Problem 71 - Chemistry The Molecular Science

Question

For the reaction $$ 2 \mathrm{NO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{NOCl}(\mathrm{g}) $$ the currently accepted mechanism is $$ \mathrm{NO}+\mathrm{Cl}_{2} \rightleftharpoons \mathrm{NOCl}_{2} $$ fast $$ \mathrm{NOCl}_{2}+\mathrm{NO} \longrightarrow 2 \mathrm{NOCl} $$ slow (a) Determine the rate law for this mechanism. (Be sure to express it in terms of concentrations of reactants or products of the overall reaction, not in terms of intermediates.) (b) Suggest another mechanism that agrees with the same rate law. (c) Suggest another mechanism that does not agree with the same rate law.

Step-by-Step Solution

Answer

(a) Rate law: \( \text{Rate} = k [\mathrm{NO}]^2[\mathrm{Cl}_2] \). (b) Similar mechanism: \( \mathrm{NO}^* + \mathrm{Cl}_2 \rightarrow 2 \mathrm{NOCl} \) (slow). (c) Different mechanism: \( \mathrm{N}_2\mathrm{O}_2 + \mathrm{Cl}_2 \rightarrow 2 \mathrm{NOCl} \) (fast).

Step 1: Identify the Slow Step

In a reaction mechanism, the slow step often determines the rate law. Here, the slow step is \( \mathrm{NOCl}_{2} + \mathrm{NO} \rightarrow 2 \mathrm{NOCl} \). This step is critical for finding the rate law.

Step 2: Determine the Rate of the Slow Step

The rate law for the slow step is determined by its reactants. For this step, the rate can be expressed as: \[\text{Rate}_{\text{slow}} = k_2 [\mathrm{NOCl}_{2}] [\mathrm{NO}]\] where \( k_2 \) is the rate constant for the slow step.

Step 3: Eliminate the Intermediate

Since \( \text{NOCl}_2 \) is an intermediate, we cannot have it in the final rate law. From the fast equilibrium \( \mathrm{NO} + \mathrm{Cl}_2 \rightleftharpoons \mathrm{NOCl}_2 \), we can express \([\mathrm{NOCl}_2]\) in terms of the other reactants:\[K = \frac{[\mathrm{NOCl}_2]}{[\mathrm{NO}][\mathrm{Cl}_2]}\]Thus, \([\mathrm{NOCl}_2] = K[\mathrm{NO}][\mathrm{Cl}_2]\), where \( K \) is the equilibrium constant.

Step 4: Substitute Intermediate with Reactants

Substitute \([\mathrm{NOCl}_2]\) in the rate law of the slow step:\[\text{Rate} = k [\mathrm{NO}][\mathrm{Cl}_2][\mathrm{NO}]\]This simplifies to:\[\text{Rate} = k [\mathrm{NO}]^2[\mathrm{Cl}_2]\]where \( k = k_2 K \) is the overall rate constant.

Step 5: State the Rate Law for the Reaction

The rate law for the reaction based on the given mechanism is:\[\text{Rate} = k [\mathrm{NO}]^2[\mathrm{Cl}_2]\]

Step 6: Propose an Alternate Mechanism Consistent with the Rate Law

To propose another mechanism that matches the rate law, imagine a two-step reaction where both involve \( \mathrm{NO} \) and \( \mathrm{Cl}_2 \) interacting with the overall second step also focusing on \( [\mathrm{NO}^2][\mathrm{Cl}_2] \). For example:1. \( 2 \mathrm{NO} \rightarrow \mathrm{NO}_2 + \mathrm{NO}^* \) (fast equilibrium)2. \( \mathrm{NO}^* + \mathrm{Cl}_2 \rightarrow 2 \mathrm{NOCl} \) (slow)Ensure intermediates cancel for the same rate expression.

