Question

In a hydrocarbon solution, the gold compound \(\left(\mathrm{CH}_{3}\right)_{3} \mathrm{AuPH}_{3}\) decomposes into ethane \(\left(\mathrm{C}_{2} \mathrm{H}_{6}\right)\) and a different gold compound, \(\left(\mathrm{CH}_{3}\right) \mathrm{AuPH}_{3} .\) The following mechanism has been proposed for the decomposition of \(\left(\mathrm{CH}_{3}\right)_{3} \mathrm{AuPH}_{3}:\) Step 1: \(\left(\mathrm{CH}_{3}\right)_{3} \mathrm{AuPH}_{3} \underset{k-1}{\stackrel{k_{1}}{\rightleftharpoons}}\left(\mathrm{CH}_{3}\right)_{3} \mathrm{Au}+\mathrm{PH}_{3} \quad\) (fast) Step 2: \(\left(\mathrm{CH}_{3}\right)_{3} \mathrm{Au} \stackrel{\mathrm{k}_{2}}{\longrightarrow} \mathrm{C}_{2} \mathrm{H}_{6}+\left(\mathrm{CH}_{3}\right) \mathrm{Au} \quad\) (slow) Step 3: \(\left(\mathrm{CH}_{3}\right) \mathrm{Au}+\mathrm{PH}_{3} \stackrel{k_{3}}{\longrightarrow}\left(\mathrm{CH}_{3}\right) \mathrm{AuPH}_{3} \quad\) (fast) (a) What is the overall reaction? (b) What are the intermediates in the mechanism? (c) What is the molecularity of each of the elementary steps? (d) What is the rate-determining step? (e) What is the rate law predicted by this mechanism? (f) What would be the effect on the reaction rate of adding \(\mathrm{PH}_{3}\) to the solution of \(\left(\mathrm{CH}_{3}\right)_{3} \mathrm{AuPH}_{3} ?\)

Short answer

The overall reaction is \((CH_3)_3AuPH_3 \longrightarrow C_2H_6+(CH_3)AuPH_3\). The intermediates are \((CH_3)_3Au\) and \((CH_3)Au\). The molecularity of steps 1, 2, and 3 are bi-molecular, uni-molecular, and bi-molecular, respectively. The rate-determining step is step 2, and the predicted rate law is \(Rate= k_2\left(\frac{K_1[(CH_3)_3AuPH_3]}{[PH_3]}\right)\). Adding PH3 to the solution will decrease the reaction rate.

Step-by-step solution

Overall Reaction

To find the overall reaction, we need to add up all the elementary steps in the mechanism and cancel out any species which appear on both the reactants and products side. Step 1: \((CH_3)_3AuPH_3\rightleftharpoons (CH_3)_3Au + PH_3\) (fast) Step 2: \((CH_3)_3Au\longrightarrow C_2H_6 + (CH_3)Au\) (slow) Step 3: \((CH_3)Au + PH_3\longrightarrow (CH_3)AuPH_3\) (fast) Summing up the steps, we have: \((CH_3)_3AuPH_3 \longrightarrow C_2H_6+(CH_3)AuPH_3\) Upon canceling out common species, we get the overall reaction: \((CH_3)_3AuPH_3 \longrightarrow C_2H_6+(CH_3)AuPH_3\)

Intermediates

Intermediates are the species that are formed during reaction steps and then consumed in subsequent steps. We can identify the intermediates by examining each step in the mechanism. In this case, the intermediates are \((CH_3)_3Au\) and \((CH_3)Au\).

Molecularity of Elementary Steps

Molecularity is the number of molecules or ions that participate in a reaction step. In this case, we have three elementary steps: 1. Step 1: 1 reactant molecule and 2 product molecules (bi-molecular) 2. Step 2: 1 reactant molecule and 2 product molecules (uni-molecular) 3. Step 3: 2 reactant molecules and 1 product molecule (bi-molecular)

Rate-Determining Step

The rate-determining step is the slowest step in the mechanism, which determines the overall reaction rate. In this case, step 2 is the slowest step: \((CH_3)_3Au\longrightarrow C_2H_6 + (CH_3)Au\). Therefore, the rate-determining step is step 2.

Rate Law Prediction

Since the rate-determining step is step 2, the rate law can be written using the concentrations of species involved in step 2: Rate Law: \(Rate = k_2[(CH_3)_3Au]\) However, the concentration of an intermediate is not directly measurable. So, we need to express the concentration of intermediate \((CH_3)_3Au\) in terms of other species using the equilibrium constant \(K_1\) from step 1: \(K_1 = \frac{[(CH_3)_3Au][PH_3]}{([(CH_3)_3AuPH_3])}\) Solving for the intermediate concentration: \([(CH_3)_3Au] = \frac{K_1[(CH_3)_3AuPH_3]}{[PH_3]}\) Substitute into the rate law: \(Rate= k_2\left(\frac{K_1[(CH_3)_3AuPH_3]}{[PH_3]}\right)\)

Effect of Adding PH3

To understand the effect of adding PH3 on the reaction rate, analyze the rate law expression: \(Rate= k_2\left(\frac{K_1[(CH_3)_3AuPH_3]}{[PH_3]}\right)\) As we can see from the rate law, the reaction rate is inversely proportional to the concentration of PH3. If we add more PH3 to the solution, the reaction rate will decrease.

Key concepts

Rate-Determining Step

In a series of reactions, the rate-determining step stands out as the bottleneck, dictating the pace of the overall process. Since reactions can occur in multiple steps, each with its own speed, the rate-determining step is typically the slowest one. Imagine it as the point in a line of traffic that holds everything up.
In the given reaction mechanism, step 2 is identified as the slowest step:

• \((\text{CH}_3)_3\text{Au} \longrightarrow \text{C}_2\text{H}_6 + (\text{CH}_3)\text{Au}\)

Because this step is slow, it controls the overall speed at which the products are formed. Even if all other steps occur rapidly, the process can only proceed as fast as this slowest step allows.
Identifying the rate-determining step is crucial because it informs us about potential targets for reaction optimization or intervention.

Reaction Intermediates

Reaction intermediates are little-known contributors in chemical reactions. These species form during one step of a reaction and are consumed in subsequent steps. They are not present in the overall balanced equation, as they do not constitute final products nor initial reactants.
In this mechanism, we encounter the intermediates:

• \((\text{CH}_3)_3\text{Au}\)
• \((\text{CH}_3)\text{Au}\)

These intermediates are transient and typically exist in low concentrations—even if their role is crucial to the pathway. As the process moves from step to step, these intermediates allow the transformation of reactants into products by bridging the steps.
Understanding intermediates help chemists develop insight into how reactions proceed and can lead to the discovery of more efficient pathways for chemical changes.

Molecularity

Molecularity refers to the number of molecules or ions participating in a particular elementary step of a reaction. It's an intrinsic property that describes how reactants collide or come together to progress the reaction. Each elementary step in a mechanism can have a varying molecularity.
In the given mechanism, each step has a different molecularity:

• **Step 1:** Bi-molecular, involves a single molecule dissociating into two products, indicating a two-species interaction.
• **Step 2:** Uni-molecular, with one reactant converting into products, suggesting a single-molecule transformation.
• **Step 3:** Bi-molecular, where two reactants combine to form products, denoting an interaction involving two collision partners.

Molecularity helps provide a dynamic picture of how reactions progress on a molecular level, explaining the kinetics of elementary steps. Unlike reaction order, which is determined experimentally, molecularity is a theoretical construct specific to each elementary step. Recognizing the type of molecularity allows us to understand how variations in concentration could affect each processing phase.