Question

A complex reaction: \(2 \mathrm{X}+\mathrm{Y} \rightarrow \mathrm{Z}\), takes place in two steps $$ \begin{array}{l} \mathrm{X}+\mathrm{Y} \stackrel{K_{1}}{\longrightarrow} 2 \mathrm{~W} \\ \mathrm{X}+2 \mathrm{~W} \stackrel{K_{2}}{\longrightarrow} \mathrm{Z} \end{array} $$ If \(K_{1}

Short answer

The overall order of the reaction will be 2.

Step-by-step solution

- Understand the mechanism

Identify and understand the given mechanism of the complex reaction which takes place in two steps. The first step is \( X + Y \stackrel{K_1}{\longrightarrow} 2 W \), and the second step is \( X + 2 W \stackrel{K_2}{\longrightarrow} Z \) with \( K_1 < K_2 \).

- Determine the rate-determining step

Based on the inequality \( K_1 < K_2 \), the first step is slower and therefore, is the rate-determining step.

- Establish the overall order of the reaction

The overall order of the reaction is determined by the molecularity of the rate-determining step. In the first step, one molecule of \( X \) reacts with one molecule of \( Y \) which indicates that the order of the reaction is 1 (for \( X \) and 1 (for \( Y \), giving a total order of 2.

Key concepts

Chemical Kinetics

Chemical kinetics is the branch of physical chemistry that concerns itself with understanding the rates of chemical reactions. It is crucial for predicting how quickly a reaction will occur under various conditions and for determining the necessary conditions for a reaction to take place.

One of the key aspects of chemical kinetics is the rate of reaction, which measures how fast reactants are converted into products. This rate can be influenced by several factors, such as temperature, pressure, and the presence of catalysts. To quantify the rate, rate laws are used, which are mathematical equations that relate the rate of a reaction to the concentration of its reactants, typically in the form of rate = k [A]^n [B]^m, where k is the rate constant, [A] and [B] are the concentrations of reactants, and n and m are the orders of the reaction with respect to each reactant.

Understanding kinetics also involves exploring concepts like half-life (the time taken for half of the reactants to be consumed), activation energy (the minimum energy required for a reaction to occur), and reaction mechanisms (the detailed pathways by which reactants transform into products).

Rate-Determining Step

The rate-determining step is often known as the 'bottleneck' of a chemical reaction. It is the slowest step in a reaction mechanism that determines the overall rate at which the reaction occurs. In a multi-step reaction, the rate-determining step has the highest activation energy barrier to overcome and therefore, sets the pace for the entire process.

In the exercise example, the given inequality, K₁ < K₂, suggests that the first step of the reaction is slower, making it the rate-determining step. This step, therefore, governs the kinetics of the reaction because subsequent steps, while possibly quicker, cannot proceed until this step is complete. Knowing the rate-determining step allows chemists to manipulate conditions to enhance the reaction rate, for instance, by adding a catalyst that specifically speeds up this slowest step without affecting the others.

Reaction Mechanism

A reaction mechanism is a detailed description of the steps through which reactants go through to form products. It provides a depiction of the sequence of elementary reactions, which are single-step processes with a single transition state that lead to a chemical reaction. Understanding the mechanism helps chemists to visualize and explain not only the path taken by the molecules but also the rate and the outcome of the reaction.

In our example, the reaction mechanism involves two elementary steps. Each step involves a distinct collision event and transition state. The first step involves the collision between molecules X and Y, forming an intermediate, while the second step includes a subsequent collision between X and the intermediate formed in the first step, resulting in the final product Z. The intermediates and transition states are often high-energy species that are not isolated but are crucial for understanding the reaction progress.

Molecularity

Molecularity refers to the number of reactant molecules involved in an elementary step of a reaction mechanism and is always an integer value. Commonly, reactions are either unimolecular, bimolecular, or termolecular, involving one, two, or three molecules, respectively. The molecularity of an elementary step is directly related to the order of reaction for that step as it indicates the power to which the concentration of the reactant is raised in the rate law.

The exercise example shows that the rate-determining step is the bimolecular reaction between one molecule of X and one molecule of Y (X + Y). This directly demonstrates that the overall order of the reaction is second-order, being the sum of the individual molecularity contributions from the reactants in the rate-determining step, which is 2 in this case (1 from X and 1 from Y). This molecularity, and hence the order of the reaction, help in determining the concentration dependence and the kinetics of the reaction process.