Problem 219 - A Complete Resource Book in Chemistry for JEE Main

Question

Consider the reaction: \(\mathrm{Cl}_{2}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{~S}(\mathrm{aq}) \longrightarrow \mathrm{S}(\mathrm{s})+2 \mathrm{H}^{+}(\mathrm{aq})+2 \mathrm{Cl}^{-}(\mathrm{aq})\) The rate equation for this reaction is Rate \(=\mathrm{k}\left[\mathrm{Cl}_{2}\right]\left[\mathrm{H}_{2} \mathrm{~S}\right]\) Which of these mechanisms is/are consistent with this rate equation? [2010] (1) \(\mathrm{Cl}_{2}+\mathrm{H}_{2} \mathrm{~S} \longrightarrow \mathrm{H}^{+}+\mathrm{Cl}^{-}+\mathrm{Cl}^{+}+\mathrm{HS}^{-}\)(slow) \(\mathrm{Cl}^{+}+\mathrm{HS}^{2} \longrightarrow \mathrm{H}^{+}+\mathrm{Cl}^{-}+\mathrm{S}\) (fast) (2) \(\mathrm{H}_{2} \mathrm{~S} \Leftrightarrow \mathrm{H}^{+}+\mathrm{HS}^{-}\)(fast equilibrium) \(\mathrm{Cl}_{2}+\mathrm{HS}^{-} \longrightarrow 2 \mathrm{Cl}^{-}+\mathrm{H}^{+}+\mathrm{S}\) (slow) (a) 2 only (b) Both 1 and 2 (c) Neither 1 nor 2 (d) 1 only

Step-by-Step Solution

Answer

Mechanism 1 only matches the rate equation.

Step 1: Understand the Rate Equation

The given rate equation is \(\text{Rate} = k[\text{Cl}_2][\text{H}_2 \text{~S}]\). This indicates that the reaction is first-order with respect to \(\text{Cl}_2\) and \(\text{H}_2 \text{~S}\), meaning both of these substances are involved in the rate-determining step of the mechanism.

Step 2: Check Mechanism 1

In Mechanism 1, the slow step is \(\text{Cl}_2 + \text{H}_2 \text{~S} \rightarrow \text{H}^+ + \text{Cl}^- + \text{Cl}^+ + \text{HS}^-\). As the slow step is the rate-determining step and involves both \(\text{Cl}_2\) and \(\text{H}_2 \text{~S}\), this matches the requirement of the rate equation \(\text{Rate} = k[\text{Cl}_2][\text{H}_2 \text{~S}]\).

Step 3: Check Mechanism 2

Mechanism 2 consists of two steps. The first step is a fast equilibrium: \(\text{H}_2 \text{~S} \Leftrightarrow \text{H}^+ + \text{HS}^-\), and the slow step is \(\text{Cl}_2 + \text{HS}^- \rightarrow 2 \text{Cl}^- + \text{H}^+ + \text{S}\). Although \(\text{Cl}_2\) appears in the slow step, \(\text{H}_2 \text{~S}\) does not. Instead, \(\text{HS}^-\) comes from the fast equilibrium step. This means \(\text{H}_2 \text{~S}\) affects the equilibrium, but not directly the slow step, thus not matching the rate law.

Step 4: Evaluate Which Mechanism Matches the Rate Law

Since Mechanism 1 correctly aligns with the rate equation by having \(\text{Cl}_2\) and \(\text{H}_2 \text{~S}\) in the slow step, and Mechanism 2 does not, Mechanism 1 is consistent with the rate equation.

Key Concepts

Rate Equation

In chemical kinetics, the rate equation, also known as the rate law, is an equation that links the rate of a chemical reaction to the concentration of the reactants. For the reaction given in the exercise, the rate equation is \(\text{Rate} = k[\text{Cl}_2][\text{H}_2 \text{~S}]\). This form of the rate equation tells us that the reaction rate depends directly on the concentration of both \(\text{Cl}_2\) and \(\text{H}_2 \text{~S}\).
The constant \(k\) is the rate constant and varies with temperature but not with concentration. Understanding the rate equation is crucial because it provides valuable insights into how changes in the concentrations of reactants will affect the rate at which products are formed. This is especially important in industrial and laboratory settings where controlling reaction speed is often necessary.
A reaction order, indicated by the exponents in the rate equation, directly correlates with the effect of concentration changes. Here, each reactant is raised to the first power, indicating each has a direct and proportional effect on the rate.

Reaction Mechanism

A reaction mechanism gives a step-by-step sequence of elementary reactions by which overall chemical change occurs. Mechanisms are crucial because they provide detailed information about the path taken by molecules during chemical reactions. In the given exercise, there are two proposed mechanisms. Each mechanism contains one or more elementary steps that occur sequentially.
It's important to note that not all steps in a mechanism have the same impact on the speed of the overall reaction. Some steps can be fast, happening instantly compared to others. Understanding the proposed mechanisms allows chemists to deduce which steps are involved in determining the rate at which products are formed.
When evaluating mechanisms, consistency with the observed rate equation is key. If a proposed mechanism’s rate-determining step aligns with the rate equation, it suggests that the mechanism could be plausible. Otherwise, it might be incorrect or incomplete.

Rate-Determining Step

The rate-determining step is the slowest step in a reaction mechanism. It is often compared to the narrowest part of a bottleneck in a series of processes. This step essentially controls the overall reaction rate, as it forms the rate-limiting barrier.
In the exercise, for Mechanism 1, the slow step is \(\text{Cl}_2 + \text{H}_2 \text{~S} \rightarrow \text{H}^+ + \text{Cl}^- + \text{Cl}^+ + \text{HS}^-\). Since both reactants in the rate equation appear in this slow step, it supports that Mechanism 1 could be correct. Conversely, in Mechanism 2, the slow step does not involve \(\text{H}_2 \text{~S}\) directly but instead proceeds from an intermediate \(\text{HS}^-\).
Recognizing the rate-determining step is important because it shows how different pathways can lead to the same product but with different speeds. Understanding this helps in designing methods to control reactions more efficiently.

Order of Reaction

The order of reaction refers to the power to which the concentration of a reactant is raised in the rate equation. It provides insight into how each reactant concentration affects the overall reaction rate. For the rate equation \(\text{Rate} = k[\text{Cl}_2][\text{H}_2 \text{~S}]\), the reaction is first-order with respect to \(\text{Cl}_2\) and first-order with respect to \(\text{H}_2 \text{~S}\).
This means that if the concentration of \(\text{Cl}_2\) is doubled, the rate of the reaction will double, and the same applies separately to \(\text{H}_2 \text{~S}\). The overall order of the reaction is the sum of the powers, making this a second-order reaction overall.
Understanding the order helps predict behavior under various conditions and is fundamental for optimizing conditions in both experimental and industrial processes. It is also helpful in determining the units for the rate constant \(k\), which vary depending on the reaction order.