Question

For the reversible reaction \(\mathrm{A}+\mathrm{B} \rightleftharpoons \mathrm{C}+\mathrm{D},\) the enthalpy change of the forward reaction is \(+21 \mathrm{kJ} / \mathrm{mol}\) The activation energy of the forward reaction is \(84 \mathrm{kJ} / \mathrm{mol}.\) (a) What is the activation energy of the reverse reaction? (b) In the manner of Figure 14-10, sketch the reaction profile of this reaction.

Short answer

The activation energy of the reverse reaction is \(105 \mathrm{kJ/mol}\). The reaction profile will show the reactants starting at one energy level, progressing to a peak representing the forward activation energy of \(84 \mathrm{kJ/mol}\), and then descending to a higher energy level for the products, representing the positive enthalpy change. The curve for the reverse reaction will do the opposite, starting from the products' energy level, going up to a peak for the reverse activation energy of \(105 \mathrm{kJ/mol}\), and then ending at the original reactants' energy level.

Step-by-step solution

Calculate the Activation Energy for Reverse Reaction

To calculate the activation energy for the reverse reaction, we use the relationship between the activation energies for the forward and reverse reactions and the enthalpy change. This relationship is defined by the equation: \( E_a^{reverse} = E_a^{forward} + \Delta H \), where \(\Delta H\) is the enthalpy change of the reaction. Using the given values for the forward activation energy, \(84 \mathrm{kJ/mol}\), and the enthalpy change, \(+21 \mathrm{kJ/mol}\), the activation energy for the reverse reaction can be calculated as: \( E_a^{reverse} = 84 \mathrm{kJ/mol} + 21 \mathrm{kJ/mol} = 105 \mathrm{kJ/mol}\).

Draw the Reaction Profile

The reaction profile is a diagram that shows the energy of the reactants and products and the activation energy for the forward and reverse reactions. On the y-axis is the energy, and on the x-axis is the reaction progress. The reactants \(A + B\) start at a certain energy level, and the products \(C + D\) end at a higher energy level due to the positive enthalpy change (\(+21 \mathrm{kJ/mol}\)). The peak of the forward reaction represents the activation energy of the forward reaction (\(84 \mathrm{kJ/mol}\)). The peak of the reverse reaction (from products to reactants) represents the activation energy of the reverse reaction (\(105 \mathrm{kJ/mol}\)). So, the forward reaction curve starts at the energy level of the reactants, goes up to the forward reaction activation energy, and then comes down to the energy level of the products. The reverse reaction curve starts at the energy level of the products, goes up to the reverse reaction activation energy, and then comes down to the energy level of the reactants.

Key concepts

Enthalpy Change

Enthalpy change (\( \Delta H \)) is a crucial concept in thermodynamics and plays a significant role in understanding chemical reactions. It represents the heat absorbed or released during a chemical reaction at constant pressure. In the context of the given reversible reaction \( \mathrm{A} + \mathrm{B} \rightleftharpoons \mathrm{C} + \mathrm{D} \), the enthalpy change of the forward reaction is given as \(+21 \mathrm{kJ/mol}\). This means the reaction absorbs energy, making it an endothermic process.

For any reaction:

• If \( \Delta H \) is positive, the reaction is endothermic and absorbs heat.
• If \( \Delta H \) is negative, the reaction is exothermic and releases heat.

Understanding the enthalpy change helps predict whether a reaction will occur spontaneously under given conditions. It also provides insight into the energy requirements of the reaction, impacting reaction rates, equilibrium, and practical applications, such as energy generation and conservation.

Reversible Reaction

A reversible reaction is one that can proceed in both directions: from reactants to products and vice versa. In the reaction \( \mathrm{A} + \mathrm{B} \rightleftharpoons \mathrm{C} + \mathrm{D} \), both the formation of \( \mathrm{C} \) and \( \mathrm{D} \) from \( \mathrm{A} \) and \( \mathrm{B} \), and the reverse process, can occur.

Reversible reactions reach a state of equilibrium where the rate of the forward reaction equals the rate of the reverse reaction, and the concentrations of reactants and products remain constant over time. This state is described by the equilibrium constant, \( K \):

• The value of \( K \) indicates the extent to which a reaction proceeds in either direction.
• A high \( K \) value suggests that products are favored at equilibrium, while a low \( K \) value indicates the favorability of reactants.

Reversible reactions are significant in various chemical processes and are key to understanding dynamic systems that involve competition between formation and decomposition of products.

Reaction Profile

A reaction profile is a graphical depiction of how the energy of a chemical system changes as a reaction proceeds. It's a valuable tool for visualizing the energy changes involved in both forward and reverse reactions, especially in complex processes like the reaction \( \mathrm{A} + \mathrm{B} \rightleftharpoons \mathrm{C} + \mathrm{D} \).

The x-axis of a reaction profile typically represents the progress of the reaction, while the y-axis indicates the energy level:

• For the forward reaction, the profile illustrates the climb in energy from the reactants \( \mathrm{A} \) and \( \mathrm{B} \) to the activation energy and then a drop to the energy level of products \( \mathrm{C} \) and \( \mathrm{D} \).
• For the reverse reaction, it shows the analogous process starting from the products.

The highest point on this curve represents the activation energy, and it varies for each direction:

• Forwards: \(84 \mathrm{kJ/mol}\)
• Reverse: \(105 \mathrm{kJ/mol}\)

By studying this profile, students gain insights into the energy barriers of reactions, helping to explain why certain reactions occur faster than others or under specific conditions.