91Ó°ÊÓ

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. \(\quad \mathrm{NO}_{2}(g) \stackrel{\text { slow }}{\longrightarrow} \mathrm{NO}(g)+\mathrm{O}(g)\) \(\mathrm{O}(g)+\mathrm{NO}_{2}(g) \stackrel{\text { fast }}{\longrightarrow} \mathrm{NO}(g)+\mathrm{O}_{2}(g)\) b. \(\mathrm{NO}_{2}(g)+\mathrm{NO}_{2}(g) \stackrel{\text { fast }}{\longrightarrow} \mathrm{N}_{2} \mathrm{O}_{4}(g)\) \(\mathrm{N}_{2} \mathrm{O}_{4}(g) \frac{\mathrm{slow}}{\mathrm{}_{\mathrm{fast}}} \mathrm{NO}(g)+\mathrm{NO}_{3}(g)\) \(\mathrm{NO}_{3}(g)+\mathrm{O}_{2}(g)\) c. \(\mathrm{NO}_{2}(g)+\mathrm{NO}_{2}(g) \stackrel{\text { slow }}{\longrightarrow} \mathrm{NO}(g)+\mathrm{NO}_{3}(g)\) \(\mathrm{NO}_{3}(g) \stackrel{\text { fast }}{\longrightarrow} \mathrm{NO}(g)+\mathrm{O}_{2}(g)\)

Step by step solution

01

Identify the rate-determining step

The rate-determining step is the slowest step in a reaction mechanism. We can see that a. and c. have a slow step first, and b. has a fast step first followed by a slow step.

02

Rate Laws for Mechanism a.

In this mechanism, the first step is the rate-determining step. Rate = k\(_{1}\left[\mathrm{NO}_{2}\right]\) The second step does not affect the rate law.

03

Rate Laws for Mechanism b.

First, we need to apply the steady-state approximation to the intermediate N\(_2\)O\(_4\). At a steady state, the rate of formation of N\(_2\)O\(_4\) equals the rate of consumption: k\(_{1}\left[\mathrm{NO}_{2}\right]^{2} = k_{-1}\left[\mathrm{N}_{2}\mathrm{O}_{4}\right]+k_{2}\left[\mathrm{N}_{2}\mathrm{O}_{4}\right]\) Divide both sides by k\(_{1}\left[\mathrm{NO}_{2}\right]^{2}\): \(\frac{\left[\mathrm{N}_{2}\mathrm{O}_{4}\right]}{\left[\mathrm{NO}_{2}\right]^{2}} = \frac{k_{1}}{k_{-1}+k_{2}}\) Since the second step is rate-determining, we can write the rate law as: Rate = k\(_{2}\left[\mathrm{N}_{2}\mathrm{O}_{4}\right]\) Substitute the expression for \(\left[\mathrm{N}_{2}\mathrm{O}_{4}\right]\): Rate = k\(_{2}\frac{k_{1}}{k_{-1}+k_{2}}\left[\mathrm{NO}_{2}\right]^{2}\)

04

Rate Laws for Mechanism c.

In this mechanism, the first step is the rate-determining step. Rate = k\(_{1}\left[\mathrm{NO}_{2}\right]^{2}\) The second step does not affect the rate law.

05

Compare rate laws with Thermal and Photochemical decomposition

Now we can compare the derived rate laws with the given rate laws. - Mechanism a: Rate = k\(_{1}\left[\mathrm{NO}_{2}\right]\) is consistent with the Photochemical decomposition rate law. - Mechanism b: Rate = k\(_{rdf}\left[\mathrm{NO}_{2}\right]^{2}\), where k\(_{rdf} = k_{2}\frac{k_{1}}{k_{-1}+k_{2}}\), is still consistent with the Thermal decomposition rate law since it's quadratic in \(\left[\mathrm{NO}_{2}\right]\). - Mechanism c: Rate = k\(_{1}\left[\mathrm{NO}_{2}\right]^{2}\) is consistent with the Thermal decomposition rate law.

06

Conclusion

Based on the rate law comparison, the following conclusions can be made: - Mechanism a is possible for the Photochemical decomposition of NO\(_2\). - Mechanism b is possible for the Thermal decomposition of NO\(_2\). - Mechanism c is possible for the Thermal decomposition of NO\(_2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Thermal Decomposition

Thermal decomposition is a process where a chemical compound breaks down into simpler substances when heated. It is an endothermic process, meaning it requires heat energy to proceed. The example given in the exercise deals with the decomposition of nitrogen dioxide \(\text{NO}_2\). Here, the rate law for the thermal decomposition of \(\text{NO}_2\) is determined to be \(\text{Rate} = k[\text{NO}_2]^2\). This quadratic dependence suggests that two molecules of \(\text{NO}_2\) are involved in the rate-determining step, which is often the slowest step of the reaction. Thus, mechanisms that yield this rate law, like mechanism b and c, are consistent with a thermal decomposition pathway. Understanding thermal decomposition is crucial as it helps explain how reactions proceed with heat and how energy changes affect chemical transformations.

Photochemical Decomposition

Photochemical decomposition involves the breakdown of a compound using light energy. It plays a vital role in various natural and industrial processes, such as photosynthesis and the degradation of pollutants. In the given problem, photochemical decomposition of \(\text{NO}_2\) is characterized by the rate law \(\text{Rate} = k[\text{NO}_2]\). The linear dependence suggests that the decomposition involves a single molecule of \(\text{NO}_2\) getting excited by light to undergo a reaction. Mechanism a is consistent with this type of decomposition because the initial step, influenced by light energy, involves a singular \(\text{NO}_2\) molecule, aligning with the first-order rate law. Photochemical reactions are remarkable as they utilize light, typically making them occur faster and requiring less energy input compared to thermal processes.

Rate-Determining Step

The rate-determining step is a concept crucial for understanding chemical kinetics. It's the slowest step in a reaction mechanism that controls the overall reaction rate. Analogous to the narrowest part of a funnel, it determines how fast reactants are converted into products. In our example, for mechanism a, the slow step is the initial photochemical reaction step: \(\text{NO}_2 \to \text{NO} + \text{O}\). This aligns with the first-order rate law indicative of photochemical decomposition. For mechanism b and c, a similar concept applies to thermal decomposition, with the slow transformation of a dinitrogen tetroxide intermediate or a direct \(\text{NO}_2\) bifurcation facilitating the quadratic rate law in \(\text{NO}_2\). Understanding which step is the rate-determining is crucial for predicting reaction rates and designing efficient chemical processes.

Steady-State Approximation

The steady-state approximation is a simplification used when dealing with complex reaction mechanisms, particularly those involving intermediates. It assumes that the concentration of an intermediate remains constant over the course of the reaction. This assumption applies when the formation and consumption rates of the intermediate balance out. In the example, mechanism b uses the steady-state approximation for \(\text{N}_2\text{O}_4\). By setting its formation rate equal to its consumption rate, we derive an expression that simplifies into the observed rate law \(\text{Rate} = k[\text{NO}_2]^2\). This approach allows chemists to bypass the complexities introduced by transient species in a mechanism, leading to a clearer understanding of the main transformation pathway of interest. Thus, the steady-state approximation becomes a powerful tool in studying reaction kinetics with intermediates.