Step 7: Suggest a Mechanism Inconsistent with the Rate Law

An inconsistent mechanism could propose a direct two-step pathway without formation of a \( \mathrm{NOCl}_2 \) intermediate. For example:1. \( 2\mathrm{NO} \rightarrow \mathrm{N}_2\mathrm{O}_2 \) (slow)2. \( \mathrm{N}_2\mathrm{O}_2 + \mathrm{Cl}_2 \rightarrow 2 \mathrm{NOCl} \) (fast)This mechanism results in a different rate law expression \( \text{Rate} = k [\mathrm{NO}]^2 \), which does not match the given rate law.

Key Concepts

Reaction Mechanism

A reaction mechanism outlines the step-by-step sequence of elementary reactions by which an overall chemical change occurs. It provides insight into which bonds are broken and formed and the order in which the process unfolds. In our given reaction, the mechanism involves two steps:

• The first step, which is rapid, forms an intermediate, \( \mathrm{NOCl}_2 \), from \( \mathrm{NO} \) and \( \mathrm{Cl}_2 \).
• The second step is slower and involves the transformation of \( \mathrm{NOCl}_2 \) and \( \mathrm{NO} \) into the final product \( \mathrm{NOCl} \).

The slow step is often referred to as the rate-determining step because it governs the rate at which the overall reaction occurs. This means the rate of this slowest step dictates the overall speed of the entire reaction.

Intermediate Species

Intermediate species are transient entities formed during the multi-step reaction processes. They are not present in the overall balanced equation but appear in the reaction mechanism. In this problem, \( \mathrm{NOCl}_2 \) is an intermediate. This species results from the initial reaction stage and is swiftly consumed in the subsequent step. Intermediates are crucial because they often participate in the rate-determining step.

Importantly, when writing the rate law, intermediates cannot appear. This is why we need to express the concentration of \( \mathrm{NOCl}_2 \) in terms of stable species using equilibrium expressions or other means. By utilizing the fast equilibrium step, we substitute intermediates to simplify the rate law to involve only reactants such as \( \mathrm{NO} \) and \( \mathrm{Cl}_2 \).

Equilibrium Constant

The equilibrium constant \( K \) provides a quantitative measure of the ratio of concentrations of products to reactants at equilibrium for a reversible reaction. In this mechanism, the initial reaction to form \( \mathrm{NOCl}_2 \) is at equilibrium:

• \( \mathrm{NO} + \mathrm{Cl}_2 \rightleftharpoons \mathrm{NOCl}_2 \)

For this step, \( K = \frac{[\mathrm{NOCl}_2]}{[\mathrm{NO}][\mathrm{Cl}_2]} \).

The equilibrium constant allows us to solve for the intermediate concentration \([\mathrm{NOCl}_2]\), which can then be used to substitute in the rate equation for the slow step. The expression for \( [\mathrm{NOCl}_2] \) becomes \( K[\mathrm{NO}][\mathrm{Cl}_2] \), helping to rebuild the rate law without intermediates, focusing on concentrations of initial reactants.

Kinetics

Kinetics is the branch of chemistry concerned with the rates of chemical reactions. It explores how different variables such as temperature, concentration, and catalysts influence the speed of a reaction. In this context, understanding kinetics means grasping how the steps of a mechanism contribute to the overall reaction rate.

The rate law formula extracted from the mechanism gives us a quantitative view of how changes in reactant concentrations affect the reaction rate. For the current problem, ensuring we obey the principles of reaction kinetics helped determine that the reaction rate is governed by the concentrations of \( [\mathrm{NO}]^2 \) and \( [\mathrm{Cl}_2] \). Therefore, this determines the kinetic behavior of the entire reaction.

Rate Law Expression

The rate law expression represents the relationship between the concentration of reactants and the rate of reaction. For the given reaction, it was derived from the slow, rate-determining step:

• Initial formulation: \( \text{Rate} = k_2 [\mathrm{NOCl}_2][\mathrm{NO}] \)
• After substituting the intermediate: \( \text{Rate} = k [\mathrm{NO}]^2[\mathrm{Cl}_2] \)

The rate law describes how the reaction rate depends on the concentration of the reactants with powers corresponding to their coefficients in the overall rate-determining step once intermediates are substituted. The coefficients in the rate law provide the order of the reaction for each reactant, together giving the overall reaction order. This allows chemists to predict how changes in concentrations will affect the speed of the reaction.