Question

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

Short answer

The enthalpy change is \(-126.4 \text{ kJ/mol}\). The energy diagram shows a drop of 126.4 kJ from reactants to products.

Step-by-step solution

Understand the Known Reactions

We are given two known reactions with their enthalpy changes:1. \( \text{CH}_4 + 2 \text{O}_2 \rightarrow \text{CO}_2 + 2 \text{H}_2\text{O} \) with \( \Delta H = -802.4 \text{ kJ/mol} \).2. \( \text{CH}_3\text{OH} + \frac{3}{2} \text{O}_2 \rightarrow \text{CO}_2 + 2 \text{H}_2\text{O} \) with \( \Delta H = -676 \text{ kJ/mol} \).

Identify Target Reaction and Apply Hess's Law

The target is to find \( \Delta H \) for \( \text{CH}_4 + \frac{1}{2} \text{O}_2 \rightarrow \text{CH}_3\text{OH} \). According to Hess's Law, we can perform algebraic operations on the known reactions to achieve the target reaction.

Reverse the Second Reaction to Match Products with Target

Reverse the second reaction to form \( \text{CH}_3\text{OH} \) on the product side:1. \( \text{CO}_2 + 2 \text{H}_2\text{O} \rightarrow \text{CH}_3\text{OH} + \frac{3}{2} \text{O}_2 \) This reversal changes the sign of \( \Delta H \) to \(+676 \text{ kJ/mol} \).

Add Reactions to Obtain Target Reaction

Add the first reaction and the reversed second reaction:1. First: \( \text{CH}_4 + 2 \text{O}_2 \rightarrow \text{CO}_2 + 2 \text{H}_2\text{O} \)2. Reversed second: \( \text{CO}_2 + 2 \text{H}_2\text{O} \rightarrow \text{CH}_3\text{OH} + \frac{3}{2} \text{O}_2 \)After canceling common terms, this results in: \( \text{CH}_4 + \frac{1}{2}\text{O}_2 \rightarrow \text{CH}_3\text{OH} \) with \( \Delta H = -802.4 + 676 = -126.4 \text{ kJ/mol} \).

Energy Level Diagram Explanation

Draw an energy level diagram with the energy levels of \( \text{CH}_4 + 2 \text{O}_2 \) and \( \text{CH}_3\text{OH} + \frac{3}{2} \text{O}_2 \) based on calculated and given \( \Delta H \) values. Indicate the transition from reactants to products showing the decrease in energy by 126.4 kJ when converting \( \text{CH}_4 + \frac{1}{2}\text{O}_2 \rightarrow \text{CH}_3\text{OH} \).

Key concepts

Enthalpy Change

The enthalpy change (\(\Delta H\)) of a reaction is the heat absorbed or released under constant pressure. It is a central concept in understanding energy transformations in chemical reactions.
Enthalpy describes the energy content of substances, and during a reaction, these content changes are observed as an enthalpy change. This change can be either:

• Exothermic: where heat is released, and \(\Delta H\) is negative.
• Endothermic: where heat is absorbed, and \(\Delta H\) is positive.

For instance, in the Hess's Law exercise provided, the enthalpy change for two reactions is given. The change must be calculated for a specific target reaction. Using Hess's Law, we can determine the overall enthalpy change by appropriately manipulating and combining known reactions.
To determine the enthalpy change for the reaction \(\text{CH}_4 + \frac{1}{2} \text{O}_2 \rightarrow \text{CH}_3\text{OH}\), we used the known reactions and applied algebraic manipulation processes to find that the enthalpy change is \(-126.4 \text{ kJ/mol}\). This indicates that when methane reacts with half a mole of oxygen to form methanol, \(126.4 \text{ kJ/mol}\) of energy are released.

Energy Level Diagram

An energy level diagram is a graphical representation of the energy changes during a chemical reaction.
It visually depicts the relative energy levels of reactants and products on a vertical scale, showing the energy pathway during the transformation.
This kind of diagram helps in understanding:

• The energy level of reactants and products.
• Whether a reaction is exothermic or endothermic.
• The magnitude of energy change during the reaction, represented by the difference in height between reactants and products.

In the exercise, after calculating the enthalpy change, the energy level diagram was drawn to show this change. Reactants like \(\text{CH}_4 + 2 \text{O}_2\) start at a certain energy level, and the products \(\text{CH}_3\text{OH} + \frac{3}{2} \text{O}_2\) are at a lower level, representing the release of \(126.4 \text{ kJ/mol}\).
The diagram clearly illustrates the transition from high-energy reactants to lower-energy products, making it easier to comprehend the energy drop that occurs when methane and oxygen react to form methanol.

Chemical Reactions

In chemistry, a reaction occurs when substances (reactants) undergo changes to form new substances (products) with different properties.
Each reaction involves breaking bonds in reactants and forming bonds in products, accompanied by an energy change. This is why calculating enthalpy change and drawing energy level diagrams are important, they provide insight into the energy required or released during these transformations.
In the example provided, the target reaction involves methanol formation from methane and oxygen. This transformation is achieved by adjusting the known reactions using Hess's Law.

• This reaction showcases how multiple steps in a chemical equation can be combined, highlighting Hess's Law's invaluable role in simplifying complex reactions to study net enthalpy changes.
• It also illustrates how reversing reactions or changing coefficients can achieve a desired equation without experimental measurement, only theoretical calculation.

Understanding chemical reactions, energy changes, and how they interrelate allows us to predict behavior in extensive chemical processes, from industrial synthesis to environmental phenomena. This basis helps bridge theoretical chemistry with practical application.