Question

Consider this overall reaction, which is experimentally observed to be second order in AB and zero order in C: $$ \mathrm{AB}+\mathrm{C} \longrightarrow \mathrm{A}+\mathrm{BC} $$ Is the following mechanism valid for this reaction? $$ \begin{array}{ll} \mathrm{AB}+\mathrm{AB} \longrightarrow \mathrm{AB}_{2}+\mathrm{A} & \text { Slow } \\ \mathrm{AB}_{2}+\mathrm{C} \longrightarrow \mathrm{AB}+\mathrm{BC} & \text { Fast } \end{array} $$

Short answer

The mechanism is valid because the rate-determining step is consistent with the experimentally determined second-order dependence on AB and zero-order dependence on C.

Step-by-step solution

Compare the Rate Law to the Proposed Mechanism

To determine the validity of the mechanism, first write down the rate law based on the overall reaction's order with respect to each reactant. The given overall reaction is second order in AB and zero order in C, which implies a rate law of the form: Rate = k[AB]^2[C]^0. Since the concentration of C does not influence the rate, it can be simplified to Rate = k[AB]^2. Next, examine the proposed mechanism for consistency with this rate law.

Analyze the Elementary Steps

The given mechanism consists of two elementary steps. According to the mechanism, the first (slow) step is a reaction involving two AB molecules, while the second (fast) step involves AB2 and C. For a mechanism to be valid, the slowest step (the rate-determining step) must be consistent with the experimentally determined rate law.

Determine the Rate-Determining Step

The rate-determining step is typically the slowest step in the mechanism because it limits the overall rate of the reaction. In the proposed mechanism, the first step is identified as slow and therefore is presumed to be the rate-determining step.

Match the Rate Law with the Rate-Determining Step

The rate law derived from the rate-determining step of the proposed mechanism is Rate = k'[AB]^2 since two molecules of AB are involved and there is no C in the slow step. This rate law corresponds directly with the experimentally determined rate law Rate = k[AB]^2.

Validate or Invalidate the Mechanism

Since the rate law deduced from the rate-determining step of the proposed mechanism aligns with the experimentally observed rate law, the mechanism is likely to be valid.

Key concepts

Reaction Order

Understanding the rate at which chemical reactions occur is essential for predicting how a system will change over time. Reaction order is one such concept that provides insight into the relationship between reactant concentration and reaction rate. Put simply, reaction order tells us how the concentration of reactants affects the speed of the reaction.

An order of reaction can be zero, first, second, or higher, and this is determined empirically; that means it's found out through experiments rather than theoretical predictions. For instance, a second-order reaction in regard to a reactant AB means that if the concentration of AB doubles, the reaction rate increases by a factor of four (since 2² = 4). Conversely, a zero-order reaction, like in the case with reactant C, indicates that changes in C’s concentration have no impact on the rate of the reaction.

In the exercise above, the rate law expression indicates that the overall reaction is second order in AB (as the rate is directly proportional to the square of AB's concentration) and zero order in C (as its concentration doesn't affect the rate). This concurrence between the rate law and the reaction order is crucial for validating the proposed mechanism.

Rate-Determining Step

In the context of chemical kinetics, the concept of a rate-determining step is akin to a bottleneck in a production process. It is the slowest step in a reaction mechanism that sets the pace for the overall reaction. Identifying the rate-determining step is key to understanding and modeling how a reaction proceeds.

The rate-determining step in the proposed reaction mechanism is the first elementary step, which involves the interaction of two AB molecules. Since this step is the slowest, it consequently governs the reaction rate. Any subsequent step, no matter how fast, cannot occur until the first step is completed.

This concept is beautifully illustrated in the exercise. Despite there being a second 'fast' step, the overall reaction cannot proceed any faster than the slowest 'slow' step allows. By comparing the rate law that the rate-determining step implies (\( k'[AB]^2 \) with no dependence on C) to the experimentally determined rate law (\( k[AB]^2 \)), it's clear that they are in agreement, which suggests that the proposed mechanism is plausible.

Elementary Steps

To appreciate the complexity of chemical reactions, it is necessary to dissect them into their most basic events known as elementary steps. These are individual reactions that occur within a larger, composite reaction mechanism. Each elementary step is characterized by a distinct transition state and can usually be classified as unimolecular, bimolecular, or termolecular, based on the number of molecules involved.

The overall reaction mechanism is often the sum of these elementary steps, which may proceed with varying speeds. In the exercise provided, there are two elementary steps. The first involves two molecules of AB and is designated as the slow step. The second, involves the molecule AB2 and C and is considerably faster.

When we analyze the validity of a mechanism, we look at each elementary step to ensure they align with the observed kinetics. In this case, the first elementary step consists solely of molecules of AB reacting, which aligns with the reaction being second order in AB if it's considered as the rate-determining step. This synthesis of information from the elementary steps is vital in building an accurate kinetic model of the reaction.