Problem 160 - A Complete Resource Book in Chemistry for JEE Main

Question

For the reaction, \(2 \mathrm{NO}+\mathrm{Cl}_{2} \longrightarrow 2 \mathrm{NOCl}\) The following mechanism has been proposed \(\mathrm{NO}+\rightleftharpoons \mathrm{Cl} \quad \mathrm{NOCl}_{2}\) (fast) \(\mathrm{NOCl}_{2}+\mathrm{NO} \longrightarrow 2 \mathrm{NOCl}\) (slow) (a) Rate \(=\mathrm{k}[\mathrm{NO}]\left[\mathrm{Cl}_{2}\right]^{2}\) (b) Rate \(=\mathrm{k}[\mathrm{NO}]^{2}\left[\mathrm{Cl}_{2}\right]\) (c) Rate \(=\mathrm{k}[\mathrm{NOCl}]^{2}\) (d) Rate \(=\mathrm{k}\left[\mathrm{NOCl}_{2}\right][\mathrm{NO}]\)

Step-by-Step Solution

Answer

The correct rate expression is (b) \( = k[\mathrm{NO}]^2[\mathrm{Cl}_2] \).

Step 1: Identify the Rate-Determining Step

Examine the given mechanism and identify the slow step, which governs the overall rate of the reaction. For the provided mechanism, the rate-determining step is \( \mathrm{NOCl}_{2} + \mathrm{NO} \longrightarrow 2 \mathrm{NOCl} \) because it is labeled as slow.

Step 2: Write the Rate Law for the Slow Step

Write the rate law for the rate-determining step based on the reactants involved. The rate law for the slow step is \( \text{Rate} = k_{2} [\mathrm{NOCl}_{2}][\mathrm{NO}] \).

Step 3: Express the Intermediate Using Equilibrium of Fast Step

Since \( \mathrm{NOCl}_{2} \) is an intermediate, express it in terms of the reactants. From the fast equilibrium step, we have \( \mathrm{NO} + \mathrm{Cl}_{2} \rightleftharpoons \mathrm{NOCl}_{2} \), giving \( K = \frac{[\mathrm{NOCl}_2]}{[\mathrm{NO}][\mathrm{Cl}_2]} \), or \( [\mathrm{NOCl}_2] = K [\mathrm{NO}][\mathrm{Cl}_2] \).

Step 4: Substitute Intermediate in Rate Law

Substitute \( [\mathrm{NOCl}_2] \) in the rate law of the slow step using the expression from the fast step equilibrium. This gives: \( \text{Rate} = k_2 (K [\mathrm{NO}][\mathrm{Cl}_2])[\mathrm{NO}] = k' [\mathrm{NO}]^2 [\mathrm{Cl}_2] \), where \( k' = k_2 K \).

Step 5: Determine the Correct Rate Expression

Compare the rate expression obtained in Step 4 with the provided options. Based on our derived expression \( \text{Rate} = k' [\mathrm{NO}]^2 [\mathrm{Cl}_2] \), the matching choice is (b).

Key Concepts

Rate-Determining Step

In any complex reaction, not all steps occur at the same pace. The rate-determining step is essentially the slowest step, controlling the overall timing of the reaction, analogous to a traffic bottleneck. For the chemical equation provided, the slow step or the rate-determining step is \( \text{NOCl}_2 + \text{NO} \longrightarrow 2 \text{NOCl} \). This step dictates the speed of the entire reaction.
Understanding which step is the rate-determining step is crucial because it indicates which molecules' concentrations will appear in the rate law. By focusing on this slowest step, we can unravel the dynamics driving the reaction forward.

Rate Law

Rate laws serve as a mathematical expression capturing the rate of reaction concerning the concentration of reactants. For the rate-determining step in our mechanism, the rate law forms as follows: \( \text{Rate} = k_2 [\text{NOCl}_2][\text{NO}] \).
The derived expression helps predict how changing concentrations can affect the reaction rate. It also provides insights into the order of reaction with respect to each reactant, showing a direct squared relationship with \( [\text{NO}] \) in the final rate law derived: \( \text{Rate} = k' [\text{NO}]^2 [\text{Cl}_2] \).

• The power to which reactant concentrations are raised in the rate law gives the reaction order with respect to each species.
• Overall reaction order can reveal much about reaction mechanisms.

Chemical Kinetics

Chemical kinetics explores the speed or rate of a chemical reaction and the factors affecting this rate. It combines understanding from the rate-determining step and rate laws.

• Focuses on the pathways and time taken for reactants to transform into products.
• Aids in the design of industrial processes by optimizing reaction conditions.

For the provided reaction, kinetics remind us that not just the surface reaction, but each mechanistic step must be understood thoroughly to control and predict behavior.

Intermediate Species

Intermediate species arise transiently within a reaction mechanism; they are created and consumed during the reaction process. They don't appear in the overall balanced equation of the reaction but play a crucial role in connecting various steps.
In the given mechanism, \( \text{NOCl}_2 \) acts as an intermediate, formed in the fast step and consumed in the slow, rate-determining step. By expressing it in terms of the reactants via equilibrium expressions, we can relate the reaction's intermediate equilibria to the working mechanism.

• Intermediates provide insight into the reaction's complexity.
• Deep analysis of intermediates leads to more accurate chemical modeling.