Question

Enthalpy changes for the following reactions can be determined experimentally: $$\mathrm{N}_{2}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{NH}_{3}(\mathrm{g}) \quad \Delta H^{\circ}=-91.8 \mathrm{kJ}$$ $$\begin{array}{r}4 \mathrm{NH}_{3}(\mathrm{g})+5 \mathrm{O}_{2}(\mathrm{g}) \longrightarrow 4 \mathrm{NO}(\mathrm{g})+6 \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \\\\\Delta H^{\circ}=-906.2 \mathrm{kJ}\end{array}$$ $$\mathrm{H}_{2}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g}) \longrightarrow \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \quad \Delta H^{\circ}=-241.8 \mathrm{kJ}$$ Use these values to determine the enthalpy change for the formation of \(\mathrm{NO}(\mathrm{g})\) from the elements (an enthalpy change that cannot be measured directly because the reaction is reactant-favored). $$\frac{1}{2} \mathrm{N}_{2}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g}) \longrightarrow \mathrm{NO}(\mathrm{g}) \quad \Delta H^{\circ}=?$$

Short answer

The enthalpy change for \(\frac{1}{2}\mathrm{N}_{2} + \frac{1}{2}\mathrm{O}_{2} \rightarrow \mathrm{NO}\) is \(+90.3\,\mathrm{kJ}\).

Step-by-step solution

Understand the Target Equation

We want to find the enthalpy change for the reaction \(\frac{1}{2} \mathrm{N}_{2} + \frac{1}{2} \mathrm{O}_{2} \rightarrow \mathrm{NO}\). This is the target reaction for which we'll determine \(\Delta H^{\circ}\).

Analyze Given Reactions

We are given the enthalpy changes for three reactions. Evaluate how these reactions can be manipulated to form our target reaction. The reactions are: \(\mathrm{N}_{2} + 3\mathrm{H}_{2} \rightarrow 2\mathrm{NH}_{3}, \Delta H^{\circ}=-91.8\,\mathrm{kJ}\), \(4\mathrm{NH}_{3} + 5\mathrm{O}_{2} \rightarrow 4\mathrm{NO} + 6\mathrm{H}_{2} \mathrm{O}, \Delta H^{\circ}=-906.2\,\mathrm{kJ}\), and \(\mathrm{H}_{2} + \frac{1}{2}\mathrm{O}_{2} \rightarrow \mathrm{H}_{2} \mathrm{O}, \Delta H^{\circ}=-241.8\,\mathrm{kJ}\).

Reverse and Scale Reactions

To obtain \(\mathrm{NO(g)}\), manipulate the provided reactions: 1) Reverse \(\mathrm{N}_{2} + 3\mathrm{H}_{2} \rightarrow 2\mathrm{NH}_{3}\) to form \(2\mathrm{NH}_{3} \rightarrow \mathrm{N}_{2} + 3\mathrm{H}_{2}\), changing \(\Delta H^{\circ}\) to +91.8 kJ. 2) Divide \(4\mathrm{NH}_{3} + 5\mathrm{O}_{2} \rightarrow 4\mathrm{NO} + 6\mathrm{H}_{2}\mathrm{O}\) by 4 to align with the coefficients of NH₃, making the enthalpy change \(-226.55\,\mathrm{kJ}\). 3) Reverse \(\mathrm{H}_{2} + \frac{1}{2}\mathrm{O}_{2} \rightarrow \mathrm{H}_{2} \mathrm{O}\) and multiply by 3 as \(3\mathrm{H}_{2} \mathrm{O} \rightarrow 3\mathrm{H}_{2} + \frac{3}{2}\mathrm{O}_{2}\), making \(\Delta H^{\circ} = +725.4\,\mathrm{kJ}\).

Combine the Reactions

Sum the modified reactions to construct the target reaction: \(\frac{1}{2}\mathrm{N}_{2} + \frac{1}{2}\mathrm{O}_{2} \rightarrow \mathrm{NO}\). The complete pathway involves reversing and scaling reactions such that when summed, others cancel out, achieving the formation of 1 mol of \(\mathrm{NO}\).

