Question

In this chapter, we have seen a number of reactions in which a single reactant forms products. For example, consider the following first-order reaction: \(\mathrm{CH}_{3} \mathrm{NC}(g) \longrightarrow \mathrm{CH}_{3} \mathrm{CN}(g)\) However, we also learned that gas-phase reactions occur through collisions. a. One possible explanation for how this reaction occurs is that two molecules of \(\mathrm{CH}_{3} \mathrm{NC}\) collide with each other and form two molecules of the product in a single elementary step. If that were the case, what reaction order would you expect? b. Another possibility is that the reaction occurs through more than one step. For example, a possible mechanism involves one step in which the two \(\mathrm{CH}_{3} \mathrm{NC}\) molecules collide, resulting in the "activation" of one of them. In a second step, the activated molecule goes on to form the product. Write down this mechanism and determine which step must be rate determining in order for the kinetics of the reaction to be first order. Show explicitly how the mechanism predicts first-order kinetics.

Short answer

For the reaction to be first-order, the rate-determining step must involve the breakdown of an activated intermediate, \(\mathrm{CH}_3\mathrm{NC}^*\), into the product in a fast pre-equilibrium step followed by a slow unimolecular decomposition of \(\mathrm{CH}_3\mathrm{NC}^*\).

Step-by-step solution

Reaction Order for a Single Elementary Step

For an elementary step involving two molecules of \(\mathrm{CH}_3\mathrm{NC}\) colliding, the order of the reaction would be determined by the molecularity of the step. In this scenario, two molecules collide, making it a bimolecular reaction. Therefore, if this were the sole step, the reaction would be second-order, following the rate law: Rate = k[\(\mathrm{CH}_3\mathrm{NC}\)]^2.

Proposing a Two-Step Mechanism

Consider a mechanism with two separate steps. First, a collision between two \(\mathrm{CH}_3\mathrm{NC}\) molecules leading to the activation of one of them, and then, the activated \(\mathrm{CH}_3\mathrm{NC}\) molecule going on to form \(\mathrm{CH}_3\mathrm{CN}\).

Writing the Mechanism Steps

The proposed mechanism can be written as follows: Step 1: \(\mathrm{CH}_3\mathrm{NC} + \mathrm{CH}_3\mathrm{NC} \rightarrow \mathrm{CH}_3\mathrm{NC}^* + \mathrm{CH}_3\mathrm{NC}\) (activation step). Step 2: \(\mathrm{CH}_3\mathrm{NC}^* \rightarrow \mathrm{CH}_3\mathrm{CN}\) (product formation step). Where \(\mathrm{CH}_3\mathrm{NC}^*\) represents the activated species.

Identifying the Rate-Determining Step

The rate-determining step (slow step) controls the kinetics of the overall reaction. If Step 1 is slow and Step 2 is fast, then the kinetics of Step 1 will determine the reaction order. Since Step 1 involves two \(\mathrm{CH}_3\mathrm{NC}\) molecules, it appears to lead to second-order kinetics, conflicting with the given first-order reaction.

Justifying First-Order Kinetics

For the reaction to be first-order, the rate must be dependent on the concentration of a single reactant. A possible scenario to justify first-order kinetics is if the activation step is in rapid equilibrium and the formation of the product (Step 2) is the slow, rate-determining step. In this case, the concentration of activated \(\mathrm{CH}_3\mathrm{NC}^*\) becomes constant and the rate depends solely on the breakdown of \(\mathrm{CH}_3\mathrm{NC}^*\) into product, which is first-order with respect to \(\mathrm{CH}_3\mathrm{NC}^*\).

Key concepts

Reaction Order

Understanding the reaction order is crucial for predicting how the concentration of reactants affects the rate of a chemical reaction. In simple terms, the reaction order is the power to which the concentration of a reactant is raised in the rate equation. For instance, in a first-order reaction, the rate is directly proportional to the concentration of one reactant, denoted by the rate law: Rate = k[A], where 'k' is the rate constant and [A] is the concentration of the reactant. If a reaction's rate is dependent on the concentration of two reactants, say A and B, and is second-order with respect to both, the rate law would be Rate = k[A][B].

If we consider the elementary reaction where \(\text{CH}_3\text{NC}\) converts to \(\text{CH}_3\text{CN}\), and assume it proceeds in one step involving two \(\text{CH}_3\text{NC}\) molecules, the law would be second-order, expressed as Rate = k[\(\text{CH}_3\text{NC}\)]^2. However, since the observed kinetics are first-order, this suggests the reaction mechanism must involve more complexity than a single bimolecular collision.

Reaction Mechanism

The reaction mechanism offers a step-by-step sequence of elementary reactions that describe the pathway from reactants to products. It details not only the actual steps involved but also the transition states and intermediates formed during the course of the reaction. A proposed mechanism needs to comply with the experimental rate law. For the isomerization of \(\text{CH}_3\text{NC}\), it's believed that the first-order kinetics imply a multi-step mechanism, as a simple bimolecular collision would result in second-order kinetics.

By analyzing each step separately, one can propose that the initial activation of one \(\text{CH}_3\text{NC}\) molecule does not directly form the product but rather leads to a reactive intermediate. This intermediate then decomposes in a separate step to produce the final product. This differentiation of steps helps reconcile the observed first-order reaction kinetics with the mechanism where multiple molecular interactions occur.

Rate-Determining Step

Within a reaction mechanism, the rate-determining step (RDS) is the slowest step that controls the overall rate of the reaction. It is the bottleneck through which all other reactions must pass, and thus sets the pace for the entire process. If the first step in a proposed mechanism, like the bimolecular collision of \(\text{CH}_3\text{NC}\) molecules, is the RDS, then the reaction rate would be expected to depend on the concentration of both reacting molecules and show second-order kinetics.

However, to align with first-order kinetics, the rate must only depend on one reactant's concentration. This can occur if the first step occurs rapidly and reaches an equilibrium, creating a steady concentration of an activated intermediate. The decomposition of this intermediate would then be the RDS, making the rate of reaction first-order with respect to the concentration of just the intermediate, not the concentration of \(\text{CH}_3\text{NC}\). This is an elegant way of explaining why a reaction involving two reacting molecules may display first-order kinetics.

Molecularity

Molecularity refers to the number of reactant particles involved in one elementary step of a reaction and is always a whole number. A reaction is unimolecular if it involves a single molecule, bimolecular if two molecules are involved, and termolecular if it involves three. It's important to note that molecularity can be determined only for elementary reaction steps and not for the overall reaction, which might be composed of several elementary steps.

In the example discussed, if the activation step of \(\text{CH}_3\text{NC}\) is assumed to be the only elementary step, its molecularity would be two because two molecules of \(\text{CH}_3\text{NC}\) collide. However, molecularity alone cannot always determine the overall reaction order because the rate law also depends on whether the mechanism includes a series of steps, and how quickly each of those steps takes place.

Chemical Kinetics

Chemical kinetics is the field of chemistry that deals with the rates of chemical reactions and the mechanisms by which these reactions proceed. It's not just about how fast a reaction occurs but also understanding the steps involved in the transformation from reactants to products. Factors like temperature, pressure, and the presence of catalysts can dramatically affect these rates.

In studying the kinetics of the \(\text{CH}_3\text{NC}\) to \(\text{CH}_3\text{CN}\) reaction, one would have to consider the energy barriers for each step, the stability of the intermediates, and how either of these could be changed to alter the reaction's speed. The primary goal of chemical kinetics is to establish the rate law, identify the order of reaction, and uncover the detailed reaction mechanism, which together reveal the answers to 'how' and 'why' a chemical reaction occurs as it does.