Problem 33 - Chemistry Matter and Change

Question

Challenge \(\Delta H\) for the following reaction is \(-1789 \mathrm{kJ}\) . Use this and Equation a to determine \(\Delta H\) for Equation \(\mathbf{b} .\) $$\begin{array}{l}{4 \mathrm{Al}(\mathrm{s})+3 \mathrm{MnO}_{2}(\mathrm{s}) \rightarrow 2 \mathrm{Al}_{2} \mathrm{O}_{3}(\mathrm{s})+3 \mathrm{Mn}(\mathrm{s}) \Delta H=-1789 \mathrm{k} J} \\ {\text { a. } 4 \mathrm{Al}(\mathrm{s})+3 \mathrm{O}_{2}(\mathrm{g}) \rightarrow 2 \mathrm{Al}_{2} \mathrm{O}_{3}(\mathrm{s}) \Delta H=-3352 \mathrm{kJ}} \\\ {\text { b. } \mathrm{Mn}(\mathrm{s})+\mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{MnO}_{2}(\mathrm{s}) \Delta H=?}\end{array}$$

Step-by-Step Solution

Answer

The enthalpy change \(\Delta H\) for Equation (b) is \(521 \text{kJ}\).

Step 1: Understanding the Given Equations

Equation given in the challenge: \(4 \text{Al}(s) + 3 \text{MnO}_2(s) \rightarrow 2 \text{Al}_2\text{O}_3(s) + 3 \text{Mn}(s), \Delta H = -1789 \text{kJ}\). Equation (a): \(4 \text{Al}(s) + 3 \text{O}_2(g) \rightarrow 2 \text{Al}_2\text{O}_3(s), \Delta H = -3352 \text{kJ}\). Equation (b): \(\text{Mn}(s) + \text{O}_2(g) \rightarrow \text{MnO}_2(s), \Delta H = ?\). Your task is to find \(\Delta H\) for equation b.

Step 2: Write the Reaction Expressions

Let's express the reaction enthalpies: \(\Delta H_1\) for the challenge equation, \(\Delta H_2\) for equation (a), and the unknown \(\Delta H_3\) for equation (b).

Step 3: Formulate the Relationship using Hess's Law

According to Hess's Law, the enthalpy change for a reaction is the sum of the enthalpy changes of the individual steps. This gives us: \(\Delta H_1 = \Delta H_2 + 3 \Delta H_3\). Substituting the known values: \(-1789 \text{kJ} = -3352 \text{kJ} + 3 \Delta H_3\).

Step 4: Solve for \(\Delta H_3\)

Rearrange the equation from step 3 to solve for \(\Delta H_3\): \[3 \Delta H_3 = -1789 + 3352\] \[3 \Delta H_3 = 1563 \text{kJ}\] Therefore, \[\Delta H_3 = \frac{1563}{3} = 521 \text{kJ}\].

Key Concepts

Understanding Enthalpy Change

Enthalpy change, denoted as \( \Delta H \), is a fundamental concept in thermochemistry that refers to the heat absorbed or released during a chemical reaction at constant pressure. It is a crucial part of studying energy exchanges in reactions.

When \( \Delta H \) is negative, the reaction is exothermic, meaning it releases heat, whereas a positive \( \Delta H \) indicates an endothermic reaction where heat is absorbed. Enthalpy changes are often expressed in kilojoules (kJ) and can provide insight into the energetics of a chemical transformation. For instance, in the given challenge, the overall enthalpy change for a specific reaction was \( -1789 \text{kJ} \), indicating an exothermic process.

In practice, enthalpy changes are critical for predicting the feasibility and spontaneity of reactions, as well as for calculating energy requirements in industrial processes. Moreover, understanding \( \Delta H \) helps chemists design reaction pathways with desired energy profiles.

Chemical Reactions and Hess's Law

Chemical reactions involve the rearrangement of atoms to form new substances, and they often involve energy changes. Hess's Law is a powerful principle in thermochemistry that states the total enthalpy change for a reaction is the same, regardless of the pathway the reaction takes. This principle allows us to calculate enthalpy changes for reactions where direct measurement might be impractical.

In the original exercise, Hess's Law is applied to determine the enthalpy change for Equation (b) using known values from related reactions. The enthalpy changes of the given reactions were combined to solve for the unknown \( \Delta H \) of Equation (b), demonstrating how Hess's Law is a valuable tool for piecing together thermodynamic data.

This principle highlights the importance of chemical reactions being continuous sequences of steps where individual reaction enthalpies add up to give a total result. Hence, it is crucial for studying complex reactions, as it allows for the combination of known steps into a single pathway of desired criteria. It acts like a puzzle, where knowing certain pieces lets you deduce what's missing.

Discovering Therochemistry

Thermochemistry is the study of the energy and heat changes associated with chemical reactions. It combines principles of chemistry and thermodynamics, offering insights into energy transfers occurring during reactions.

At the heart of thermochemistry are concepts like enthalpy changes, heat capacity, and calorimetry, which chemists use to quantify the energy changes in a system. By understanding thermochemistry, one can predict how energy will flow in reactions, providing essential data for industries that require energy-efficient processes.

The scope of thermochemistry extends to the design of chemical processes, environmental studies, and even in the development of new materials and fuels. Within the original problem, thermochemistry plays a role in relating the energy content of substances before and after a reaction, as demonstrated by calculating the change in \( \Delta H \) through Hess's Law. In essence, thermochemistry can be seen as a bridge to understanding the energetic footprints of chemical phenomena.