Question

Given the following data $$ \begin{aligned} 2 \mathrm{O}_{3}(g) & \longrightarrow 3 \mathrm{O}_{2}(g) & \Delta H &=-427 \mathrm{~kJ} \\ \mathrm{O}_{2}(g) & \longrightarrow 2 \mathrm{O}(g) & \Delta H &=495 \mathrm{~kJ} \\ \mathrm{NO}(g)+\mathrm{O}_{3}(g) & \longrightarrow \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g) & \Delta H=-199 \mathrm{~kJ} \end{aligned} $$ calculate \(\Delta H\) for the reaction $$ \mathrm{NO}(g)+\mathrm{O}(g) \longrightarrow \mathrm{NO}_{2}(g) $$

Short answer

The enthalpy change for the reaction \(NO(g) + O(g) \rightarrow NO_{2}(g)\) is \(-907.5\:kJ\).

Step-by-step solution

Analyze the given reactions

List the given reactions and their enthalpy changes: 1. \(2\: O_{3}(g) \rightarrow 3\:O_{2}(g)\) and \(\Delta H_{1} = -427\:kJ\) 2. \(O_{2}(g) \rightarrow 2\:O(g)\) and \(\Delta H_{2} = 495\:kJ\) 3. \(NO(g) + O_{3}(g) \rightarrow NO_{2}(g) + O_{2}(g)\) and \(\Delta H_{3} = -199\:kJ\) Our target reaction is: \(NO(g) + O(g) \rightarrow NO_{2}(g)\)

Manipulate the given reactions

We'll now create a series of manipulations to the given reactions to obtain the desired reaction: 1. Divide reaction 1 by 2: \(\frac{1}{2}(2 O_{3}(g) \rightarrow 3 O_{2}(g))\); and likewise, divide \(\Delta H_{1}\) by 2: \(\frac{1}{2}(-427\:kJ) = -213.5\:kJ\). This results in the new reaction 1: \(O_{3}(g) \rightarrow \frac{3}{2} O_{2}(g)\) and \(\Delta H_{1'} = -213.5\:kJ\) 2. Reverse reaction 2: \(2 O(g) \rightarrow O_{2}(g)\) and, consequently, reverse the sign of \(\Delta H_{2}\): \(-495\:kJ\)

Combine the manipulated reactions

Now, add the manipulated reactions to obtain the target reaction: \(O_{3}(g) \rightarrow \frac{3}{2} O_{2}(g)\) (\(\Delta H_{1'} = -213.5\:kJ\)) + \(2 O(g) \rightarrow O_{2}(g)\) (\(\Delta H_{2'} = -495\:kJ\)) + \(NO(g) + O_{3}(g) \rightarrow NO_{2}(g) + O_{2}(g)\) (\(\Delta H_{3} = -199\:kJ\)) \(\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\) \(NO(g) + O(g) \rightarrow NO_{2}(g)\) Next, sum up the respective \(\Delta H\) values: \(\Delta H = \Delta H_{1'} + \Delta H_{2'} + \Delta H_{3}\)

Calculate the enthalpy change for the target reaction

Utilizing the formula above, calculate the enthalpy change for the target reaction: \(\Delta H = (-213.5\:kJ) + (-495\:kJ) + (-199\:kJ)\) \(\Delta H = -907.5\:kJ\) So, the enthalpy change for the reaction \(NO(g) + O(g) \rightarrow NO_{2}(g)\) is \(-907.5\:kJ\).

Key concepts

Enthalpy Change

Enthalpy change is a fundamental concept in thermochemistry, representing the heat absorbed or released during a chemical reaction at constant pressure. It is denoted by \( \Delta H \), and can be positive or negative depending on whether the reaction absorbs or releases energy. In an exothermic reaction, energy is released, and \( \Delta H \) is negative. Conversely, in an endothermic reaction, energy is absorbed, making \( \Delta H \) positive.

Understanding enthalpy change involves analyzing different reactions. For example, when ozone (\( O_3 \)) decomposes to produce oxygen gas (\( O_2 \)), it releases energy, resulting in a negative \( \Delta H \). This is crucial because combining reactions to determine \( \Delta H \) for a new reaction is at the core of Hess's Law, which states that the total enthalpy change for a reaction is the sum of all enthalpy changes of the individual steps leading to it.

Chemical Reactions

Chemical reactions involve the rearrangement of atoms to form new substances. They represent a transformation from reactants to products, often accompanied by energy changes. These reactions can be classified based on their energy changes, such as exothermic (releasing energy) or endothermic (absorbing energy).

In the given exercise, several reactions are presented with their respective enthalpy changes. Each step describes a transformation that either releases or absorbs energy, marked by \( \Delta H \). For instance, the decomposition process of \( O_3 \) into \( O_2 \) and the reaction of \( NO \) with \( O_3 \) show differing energy profiles. By manipulating these reactions, such as reversing or multiplying them, a chemist can craft pathways that result in a desired target reaction. This manipulation is essential for understanding reaction thermodynamics and plays a pivotal role in illustrating the broader concept of Hess's Law.

Thermochemistry

Thermochemistry is the study of the energy changes that occur during chemical reactions. It focuses on examining how heat energy is absorbed or released. This branch of chemistry applies mathematical calculations to understand and predict the energy requirements or releases in chemical processes.

A key component of thermochemistry is Hess's Law, which allows for the calculation of enthalpy changes in complex reactions. By using the enthalpy changes given for certain reactions, one can calculate the \( \Delta H \) for new reactions by appropriately combining these known reactions. In the exercise, manipulation of reaction equations allowed for the determination of the enthalpy change of a target reaction that wasn't directly measured.

Understanding thermochemistry helps in fields like material science, environmental studies, and industrial chemistry, where energy efficiency and detailed control over reaction conditions are crucial. Overall, the mathematical and practical insights provided by thermochemistry are invaluable for predicting the feasibility and impact of chemical reactions.