Problem 77 - Chemistry A Molecular Approach

Question

Consider this three-step mechanism for a reaction: \(\mathrm{Cl}_{2}(g) \underset{k_{2}}{\stackrel{k_{1}}{\rightleftarrows}} 2 \mathrm{Cl}(g)\) Fast \(\mathrm{Cl}(g)+\mathrm{CHCl}_{3}(g) \longrightarrow \mathrm{HCl}(g)+\mathrm{CCl}_{3}(g) \quad\) Slow \(\mathrm{Cl}(g)+\mathrm{CCl}_{3}(g) \longrightarrow \mathrm{CCl}_{4}(g)\) Fast a. What is the overall reaction? b. Identify the intermediates in the mechanism. c. What is the predicted rate law?

Step-by-Step Solution

Answer

a. The overall reaction is \(\mathrm{Cl}_{2}(g) + \mathrm{CHCl}_{3}(g) \longrightarrow \mathrm{HCl}(g) + \mathrm{CCl}_{4}(g)\). b. The intermediates are \(\mathrm{Cl}(g)\) and \(\mathrm{CCl}_{3}(g)\). c. The predicted rate law is \(\text{rate} = k[\mathrm{Cl}][\mathrm{CHCl}_{3}]\), which after substitution for \(\mathrm{Cl}\) using the equilibrium expression from step 1 becomes \(\text{rate} = k' [\mathrm{Cl}_{2}]^{1/2}[\mathrm{CHCl}_{3}]\).

Step 1: Determine the Overall Reaction

To find the overall reaction, add up all the steps of the mechanism and cancel out species that appear on both sides of the reaction arrow as reactants and products. These canceled species are intermediates, not present in the overall reaction.

Step 2: Identify the Intermediates

Intermediates are species that are produced in one step of a mechanism and consumed in a subsequent step. They do not appear in the overall reaction formula.

Step 3: Derive the Rate Law

The rate law for a reaction is determined by the slow step of the mechanism. Write the rate law by using the concentrations of the reactants in the slow step. In cases where intermediates are involved, use the fast pre-equilibrium step to express the intermediates in terms of only initial reactants.

Key Concepts

Reaction Mechanism

Understanding the reaction mechanism is key to grasping how chemical reactions occur. It is essentially the microscopic pathway or series of steps through which reactants transform into products. Each step is characterized by its own speed: fast or slow. In the given exercise, the overall chemical transformation occurs through a three-step mechanism, involving two fast steps, and one slow, rate-determining step.

The significance of the mechanism lies in its ability to detail the order of bond breakage and formation, along with the changes in molecular structure during the reaction. In a typical mechanism, you will find that the actual sequence of events can be quite complex and may include the creation of unstable, short-lived species called intermediates. These mechanisms are frequently used to predict the rate law of a reaction, a critical aspect in the field of chemical kinetics.

Reaction Intermediates

In the context of a chemical mechanism, reaction intermediates are species that appear temporarily in the course of a reaction sequence. They are produced in one step and consumed in another and are distinct from reactants or products which are present at the start or end of the reaction, respectively. In our exercise, intermediates are identified by analyzing the mechanism and noting species that are both formed and subsequently consumed as the reaction proceeds.

These intermediates do not appear in the overall balanced chemical equation for the reaction, as seen in the exercise where you were asked to cancel out species appearing on both sides of the arrows when outlining the overall reaction. Understanding intermediates is crucial because their fleeting existence can impact the rate and outcome of a reaction and also aid in identifying the rate law.

Rate Law

The rate law provides a mathematical relationship between the concentration of reactants and the rate of the reaction. It is determined by the slowest step in the reaction mechanism, also known as the rate-determining or rate-limiting step. This step acts as a bottleneck to the overall reaction speed since it is the slowest to occur.

As outlined in the exercise, to deduce the rate law, focus solely on the reactants involved in this slow step, rather than intermediate species or products. However, if an intermediate appears in the slow step, it's essential to express it in terms of the initial reactants through the relationships established by earlier, faster steps. This is achieved through algebraic manipulation based on the fast pre-equilibrium conditions prior to the slow step. Consequently, the rate law derived from the mechanism not only tells us about the speed of a reaction but also hints at the concentration dependencies and the order of reaction, which are pivotal for any further kinetics studies.