Problem 75 - Chemistry and Chemical Reactivity

Question

A reaction that occurs in our atmosphere is the oxidation of NO to the brown gas \(\mathrm{NO}_{2}\) $$ 2 \mathrm{NO}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \rightarrow 2 \mathrm{NO}_{2}(\mathrm{g}) $$ The mechanism of the reaction is thought to be Step \(1: \quad 2 \mathrm{NO}(\mathrm{g}) \rightleftharpoons \mathrm{N}_{2} \mathrm{O}_{2}(\mathrm{g})\) rapidly established equilibrium Step \(2: \quad \mathrm{N}_{2} \mathrm{O}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \rightarrow 2 \mathrm{NO}_{2}(\mathrm{g}) \quad\) slow Which is the rate determining step? Is there an intermediate in the reaction? If this is the correct mechanism for this reaction, what is the experimentally determined rate law?

Step-by-Step Solution

Answer

Rate-determining step: Step 2. Intermediate: \(\mathrm{N}_{2}\mathrm{O}_{2}\). Rate law: \(\text{Rate} = k'[\mathrm{NO}]^2[\mathrm{O}_{2}]\).

Step 1: Identify the Rate-Determining Step

The rate-determining step is the slowest step in the reaction mechanism, which controls the overall rate of the reaction. In this case, Step 2, \( \mathrm{N}_{2}\mathrm{O}_{2}(\mathrm{g}) + \mathrm{O}_{2}(\mathrm{g}) \rightarrow 2\mathrm{NO}_{2}(\mathrm{g}) \), is described as slow, while Step 1 is a rapidly established equilibrium. Therefore, Step 2 is the rate-determining step.

Step 2: Identify the Reaction Intermediate

A reaction intermediate is a species that is formed in one step of a reaction mechanism and consumed in another. In this mechanism, \( \mathrm{N}_{2}\mathrm{O}_{2}(\mathrm{g}) \) is formed in Step 1 and consumed in Step 2. Therefore, \( \mathrm{N}_{2}\mathrm{O}_{2}(\mathrm{g}) \) is an intermediate in this reaction.

Step 3: Write the Experimentally Determined Rate Law

The rate law for a reaction is based on the rate-determining step, not the overall balanced equation. Since Step 2 is the rate-determining step, the rate law is proportional to the concentration of the reactants in this step, \( [\mathrm{N}_{2}\mathrm{O}_{2}][\mathrm{O}_{2}] \). However, \( \mathrm{N}_{2}\mathrm{O}_{2} \) is an intermediate and not present in the overall reaction equation. We must express its concentration in terms of the reactants. From Step 1, we have the equilibrium \( 2\mathrm{NO} \rightleftharpoons \mathrm{N}_{2}\mathrm{O}_{2} \), suggesting \( [\mathrm{N}_{2}\mathrm{O}_{2}] = K[\mathrm{NO}]^2 \) where \( K \) is the equilibrium constant. Substituting this into the rate law gives \( \text{Rate} = k[\mathrm{N}_{2}\mathrm{O}_{2}][\mathrm{O}_{2}] = kK[\mathrm{NO}]^2[\mathrm{O}_{2}] \). The experimentally determined rate law thus is \( \text{Rate} = k'[\mathrm{NO}]^2[\mathrm{O}_{2}] \), where \( k' = kK \).

Key Concepts

Oxidation Reactions

Oxidation reactions are fundamental to understanding chemical processes, particularly those occurring in the atmosphere. In an oxidation reaction, a substance loses electrons, often involving an increase in the oxidation state of an element. This change facilitates the transformation of molecules, as seen in the oxidation of nitric oxide (NO) to nitrogen dioxide ( NO_2 ). In this reaction, oxygen ( O_2 ) acts as the oxidizing agent by gaining the electrons lost by NO, forming the brown gas NO_2 . Understanding this process is crucial when studying atmospheric chemistry because oxidation reactions significantly affect air quality and chemical equilibrium in the environment. They can produce pollutants, such as NO_2 , which contribute to smog and acid rain. By examining these reactions, scientists can develop strategies to reduce pollution and its adverse effects on health and ecosystems.

Rate-Determining Step

In any reaction mechanism, the rate-determining step holds the key to understanding the overall reaction rate. Think of it as the slowest runner in a relay race, setting the pace for the entire team. It's the step that takes the longest time to complete, effectively bottlenecking the whole process. For the NO oxidation reaction, the slow step is when N_2O_2 and O_2 react to form 2NO_2 . Because this process is slower than the rapidly establishing equilibrium in the first step, it dictates the speed of the overall reaction. Identifying the rate-determining step is crucial for developing accurate rate laws and predicting how reaction conditions will impact the rate. By focusing scientific efforts here, chemists can manipulate factors, such as concentration and temperature, to optimize reaction rates in industrial and laboratory settings.

Reaction Intermediates

Reaction intermediates are transient species that appear temporarily during a chemical reaction. They are produced in one step and get consumed in another, never appearing in the overall balanced equation. These intermediates facilitate the progression of the reaction through complex multi-step pathways. In the oxidation of NO, the intermediate N_2O_2 is formed in the initial equilibrium step, where two NO molecules combine. This intermediate is then consumed in the subsequent slow step to produce 2NO_2 . Understanding intermediates is vital as they often dictate the reaction's pathway and stability. While intermediates are hard to detect directly because of their fleeting existence, they provide invaluable insight into the mechanisms of chemical reactions. This knowledge can be applied in designing chemical processes, such as synthesizing desired products or mitigating side reactions.

Experimental Rate Laws

Experimental rate laws provide a mathematical description of how the concentration of reactants affects the rate of a reaction. They are derived from experimental data and relate to the rate-determining step of the mechanism rather than the overall reaction. For the oxidation of NO to NO_2 , the rate is dependent on the concentrations involved in the slow step. Initially, the rate would seem to depend on [N_2O_2][O_2] . However, since N_2O_2 is an intermediate made from NO in a rapidly established equilibrium, we can express its concentration in terms of NO. The concentration of the intermediate N_2O_2 , as derived from equilibrium principles, is [N_2O_2] = K[NO]^2 , where K is the equilibrium constant. Substituting this into the rate expression leads to the experimentally determined rate law: Rate = k'[NO]^2[O_2] , with k' being a product of the rate constant k and the equilibrium constant K. Understanding experimental rate laws is essential for predicting how changes in conditions will affect reactions, which is vital for chemical manufacturing, environmental modeling, and more.