Problem 73 - Chemistry The Molecular Science

Question

The mechanism for the reaction of \(\mathrm{CH}_{3} \mathrm{OH}\) and \(\mathrm{HBr}\) is believed to involve two steps. The overall reaction is exothermic. $$ \text { Step 1: } \mathrm{CH}_{3} \mathrm{OH}+\mathrm{H}^{+} \rightleftharpoons \mathrm{CH}_{3} \mathrm{OH}_{2}^{+} \text {fast, endothermic } $$ $$ \text { Step } 2: \mathrm{CH}_{3} \mathrm{OH}_{2}^{+}+\mathrm{Br}^{-} \longrightarrow \mathrm{CH}_{3} \mathrm{Br}+\mathrm{H}_{2} \mathrm{O} $$ slow (a) Write an equation for the overall reaction. (b) Draw a reaction energy diagram for this reaction. (c) Show that the rate law for this reaction is $$ \text { Rate }=k\left[\mathrm{CH}_{3} \mathrm{OH}\right]\left[\mathrm{H}^{+}\right]\left[\mathrm{Br}^{-}\right] $$

Step-by-Step Solution

Answer

Overall: \(\mathrm{CH}_3\mathrm{OH} + \mathrm{HBr} \rightarrow \mathrm{CH}_3\mathrm{Br} + \mathrm{H}_2\mathrm{O}\). Diagram shows exothermic profile. Rate is \(k[\mathrm{CH}_3\mathrm{OH}][\mathrm{H}^+][\mathrm{Br}^-]\).

Step 1: Identify Overall Reaction

The overall reaction involves the reactants and the final products of the reaction mechanism. From the provided steps: Step 1 and Step 2, we know that \(\mathrm{CH}_3\mathrm{OH}\) reacts with \(\mathrm{H}^{+}\) and \(\mathrm{Br}^{-}\) to form \(\mathrm{CH}_3\mathrm{Br}\) and \(\mathrm{H}_2\mathrm{O}\). Thus, the balanced equation for the overall reaction is: \[\mathrm{CH}_3\mathrm{OH} + \mathrm{HBr} \rightarrow \mathrm{CH}_3\mathrm{Br} + \mathrm{H}_2\mathrm{O}\].

Step 2: Reaction Energy Diagram

Draw an energy diagram where the y-axis represents energy and the x-axis represents the reaction progress. Start with reactants \(\mathrm{CH}_3\mathrm{OH}\) and \(\mathrm{H}^{+}\) at a certain energy level. Show an energy increase for the endothermic step (Step 1). Indicate a transition to an intermediate state (\(\mathrm{CH}_3\mathrm{OH}_2^{+}\)). Next, show a second curve with a smaller energy barrier due to the slow, exothermic reaction (Step 2), resulting in the products \(\mathrm{CH}_3\mathrm{Br}\) and \(\mathrm{H}_2\mathrm{O}\). The final energy level of the products should be lower than the initial reactants, highlighting the overall exothermic nature.

Step 3: Derive Rate Law from Mechanism

From the given mechanism, Step 1 is fast and reversible while Step 2 is slow and therefore rate-determining. Assuming a fast equilibrium in Step 1 for \(\mathrm{CH}_3\mathrm{OH}_2^{+}\), the concentration of \(\mathrm{CH}_3\mathrm{OH}_2^{+}\) can be expressed in terms of \(\mathrm{CH}_3\mathrm{OH}\) and \(\mathrm{H}^{+}\) using equilibrium constants. The rate of the overall reaction is determined by the rate of Step 2: \[\text{Rate} = k_2[\mathrm{CH}_3\mathrm{OH}_2^{+}][\mathrm{Br}^{-}]\]. Applying the steady-state assumption or assuming equilibrium in Step 1 allows us to express \([\mathrm{CH}_3\mathrm{OH}_2^{+}]\) in terms of \([\mathrm{CH}_3\mathrm{OH}]\) and \([\mathrm{H}^{+}]\), leading to: \[\text{Rate} = k[\mathrm{CH}_3\mathrm{OH}][\mathrm{H}^{+}][\mathrm{Br}^{-}]\].

