Problem 68 - General Chemistry: Principles and Modern Applications

Question

The mechanism proposed for the reaction of \(\mathrm{H}_{2}(\mathrm{g})\) and \(\mathrm{I}_{2}(\mathrm{g})\) to form \(\mathrm{HI}(\mathrm{g})\) consists of a fast reversible first step involving \(\mathrm{I}_{2}(\mathrm{g})\) and \(\mathrm{I}(\mathrm{g}),\) followed by a slow step. Propose a two-step mechanism for the reaction \(\mathrm{H}_{2}(\mathrm{g})+\mathrm{I}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{HI}(\mathrm{g}),\) which is known to be first order in \(\mathrm{H}_{2}\) and first order in \(\mathrm{I}_{2}.\)

Step-by-Step Solution

Answer

A proposed mechanism that fulfills the given conditions is: Step 1 (Fast): \( I_{2}(g) \longleftrightarrow 2I(g) \), Step 2 (Slow): \( H_{2}(g) + I(g) \rightarrow 2HI(g) \). Adding these reactions gives the original reaction ( \( H_{2}(g) + I_{2}(g) \rightarrow 2HI(g) \) ) and the first order dependencies on \( H_{2} \) and \( I_{2} \) are observed as expected.

Step 1: Propose possible fast and reversible reaction

We start with the first step which is the fast reversible reaction involving \( I_{2}(g) \) and \( I(g) \), we write this as: \( I_{2}(g) \longleftrightarrow 2I(g) \)

Step 2: Propose slow reaction

Then we need to propose a reaction that is slow. Given that the overall reaction involves \( H_{2} \) and \( I_{2} \), and that \( I(g) \) is generated in our fast step, a logical second slow step could be the reaction between \( H_{2} \) and \( I(g) \): \( H_{2}(g) + I(g) \rightarrow 2HI(g) \)

Step 3: Validate the proposed mechanism

The overall reaction from adding the two steps yields: \( H_{2}(g) + I_{2}(g) \rightarrow 2HI(g) \) This matches the overall reaction given in the problem. Also the rate law of the rate-determining step is first order in \( H_{2} \) and first order in \( I_{2} \), which is as required.

Key Concepts

Rate-Determining Step

In a multi-step reaction mechanism, the rate-determining step plays a crucial role. It's the slowest step in a sequence of reactions and acts like a bottleneck, limiting the overall reaction rate.
The rate-determining step is determined by comparing the reaction rates of each step. Typically, it involves the highest energy barrier compared to other steps. This step directly influences the rate law of the overall reaction.
For our case with the reaction between \(H_2(g)\) and \(I_2(g)\), the slow formation of \(HI(g)\) from \(H_2(g)\) and \(I(g)\) is proposed as the rate-determining step.

• This means that even though the first step is fast and reversible, the overall reaction speed is dictated by this slow process.
• The rate law derived will align with the components involved in the slowest step.

Understanding this concept helps to control and optimize chemical reactions effectively.

Fast Reversible Reactions

Fast reversible reactions are those where the reactants quickly convert to products and back to reactants with little resistance.
These types of reactions help set up intermediate species for the subsequent steps.
In our proposed mechanism, the fast reversible step involves \(I_2(g)\) splitting into \(2I(g)\).

• This reaction happens swiftly, allowing \(I(g)\) to be available for the slow, rate-determining step.
• Despite their speed, these steps don't affect the overall rate law but are essential for the mechanism to proceed.

This mechanism works cohesively, allowing the buildup of intermediates that participate in slower steps.

First-Order Reactions

First-order reactions depend linearly on the concentration of one reactant.
This means the rate of the reaction is directly proportional to the concentration of a single reactant, and the rate law can be written as:
\[\text{Rate} = k[A]\]where \(k\) is the rate constant and \([A]\) is the concentration of the reactant.
In our reaction mechanism, the overall reaction is first order in both \(H_2\) and \(I_2\).

• This indicates that doubling the concentration of either reactant will double the rate of production of \(HI(g)\).
• The rate expression aligns with the concentration dependence observed in the rate-determining step.

Grasping how first-order reactions function is key for predicting how changes in concentration affect reaction rates.