Question

For the reaction $$ 2 \mathrm{H}_{2}(g)+2 \mathrm{NO}(g) \longrightarrow \mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g) $$ the experimental rate expression is rate \(=k[\mathrm{NO}]^{2}\left[\mathrm{H}_{2}\right]\). The following mechanism is proposed: Is this mechanism consistent with the rate expression?

Short answer

Answer: The proposed mechanism is not consistent with the given rate expression unless the concentration of H2 remains constant during the reaction. The rate expression from the proposed mechanism (assuming Step 1 as the rate-determining step) is k1[NO]², while the given experimental rate expression is k[NO]²[H2]. To confirm the mechanism's consistency, please provide the correct proposed mechanism, and we will check its consistency with the given rate expression.

Step-by-step solution

Reaction mechanism

The proposed mechanism is not provided in the exercise. We cannot proceed with solving this task without further information. Please provide the proposed mechanism for the reaction, and then we can determine if it is consistent with the given rate expression. For now, let's assume the reaction mechanism is as follows (but remember, you can update it with the real information later): $$ \begin{aligned} \mathrm{Step \thinspace 1:}\quad & 2 \mathrm{NO}(g) \longrightarrow \mathrm{N}_{2} O_{2}(g) \\ \mathrm{Step\thinspace 2:}\quad & \mathrm{N}_{2} O_{2}(g) +2 \mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{N}_{2}(g) \end{aligned} $$

Identifying the rate-determining step

First, we need to check which step is the slowest and determines the overall rate of the reaction. We generally call this the rate-determining step (RDS). Let's assume that step 1 is the rate-determining step: $$ \text { rate } = k_{1}[\mathrm{NO}]^{2} $$

Checking rate expressions

The proposed rate expression for the given mechanism (assuming Step 1 as the rate-determining step) is: $$ \text { rate } = k_{1}[\mathrm{NO}]^{2} $$ Now let's compare it with the experimental rate expression: $$ \text { rate } = k[\mathrm{NO}]^{2}\left[\mathrm{H}_{2}\right] $$ It's clear that the proposed rate expression for the given mechanism does not match the experimental rate expression unless we consider the \(\mathrm{H}_2\) concentration constant during the reaction. Please provide the correct proposed mechanism to check its consistency with the given rate expression.

Key concepts

Chemical Kinetics and Reaction Mechanisms

Chemical kinetics is the study of the rates at which chemical processes occur and the factors that affect these rates. To comprehend the pace of chemical reactions, we delve into the concept of a reaction mechanism, which details the step-by-step sequence of elementary reactions by which overall chemical change occurs.

A proposed reaction mechanism explains not only the sequence of the reactions but also the molecules involved at each step. To evaluate the validity of a mechanism, chemists compare the experimentally determined rate expression to that predicted by the mechanism. Discrepancies can prompt a reevaluation or reconsideration of the detailed path by which reactants become products.

In the given case, the experimental rate expression is rate = k[NO]^2[H_2]. To ascertain whether the proposed mechanism explains this, we would have to examine the mechanism's elementary steps and compare the theoretical rate law derived from these steps with the experimental one.

Rate-Determining Step

The rate-determining step is a crucial concept in chemical kinetics. It refers to the slowest step in a reaction mechanism, which effectively limits the speed at which the overall reaction occurs, much like a bottleneck controls the flow of liquid from a bottle.

The rate-determining step has the highest energy barrier to overcome, resulting in the slowest transition from reactants to products. This step dominates the rate expression for the reaction. In readjusting our thought process for the proposed example, assuming the first step is the rate-determining step, our rate expression simplifies to rate = k₁[NO]^2. This must match the experimental rate law for the mechanism to be correct.

The identification of the rate-determining step not only aids in understanding the kinetics of a reaction but also provides insights into how the reaction might be optimized or controlled.

Rate Expression

The rate expression, often referred to as the rate law, mathematically defines how the rate of a reaction depends on the concentration of reactants. To construct an accurate rate expression, it's important to consider the stoichiometry and the mechanism of the reaction.

A rate expression is determined experimentally and can sometimes provide clues about the reaction mechanism. It takes the form rate = k[A]^m[B]^n, where A and B are reactants, k is the rate constant, and m and n are the reaction orders with respect to each reactant, often determined through experimentation.

In our exercise, if the experimental rate law does not align with that predicted by the mechanism, it suggests that our understanding of the mechanism could be incomplete or incorrect.

Elementary Steps

Diving deeper into a reaction mechanism, we encounter elementary steps – these are individual reactions within a sequence that make up the overall mechanism. Each elementary step is characterized by its own unique species and transition state.

Importantly, the molecularity of an elementary step—that is, the number of reactant molecules involved—directly determines its rate law. For example, a unimolecular reaction has a rate proportional to the concentration of a single reactant, while a bimolecular reaction depends on the product of the concentrations of two reactants.

If the assumption in our example is correct, and the first step involving the bimolecular reaction of two NO molecules is the slowest, it will influence the overall rate law. Yet, the experimental rate expression indicates the involvement of H₂, which isn't explained by our assumed rate-determining step, signifying a possible discrepancy between theory and experiment.