Question

The following mechanism has been proposed for the gasphase reaction of chloroform (CHCl_ ) and chlorine:$$\begin{array}{l}{\text { Step } 1 : \mathrm{Cl}_{2}(g) \frac{k_{1}}{k_{-1}} 2 \mathrm{Cl}(g) \text { (fast) }} \\\ {\text { Step } 2 : \mathrm{Cl}(g)+\mathrm{CHCl}_{3}(g) \stackrel{k_{2}}{\longrightarrow} \mathrm{HCl}(g)+\mathrm{CCl}_{3}(g) \text { (slow) }}\end{array}$$ $$ { Step } \quad3 : \quad \mathrm{Cl}(g)+\mathrm{CCl}_{3}(g) \stackrel{k_{3}}{\longrightarrow} \mathrm{CCl}_{4} \quad(\text { fast })$$ (a) What is the overall reaction? (b) What are the intermedi- ates in the mechanism? (c) What is the molecularity of each of the elementary reactions? (d) What is the rate-determining step? (e) What is the rate law predicted by this mechanism? (Hint: The overall reaction order is not an integer.)

Short answer

a) Overall reaction: \(Cl_2(g) + CHCl_3(g) ⟶ HCl(g) + CCl_4\) b) Intermediates: Cl(g) and CCl₃(g) c) Molecularity: unimolecular (Step 1), bimolecular (Step 2), bimolecular (Step 3) d) Rate-determining step: Step 2: Cl(g) + CHCl₃(g) ⟶ HCl(g) + CCl₃(g) e) Rate law: \(Rate = k\sqrt{[Cl_2]}[CHCl_3]\)

Step-by-step solution

(Step 1: Adding the elementary reactions to find the overall reaction)

We can find the overall reaction by adding up all the elementary reactions and cancelling out the species that appear on both sides of the reaction: Step 1: Cl₂(g) ⇌ 2 Cl(g) Step 2: Cl(g) + CHCl₃(g) ⟶ HCl(g) + CCl₃(g) Step 3: Cl(g) + CCl3(g) ⟶ CCl₄ Now we add up the reactions, cancelling intermediates (in this case, Cl and CCl₃) which appear on both sides: Cl₂(g) + CHCl₃(g) ⟶ HCl(g) + CCl₄ Hence, the overall reaction is: \(Cl_2(g) + CHCl_3(g) ⟶ HCl(g) + CCl_4\)

(Step 2: Identifying the intermediates in the mechanism)

The intermediates are the chemical species that are produced in one elementary reaction and consumed in another. In this mechanism, the intermediates are Cl(g) and CCl₃(g), as they appear on both sides of the reactions.

(Step 3: Determining the molecularity of each of the elementary reactions)

Molecularity describes the number of molecules participating in an elementary reaction: Step 1: Cl₂(g) ⇌ 2 Cl(g) (unimolecular as one molecule, Cl₂, is involved) Step 2: Cl(g) + CHCl₃(g) ⟶ HCl(g) + CCl₃(g) (bimolecular because two molecules, Cl and CHCl₃, are involved) Step 3: Cl(g) + CCl3(g) ⟶ CCl₄ (bimolecular because two molecules, Cl and CCl₃, are involved)

(Step 4: Identifying the rate-determining step)

The rate-determining step is the slowest step in the mechanism. In this mechanism, Step 2 is mentioned as slow, so this is the rate-determining step: Step 2: Cl(g) + CHCl₃(g) ⟶ HCl(g) + CCl₃(g)

(Step 5: Predicting the rate law for the mechanism)

