Problem 37 - Chemistry Matter and Change

Question

Challenge Two enthalpy of formation equations, a and b, combine to form the equation for the reaction of nitrogen oxide and oxygen. The product of the reaction is nitrogen dioxide: \(\mathrm{NO}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{NO}_{2}(\mathrm{g}) \Delta H_{\mathrm{mn}}^{\circ}=-58.1 \mathrm{kJ}\) $$\begin{array}{l}{\text { a. } \frac{1}{2} \mathrm{N}_{2}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{NO}(\mathrm{g}) \Delta H_{\mathrm{f}}^{\circ}=91.3 \mathrm{kJ}} \\\ {\text { b. } \frac{1}{2} \mathrm{N}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{NO}_{2}(\mathrm{g}) \Delta H_{\mathrm{f}}^{\circ}=?} \\ {\text { What is } \Delta H_{\mathrm{f}}^{\circ} \text { for Equation } \mathrm{b} ?}\end{array}$$

Step-by-Step Solution

Answer

\( \Delta H_{\mathrm{f}}^{\circ} = 33.2 \, \mathrm{kJ} \) for reaction b.

Step 1: Understanding the Equations

We are given two formation reactions for nitrogen oxide to nitrogen dioxide (NO to NO₂), as well as their enthalpies of formation. Equation (a) is \( \frac{1}{2} \mathrm{N}_{2} + \frac{1}{2} \mathrm{O}_{2} \rightarrow \mathrm{NO} \) with \( \Delta H_{\mathrm{f}}^{\circ} = 91.3 \, \mathrm{kJ} \). Equation (b) is \( \frac{1}{2} \mathrm{N}_{2} + \mathrm{O}_{2} \rightarrow \mathrm{NO}_{2} \) with an unknown \( \Delta H_{\mathrm{f}}^{\circ} \). We also have a reaction equation \( \mathrm{NO} + \frac{1}{2} \mathrm{O}_{2} \rightarrow \mathrm{NO}_{2} \) with \( \Delta H_{\mathrm{rxn}}^{\circ} = -58.1 \, \mathrm{kJ} \).

Step 2: Combine and Manipulate Equations

The task involves using the given enthalpy of the reaction \( \mathrm{NO} + \frac{1}{2} \mathrm{O}_{2} \rightarrow \mathrm{NO}_{2} \) and enthalpy from equation (a) to find the enthalpy in equation (b). For this, use the Hess's Law which states \( \Delta H_{\mathrm{rxn}}^{\circ} = \Delta H_{\mathrm{b}}^{\circ} - \Delta H_{\mathrm{a}}^{\circ} \). Substitute given values into this expression to solve for the unknown.

Step 3: Solving Using Hess's Law

Substitute the enthalpy values into the Hess's Law equation: \[ -58.1 = \Delta H_{\mathrm{b}}^{\circ} - 91.3 \]. Solve for \( \Delta H_{\mathrm{b}}^{\circ} \) to get \( \Delta H_{\mathrm{b}}^{\circ} = -58.1 + 91.3 \). Calculate this to find the unknown enthalpy of formation.

Step 4: Calculation

Calculate as follows: \[ \Delta H_{\mathrm{b}}^{\circ} = 33.2 \, \mathrm{kJ} \]. This is the enthalpy change for the formation of \( \mathrm{NO}_{2} \) from \( \frac{1}{2} \mathrm{N}_{2} \) and \( \mathrm{O}_{2} \).

Key Concepts

Hess's Law

Hess's Law is a principle in chemical thermodynamics that provides a way to calculate the enthalpy change of a chemical reaction. According to Hess's Law, if a reaction is carried out in a series of steps, the total enthalpy change is the sum of the enthalpy changes for each individual step. This principle allows us to determine the enthalpy change even when it is impractical to measure it directly.
In the context of the given problem, Hess's Law helps us find the unknown enthalpy of formation for reaction equation b: \( \frac{1}{2} \text{N}_2 + \text{O}_2 \rightarrow \text{NO}_2 \). By manipulating the given reactions and their enthalpy values, Hess's Law enables us to deduce the required enthalpy change for this reaction.

Reaction Enthalpy

Reaction enthalpy, denoted as \( \Delta H_{\text{rxn}} \), represents the heat change during a chemical reaction at constant pressure. It indicates whether the reaction is exothermic (releases heat) or endothermic (absorbs heat). This is crucial for understanding the energy flow in reactions.
In this exercise, we're given the reaction enthalpy \( \Delta H_{\text{rxn}}^\circ \) of \(-58.1 \) kJ for the formation of nitrogen dioxide from nitrogen monoxide and oxygen. This tells us that the formation of \( \text{NO}_2 \) from \( \text{NO} \) and \( \frac{1}{2} \text{O}_2 \) releases energy, making it an exothermic reaction. Using this information in combination with Hess's Law, we are able to solve for the unknown enthalpy of formation for the second reaction equation.

Nitrogen Dioxide Formation

The formation of nitrogen dioxide (\( \text{NO}_2 \)) is a key reaction in atmospheric chemistry and industrial processes. In this exercise, the focus is on understanding the energy changes associated with the formation of \( \text{NO}_2 \) from different precursor reactions. This involves breaking down reactions into simpler steps to evaluate their enthalpies of formation.
The reaction \( \text{NO} + \frac{1}{2} \text{O}_2 \rightarrow \text{NO}_2 \) is particularly interesting because it showcases the application of Hess's Law in determining the enthalpy change. By understanding the step-by-step transformation from nitrogen and oxygen gases to \( \text{NO} \) and \( \text{NO}_2 \), students gain insights into how energy is absorbed and released during these chemical processes.

Chemical Thermodynamics

Chemical thermodynamics is the branch of chemistry that deals with the interrelation of heat and work with chemical reactions. It provides the tools and concepts needed to quantify the energy changes that occur during chemical processes.
In this context, the exercise deals with applying principles of chemical thermodynamics to calculate enthalpy changes, specifically using Hess's Law as a tool. The enthalpy changes tell us about the stability and energy requirements of different chemical species. Understanding these concepts helps predict how and why reactions happen, making it a vital part of studying chemical reactions and directing industrial applications efficiently.