Question

For the reaction \(\mathrm{X}_{2}+\mathrm{Y}+\mathrm{Z} \longrightarrow \mathrm{XY}+\mathrm{XZ},\) it is found that doubling the concentration of \(\mathrm{X}_{2}\) doubles the reaction rate, tripling the concentration of \(Y\) triples the rate, and doubling the concentration of \(Z\) has no effect. (a) What is the rate law for this reaction? (b) Why is it that the change in the concentration of \(Z\) has no effect on the rate? (c) Suggest a mechanism for the reaction that is consistent with the rate law.

Short answer

(a) Rate law: \(\text{rate} = k[\mathrm{X}_2][\mathrm{Y}]\). (b) \(\mathrm{Z}\) does not affect the rate because it is not in the rate-determining step. (c) Two-step mechanism: slow \(\mathrm{X}_2 + \mathrm{Y} \rightarrow\) intermediate, fast \(\text{intermediate} + \mathrm{Z} \rightarrow \mathrm{XY} + \mathrm{XZ}\).

Step-by-step solution

Understanding the Rate Law

For the reaction \(\mathrm{X}_{2} + \mathrm{Y} + \mathrm{Z} \longrightarrow \mathrm{XY} + \mathrm{XZ},\) we need to write the rate law that fits the given changes in reaction rates. The rate law is generally expressed as \(\text{rate} = k[\mathrm{X}_2]^a[\mathrm{Y}]^b[\mathrm{Z}]^c\). We will determine the exponents \(a\), \(b\), and \(c\) based on the experimental observations.

Determining Exponent 'a' for X2

Doubling the concentration of \(\mathrm{X}_2\) doubles the reaction rate. This indicates that \(a = 1\) because the rate is directly proportional to the concentration of \(\mathrm{X}_2\).

Determining Exponent 'b' for Y

Tripling the concentration of \(\mathrm{Y}\) triples the reaction rate. This suggests that \(b = 1\), showing that the rate is directly proportional to the concentration of \(\mathrm{Y}\) as well.

Determining Exponent 'c' for Z

Doubling the concentration of \(\mathrm{Z}\) has no effect on the reaction rate, indicating that \(c = 0\). This means \(\mathrm{Z}\) is not involved in the rate-determining step.

Writing the Rate Law

Using the values of exponents derived, the rate law for the reaction is \( \text{rate} = k[\mathrm{X}_2]^1[\mathrm{Y}]^1[\mathrm{Z}]^0\), which simplifies to \( \text{rate} = k[\mathrm{X}_2][\mathrm{Y}]\).

Explanation for Z's Effect

The concentration of \(\mathrm{Z}\) has no effect on the rate because it does not appear in the rate law. This indicates that \(\mathrm{Z}\) does not participate in the rate-determining step, which is the slowest step in the reaction mechanism.

Proposing a Reaction Mechanism

A suitable mechanism that aligns with the rate law is a two-step mechanism: the first step (rate-determining) could involve \(\mathrm{X}_2\) and \(\mathrm{Y}\) forming an intermediate \(\mathrm{XY}\), and the second fast step involving \(\mathrm{XY}\) reacting with \(\mathrm{Z}\) to form \(\mathrm{XZ}\). The slow step defines the rate, consistent with the rate law \( k[\mathrm{X}_2][\mathrm{Y}] \).

Key concepts

Rate Law

The rate law of a chemical reaction is a mathematical expression that describes the relationship between the concentration of reactants and the rate of the reaction. For the given reaction \(\mathrm{X}_{2} + \mathrm{Y} + \mathrm{Z} \longrightarrow \mathrm{XY} + \mathrm{XZ},\)the rate law provides insights into how changes in concentrations affect the reaction rate. In this reaction, experiments show us the following:

• Doubling concentration of \(\mathrm{X}_2\) doubles the rate.
• Tripling concentration of \(\mathrm{Y}\) triples the rate.
• Doubling concentration of \(\mathrm{Z}\) has no effect on the rate.

This tells us that the reaction is first order with respect to \(\mathrm{X}_2\) and \(\mathrm{Y}\), and zero order with respect to \(\mathrm{Z}\). Mathematically, the rate law can be written as:\[\text{rate} = k[\mathrm{X}_2]^1[\mathrm{Y}]^1[\mathrm{Z}]^0\]This simplifies to \(\text{rate} = k[\mathrm{X}_2][\mathrm{Y}]\) indicating that the concentration of \(\mathrm{Z}\) does not directly affect the rate.

Reaction Mechanism

The reaction mechanism provides a step-by-step description of how reactants turn into products. For a reaction such as \(\mathrm{X}_{2} + \mathrm{Y} + \mathrm{Z} \longrightarrow \mathrm{XY} + \mathrm{XZ},\)it's possible that it does not occur in a single step. Instead, it may involve multiple stages with intermediates.
A plausible mechanism that aligns with the rate law might include:

• Slow Step (Rate-Determining): \(\mathrm{X}_2 + \mathrm{Y} \longrightarrow \mathrm{XY} \)
• Fast Step: \(\mathrm{XY} + \mathrm{Z} \longrightarrow \mathrm{XZ} \)

The first step is considered the slowest and thus controls the reaction rate, confirming why the concentration of \(\mathrm{Z}\) has no impact on the initial rate. The second fast step involves \(\mathrm{XY}\), an intermediate consuming \(\mathrm{Z}\) quickly to form the final products \(\mathrm{XY}\) and \(\mathrm{XZ}\). Because \(\mathrm{Z}\) is consumed in a fast reaction, it does not influence the rate law.

Concentration Effect

Concentration of reactants can significantly affect the rate of chemical reactions. In our studied reaction, analyses show:- Increasing concentration of \(\mathrm{X}_2\) and \(\mathrm{Y}\) increases the rate proportionally, meaning these reactants directly impact how quickly products form.- Doubling \(\mathrm{Z}\) does not change the rate, pointing out it is not involved in the rate-limiting step.
Let's dive deeper:

• For reactants where the concentration change leads to rate change, they are involved in the slow, controlling step of the mechanism.
• If a concentration change doesn't affect the rate, the reactant may engage in faster subsequent steps.

Therefore, the concentration effect in the given reaction suggests that the real bottleneck occurs between \(\mathrm{X}_2\) and \(\mathrm{Y}\), while \(\mathrm{Z}\) only participates later, after the rate-determining step completes.