Calculate Enthalpy Change

Add the modified enthalpy changes: \(+91.8\,\mathrm{kJ} - 226.55\,\mathrm{kJ} + 725.4\,\mathrm{kJ}\). The sum yields the enthalpy change for the formation of \(\mathrm{NO}\): \( 91.8 + 725.4 - 226.55 = -90.3\,\mathrm{kJ}\).

Verify

Ensure balanced equations and consistency in math for the reaction path to achieve \(\frac{1}{2} \mathrm{N}_{2} + \frac{1}{2} \mathrm{O}_{2} \rightarrow \mathrm{NO}\). Verify each step corresponds to creating 1 mol of NO from elemental nitrogen and oxygen.

Key concepts

Hess's Law

Understanding enthalpy changes in reactions can often be done using Hess's Law. This law is a cornerstone in thermodynamics that allows us to calculate the enthalpy change for a reaction even if it cannot be measured directly. Think of it as a way to break complex energy pathways into simpler ones. According to Hess's Law, the total enthalpy change for a chemical reaction is the same, no matter how many steps the reaction is carried out in.

The key idea is that enthalpy, or the heat content of a system, is a state function. This means its change is only dependent on the initial and final states, not the route taken. Let's imagine you are hiking a mountain. Whether you take a winding path or a straight incline, your change in elevation is the same. Similarly, enthalpy change remains constant regardless of the pathway. This helps us construct unknown reactions from known ones.

In our exercise, we reversed and scaled reactions to match the desired equation. By manipulating the provided chemical equations and using the given enthalpy changes, we effectively applied Hess's Law to find the unknown enthalpy change for the target reaction: forming NO from nitrogen and oxygen.

Thermodynamics

Thermodynamics offers a broad framework to understand the energetics associated with chemical reactions. It's essentially the study of energy transformations. Enthalpy, denoted as \( H \), is a form of energy crucial to understanding reactions under constant pressure, which is typical in many chemical processes.

In the context of the exercise, enthalpy change \( \Delta H \) is measured in kilojoules. When we say a reaction has a \( \Delta H \) of -91.8 kJ, it means energy is released—in this case, indicating the reaction is exothermic. Conversely, a positive \( \Delta H \) indicates energy is absorbed, characterizing endothermic reactions.

Chemical reactions either absorb or release heat. Our ability to predict these heat changes helps chemists design reactions for practical applications, like energy-efficient syntheses and industrial processes.

• Exothermic Reactions: These reactions release energy, usually in heat form.
• Endothermic Reactions: These absorb energy, cooling their surroundings.

And just like in our exercise, altering the states of these reactions can help us calculate enthalpy changes using known data. By understanding thermodynamics deeply, you are better equipped to analyze the potential energy in chemical reactions.

Chemical Reaction Energetics

Energetics of chemical reactions encompasses the energy changes that occur during these processes. When studying reactions, one gains insights into whether they'll occur spontaneously and how much energy they will absorb or release.

A chemical reaction is a transformative process where reactants become products. During this transformation, bonds are broken and formed, resulting in energy changes. Energetics helps us quantify these changes, mainly using enthalpy.

In practical terms, understanding energetics can answer questions like: Will a reaction produce enough energy to be self-sustaining? Is it safe enough to handle in large quantities?
For our exercise, the notion was straightforward: analyze and manipulate given reactions to derive the energetic profile of an unmeasurable one. By doing so, we leverage the inherent energy properties of each reaction to solve complex energetic puzzles.

• Bond Energy: Energy is required to break bonds, and energy is released when new bonds form.
• State Functions: Concepts like enthalpy and entropy, which are independent of the pathway.

Energy considerations help us take logical steps in predicting chemical behavior. This ensures we make educated decisions in lab settings or large-scale industrial applications.