Problem 56 - Chemistry: The Central Science

Question

For the elementary process \(\mathrm{N}_{2} \mathrm{O}_{5}(g) \longrightarrow \mathrm{NO}_{2}(g)+\mathrm{NO}_{3}(g)\) the activation energy \(\left(E_{a}\right)\) and overall \(\Delta E\) are \(154 \mathrm{~kJ} / \mathrm{mol}\) and \(136 \mathrm{~kJ} / \mathrm{mol}\), respectively. (a) Sketch the energy profile for this reaction, and label \(E_{a}\) and \(\Delta E\). (b) What is the activation energy for the reverse reaction?

Step-by-Step Solution

Answer

The activation energy for the reverse reaction is \(E_a^{\text{reverse}} = 290 \text{ kJ/mol}\).

Step 1: (Step 1: Sketch the Energy Profile)

To sketch the energy profile for the reaction, we will first draw the energy levels for the reactants and products. The reactants have higher energy than the products, with the difference being \(\Delta E\). Then, we will draw the "hill" representing the activation energy barrier.

Step 2: (Step 2: Label Ea and ΔE)

Now that we have the energy profile, let's label the activation energy (\(E_a\)) and change in energy (\(\Delta E\)). The activation energy is the difference between the energy of the reactants and the highest point on the energy profile (the "hill"), while \(\Delta E\) is the difference between the energy of the reactants and the energy of the products.

Step 3: (Step 3: Use The Relationship Between Activation Energies and Change in Energy)

To find the activation energy for the reverse reaction, we can use the following relationship: \( E_a^{\text{reverse}} = E_a^{\text{forward}} + \Delta E \) We are given the activation energy and change in energy for the forward reaction: \( E_a^{\text{forward}} = 154 \text{ kJ/mol} \) and \( \Delta E = 136 \text{ kJ/mol} \)

Step 4: (Step 4: Calculate the Activation Energy for the Reverse Reaction)

Now, we can plug in the values for the activation energy and change in energy for the forward reaction into the relationship: \( E_a^{\text{reverse}} = 154 \text{ kJ/mol} + 136 \text{ kJ/mol} \) \( E_a^{\text{reverse}} = 290 \text{ kJ/mol} \) So, the activation energy for the reverse reaction is 290 kJ/mol.

Key Concepts

Elementary Process

An elementary process is a single step in a chemical reaction that involves a direct change from reactants to products.
In an elementary process, molecules either collide or decompose to form products. This process is significant because it dictates how molecules interact and ultimately determines the speed and outcome of the reaction. For instance, the simplification of the decomposition of dinitrogen pentoxide (N₂O₅) into nitrogen dioxide (NO₂) and nitrate radical (NO₃) represents such a process.

Understanding this concept helps us to analyze complex reactions by breaking them down into more manageable steps, which facilitates the calculation of reaction rates and helps us to construct more detailed mechanisms for how reactions occur on a molecular level.

Energy Profile

An energy profile is a graphical representation showing how the energy of a system changes as a chemical reaction proceeds from reactants to products.
On an energy diagram, the y-axis represents energy, while the x-axis shows the progress of the reaction. The initial energy level of the reactants and the final energy level of the products are connected by a curve, typically resembling a hill, which illustrates the energy barrier that must be overcome for the reaction to proceed, known as the activation energy (E_a). The difference in energy between the reactants and the products is depicted as the enthalpy change (∆E) of the reaction. Sketching an energy profile allows us to visualize the energy landscape of a reaction more easily.

Enthalpy Change

Enthalpy change, designated as ∆H or ∆E in some contexts, refers to the heat exchange in a chemical reaction under constant pressure.
It indicates whether a reaction is exothermic (releases energy) or endothermic (absorbs energy). If ∆E is negative, the reaction is exothermic, whereas a positive ∆E suggests an endothermic reaction. For the given elementary process, the enthalpy change is 136 kJ/mol, which means there is a release of energy as N₂O₅ decomposes to NO₂ and NO₃, indicative of an exothermic process.

Knowing the enthalpy change of a reaction helps predict the spontaneity and feasibility of a reaction under different conditions and is an essential parameter for making energy calculations in thermodynamics.

Reverse Reaction Activation Energy

The activation energy for the reverse reaction, often denoted as E_a^reverse, is not simply the opposite of the activation energy for the forward reaction (E_a^forward).
Instead, it can be calculated by adding the enthalpy change (∆E) of the forward reaction to its activation energy. For the decomposition of N₂O₅, the high activation energy for the reverse reaction (290 kJ/mol) reflects the energy required to recombine NO₂ and NO₃ back into N₂O₅. This concept is crucial in understanding chemical kinetics as it dictates the speed at which the reverse reaction will occur and also influences equilibrium dynamics in reversible reactions.