Problem 74 - Chemistry The Central Science In Si Units

Question

You have studied the gas-phase oxidation of \(\mathrm{HBr}\) by \(\mathrm{O}_{2}\) : $$ 4 \mathrm{HBr}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(g)+2 \mathrm{Br}_{2}(g) $$ You find the reaction to be first order with respect to \(\mathrm{HBr}\) and first order with respect to \(\mathrm{O}_{2}\). You propose the follow. ing mechanism: $$ \begin{aligned} \operatorname{HBr}(g)+\mathrm{O}_{2}(g) & \longrightarrow \mathrm{HOOBr}(g) \\\ \mathrm{HOOBr}(g)+\mathrm{HBr}(g) & \longrightarrow 2 \mathrm{HOBr}(g) \\ \mathrm{HOBr}(g)+\mathrm{HBr}(g) & \longrightarrow \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{Br}_{2}(g) \end{aligned} $$ (a) Confirm that the elementary reactions add to give the overall reaction. (b) Based on the experimentally determined rate law, which step is rate determining? (c) What are the intermediates in this mechanism? (d) If you are unable to detect HOBr or HOOBr among the products, does this disprove your mechanism?

Step-by-Step Solution

Answer

(a) The elementary reactions add up to the overall reaction. (b) The rate-determining step is the first step. (c) Intermediates are \( \mathrm{HOOBr} \) and \( \mathrm{HOBr} \). (d) The undetected intermediates do not disprove the mechanism.

Step 1: Identify the Overall Reaction from Elementary Steps

To ensure the elementary reactions add up to the overall reaction, start by writing down each individual elementary reaction:1. \( \operatorname{HBr}(g)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{HOOBr}(g) \)2. \( \mathrm{HOOBr}(g)+\mathrm{HBr}(g) \longrightarrow 2 \mathrm{HOBr}(g) \)3. \( \mathrm{HOBr}(g)+\mathrm{HBr}(g) \longrightarrow \mathrm{H}_{2}\mathrm{O}(g)+\mathrm{Br}_{2}(g) \)Add all the reactants and products. Upon canceling the intermediates \( \mathrm{HOOBr}(g) \) and \( \mathrm{HOBr}(g) \), the overall reaction becomes:\( 4\mathrm{HBr}(g) + \mathrm{O}_{2}(g) \longrightarrow 2\mathrm{H}_{2} \mathrm{O}(g) + 2\mathrm{Br}_{2}(g) \)This matches the given overall reaction.

Step 2: Determine the Rate-Determining Step

The experimentally determined rate law is first order in both \( \mathrm{HBr} \) and \( \mathrm{O}_{2} \), indicating that the rate-determining step should involve one molecule of each. Look at the proposed mechanism:The first elementary step is \( \operatorname{HBr}(g)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{HOOBr}(g) \), involving both \( \mathrm{HBr} \) and \( \mathrm{O}_{2} \) once, matching the rate law orders. Therefore, the first step is the rate-determining step.

Step 3: Identify Reaction Intermediates

Intermediates are species that are produced in one step and consumed in another within the mechanism, thus not appearing in the overall reaction. From the steps:- \( \mathrm{HOOBr}(g) \) is formed in the first step and consumed in the second.- \( \mathrm{HOBr}(g) \) is formed in the second step and consumed in the third.Thus, \( \mathrm{HOOBr}(g) \) and \( \mathrm{HOBr}(g) \) are intermediates.

Step 4: Assess the Impact of Undetected Intermediates

Intermediates do not appear in the overall reaction and are often present at very low concentrations, potentially escaping detection. Their absence among the final products does not disprove the mechanism. The mechanism focuses on path and transitions, not the final detectable products. Therefore, the inability to detect \( \mathrm{HOOBr} \) or \( \mathrm{HOBr} \) does not invalidate the proposed mechanism.

Key Concepts

Rate Law

Understanding the rate law is crucial for analyzing chemical reactions. It provides a mathematical relationship between the concentration of reactants and the rate of the reaction. In this exercise, the rate law for the oxidation of \(\text{HBr} \) by \(\text{O}_2 \) is given as first-order with respect to both \(\text{HBr} \) and \(\text{O}_2 \).
This means the rate of reaction doubles when the concentration of either \(\text{HBr} \) or \(\text{O}_2 \) is doubled, keeping the other constant. The rate law is expressed mathematically as:
\[\text{Rate} = k[\text{HBr}][\text{O}_2]\]
where \(k\) is the rate constant, a constant for a specific reaction at a given temperature. Recognizing the order of the reaction with respect to each reactant is pivotal for predicting how changes in conditions affect the reaction rate.

Elementary Reactions

Elementary reactions represent the simplest steps within a complex mechanism. Each elementary step involves a single reaction process, where reactants form products directly without any intermediates. In the proposed mechanism for the oxidation of \(\text{HBr} \), we observe three elementary reactions:

• \(\text{HBr}(g) + \text{O}_2(g) \to \text{HOOBr}(g) \)
• \(\text{HOOBr}(g) + \text{HBr}(g) \to 2 \text{HOBr}(g) \)
• \(\text{HOBr}(g) + \text{HBr}(g) \to \text{H}_2\text{O}(g) + \text{Br}_2(g) \)

The combination of these individual steps results in the overall balanced chemical reaction. Elementary reactions provide insight into the molecular-level events occurring within a mechanism. These details help chemists understand the intricacies of how reactants are transformed into products through a series of succinct yet powerful transformation events.

Reaction Intermediates

Reaction intermediates are species formed and then consumed during the course of a chemical reaction. They do not appear in the overall reaction equation. In our mechanism, we identify two intermediates:

• \(\text{HOOBr}(g) \) which is produced in the first step and consumed in the second
• \(\text{HOBr}(g) \) which is produced in the second step and consumed in the third

These intermediates are crucial for the progression of the reaction but are often transient and exist in low concentrations. This can make them challenging to detect experimentally. Their roles, however, are vital for ensuring the proposed mechanism aligns with the observed kinetics and stoichiometry. Detecting intermediates can provide strong evidence for a reaction mechanism, but failure to detect them does not necessarily invalidate the proposed steps.

Rate-Determining Step

The rate-determining step, often referred to as the "bottleneck" of a reaction mechanism, is the slowest step that limits the overall reaction rate. It is crucial because it dictates how quickly a reaction progresses.
In the oxidation of \(\text{HBr} \) by \(\text{O}_2 \), the rate-determining step is the first elementary reaction where one \(\text{HBr} \) molecule reacts with one \(\text{O}_2 \) molecule to form \(\text{HOOBr} \). This step is consistent with the experimentally determined rate law, being first order in both \(\text{HBr} \) and \(\text{O}_2 \).
Understanding which step in a mechanism is rate-determining helps in manipulating and improving reaction conditions to enhance the rate of the slowest step, thereby speeding up the entire reaction process efficiently. Recognizing the rate-determining step is key to understanding the overall kinetics of the reaction.