Problem 24 - Basic Principles of Organic Chemistry

Question

An alternative mechanism for \(E 2\) elimination is the following: $$ \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{Cl}+\mathrm{OH}^{\ominus} \stackrel{\text { fast }}{\rightleftharpoons} \ominus: \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{Cl}+\mathrm{H}_{2} \mathrm{O} \stackrel{\text { slow }}{\longrightarrow} \mathrm{CH}_{2}=\mathrm{CH}_{2}+\mathrm{Cl}^{\ominus} $$ a. Would this mechanism lead to overall second-order kinetics with respect to the concentrations of \(\mathrm{OH}^{\ominus}\) and ethyl chloride? Explain. b. This mechanism as written has been excluded for several halides by carrying out the reaction in deuterated solvents such as \(\mathrm{D}_{2} \mathrm{O}\) and \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OD}\). Explain how such experiments could be relevant to the reaction mechanism.

Step-by-Step Solution

Answer

Yes, the mechanism leads to second-order kinetics. Deuterated solvents rule out proton transfers in the rate-determining step.

Step 1: Analyzing the Mechanism

The given mechanism involves a fast reversible reaction followed by a slow irreversible reaction. The fast step forms an intermediate ion complex: \[\begin{align*}\mathrm{CH}_{3}\mathrm{CH}_{2}\mathrm{Cl} + \mathrm{OH}^{\ominus} &\rightleftharpoons \ominus:\mathrm{CH}_{2}\mathrm{CH}_{2}\mathrm{Cl} + \mathrm{H}_{2}\mathrm{O} \quad (\text{fast}) \\ominus:\mathrm{CH}_{2}\mathrm{CH}_{2}\mathrm{Cl} &\rightarrow \mathrm{CH}_{2}=\mathrm{CH}_{2} + \mathrm{Cl}^{\ominus} \quad (\text{slow})\end{align*}\]

Step 2: Determining the Rate-Determining Step

The rate-determining step is the slow step, which involves the decomposition of the intermediate ion. Since the fast step establishes an equilibrium, we assume the concentration of the intermediate is determined by the equilibrium between the starting materials and the intermediate.

Step 3: Deriving the Rate Law

The rate of the reaction is determined by the slow step. Generally, for the elimination step, the rate \(\text{Rate} = k[\ominus: \mathrm{CH}_{2}\mathrm{CH}_{2}\mathrm{Cl}]\). From the equilibrium of the fast step, we can deduce that \([\ominus :\mathrm{CH}_{2}\mathrm{CH}_{2}\mathrm{Cl}] = K[\mathrm{CH}_{3}\mathrm{CH}_{2}\mathrm{Cl}][\mathrm{OH}^{\ominus}]\) where \(K\) is the equilibrium constant.

Step 4: Finding the Overall Kinetics Order

Substituting the expression for the intermediate concentration into the rate law, we find \[\text{Rate} = kK[\mathrm{CH}_{3}\mathrm{CH}_{2}\mathrm{Cl}][\mathrm{OH}^{\ominus}]\]This shows that the reaction is second-order overall, with first-order dependence on both ethyl chloride and hydroxide ion.

Step 5: Analyzing Deuterated Solvent Impact

Deuterated solvents might affect the rate if there's an exchangeable proton in the mechanism. In the slower step, no hydrogen transfers relevant to the solvent would suggest a solvent kinetic isotope effect. Thus, lack of such effects indicates no involvement of solvent hydrogen in the rate-determining step.

Key Concepts

Second-Order Kinetics

Second-order kinetics refers to reactions whose rate depends on the concentration of two different reactant species, each impacting the rate. In the case of the E2 elimination mechanism provided in the exercise, we see a second-order overall rate law. This means that the reaction rate is influenced equally by both the concentration of ethyl chloride (\(\mathrm{CH}_3\mathrm{CH}_2\mathrm{Cl}\)) and hydroxide ion (\(\mathrm{OH}^{\ominus}\)).
The intermediate step in the mechanism helps us understand this dependence. The fast step creates an intermediate that the slow step uses to proceed further.

• This intermediate concentration is crucial as it links back to the initial reactants when the equilibrium is established.
• The rate law derived from the slow step includes both reactants, signifying their direct influence on reaction speed.
• Knowing each reactant's effect helps in predicting and controlling reaction outcomes effectively.

Understanding this aspect is crucial because it helps predict how a reaction will respond if you increase or decrease the concentration of one reactant—in short, understanding second-order kinetics can be a powerful tool in chemistry.

Rate-Determining Step

The rate-determining step is key to understanding the overall speed of a chemical reaction, since it is the slowest step of the mechanism. In the E2 elimination example provided, the rate-determining step is the second step, where the intermediate decomposes to form the final products.

• This step is slower than the previous, fast equilibrium reaction, and therefore acts as a bottleneck for the overall reaction.
• Since the reaction is determined by this slow step, all energy barriers and transition states come into play at this point, making it critical to the progression of the reaction.

To illustrate this, think of the rate-determining step as the point in a queue that moves the slowest. No matter how fast people get into the queue, they cannot finish faster than the slowest server. Understanding which step is rate-determining provides insight into where one might intervene with catalysts or optimize conditions to accelerate the reaction.

Deuterated Solvents Impact

Deuterated solvents are special liquids where hydrogen atoms are replaced by deuterium, a heavier isotope of hydrogen. These solvents are useful in studying reaction mechanisms, including E2 eliminations, because their different physical properties can influence the reaction's progression.
The exercise highlights using deuterated solvents like \(\mathrm{D}_2\mathrm{O}\) and \(\mathrm{C}_2\mathrm{H}_5\mathrm{OD}\) to probe the mechanism. Here's why they're relevant:

• If a reaction step involves transferring or exchanging hydrogen with the solvent, changing hydrogen to deuterium could shift reaction rates—often referred to as a kinetic isotope effect.
• Observing no change in reaction rates with deuterated solvents implies that no hydrogen exchange with the solvent happens in the rate-determining step, confirming or refuting parts of the proposed mechanism.
• This effect allows chemists to rule out potential pathways and validate or refine their understanding of which steps may be involved in the reaction.

By employing deuterated solvents, one can unveil intricate details of the mechanism that are otherwise hidden, leading to a more thorough understanding of chemical processes.