Problem 64 - Chemistry An Atoms First Approach

Question

Draw a rough sketch of the energy profile for each of the following cases: a. \(\Delta E=+10 \mathrm{kJ} / \mathrm{mol}, E_{\mathrm{a}}=25 \mathrm{kJ} / \mathrm{mol}\) b. \(\Delta E=-10 \mathrm{kJ} / \mathrm{mol}, E_{\mathrm{a}}=50 \mathrm{kJ} / \mathrm{mol}\) c. \(\Delta E=-50 \mathrm{kJ} / \mathrm{mol}, E_{\mathrm{a}}=50 \mathrm{kJ} / \mathrm{mol}\)

Step-by-Step Solution

Answer

For each case, draw the energy profile using a graph with the x-axis representing the progress of the reaction and the y-axis representing the energy. Identify the initial and final energy levels based on the given \(\Delta E\), and mark the transition state energy level by adding the activation energy, \(E_\text{a}\), to the initial energy level. Then, sketch curves connecting these points: a. Initial energy level below final energy level, transition state at \(25 \mathrm{kJ/mol}\) above the initial level. b. Initial energy level above final energy level, transition state at \(50 \mathrm{kJ/mol}\) above the initial level. c. Initial energy level above final energy level, transition state at \(50 \mathrm{kJ/mol}\) above the initial level.

Step 1: Set up the axes

Set up a graph with the x-axis representing the progress of the reaction and the y-axis representing the energy.

Step 2: Identify initial and final energy levels

Since the change in energy, \(\Delta E\), is positive, we know that the final energy level will be higher than the initial energy level. Mark these levels on the y-axis.

Step 3: Identify the activation energy

The activation energy, \(E_\text{a}\), is given as \(25 \mathrm{kJ/mol}\). Calculate the energy level of the transition state by adding the activation energy to the initial energy level. Mark this level on the y-axis.

Step 4: Sketch the energy profile

Draw a curve from the initial energy level to the transition state energy level, and then another curve from the transition state energy level to the final energy level. Case b: \(\Delta E = -10 \mathrm{kJ/mol}, E_{\text{a}} = 50 \mathrm{kJ/mol}\)

Step 2: Identify initial and final energy levels

Since the change in energy, \(\Delta E\), is negative, we know that the final energy level will be lower than the initial energy level. Mark these levels on the y-axis.

Step 3: Identify the activation energy

The activation energy, \(E_\text{a}\), is given as \(50 \mathrm{kJ/mol}\). Calculate the energy level of the transition state by adding the activation energy to the initial energy level. Mark this level on the y-axis.

Step 4: Sketch the energy profile

Draw a curve from the initial energy level to the transition state energy level, and then another curve from the transition state energy level to the final energy level. Case c: \(\Delta E = -50 \mathrm{kJ/mol}, E_{\text{a}} = 50 \mathrm{kJ/mol}\)

Step 2: Identify initial and final energy levels

Since the change in energy, \(\Delta E\), is negative, we know that the final energy level will be lower than the initial energy level. Mark these levels on the y-axis.

Step 3: Identify the activation energy

The activation energy, \(E_\text{a}\), is given as \(50 \mathrm{kJ/mol}\). Calculate the energy level of the transition state by adding the activation energy to the initial energy level. Mark this level on the y-axis.

Step 4: Sketch the energy profile

Draw a curve from the initial energy level to the transition state energy level, and then another curve from the transition state energy level to the final energy level. After following these steps for each case, you will have a rough sketch of the energy profile for each situation.

Key Concepts

Activation Energy

Imagine you are starting a journey and need a boost of energy to reach the top of a hill before continuing downhill. Activation energy, often symbolized as \( E_a \), is similar in chemical reactions. It is the energy needed for reactants to reach the transition state, which can be thought of as the top of our hill. It's the minimum amount of energy that must be provided to convert reactants into products through a reaction.

This energy often comes from heat, collisions between molecules, or other sources, and it determines the rate at which a reaction will proceed. Lower activation energy means a faster reaction at a given temperature. Think of it like needing less energy to pedal up a less steep hill.

In our exercise, the activation energy for the first case is \( 25 \text{kJ/mol} \), which means the molecules need \( 25 \text{kJ} \) of energy per mole to reach the transition state. For cases b and c, the activation energy is \( 50 \text{kJ/mol} \), indicating a steeper hill to climb. These values are crucial as they help in sketching the energy profile.

Reaction Energy Change

The reaction energy change, represented as \( \Delta E \), is the difference in energy levels between reactants and products. It tells us whether a reaction is endothermic or exothermic.

• Endothermic reactions have a positive \( \Delta E \). They absorb energy (like climbing uphill and ending higher). In the first case (\( \Delta E = +10 \text{kJ/mol} \)), the products have higher energy than the reactants.
• Exothermic reactions have a negative \( \Delta E \). They release energy (akin to starting high and ending low, like descending a hill). Cases b and c (\( \Delta E = -10 \text{ and } -50 \text{kJ/mol} \) respectively) show the products at a lower energy than the reactants.

The knowledge of \( \Delta E \) helps us predict the energy level differences between the initial and final states, crucial for sketching energy profiles.

Transition State Energy

The transition state is like a peak every molecule must cross during a reaction, often represented as a high point on an energy profile diagram. It is a temporary and unstable state containing the highest energy level during the reaction pathway.

The energy of the transition state can be calculated by adding the activation energy \( E_a \) to the energy of the reactants. For instance, in the first scenario where \( E_a = 25 \text{kJ/mol} \) and assuming the initial energy is considered zero for simplicity, the transition state energy is also \( 25 \text{kJ/mol} \).

Understanding the transition state's energy is vital as it dictates how easy or difficult it is for reactants to convert into products. It also helps us determine the overall look of the energy profile, showcasing the "hurdle" each reaction must overcome. This hurdle affects the speed and success of chemical reactions, making it an essential aspect of reaction kinetics.