step sequence of elementary reactions by which overall chemical change occurs. They provide a detailed picture of the individual steps and how they relate to observed reaction rates.
Chemical reactions often comprise several steps, each with different speeds, usually described as 'slow' or 'fast'.
The slowest step, known as the rate-determining step, plays a crucial role in determining the overall reaction rate and corresponding rate law.

• The choice of mechanism is based on aligning the rate law derived from experimental data with the theoretical prediction from proposed mechanisms.
• For example, if experimental data suggests a second-order reaction, valid mechanisms must involve interactions between two reactant molecules in the rate-determining step.

This systematic approach helps chemists design reactions and predict their behaviors under varied conditions.