Problem 43 - Chemistry and Chemical Reactivity

Question

The enthalpy changes for the following reactions can be measured: $$\begin{aligned}&\mathrm{CH}_{4}(\mathrm{g})+2 \mathrm{O}_{2}(\mathrm{g}) \longrightarrow \mathrm{CO}_{2}(\mathrm{g})+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})\\\&&\Delta H^{\circ}=-802.4 \mathrm{kJ}\end{aligned}$$ $$\begin{aligned}&\mathrm{CH}_{3} \mathrm{OH}(\mathrm{g})+\frac{3}{2} \mathrm{O}_{2}(\mathrm{g}) \longrightarrow \mathrm{CO}_{2}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\\\&&\Delta H^{\circ}=-676 \mathrm{kJ} \end{aligned}$$ (a) Use these values and Hess's law to determine the enthalpy change for the reaction $$\mathrm{CH}_{4}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g}) \longrightarrow \mathrm{CH}_{3} \mathrm{OH}(\mathrm{g})$$ (b) Draw an energy level diagram that shows the relationship between the energy quantities involved in this problem.

Step-by-Step Solution

Answer

The enthalpy change is \(-126.4 \mathrm{kJ}\); the reaction is exothermic.

Step 1: Write the Target Reaction

The target reaction for which we need to find the enthalpy change is given as follows: $$\mathrm{CH}_{4}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g}) \longrightarrow \mathrm{CH}_{3} \mathrm{OH}(\mathrm{g})$$. This is the reaction we want to derive using the given reactions.

Step 2: Use Hess's Law

Hess's Law states that the total enthalpy change for a reaction is the same, regardless of the number of steps in which the reaction is carried out. We can manipulate the given reactions and their enthalpy changes to obtain the enthalpy change for the target reaction.

Step 3: Reverse the Methanol Combustion Reaction

To obtain methanol on the right side, reverse the methanol combustion reaction: $$\mathrm{CO}_{2}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \rightarrow \mathrm{CH}_{3} \mathrm{OH}(\mathrm{g})+\frac{3}{2} \mathrm{O}_{2}(\mathrm{g})$$. The enthalpy change will be the opposite sign: \( \Delta H^{\circ} = +676 \mathrm{kJ} \).

Step 4: Combine with Methane Combustion Reaction

Next, we combine the reversed methanol reaction with the methane combustion reaction $$\mathrm{CH}_{4}(\mathrm{g})+2 \mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{CO}_{2}(\mathrm{g})+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})$$, \( \Delta H^{\circ} = -802.4 \mathrm{kJ} \). Add both reactions together. After adding: $$\mathrm{CH}_{4}(\mathrm{g})+\left(2 - \frac{3}{2} \right)\mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{CH}_{3} \mathrm{OH}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g})$$.

Step 5: Simplify and Calculate Enthalpy Change

Simplify the reaction to match the target reaction: $$\mathrm{CH}_{4}(\mathrm{g})+\frac{1}{2}\mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{CH}_{3} \mathrm{OH}(\mathrm{g})$$. Calculate the net enthalpy change using Hess's Law: \( \Delta H^{\circ} = (-802.4 \mathrm{kJ}) + (676 \mathrm{kJ}) = -126.4 \mathrm{kJ} \).

Step 6: Draw Energy Level Diagram

Create an energy level diagram: start with \(\mathrm{CH}_{4}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g})\) at a higher energy level, then show a downward arrow to \(\mathrm{CH}_{3} \mathrm{OH}(\mathrm{g})\) to indicate the release of energy (\(-126.4 \mathrm{kJ}\)), indicating it is an exothermic reaction.

Key Concepts

Enthalpy Change

Enthalpy change is a concept that explains how much heat energy is absorbed or released during a chemical reaction. When we talk about enthalpy changes, we express them as \(\Delta H\), where \(H\) represents enthalpy.
The value of \(\Delta H\) tells us whether the reaction is exothermic or endothermic.- **Exothermic Reactions:** - These release heat, resulting in a negative \(\Delta H\). - In our example reaction, \( \Delta H^{\circ} = -126.4 \text{kJ} \) indicates it's an exothermic process.- **Endothermic Reactions:** - These absorb heat, resulting in a positive \(\Delta H\).
The key takeaway is that the overall enthalpy change for a multi-step reaction depends only on the initial and final states of the system.This principle is crucial because it allows us to calculate \(\Delta H\) for reactions indirectly using Hess's Law when we can't measure them directly.

Energy Level Diagram

An energy level diagram helps visualize the energy changes that occur during a chemical reaction. It serves as a graphical representation of the enthalpy changes and helps to quickly picture if a reaction is exothermic or endothermic at a glance.
For the given exercise, an energy level diagram would show:- **Starting Point:** - Represents the reactants, in this case, \(\mathrm{CH}_4(\mathrm{g}) + \frac{1}{2} \mathrm{O}_2(\mathrm{g})\).
We place it at a higher energy level to indicate that the reactants have more enthalpy.- **Ending Point:** - Represents the product, methanol \(\mathrm{CH}_3\mathrm{OH}(\mathrm{g})\).
This is at a lower energy level compared to the reactants, showcasing that energy is released.- **Arrow Indicating Energy Change:** - Arrows drawn from the initial higher energy level to the final lower level represent the release of energy\[(-126.4 \text{kJ})\].In essence, energy level diagrams simplify the understanding of energy transfers in reactions, making them invaluable tools in chemistry education.

Combustion Reactions

Combustion reactions are a type of exothermic reaction where a substance (often a carbon-based fuel) reacts with oxygen, releasing heat and forming combustion products.
In this case, the initial given reactions both involve combustion:- **Methane Combustion:** - The reaction \( \mathrm{CH}_4(\mathrm{g}) + 2 \mathrm{O}_2(\mathrm{g}) \rightarrow \mathrm{CO}_2(\mathrm{g}) + 2 \mathrm{H}_2\mathrm{O}(\mathrm{g}) \) releases \(-802.4 \text{kJ}\).- **Methanol Combustion:** - Similarly, \( \mathrm{CH}_3\mathrm{OH}(\mathrm{g}) + \frac{3}{2} \mathrm{O}_2(\mathrm{g}) \rightarrow \mathrm{CO}_2(\mathrm{g}) + \mathrm{H}_2\mathrm{O}(\mathrm{g}) \) releases \(-676 \text{kJ}\).Key features of combustion reactions include:- Rapid energy release as heat.- Production of combustion gases, typically \(\mathrm{CO}_2\) and \(\mathrm{H}_2\mathrm{O}\).These reactions are foundational in various applications, from fueling engines to generating heat, illustrating their importance in everyday life and chemical energy studies.