Key Concepts

Exothermic Reaction

In chemistry, an exothermic reaction is a process that releases energy, usually in the form of heat, to its surroundings. This is why such reactions often result in an increase in temperature. In the case of the reaction between \( \mathrm{CH}_3\mathrm{OH} \) and \( \mathrm{HBr} \), the overall process is exothermic.

This means that while the system progresses from reactants like \( \mathrm{CH}_3\mathrm{OH} \) and \( \mathrm{HBr} \) to the products \( \mathrm{CH}_3\mathrm{Br} \) and \( \mathrm{H_2O} \), it releases energy to the surrounding environment.

• The energy released is often visualized in an energy diagram where the final energy state of the products is lower than that of the reactants.
• This lower energy state is what defines the exothermic nature of the overall reaction.

Understanding the nature of exothermic reactions helps to predict the feasibility and conditions under which the reaction might occur spontaneously.

Reaction Kinetics

Reaction kinetics explores how and why reactions occur at certain rates. It involves analyzing the steps or mechanism the reaction undergoes to form the final products.

For the reaction of \( \mathrm{CH}_3\mathrm{OH} \) and \( \mathrm{HBr} \), it happens in two steps. The first step is a fast, endothermic process forming an intermediate \( \mathrm{CH}_3\mathrm{OH}_2^{+} \). The second step is slower and therefore rate-determining, meaning it governs the reaction's overall rate.

• The speed of each step contributes to the overall kinetics of the reaction.
• Understanding which step is slower helps in identifying the rate-limiting factor and provides insight for increasing the reaction rate through catalysts or temperature changes if necessary.

Studying these kinetics is essential to manipulate and control reactions in chemical processes and industrial applications.

Rate Laws

Rate laws describe how the concentration of reactants affects the speed of a reaction. They are usually expressed as equations that relate the rate of a reaction to the concentration of its reactants.

For the reaction between \( \mathrm{CH}_3\mathrm{OH} \) and \( \mathrm{HBr} \), the rate law is given by: \[ \text{Rate} = k[\mathrm{CH}_3\mathrm{OH}][\mathrm{H}^{+}][\mathrm{Br}^{-}] \] where \( k \) is the rate constant, and the exponents indicate the order of the reaction with respect to each reactant.

• This rate law shows a direct dependence of the reaction rate on the concentration of all three reactants involved.
• It can be deduced from the mechanism, considering that the second, slow step determines the rate.

Knowing this relationship allows chemists to predict how varying concentrations will affect the reaction speed, which is useful for controlling reaction conditions.

Energy Diagrams

Energy diagrams are graphical representations that show the change in energy as a chemical reaction progresses. These diagrams illustrate the energy levels of reactants, intermediates, and products.

In our example of \( \mathrm{CH}_3\mathrm{OH} \) and \( \mathrm{HBr} \), the energy diagram would begin with the initial reactants at a higher energy level. As the reaction proceeds through the endothermic step, there is a rise in energy—an energy barrier that the reactants must overcome.

• This is followed by a drop in energy representing the exothermic nature of the final step leading to the formation of products.
• The final energy level is lower than the initial, marking the overall reaction as exothermic.

These diagrams are helpful to visualize the energy changes occurring during a reaction, allowing for insights into stability and reaction spontaneity.

Molecular Interactions

Molecular interactions are forces that act between molecules, affecting their reactions. These interactions are crucial in understanding the mechanism of how reactants transform into products.

During the reaction of \( \mathrm{CH}_3\mathrm{OH} \) and \( \mathrm{HBr} \), initial molecular interactions enable the formation of \( \mathrm{CH}_3\mathrm{OH}_2^{+} \). This intermediate is stabilized through interactions with \( \mathrm{H}^{+} \). In the subsequent step, interactions between \( \mathrm{CH}_3\mathrm{OH}_2^{+} \) and \( \mathrm{Br}^{-} \) lead to the final product formation.

• These interactions determine the course and speed of the reaction.
• They also help in the stabilization of transition states and intermediates.

Comprehending these molecular interactions provides a deeper insight into the steps and energies involved in chemical reactions, aiding in designing better reaction conditions and new synthetic pathways.