The rate-determining step is the slowest step, so the rate law is determined by this step. Using the balanced equation for Step 2, the rate law for this mechanism would be: \(Rate = k_2[Cl][CHCl_3]\) However, Cl is an intermediate, so we must find its concentration using the equilibrium established in Step 1 (fast equilibrium step): \(K_{eq} = \frac{k_1}{k_{-1}} = \frac{[Cl]^2}{[Cl_2]}\) Solving for [Cl] in terms of [Cl₂]: \([Cl] = \sqrt{K_{eq}[Cl_2]}\) Now we can substitute the expression for [Cl] in the rate law: \(Rate = k_2\sqrt{K_{eq}[Cl_{2}]}[CHCl_3]\) Now the rate law is expressed in terms of reactants and constants only: \(Rate = k\sqrt{[Cl_2]}[CHCl_3]\) where \(k=k_2\sqrt{K_{eq}}\) In summary: a) Overall reaction: \(Cl_2(g) + CHCl_3(g) ⟶ HCl(g) + CCl_4\) b) Intermediates: Cl(g) and CCl₃(g) c) Molecularity: unimolecular (Step 1), bimolecular (Step 2), bimolecular (Step 3) d) Rate-determining step: Step 2: Cl(g) + CHCl₃(g) ⟶ HCl(g) + CCl₃(g) e) Rate law: \(Rate = k\sqrt{[Cl_2]}[CHCl_3]\)

Key concepts

Elementary Reactions

An elementary reaction is a basic single-step chemical reaction where reactants convert directly into products, without undergoing a complex sequence of events. Each of these reactions has specific reactants and products, occurring under a particular set of conditions. They are the foundational building blocks of complex reaction mechanisms that we often see in chemistry.
For instance, the gas-phase reaction between chloroform ($CHCl\_3$) and chlorine ($Cl\_2$) consists of a series of elementary steps. Understanding each of these steps provides insight into how reactants transform into products over the course of the overall reaction.

Rate-Determining Step

The rate-determining step (RDS) acts as the 'traffic jam' in a chemical reaction mechanism. It's the slowest step in the sequence of events that occur during a chemical reaction. Identifying the rate-determining step is crucial for understanding the kinetics of a reaction, as it directly influences the overall rate at which the reaction proceeds. In our example with chloroform and chlorine, Step 2 is identified as the rate-determining step because it's marked as 'slow', meaning it dictates the time scale over which the reaction occurs. Thus, when calculating the reaction rate or discerning kinetic properties, we focus on the reactants and conditions that affect this rate-limiting step.

Reaction Intermediates

Reaction intermediates are species that are formed in one step of a reaction mechanism and consumed in subsequent steps. They are neither the initial reactants nor the final products of the overall chemical reaction. Instead, they are temporary, often unstable, and serve as a bridge between the reactants and products. In the chloroform and chlorine reaction, both Cl(g) and CCl₃(g) are intermediates. They are generated in the initial and middle steps but do not appear in the final equation of the overall reaction, highlighting their transitory nature during the reaction process.

Reaction Molecularity

Reaction molecularity tells us the number of reactant particles involved in an elementary reaction. It is a direct reflection of how reactants collide and interact with each other to form products. There are different types of molecularities such as unimolecular (involving one reactant molecule), bimolecular (involving two reactant molecules), and termolecular (involving three reactant molecules). In our textbook exercise, Step 1 is a unimolecular reaction where a chlorine molecule splits into two chlorine atoms, while Steps 2 and 3 are bimolecular because they involve the collisions between two different species—Cl and either CHCl₃ or CCl₃.

Rate Law

The rate law is an expression that relates the rate of a reaction to the concentration of the reactants and is grounded in the rate-determining step of the reaction mechanism. It cannot be deduced only from the stoichiometry of the overall reaction but requires understanding of the reaction mechanism. In our chloroform and chlorine reaction, the rate law derived from Step 2, which is the rate-determining step, takes into account the concentration of the reactants involved in that specific step only. It's also influenced by the concentrations of any intermediates, like Cl(g), which are determined from other parts of the mechanism, such as the equilibrium condition of the fast step (Step 1).
Since intermediates are often very short-lived and hard to measure directly, the rate law can be modified using conditions from other steps where the intermediate is at equilibrium to express it solely in terms of the initial reactants and constant parameters.