Problem 56 - Chemistry The Central Science In Si Units

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

(a) Sketch an energy profile with \(E_{a} = 154 \, \text{kJ/mol}\) and \(\Delta E = 136 \, \text{kJ/mol}\) labeled appropriately. (b) \(E_{a,\text{reverse}} = 290 \, \text{kJ/mol}\).

Step 1: Understanding the Problem

We are given an elementary reaction and its associated energies: activation energy \(E_{a} = 154 \, \text{kJ/mol}\) and overall energy change \(\Delta E = 136 \, \text{kJ/mol}\). We need to sketch the energy profile of the reaction, label these energies, and find the activation energy for the reverse reaction.

Step 2: Draw the Energy Profile

Start by drawing a graph with the vertical axis representing energy and the horizontal axis representing the reaction coordinate. The graph should have a peak, indicating the transition state, and the starting energy level represents the energy of the reactants (\(\mathrm{N}_{2}\mathrm{O}_{5}\)).

Step 3: Label Activation Energy (Forward Reaction)

The activation energy \(E_{a}\) is the energy required to go from the energy level of the reactants to the peak of the transition state, which is given as \(154 \, \text{kJ/mol}\). Label this on the graph as the difference between the reactants and the highest point of the curve.

Step 4: Label Overall Energy Change

The overall energy change \(\Delta E\) represents the difference in energy between the reactants and the products. Since \(\Delta E = 136 \, \text{kJ/mol}\), this is a reduction in energy during the reaction. Label \(\Delta E\) as the difference between the reactant energy level and the product energy level.

Step 5: Calculate Activation Energy for Reverse Reaction

To find the activation energy for the reverse reaction \(E_{a,\text{reverse}}\), we need to add the overall energy change \(\Delta E\) to the activation energy of the forward reaction \(E_{a}\). This is because the reverse reaction starts at a lower energy level (the products) and moves to a higher energy level (the reactants). Thus, \(E_{a,\text{reverse}} = E_{a} + \Delta E = 154 + 136 = 290 \, \text{kJ/mol}\).

Key Concepts

Elementary Reaction

An elementary reaction is a single step process where reactants transform into products in one smooth transition. Think of it as a direct shot between molecules. Unlike complex reactions that occur in multiple stages, an elementary reaction happens in one sequence. There's no intermediate compounds—just a straight path from the starting molecules to the final ones.

In our case, the elementary reaction is: \( \mathrm{N}_{2} \mathrm{O}_{5}(g) \longrightarrow \mathrm{NO}_{2}(g) + \mathrm{NO}_{3}(g) \)

This reaction involves the breakdown of nitrogen pentoxide into simpler gases, nitrogen dioxide and nitrogen trioxide. It's important to note that elementary reactions, by definition, have only one transition state and their rate is dependent solely on the concentration of reactants.

Energy Profile

An energy profile showcases the journey of a chemical reaction in terms of energy. Imagine it as a roadmap of how energy changes as the reaction progresses from reactants to products.
Here's a basic breakdown of what you'll see on an energy profile:

• The vertical axis represents energy levels.
• The horizontal axis is the reaction coordinate, showing progression from reactants to products.
• A peak represents the transition state, where the highest energy point or the energy barrier is met.

In our exercise, the energy profile starts with the energy level of \(\mathrm{N}_2\mathrm{O}_5\). The reaction has to climb a hill to reach the peak, which is the activation energy or \(E_{a} = 154 \, \text{kJ/mol}\). The difference from the start to this peak is the amount of energy required for the reaction to proceed.

After the peak, the energy drops reaching the product level of \(\mathrm{NO}_2\) and \(\mathrm{NO}_3\). The overall energy change, \(\Delta E = 136 \, \text{kJ/mol}\), is the difference from the beginning to the endpoint, indicating energy release. You'll label both \(E_{a}\) and \(\Delta E\) on this profile.

Reverse Reaction Activation Energy

For chemical reactions, there's always a two-way street: the forward and reverse reactions. While we've determined the energy needed for the forward path, we also need to uncover the energy for the reverse.

The reverse reaction starts from where the forward reaction ends, i.e., from the products back to the reactants. Therefore, to find the activation energy of the reverse direction, we must consider both the initial activation energy and the overall energy change released in the forward reaction.

Using the chemistry behind it, the reverse activation energy \(E_{a,\text{reverse}}\) is given by the sum:

• Forward Activation Energy \(E_{a}\)
• Overall Energy Change \(\Delta E\)

Therefore, \( E_{a,\text{reverse}} = 154 \, \text{kJ/mol} + 136 \, \text{kJ/mol} = 290 \, \text{kJ/mol} \). This shows the energy required to reverse the pathway, climbing back from the product state to the higher energy reactant state. Understanding this concept aids in grasping how chemical reactions maintain a balance of energy no matter the direction they occur.